# What Does "Harmonic" Mean?



## millionrainbows

What, exactly and completely in all its senses, does the term "harmonic" mean?


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## millionrainbows

No takers? Come on, live dangerously!


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## Woodduck

The "exactly and completely in all its senses" bit probably has people checking their schedules to see how many hours they can spare. Besides, they know that if they can't find the time, you will.


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## Guest

Harmonic? Harmonic _what_? It's an adjective, isn't it, so it needs a noun to go with it.

(I promised myself I wouldn't take the bait, but...)


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## Bwv 1080

A harmonic is any member of the harmonic series. The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. It is typically applied to repeating signals, such as sinusoidal waves. A harmonic of such a wave is a wave with a frequency that is a positive integer multiple of the frequency of the original wave, known as the fundamental frequency. The original wave is also called the 1st harmonic, the following harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. For example, if the fundamental frequency is 50 Hz, a common AC power supply frequency, the frequencies of the first three higher harmonics are 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50 Hz.


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## millionrainbows

Bwv 1080 said:


> A harmonic is any member of the harmonic series. The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. It is typically applied to repeating signals, such as sinusoidal waves. A harmonic of such a wave is a wave with a frequency that is a positive integer multiple of the frequency of the original wave, known as the fundamental frequency. The original wave is also called the 1st harmonic, the following harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. For example, if the fundamental frequency is 50 Hz, a common AC power supply frequency, the frequencies of the first three higher harmonics are 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50 Hz.


That's correct, but incomplete. What is Bwv 1080 missing?



MacLeod said:


> Harmonic? Harmonic _what_? It's an adjective, isn't it, so it needs a noun to go with it.


So far, MacLeod is the only one to recognize the term "harmonic" as having meaning as something other than a noun.

Adjectives don't necessarily "need" nouns; they can be descriptive terms on their own, decdribing a quality that remains unspecified.


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## millionrainbows

Furthermore, "harmonic" as a noun can be used as an adjective, with a meaning more connected to its meaning as a noun than with its more common adjective meaning. The term "harmonic model" is an example of this 'new' adjective meaning.


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## isorhythm

No idea, not familiar with that term. Are you going to tell us?


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## Larkenfield

A harmonic progression is where the word is used as an adjective instead of a noun. It’s a vertical arrangement of notes.


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## millionrainbows

Larkenfield said:


> A harmonic progression is where the word is used as an adjective instead of a noun. It's a vertical arrangement of notes.


Is it? I thought "harmonic progression" described a progression of chords, or a sequence. The "harmonic" adjective refers to "harmony," not any vertical entity such as an upper partial or overtone.


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## philoctetes

The harmonic series sums up terms like 1/k, where k is an integer.

A Fourier series sums up terms like sin (kwt), where k is an integer and w is the fundamental (k=1). k>1 are the overtones. 

A harmonic oscillator is a classic problem in physics where a mass in subject to a force proportional to its displacement.


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## starthrower

It's that pretty sound that Steve Howe makes on his acoustic guitar at the beginning of Roundabout.


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## KenOC

I think it's that thing Bob Dylan plays between verses...


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## Bwv 1080

philoctetes said:


> The harmonic series sums up terms like 1/k, where k is an integer.


And like the harmonic series, this thread diverges


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## millionrainbows

Bwv 1080 said:


> And like the harmonic series, this thread diverges


Math & physics use the term as well.


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## Luchesi

How does the human brain sense the arithmetical relationships in the intervals up to the major third? A child can do it.

At the minor third the uncertainty and sadness begins because the brain can't sense (calculate) the difference between 6 and 7 times the fundamental?


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## millionrainbows

Luchesi said:


> How does the human brain sense the arithmetical relationships in the intervals up to the major third? A child can do it.
> 
> At the minor third the uncertainty and sadness begins because the brain can't sense (calculate) the difference between 6 and 7 times the fundamental?


The brain senses things relatively, so the difference in thirds is minimal. I proved this to myself by constructing a wind chime set using the Thai scale: 7 notes equally spaced across an octave, called 7-note equal temperament.

When I played melodies on it, it sounded surprisingly diatonic, because it had seven notes.


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## fluteman

I think Bwv 1080's definition is more than complete enough. "Harmonic" as an adjective should refer in some way to the concept of a frequency or frequencies that are a positive integer multiple of a fundamental frequency. It is the relationship of two or more frequencies that is (or isn't) harmonic.

In western music, there is a significant "fudge" factor in the concept of an harmonic relationship, as the equal-tempered scale has become standard, and although in that scale the "perfect" fifth is very close to a true harmonic fifth (just under 2 cents off), other intervals generally considered to be harmonic diverge much further. Our western ears have been trained to accept these as sounding "right", and traditional music of other cultures tuned to different scales typically sounds odd or dissonant to us. But it is worth remembering that there is nothing inherently natural or correct about equal-tempered harmonic intervals or relationships.


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## millionrainbows

fluteman said:


> I think Bwv 1080's definition is more than complete enough. "Harmonic" as an adjective should refer in some way to the concept of a frequency or frequencies that are a positive integer multiple of a fundamental frequency. It is the relationship of two or more frequencies that is (or isn't) harmonic.


So, it appears you are prepared to deviate from the idea of the 'natural' harmonic series, and accept _any positive integer multiple_ as a "harmonic." Some literalists would disapprove, saying that "harmonic" should properly only refer to the natural overtone series.

But that would exclude the commonly and frequently-used terms "harmonic progression" and "harmonic analysis," both of which refer to chord progressions and "harmony".

As I said in post #10, "harmonic" as an adjective in this case refers to "harmony," not any vertical entity such as an upper partial or overtone.


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## Guest

One would have thought access to a dictionary, or perhaps Wikipedia would make such threads utterly redundant. Apparently not.

https://en.wikipedia.org/wiki/Harmonic


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## fluteman

Baron Scarpia said:


> One would have thought access to a dictionary, or perhaps Wikipedia would make such threads utterly redundant. Apparently not.
> 
> https://en.wikipedia.org/wiki/Harmonic


How is that any different from what I posted, or bwv 1080, for that matter? No big secret.



millionrainbows said:


> So, it appears you are prepared to deviate from the idea of the 'natural' harmonic series, and accept any positive integer multiple as a "harmonic." Some literalists would disapprove, saying that "harmonic" should properly only refer to the natural overtone series.
> 
> But that would exclude the commonly and frequently-used terms "harmonic progression" and "harmonic analysis," both of which refer to chord progressions and "harmony".
> 
> As I said in post #10, "harmonic" as an adjective in this case refers to "harmony," not any vertical entity such as an upper partial or overtone.


One of the ways verbal language works is that descriptive terms do extra duty, i.e., are given broader or comparable meanings. Western harmony is an outgrowth or development of the harmonic series, making special use first of the unison and octave, then of the fifth, and finally of the triad. Other scales are possible, of course, but the fifth is very useful because in the twelve-tone equal tempered scale, it corresponds so closely to the natural harmonic. The total "correction" required is pretty small, only about 23 cents. But the true 12-tone equal tempered scale is a very modern development in western music, as each step in the scale is higher than the preceding one by the twelfth root of two, and only relatively recently was it possible to calculate that.

So, harmonic progression and analysis, or harmony, as applied to the equal tempered 12-tone scale, still fall under the same general umbrella as the harmonic series. These are related ideas.

All of which you already know.


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## millionrainbows

fluteman said:


> ...harmonic progression and analysis, or harmony, as applied to the equal tempered 12-tone scale, still fall under the same general umbrella as the harmonic series. These are related ideas. All of which you already know.


That's true, but the umbrella is too general for me, and much nuance of meaning is lost.


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## millionrainbows

Baron Scarpia said:


> One would have thought access to a dictionary, or perhaps Wikipedia would make such threads utterly redundant. Apparently not.


Apparently not, judging by the answers from philocetes and flute man.

As it was said years ago,



> Isn't there a more appropriate place for threads like this? Perhaps in the Community Forum or something?


I realize, after years of interacting with people who I now see as basically "too practical," or have knowledge which they have memorized, but never really "grokked" or understood on a basic level, or ruminated for hours about, or whatever, 
...that a creative person, like myself, thinks differently about the most basic things. This subject is big enough to fill pages, so I will leave it at that.

Many here keep referring back to the harmonic series, and that is too literal; it does not apply to what I am saying. 'Inharmonic" is defined as sound which departs from the whole-numbered harmonic series, and my harmonic models do not use the harmonic series literally, but as a model. There seems to be a problem with "abstracting" this principle out from the actual, literal harmonic series.

I understand; "divide and conquer" by fragmenting everything up into a literal series of unconnected facts in time, and then insisting that everything fit this template, with no abstraction, no extrapolation, no creativity...


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## Bwv 1080

All instruments are inharmonic to some degree (the partials deviate from whole number multiples), with some pitched percussion completely inharmonic.

The most purely harmonic composers would be the Spectralists?


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## SixFootScowl

MacLeod said:


> Harmonic? Harmonic _what_? *It's an adjective, isn't it, so it needs a noun to go with it.
> *


Oh, like as in HARMONIC DAMPER:


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## millionrainbows

Bwv 1080 said:


> All instruments are inharmonic to some degree (the partials deviate from whole number multiples), with some pitched percussion completely inharmonic.
> 
> The most purely harmonic composers would be the Spectralists?


Yes, for literalists like you. Other music can create "harmonic models" by using scales. These scales can contain any notes, which do not need to reflect the harmonic series except in the way that they create an hierarchy of relations to the "fundamental" or key note of the scale.
Is anybody here capable of that level of abstract thought? A scale is a harmonic model.


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## fluteman

millionrainbows said:


> Apparently not, judging by the answers from philocetes and flute man.
> 
> As it was said years ago,
> 
> I realize, after years of interacting with people who I now see as basically "too practical," or have knowledge which they have memorized, but never really "grokked" or understood on a basic level, or ruminated for hours about, or whatever,
> ...that a creative person, like myself, thinks differently about the most basic things. This subject is big enough to fill pages, so I will leave it at that.
> 
> Many here keep referring back to the harmonic series, and that is too literal; it does not apply to what I am saying. 'Inharmonic" is defined as sound which departs from the whole-numbered harmonic series, and my harmonic models do not use the harmonic series literally, but as a model. There seems to be a problem with "abstracting" this principle out from the actual, literal harmonic series.
> 
> I understand; "divide and conquer" by fragmenting everything up into a literal series of unconnected facts in time, and then insisting that everything fit this template, with no abstraction, no extrapolation, no creativity...


I'm sorry, but what a bunch of condescending $%^&. You are creative, and I and others are not? Is this the point you sought to establish by starting this thread? And you arrive at this conclusion because we dare to define some basic musical terms in the way they have been defined for us since early in our musical education? My teachers included composers who have won the Pulitzer Prize and Grammy Awards. Is your understanding of harmony that much greater or deeper than theirs?

And, by the way, I have been reading your lengthy posts about harmony here for some time. I see nothing especially original in them. Nearly all of your discussion here is based on the western, equal-tempered, 12-tone scale. Nothing wrong with that, as this forum is mainly dedicated to western music of the last 350 years or so, and that is the scale that has predominated for all but the earliest part of that period, when a few other scales not too dissimilar to it were still in common use. But you are hardly the only one here who knows that harmony is a big, many-faceted subject, even if one confines oneself to the equal-tempered, 12-tone scale. And it got even bigger in the 20th century when western classical composers began to experiment with non-western and other non-traditional ideas, or even abandon the standard 12-tone scale entirely. And of course, there is much more to music than harmony, however defined, and there always has been.

So write a book, if you have so many creative ideas so far beyond those of mere mortals. And by the way, most of my knowledge of music comes from a lifetime of playing it in orchestras and chamber music groups, singing it in choruses, or listening to it. Not from memorizing anything in a book. And I'm certainly not compulsive about dictionary definitions. Nor do I reject any music not based, or not entirely based, on traditional western scales and harmony, unlike some here, apparently. So I'll await your book. But I would be a fool if I didn't recognize that the basic ideas of the octave, which is the second harmonic, and the fifth, which is the third harmonic, underlie all of traditional western harmony.


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## mikeh375

millionrainbows said:


> Yes, for literalists like you. Other music can create "harmonic models" by using scales. These scales can contain any notes, which do not need to reflect the harmonic series except in the way that they create an hierarchy of relations to the "fundamental" or key note of the scale.
> *Is anybody here capable of that level of abstract thought? A scale is a harmonic model.*


Yes and probably more than one. This quite accurately describes how I often start composing and have found the process of extracting new rules (and even the old rules!) from unusual scales to be very fertile.


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## millionrainbows

fluteman said:


> I'm sorry, but what a bunch of condescending $%^&. You are creative, and I and others are not? Is this the point you sought to establish by starting this thread? And you arrive at this conclusion because we dare to define some basic musical terms in the way they have been defined for us since early in our musical education? My teachers included composers who have won the Pulitzer Prize and Grammy Awards. Is your understanding of harmony that much greater or deeper than theirs?


No, but as is always the problem with large, established bodies of knowledge, people forget to think, and take everything as a "given" without questioning anything. I think the British are better at this than Americans; they call it "lateral thinking."



> And, by the way, I have been reading your lengthy posts about harmony here for some time. I see nothing especially original in them. Nearly all of your discussion here is based on the western, equal-tempered, 12-tone scale. Nothing wrong with that, as this forum is mainly dedicated to western music of the last 350 years or so, and that is the scale that has predominated for all but the earliest part of that period, when a few other scales not too dissimilar to it were still in common use.


Conversely, your many deviations into microtonality tend to be distracting rather than clarifying.



> But you are hardly the only one here who knows that harmony is a big, many-faceted subject, even if one confines oneself to the equal-tempered, 12-tone scale. And it got even bigger in the 20th century when western classical composers began to experiment with non-western and other non-traditional ideas, or even abandon the standard 12-tone scale entirely. And of course, there is much more to music than harmony, however defined, and there always has been.


Then I guess my attitude as you are seeing it is my reaction to internet hostility from know-it-alls who can't stand any kind of exploring or speculation on those things they take as "givens." For example, questioning the influence of Pythagoras on Western music, our 12-note octave, and the fifth as the basis of much of it. And the inability to see scales as harmonic models, an easy-to-grasp concept of "harmonic models" based on internal scale relations of each note to a fundamental. Simple stuff like that, which doesn't need to be questioned.



> So write a book, if you have so many creative ideas so far beyond those of mere mortals. And by the way, most of my knowledge of music comes from a lifetime of playing it in orchestras and chamber music groups, singing it in choruses, or listening to it. Not from memorizing anything in a book. And I'm certainly not compulsive about dictionary definitions. Nor do I reject any music not based, or not entirely based, on traditional western scales and harmony, unlike some here, apparently. So I'll await your book.


I didn't ask for your credentials, and I think you are being overly defensive. I don't see it as "compulsive" to examine definitions. I've read several books on the idea of "zero," so I am not the only thinker who examines basic ideas in order to truly "grok" them.



> But I would be a fool if I didn't recognize that the basic ideas of the octave, which is the second harmonic, and the fifth, which is the third harmonic, underlie all of traditional western harmony.


I don't question that basic premise, but I ask that you go past, abstract that basic principle, and see that other "harmonic models" are possible, and also to realize that the division of the octave will never be "perfect."


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## fluteman

millionrainbows said:


> I didn't ask for your credentials, and I think you are being overly defensive. I don't see it as "compulsive" to examine definitions. I've read several books on the idea of "zero," so I am not the only thinker who examines basic ideas in order to truly "grok" them.
> 
> I don't question that basic premise, but I ask that you go past, abstract that basic principle, and see that other "harmonic models" are possible, and also to realize that the division of the octave will never be "perfect."


I never said anything about perfect. And I find it puzzling that somehow you think only you can be open minded, or open eared, about scales and tonal relationships. The 12-tone scale, the diatonic scale, equal temperament, etc., are all only practical and necessarily imperfect approaches to the problem of creating hierarchical systems of tones. There is nothing inevitable about them. Human music and the idea of scales or tonal hierarchies are thousands of years old, and the 12-tone scale is still a relatively recent approach. I've already discussed the "fudge factor" of equal temperament. Other systems of temperament are no more perfect. I understand that.

I usually find semantic debates boring and pointless. Words are useful if, and only to the extent that, they effectively communicate significant ideas. They are a practical tool first and foremost. Harmonic relationships, in the way I and others here have defined that concept, have been a guiding principle of human music for as long as we can determine its structure, back to ancient Greece, at least, and still are. Not the only principle, to be sure. We all know tonal hierarchies can be established on principles other than that of integer multiples of a fundamental frequency. But the concept is important enough that there is good reason, for clarity of discourse, to use the word "harmonic" with that concept in mind.


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## millionrainbows

fluteman said:


> We all know tonal hierarchies can be established on principles other than that of integer multiples of a fundamental frequency. But the concept is important enough that there is good reason, for clarity of discourse, to use the word "harmonic" with that concept in mind.


I think good argument can be made for the fifth as a basic harmonic establishing principle, since it also ties in, by projecting or "stacking" it, to the division (almost perfect) of the octave into 12 parts.

The major third? Not nearly as perfect.


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## Guest

millionrainbows said:


> I think good argument can be made for the fifth as a basic harmonic establishing principle, since it also ties in, by projecting or "stacking" it, to the division (almost perfect) of the octave into 12 parts.
> 
> The major third? Not nearly as perfect.


So now we are supposed to refer to Pythagorean temperament as "millionrainbows" temperament because you have rediscovered it?


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## Guest

millionrainbows said:


> I didn't ask for your credentials, and I think you are being overly defensive. I don't see it as "compulsive" to examine definitions. I've read several books on the idea of "zero," so I am not the only thinker who examines basic ideas in order to truly "grok" them.


Reexamining basic ideas in order to truly "grok" them is admirable and I do it all the time. Thinking that others need to know about this internal process is questionable.

Regarding the idea of temperament, the basic math is pretty simple, a lot of the apparent complexity comes from the fact that people were trying to tune harpsichords and other instruments hundreds of years ago when there was no way of measuring the frequency of sound. They had to use indirect measures, such as the beat between the harmonics of different pitches, which is further complicated by the fact that the modes of vibration of strings and other objects don't actually occur at precise integer ratios because of non-ideal behavior of the media. If we are to understand the historical tuning methods we have to unpack those peculiar signals. (Who would have defined the Pythagorean comma as a unit of frequency difference if not for the peculiar history of temperament?)

According to the dictionary:

harmony is from a Greek root meaning joint. In music harmony refers to more than one voice or note sounding simultaneously.

harmonic is an adjective meaning "relating to harmony"

in science/math a harmonic is a solution of a linear wave equation which occurs at integer multiples of the lowest frequency solution. Musical instruments produce harmonics, to the extent that their 1D sound sources respond linearly. In a musical context these harmonics can be selected on a string instrument by touching nodal points or made use of in brass instruments.

Sound sources which are not 1D (like a drum head) have a more complicated set of solutions and these are usually called "modes" rather than "harmonics." The fact that these modes are not integer multiples of a fundamental frequency makes such instruments sound "untuned."

Is there more to it than that?


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## fluteman

millionrainbows said:


> I think good argument can be made for the fifth as a basic harmonic establishing principle, since it also ties in, by projecting or "stacking" it, to the division (almost perfect) of the octave into 12 parts.
> 
> The major third? Not nearly as perfect.


Well, I said exactly that above. The traditional western system is based on the unison (first harmonic), octave (second), and fifth (third). Twelve fifths stack up to only about 23 cents shy of an octave. That's pretty convenient. But the major third and the seventh are both way off. That's why it annoys me when people glibly assume that the western 12-tone scale, and therefore conventional western harmony, are somehow inherently right and natural.


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## millionrainbows

Baron Scarpia said:


> Reexamining basic ideas in order to truly "grok" them is admirable and I do it all the time. Thinking that others need to know about this internal process is questionable.


As I said earlier, my reaction is due to internet aggression.



> Regarding the idea of temperament, the basic math is pretty simple, a lot of the apparent complexity comes from the fact that people were trying to tune harpsichords and other instruments hundreds of years ago when there was no way of measuring the frequency of sound. They had to use indirect measures, such as the beat between the harmonics of different pitches, which is further complicated by the fact that the modes of vibration of strings and other objects don't actually occur at precise integer ratios because of non-ideal behavior of the media. If we are to understand the historical tuning methods we have to unpack those peculiar signals. (Who would have defined the Pythagorean comma as a unit of frequency difference if not for the peculiar history of temperament?)


 Actually, the 12 notes of our Western octave (even before equal temperament) we arrived at by a Pythagoran-derived procedure of "stacking" or projecting fifths. After 12 cycles of fifths, the starting point almost coincided with the end point of 12; but not quite, and this is called "The Pythagoran comma."

Still, the net result was that the "circle" of the octave was closed, and the Western octave is still based on 12 notes arrived at by the 12-division of fifths. The modern tempered fifth is off by only 2 cents (a cent is 1/100th of a semitone), and this is negligible. Our Western system is still based on root movement by fifths. The fifth is the favored interval. By contrast, major thirds are sharp by a full 14 cents! That is very audible, but we seem to have gotten used to it. The fifth is the more important interval anyway, because it creates stability in triads.


> According to the dictionary:
> 
> harmony is from a Greek root meaning joint. In music harmony refers to more than one voice or note sounding simultaneously.
> 
> harmonic is an adjective meaning "relating to harmony"
> 
> in science/math a harmonic is a solution of a linear wave equation which occurs at integer multiples of the lowest frequency solution. Musical instruments produce harmonics, to the extent that their 1D sound sources respond linearly. In a musical context these harmonics can be selected on a string instrument by touching nodal points or made use of in brass instruments.
> 
> Sound sources which are not 1D (like a drum head) have a more complicated set of solutions and these are usually called "modes" rather than "harmonics." The fact that these modes are not integer multiples of a fundamental frequency makes such instruments sound "untuned."
> 
> Is there more to it than that?


(What is "1D"?)

You need to explicitly acknowledge that scales create harmonic hierarchies.

Consonance is simpler vibration, and dissonance is more complex, even chaotic vibrations.

Western tone-centered music, including folk, popular, and classical, and all forms of basic tone-centered music globally, are based on harmonic (adj.) models.

These "models" are not the harmonics themselves (used as a noun), but "harmonic models" based on divisions of the octave to "1" or a key note. This can be done with any division of the octave. It produces a "harmonic model" of ratios to "1." 

What this does is, in effect, create a "tonality" with the scale-steps (the divisions), and each step (division) will be a ratio. 

These ratios can be classified in order of their consonance (close relation to "1") or dissonance (more distant relation to "1"). 

These can be called "functions" when triads or other chords are built on them, and that "root" will have a function which is a measure of its "tonal gravity" or its tendency to "pull" or "repel" our ear to or from "home" or the key note.


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## Guest

1D is short for one dimensional.


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## mikeh375

millionrainbows said:


> ...........*You need to explicitly acknowledge that scales create harmonic hierarchies*........


Not necessarily MR. The composer's vertical organisation and invention can also determine harmonic (chordal/vertical) hierarchies, which are then reinforced by the actual music written and manipulated to create a pull one way or another to confirm any dominance.

Harmonic hierarchies are implicit within a scale, rather they are nascent - that is, vertical constructs that are not typical CP - but it is also up to the composer's imagination, invention and willingness to venture that brings order and hierarchy to any more unusual material found.


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## Guest

millionrainbows said:


> Actually, the 12 notes of our Western octave (even before equal temperament) we arrived at by a Pythagoran-derived procedure of "stacking" or projecting fifths. After 12 cycles of fifths, the starting point almost coincided with the end point of 12; but not quite, and this is called "The Pythagoran comma."
> 
> Still, the net result was that the "circle" of the octave was closed, and the Western octave is still based on 12 notes arrived at by the 12-division of fifths. The modern tempered fifth is off by only 2 cents (a cent is 1/100th of a semitone), and this is negligible. Our Western system is still based on root movement by fifths. The fifth is the favored interval. By contrast, major thirds are sharp by a full 14 cents! That is very audible, but we seem to have gotten used to it. The fifth is the more important interval anyway, because it creates stability in triads.


All well and good, but you're explaining to us that the sun rises in the east. The most basic fact of temperament is that the Pythagorean series defines a scale that almost closes and centuries were spent tweaking it in various ways.



> You need to explicitly acknowledge that scales create harmonic hierarchies.
> 
> Consonance is simpler vibration, and dissonance is more complex, even chaotic vibrations.
> 
> Western tone-centered music, including folk, popular, and classical, and all forms of basic tone-centered music globally, are based on harmonic (adj.) models.
> 
> These "models" are not the harmonics themselves (used as a noun), but "harmonic models" based on divisions of the octave to "1" or a key note. This can be done with any division of the octave. It produces a "harmonic model" of ratios to "1."
> 
> What this does is, in effect, create a "tonality" with the scale-steps (the divisions), and each step (division) will be a ratio.
> 
> These ratios can be classified in order of their consonance (close relation to "1") or dissonance (more distant relation to "1").
> 
> These can be called "functions" when triads or other chords are built on them, and that "root" will have a function which is a measure of its "tonal gravity" or its tendency to "pull" or "repel" our ear to or from "home" or the key note.


Yes, yes, yes. But at some level musicians did what sounded good to them, and the mathematical back-story was added after the fact.


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## millionrainbows

MR: Scales create harmonic hierarchies.



mikeh375 said:


> Not necessarily MR. The composer's vertical organisation and invention can also determine harmonic (chordal/vertical) hierarchies, which are then reinforced by the actual music written and _manipulated_ to create a pull one way or another to confirm any dominance.


I suppose a composer could determine an hierarchy of his own which contradicts the inherent hierarchy in the scale (the scale which also reinforces the way we actually _hear centricity and tonality__), _if he is some sort of cerebral composer.

Those kind of "willed" hierarchies imposed by a composer are not 'primary' hierarchies; they are derivative, _manipulative,_ and after-the fact.

The hierarchies I refer to are implicit in the ratios of the intervals of each note of the scale to "1" (the key note). What is it that you disagree with? The fact that these are primary, and came first?

In a scale, the pull towards a tonic is inherently determined by vertical harmonic factors, not horizontal "emphasis" by repetition or accent. That comes later.

1. minor seventh (C-Bb) 9:16
2. major seventh (C-B) 8:15
3. major second (C-D) 8:9
4. minor sixth (C-Ab) 5:8
5. minor third (C-Eb) 5:6
6. major third (C-E) 4:5
7. major sixth (C-A) 3:5
8. perfect fourth (C-F) 3:4
9. perfect fifth (C-G) 2:3
10. octave (C-C') 1:2
11. unison (C-C) 1:1

So a C major scale's horizontal functions correspond to these harmonic relations; and one can observe how these functions were derived:

I - 1:1
ii - 8:9
iii - 4:5
IV - 3:4
V - 2:3
vi - 3:5
vii - 8:15

Their importance in establishing the tonality is ranked by the order of consonance to dissonance, with smaller-number ratios being more consonant.

I - 1:1
V - 2:3
IV - 3:4
vi - 3:5
iii - 4:5
ii - 8:9
vii - 8:15

Using this model, a "function" hierarchy can be applied to any scale, after the degrees of dissonance are ranked.

Whole Tone scale: C-D-D-F#-G#-A#

C - 1:1
D -8:9
E -4:5
F#- 45:32
G# - 8:5
A# - 16:9

Whether or not you attach Roman numerals to the above is optional; but by the numbers, one can see a ranking:

C - 1:1
E -4:5
G# - 8:5
D -8:9
A# - 16:9
F#- 45:32


----------



## millionrainbows

Baron Scarpia said:


> All well and good, but you're explaining to us that the sun rises in the east. The most basic fact of temperament is that the Pythagorean series defines a scale that almost closes and centuries were spent tweaking it in various ways.


So what is it you are explaining to us? What problem do you have with the statements I made?



> Yes, yes, yes. But at some level musicians did what sounded good to them, and the mathematical back-story was added after the fact.


I'm unclear as to what you are getting at. My "mathematical back story" is really physics, and the way we hear. Consonance and dissonance are ratios, and we hear them.

Do you think these ratios are just "mathematics," and we can't hear them?

These acoustic facts are not "backstory;" they are the primary basis of tonality. To me, _you_ are thinking backwards.


----------



## Guest

millionrainbows said:


> So what is it you are explaining to us? What problem do you have with the statements I made?
> 
> I'm unclear as to what you are getting at. My "mathematical back story" is really physics, and the way we hear. Consonance and dissonance are ratios, and we hear them.
> 
> Do you think these ratios are just "mathematics," and we can't hear them?


You are posting all of this, and this implies you expect some feedback or reaction. The fact that consonance and dissonance correspond to simple and not-simple relationships between tones is obvious. Part of the content of your many, many posts in this area is restating and restating this obvious and acknowledged fact. It is clearly related to the harmonic series (a one dimensional vibrating object naturally produces a fundamental and its harmonics). It is related to ear physiology. The ear has hair cells that respond to specific frequencies and when two tones related by a simple integer ratio are sounded simultaneously their overtones overlap with each other and produce a simpler sensory experience than two tones whose overtones do not overlap with each other. That is all well, and good, and a bit boring.

When you start talking about your "harmonic models" I find it very speculative and ill defined. My point is that when Palestrina, Monteverdi, Bach, Mozart, etc, invented standard practice harmony, modulations and different ways to use them they were not guided by "harmonic models" they were guided by what sounded good to them. The reasons it sounded good had to do with physics and physiology and leaps of creativity, but I don't personally find much insight in trying to come up with "models" that justify what they did for other reasons. They did it as they did because they found it awesome. Other cultures found other organizations of sound that they likewise found awesome.


----------



## millionrainbows

Baron Scarpia said:


> You are posting all of this, and this implies you expect some feedback or reaction. The fact that consonance and dissonance correspond to simple and not-simple relationships between tones is obvious. Part of the content of your many, many posts in this area is restating and restating this obvious and acknowledged fact. It is clearly related to the harmonic series (a one dimensional vibrating object naturally produces a fundamental and its harmonics). It is related to ear physiology. The ear has hair cells that respond to specific frequencies and when two tones related by a simple integer ratio are sounded simultaneously their overtones overlap with each other and produce a simpler sensory experience than two tones whose overtones do not overlap with each other. That is all well, and good, and a bit boring.
> 
> When you start talking about your "harmonic models" I find it very speculative and ill defined. My point is that when Palestrina, Monteverdi, Bach, Mozart, etc, invented standard practice harmony, modulations and different ways to use them they were not guided by "harmonic models" they were guided by what sounded good to them. The reasons it sounded good had to do with physics and physiology and leaps of creativity, but I don't personally find much insight in trying to come up with "models" that justify what they did for other reasons. They did it as they did because they found it awesome. Other cultures found other organizations of sound that they likewise found awesome.


You seem to have lost your focus. What is speculative about a chart that shows how harmonic function is derived, and that this is _exactly congruent _with the mathematical ratios?

I - 1:1
ii - 8:9
iii - 4:5
IV - 3:4
V - 2:3
vi - 3:5
vii - 8:15

Palestrina: wasn't that before harmony and functions? He probably just did whatever he wanted to do.



> ...when Palestrina, Monteverdi, Bach, Mozart, etc, invented standard practice harmony, modulations and different ways to use them...they were guided by what sounded good to them. The reasons it sounded good had to do with physics and physiology...


Which is the same thing I just showed you.

When you say "they were not guided by 'harmonic models" that's contradictory. The models existed _because_ they were based on sound. I'm just expressing them as ratios.


----------



## Guest

millionrainbows said:


> You seem to have lost your focus. What is speculative about a chart that shows how harmonic function is derived, and that this is _exactly congruent _with the mathematical ratios?
> 
> I - 1:1
> ii - 8:9
> iii - 4:5
> IV - 3:4
> V - 2:3
> vi - 3:5
> vii - 8:15


That is the boring part, not the speculative part. The speculative and (to me) unconvincing part is the bit after "You need to explicitly acknowledge that scales create harmonic hierarchies."


----------



## millionrainbows

Baron Scarpia said:


> That is the boring part, not the speculative part. The speculative and (to me) unconvincing part is the bit after "You need to explicitly acknowledge that scales create harmonic hierarchies."


Does that shatter your illusions? Are you disappointed that music is not all "magic" and "inspiration?"


----------



## fluteman

Millionrainbows -- To use your terminology, "key areas" are established in western music not just by "root movement", but also by patterns and repetition. Rhythm, dynamics and timbre are also used by sophisticated composers to reinforce and complement the tonal system and point the way to the tonal center, often with great subtlety. It's true that over a span of six centuries or so, harmonic progressions played an ever-increasing and more dominant role in western music. But in the early 20th century, there was suddenly a bit of a roll back in that area. 

Schoenberg famously demonstrated that music need not be based on establishing tonal hierarchies at all, and some composers followed that idea in many directions. But even composers who rejected that path understood that their music did not have to be dominated by root movement away from and towards a tonal center around the circle of fifths, or at least not as dominated by that principle as most western music had become by the late 19th century. Those composers rely more on structural elements such as patterns, symmetries and repetition, and elements of music not directly related to pitch such as rhythm, dynamics and timbre, to complement their use of harmonic progressions. They also often use dissonance more extensively, and in some cases microtones, pitch bending, and tonal manipulations made possible with modern technology.

To me, the most important leader of western music away from its slavish obsession with harmonic progression or what you call root movement was not Schoenberg, as important as he was, but Stravinsky, who was to music what his good friend Picasso was to visual art. He demonstrated that, while it is possible to abandon traditional western harmony entirely in favor of serialism (and later, musique concrete, indeterminacy, and other alternative systems), much could be accomplished by simply reducing, even if only slightly, the dominant role harmonic progressions had reached in western music by the late 19th century in favor of other elements, many of which could be made to work well with, and even enhance, harmonic progressions along the circle of fifths.

That is why so many contemporary composers owe so much, almost everything, really, to Stravinsky. Even composers who quite consciously took Schoenberg's path like Boulez (who was such a disciple he didn't think he could advance his own ideas without first proclaiming "Schoenberg is dead!") very much show the influence of Stravinsky.


----------



## millionrainbows

fluteman said:


> Millionrainbows -- To use your terminology, "key areas" are established in western music not just by "root movement", but also by patterns and repetition. Rhythm, dynamics and timbre are also used by sophisticated composers to reinforce and complement the tonal system and point the way to the tonal center, often with great subtlety. It's true that over a span of six centuries or so, harmonic progressions played an ever-increasing and more dominant role in western music. But in the early 20th century, there was suddenly a bit of a roll back in that area.


Harmonic progressions, as strictly defined by CP Western harmony, are derived from the major/minor scales. If a modern composer deviated from that, and used other 'exotic' scales, diminished scales, whole tone, and the kind of things you might see in Slonimsky's Thesaurus of Scales, I don't see what's so "innovative" about that, except that it has expanded the possibilities from the limitations of the CP major/minor system. The principles I have outlined would still apply to those scales which lie outside the purview of CP tonality, and that's why I think they're important, since they are harmonic principles which can apply to areas well outside CP harmonic practice.



> To me, the most important leader of western music away from its slavish obsession with harmonic progression or what you call root movement was not Schoenberg, as important as he was, but Stravinsky, who was to music what his good friend Picasso was to visual art. He demonstrated that, while it is possible to abandon traditional western harmony entirely in favor of serialism (and later, musique concrete, indeterminacy, and other alternative systems), much could be accomplished by simply reducing, even if only slightly, the dominant role harmonic progressions had reached in western music by the late 19th century in favor of other elements, many of which could be made to work well with, and even enhance, harmonic progressions along the circle of fifths.
> 
> That is why so many contemporary composers owe so much, almost everything, really, to Stravinsky. Even composers who quite consciously took Schoenberg's path like Boulez (who was such a disciple he didn't think he could advance his own ideas without first proclaiming "Schoenberg is dead!") very much show the influence of Stravinsky.


Stravinsky was an innovator, without a doubt, and I think there were also many other innovators, like Slonimsky for his work in scales.
In Boulez' defense, I think he saw Schoenberg as an impediment to musical progress in the serial area, since Schoenberg openly acknowledged that he was a conservative in the late Romantic tradition. Boulez saw the possibilities for expansion of Schoenberg's direction, whereas Schoenberg's time was over, and he was, indeed, dead by this time.

I don't see Stravinsky as being as great an innovator as you do. Schoenberg and Boulez' ideas were more directed away from harmony itself, and in that sense are more radical than Stravinsky, who remained an expanded tonalist/harmonicist until the time of his dabbling with serialism in "Mouvements."

So I see all "harmonic" music as being tonal, or as using principles of tonality, and these are all traceable to "harmonic models" or scales of one kind or another. It doesn't matter if it's Stravinsky or jazz or forms of ethnic music. I trust you see what I am "after" now.


----------



## fluteman

millionrainbows said:


> Harmonic progressions, as strictly defined by CP Western harmony, are derived from the major/minor scales. If a modern composer deviated from that, and used other 'exotic' scales, diminished scales, whole tone, and the kind of things you might see in Slonimsky's Thesaurus of Scales, I don't see what's so "innovative" about that, except that it has expanded the possibilities from the limitations of the CP major/minor system. The principles I have outlined would still apply to those scales which lie outside the purview of CP tonality, and that's why I think they're important, since they are harmonic principles which can apply to areas well outside CP harmonic practice.
> 
> Stravinsky was an innovator, without a doubt, and I think there were also many other innovators, like Slonimsky for his work in scales.
> In Boulez' defense, I think he saw Schoenberg as an impediment to musical progress in the serial area, since Schoenberg openly acknowledged that he was a conservative in the late Romantic tradition. Boulez saw the possibilities for expansion of Schoenberg's direction, whereas Schoenberg's time was over, and he was, indeed, dead by this time.
> 
> I don't see Stravinsky as being as great an innovator as you do. Schoenberg and Boulez' ideas were more directed away from harmony itself, and in that sense are more radical than Stravinsky, who remained an expanded tonalist/harmonicist until the time of his dabbling with serialism in "Mouvements."
> 
> So I see all "harmonic" music as being tonal, or as using principles of tonality, and these are all traceable to "harmonic models" or scales of one kind or another. It doesn't matter if it's Stravinsky or jazz or forms of ethnic music. I trust you see what I am "after" now.


Yes, I think I do see what you are after. I think you greatly underestimate the significance, or potential significance, of elements of music unrelated or not directly related to scales, harmony, harmonic progressions and root movement. I think your analysis begins with a premise that is very much grounded in the traditional conventions of western music prior to the 20th century, and thus you endlessly struggle against the limitations you impose on yourself. You call yourself "innovative", but you never consider thinking outside the box you have built for yourself.

As for Stravinsky: One of Stravinsky's greatest innovations is the use of rhythm, dissonance and polytonality to convey his story in a complex and subtle, but effective way. Listen to Petrushka, and notice how dissonance (including the famous C-F# tritone and accented minor and major seconds) and halting and irregular rhythms are used, for example to portray the chaotic crowd at the Shrovetide fair, or the puppet Petrushka, which is not quite alive but not quite an inanimate piece of wood either, but rather something in between in an unstable way, to its tragic frustration. Stravinsky steals melodies from popular songs, including one he heard from the street beneath his Paris studio as he was writing the music, but they are always refracted through the prism of dissonance, polytonality and irregular rhythms.






Now, jazz is an American idiom mainly derived from the standard 19th century European tonal tradition, but with a crucial difference: a much greater emphasis on rhythm, probably due to a west African influence carried down by the descendants who were generally only one or two generations removed from the era of slavery, a period when descendants of people shipped in from western Africa were prevented from assimilating into American society and preserved African cultural traditions more than they may have otherwise. Stravinsky understood this innovative emphasis on rhythm and seized it for his own purposes. He isn't as concerned with other elements of jazz. The Ebony Concerto, written for jazz clarinet great Woody Herman, has a percussion part that often sounds far more African than jazzy, though obviously, there is nothing African about a clarinet. Listening to the opening measures, Stravinsky immediately dives into a thumping, rhythmically irregular vamp. Traditional jazz usually opens with a relatively conventional lyrical western-style melody to ground it in the familiar European tradition before the rhythmic variations take over (and harmonic variations, of course). But Stravinsky is less interested in those aspects of jazz. There is no real melody until well into the work, and even then, the rhythm section dominates with its jagged, irregular vamp. Of course, ultimately, jazzy clarinet riffs and trombone slides, a New Orleans-style funeral march, and other traditional jazz elements emerge. But rhythmic energy and variety always dominate.






It's amazing to think that when Stravinsky was born, Brahms and Tchaikovsky were still alive. In many ways, Stravinsky's music is a vastly greater departure from that of Brahms than is Schoenberg's, who as you correctly say and as Boulez understood in many important ways was conservative in the late Romantic tradition. If you free your mind and ear to listen for dissonance, polytonality and complex rhythms rather than just scales and root movement, you can hear Stravinsky's revolution. He didn't invent any of those things. The innovation was to put them at center stage rather than letting scales and harmony dominate the proceedings unchallenged. That is reflected in nearly all serious western music since then.

Alas, many here are even more narrow-minded about these things than you are. You, at least, are willing to think about scales in innovative ways. (A Thai 7-tone scale? Cool!) They seem to see the 12-tone equal tempered scale and the circle of fifths as necessarily the central elements of all music.


----------



## Luchesi

for the math people in here, here’s what I get;

derived from the C fundamental with A = 440 hz 

when they're tempered to be in the equidistant series —— and then from Nature

196.22 G 196

327.03 E 329.63

392.44 G 392

457.84 A 440

457.84 Bb 466.16 

588.66 D 587.33



19th Nervous Breakdown - it’s surprising to me that these notes below are so close

1242.72 Eb 1244.51 


---------

the rest of the higher notes in the harmonic series included below

654.06 E 659.26

719.47 F 698.46

784.88 G 783.99

850.28 Ab 830.61

850.28 A 880

981.1 B 987.77

1111.91 C# 1108.73

1177.32 D 1174.66


----------



## millionrainbows

fluteman said:


> Yes, I think I do see what you are after. I think you greatly underestimate the significance, or potential significance, of elements of music unrelated or not directly related to scales, harmony, harmonic progressions and root movement. I think your analysis begins with a premise that is very much grounded in the traditional conventions of western music prior to the 20th century, and thus you endlessly struggle against the limitations you impose on yourself. You call yourself "innovative", but you never consider thinking outside the box you have built for yourself.


I may seem in a box, but that's because I'm just talking about "tonal" or harmonic music. What other alternatives are you talking about? Stravinsky is not. Serialism and related? Microtonality?

As far as rhythmic innovation, I see Frank Zappa as more innovative than Stravinsky; he took the nested tuplets and irrational rhythms of Stockhausen and Boulez and applied them to his own tonal music, as in "The Black Page."


----------



## fluteman

millionrainbows said:


> I may seem in a box, but that's because I'm just talking about "tonal" or harmonic music. What other alternatives are you talking about? Stravinsky is not. Serialism and related? Microtonality?
> 
> As far as rhythmic innovation, I see Frank Zappa as more innovative than Stravinsky; he took the nested tuplets and irrational rhythms of Stockhausen and Boulez and applied them to his own tonal music, as in "The Black Page."


There would be no Zappa without Stravinsky. Zappa's first important classical music influence, according to him, was Edgard Varese, who in turn was heavily influenced by Stravinsky. The "irrational" rhythms of Stockhausen and Boulez are derived directly from Stravinsky. The imo best commercial recording of Stravinsky's The Rite of Spring is conducted not by Stravinsky, but by Boulez. That is no accident. Also, compare the orchestrated version of this Boulez piece, Notations II, to Stravinsky's Shrovetide Fair scene from Petrushka that I discussed above, or to his Sacrificial Dance from The Rite of Spring. Isn't the influence obvious?






You are in a box, not because you are just taking about tonal or harmonic music, but because you focus solely on one aspect of that music, whether you call it harmonic progression, root movement, or something else. Yes, it is possible to focus on that aspect of music yet move away from the conventional principle of the twelve tone circle of fifths. Easley Blackwood's 12 Microtonal Etudes is a good example. These misnamed keyboard etudes are not microtonal at all, rather they are based on equal-tempered scales of 13 through 24 notes. The overall result is to de-emphasize the dominant-tonic movement (its still there, sort of) since there are so many other intervals and distant "keys" to contend with, and produce a spooky, otherworldly feel that would go well with a high-tech science fiction movie involving aliens and other galaxies, though Blackwood cheerfully employs romantic, classical and even early baroque forms.

So that is an example of an alternative harmonic model. Obviously, the 7-tone Thai scale you cite and other traditional non-western music is based on alternative harmonic models. But for me, what most decisively separates that kind of music, or at least that of it I have heard, from traditional western music from an harmonic point of view is the lack of equal temperament.

Experiments in "harmonic models" are not only possible, but increasingly routine in this electronic age when it is easy to abandon the 12-tone equal tempered scale that traditional western instruments are designed to play. Other composers stick to the conventional 12-tone circle of fifths but wander ever further away and in more elaborate ways from the tonal center. To me, that also often imparts an exotic, dreamlike, otherworldly flavor. All well and good, but what interests me far more is music where the focus is not so entirely on harmony, though harmony may be an important element.


----------



## Luchesi

fluteman said:


> There would be no Zappa without Stravinsky. Zappa's first important classical music influence, according to him, was Edgard Varese, who in turn was heavily influenced by Stravinsky. The "irrational" rhythms of Stockhausen and Boulez are derived directly from Stravinsky. The imo best commercial recording of Stravinsky's The Rite of Spring is conducted not by Stravinsky, but by Boulez. That is no accident. Also, compare the orchestrated version of this Boulez piece, Notations II, to Stravinsky's Shrovetide Fair scene from Petrushka that I discussed above, or to his Sacrificial Dance from The Rite of Spring. Isn't the influence obvious?
> 
> 
> 
> 
> 
> 
> You are in a box, not because you are just taking about tonal or harmonic music, but because you focus solely on one aspect of that music, whether you call it harmonic progression, root movement, or something else. Yes, it is possible to focus on that aspect of music yet move away from the conventional principle of the twelve tone circle of fifths. Easley Blackwood's 12 Microtonal Etudes is a good example. These misnamed keyboard etudes are not microtonal at all, rather they are based on equal-tempered scales of 13 through 24 notes. The overall result is to de-emphasize the dominant-tonic movement (its still there, sort of) since there are so many other intervals and distant "keys" to contend with, and produce a spooky, otherworldly feel that would go well with a high-tech science fiction movie involving aliens and other galaxies, though Blackwood cheerfully employs romantic, classical and even early baroque forms.
> 
> So that is an example of an alternative harmonic model. Obviously, the 7-tone Thai scale you cite and other traditional non-western music is based on alternative harmonic models. But for me, what most decisively separates that kind of music, or at least that of it I have heard, from traditional western music from an harmonic point of view is the lack of equal temperament.
> 
> Experiments in "harmonic models" are not only possible, but increasingly routine in this electronic age when it is easy to abandon the 12-tone equal tempered scale that traditional western instruments are designed to play. Other composers stick to the conventional 12-tone circle of fifths but wander ever further away and in more elaborate ways from the tonal center. To me, that also often imparts an exotic, dreamlike, otherworldly flavor. All well and good, but what interests me far more is music where the focus is not so entirely on harmony, though harmony may be an important element.


"To me, that also often imparts an exotic, dreamlike, otherworldly flavor."

Are these responses universal or are there differences among people, in your experience?

I think all people respond in the same way, but why is that?

I've tuned pianos for many years and at this point it's not even surprising to me anymore that it's so jarring to people. I mean, there are a few people who remain comfortable. They adjust to it? You see pianos out of tune on YouTube videos. I bring this up because I just had this discussion with my violinist who has an exceptionally good ear, while I don't. Half jokingly I said I have the advantage of having a bad ear and therefore I don't accidentally transpose a pop piece while trying to work it out. I stay in the one key, while he will figure it all out and then realize he's changed the key. < I'd rather have his 'problem'>


----------



## fluteman

Luchesi said:


> "To me, that also often imparts an exotic, dreamlike, otherworldly flavor."
> 
> Are these responses universal or are there differences among people, in your experience?
> 
> I think all people respond in the same way, but why is that?
> 
> I've tuned pianos for many years and at this point it's not even surprising to me anymore that it's so jarring to people. I mean, there are a few people who remain comfortable. They adjust to it? You see pianos out of tune on YouTube videos. I bring this up because I just had this discussion with my violinist who has an exceptionally good ear, while I don't. Half jokingly I said I have the advantage of having a bad ear and therefore I don't accidentally transpose a pop piece while trying to work it out. I stay in the one key, while he will figure it all out and then realize he's changed the key. < I'd rather have his 'problem'>


Human hearing is a very complicated thing, obviously. But imo, those of us who grew up entirely within the 12-tone, equal tempered world, can have certain musical expectations, or what some call, I think somewhat inaccurately, "tastes", deeply ingrained. Those of us extensively exposed starting at a young age to classical music from the 19th century and earlier, and popular music from before the 1960s, maybe even more so than those who grew up listening to popular music from the 1960s and later. Without necessarily realizing it, we can have strong harmonic expectations, and greatly varying tolerances for dissonance or polyphony, based on our particular background and experience (and I'm not referring to formal musical training or education).

As for good ear v. bad ear, it's hard to say exactly what those terms mean. It is well known that the ear will eventually adjust to slightly out-of-tune tones if they are heard long and loud enough, and hear them as in tune. So it's hard to compare one's own hearing in different contexts, much less with the hearing of others.


----------



## BabyGiraffe

fluteman said:


> But imo, those of us who grew up entirely within the 12-tone, equal tempered world, can have certain musical expectations, or what some call, I think somewhat inaccurately, "tastes", deeply ingrained. Those of us extensively exposed starting at a young age to classical music from the 19th century and earlier, and popular music from before the 1960s, maybe even more so than those who grew up listening to popular music from the 1960s and later. Without necessarily realizing it, we can have strong harmonic expectations, and greatly varying tolerances for dissonance or polyphony, based on our particular background and experience (and I'm not referring to formal musical training or education).


Hm, classical music is not really performed in 12 equal, despite that's the theoretical framework...

Modern electronic and pop music is another story (because of midi standard and auto-tune software).

" It is well known that the ear will eventually adjust to slightly out-of-tune tones if they are heard long and loud enough, and hear them as in tune."

Considering that most 12 equal steps are not in tune unless you use some kind of 17 and 19-limit ratios, I don't see the problem.

17/16 104.955410 17th harmonic
9/8 203.910002 major whole tone
19/16 297.513016 19th harmonic
24/19 404.441985 smaller undevicesimal major third
4/3 498.044999 perfect fourth
17/12 603.000409 2nd septendecimal tritone
3/2 701.955001 perfect fifth
19/12 795.558015 undevicesimal minor sixth
32/19 902.486984 19th subharmonic
16/9 996.089998 Pythagorean minor seventh
32/17 1095.044590 17th subharmonic

About the stylistic expectations, that's not something related to the musical system.
Modern (and early) classical music as a whole cannot be easily reduced to certain temperament or melodic/harmonic formulas. (But Galant style and late baroque - can; at least in theory, actual performance with authentic historical improvisatory and melismatic gestures, instruments and tuning is something that we cannot be sure about until we discover the time machine...)


----------



## millionrainbows

BabyGiraffe said:


> Hm, classical music is not really performed in 12 equal, despite that's the theoretical framework...


Explain that, and back it up! On the surface, it sounds absurd. Classical music does not use equal temperament? What kind of wacky, microtonal, hare-brained theory will you use to back this up?

Supposedly, the ear cannot hear a 4 cent difference in pitch, so our ET fifths, only 2 cents flat, are for all intents and purposes "in tune." _I dare you to refute that!

_BTW, what is that chart suppose to represent, and what's it got to do with what we are discussing?




> Considering that most 12 equal steps are not in tune unless you use some kind of 17 and 19-limit ratios, I don't see the problem.


No. they wouldn't be in tune. They'll never be in tune with a closed octave 1:1, because 1 is not divisible by 3:2 and 4:5!

That's because the 12-note chromatic scale was derived from the projection of the fifth; thus the circle of fifths. This is an acoustically-based method.

But actually, the 12-note scale is an anomaly, an approximation, based on the attempt to close the octave after 12 cycles of 3:2 fifths.

Thus, "12" is the resulting mathematical result of this error or approximation; there is no acoustic ("tonal") reason for its existence, other than that it approximates fifths. In ET, all these fifths are 2 cents flat, to compensate for this error, and to close the octave, which would otherwise spiral onward into irrational values. *No ratio, such as the 3:2 fifth, can be divided into "1" (the octave) as a whole number (such as 12). That means no "17" or "19" cycle can accomplish this either!*

Thus, all the resulting symmetries created by "12" are mathematical in nature, and thus have a way of degrading tonality's supposed "acoustic" nature of ratios, and turning it into a mathematically/geometrically based system. Thus, the "undoing" of tonality was always inherent in the "12" based scale of Pythagoras.


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## BabyGiraffe

millionrainbows said:


> Explain that, and back it up! On the surface, it sounds absurd. Classical music does not use equal temperament? What kind of wacky, microtonal, hare-brained theory will you use to back this up?
> 
> Supposedly, the ear cannot hear a 4 cent difference in pitch, so our ET fifths, only 2 cents flat, are for all intents and purposes "in tune." _I dare you to refute that!
> 
> _


_

Fifths and fourths? I have yet to hear a song that uses only fifths and fourths, even most primitive music is usually more complex and has at least 4 pitches in the melody...

About the classical music - of course, players use adaptive tuning - basically you adjust the intervals (which aren't 12 equal and depend on the instrument) to fit the context and sound good. What is interesting is that sometimes players tune to a 12 equal melody (some a capella vocal groups also do this), but the triadic harmony is not 12 equal - using flatter major thirds or sharper minor thirds. 
It's easy to check using something like melodyne (non-realtime polyphonic autotune software that uses spectral resynthesis) how off are the intervals from the theoretical framework in any recording. (It is ironic that autotune software sometimes can ruin perfectly fine vocal takes, because the singer didn't perform to the 12 equal pitches, wow.)_


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## millionrainbows

BabyGiraffe said:


> Fifths and fourths? I have yet to hear a song that uses only fifths and fourths, even most primitive music is usually more complex and has at least 4 pitches in the melody...


You're always complaining about ET and its out-of-tune fifths...at least ET fifths give root movement some stability.


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## millionrainbows

BabyGiraffe said:


> About the classical music - of course, players use adaptive tuning - basically you adjust the intervals (which aren't 12 equal and depend on the instrument) to fit the context and sound good. What is interesting is that sometimes players tune to a 12 equal melody (some a capella vocal groups also do this), but the triadic harmony is not 12 equal - using flatter major thirds or sharper minor thirds.


Those "just" major thirds sound like crap. They need to be sharper.


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## fluteman

BabyGiraffe said:


> About the stylistic expectations, that's not something related to the musical system.
> Modern (and early) classical music as a whole cannot be easily reduced to certain temperament or melodic/harmonic formulas. (But Galant style and late baroque - can; at least in theory, actual performance with authentic historical improvisatory and melismatic gestures, instruments and tuning is something that we cannot be sure about until we discover the time machine...)


Ah, but it is, BabyGiraffe. If you've spent enough time reading the posts here, you can see certain patterns emerging. There is much hostility towards modern "classical" music, especially that which is (misleadingly, imo) called "atonal", the earliest of which is now over a century old. Renaissance and early baroque music have their adherents here, but they are not in the majority. [Edit: And there is much hostility towards "HIP" performances of that music, which include older intonation systems.] Non-western classical music also has its fans, but you don't see many here. (Though, in that case, I have to concede that the focus of talkclassical is on western classical music, and that's reasonable enough.) One of the moderators here, mmbls, who is probably moderating this thread, once pronounced that 20th century music was harder to understand and learn than 19th or 18th century music. Others here have gone even further and concluded that the 20th century was a dud for classical music compared to the 19th or 18th centuries. Read those many posts carefully, and you can only conclude that stylistic expectations are very closely related to the musical system.


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## BabyGiraffe

millionrainbows said:


> Those "just" major thirds sound like crap. They need to be sharper.


This reminds me of the anecdote about the kid with absolute pitch that was trained on a out of tune piano, wow.

Fluteman:

"Read those many posts carefully, and you can only conclude that stylistic expectations are very closely related to the musical system."

I don't know, it's hard to generalize, even back then we had plenty of diversity. It's just that few, selected composers are considered representative of the musical styles.

About generalizing from meantone or pythagorean to the all other possible musical systems as a whole - that doesn't work at all, because they can be quite different.

An example: there is a scale (playable even in 12 equal, but it's close to its just version there, so it's untempered) with defining unison vectors - 25/24 - chromatic semitone and 250/243 - maximal diesis of 49 cents. (Diatonic meantone, the European tuning is defined by 25/24 and 81/80). The resulting scale is called Romanian minor in 12 equal (or it's very close to it). (The only classical composer that I have heard using it is actually Stravinsky.)
The just version is 
0: 1/1 0.000000 unison, perfect prime
1: 27/25 133.237575 large limma, BP small semitone, Zarlino semitone
2: 5/4 386.313714 major third
3: 25/18 568.717426 classic augmented fourth
4: 3/2 701.955001 perfect fifth
5: 5/3 884.358713 major sixth, BP sixth
6: 9/5 1017.596288 just minor seventh, BP seventh
7: 2/1 1200.000000 octave

Still, just like 81/80 is the interval, tempering which leads to meantone, tempering 250/243 leads to a similar family of musical systems - like in 15 or 22, 29 and other equal tunings. It actually has major and minor thirds in the scale and is pretty good, I recommend 22 equal for testing it. The musical logic in this temperament and equal temperaments that contain it are also different.
While meantone goes like 5->7->12->19->31 etc note scales, this one goes like 7->8->15->22 etc. Modulations are different, different chord progressions, optimal notation and note naming should be also different etc.

(Any periodicity block with unison vector of 25/24 actually has major/minor thirds - that's the difference between them - it's tempered for example in 7 equal, which is neither major or minor.)


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## BabyGiraffe

millionrainbows said:


> You're always complaining about ET and its out-of-tune fifths...at least ET fifths give root movement some stability.


They could... if the fifth was 720 cents, not around 702. It's simple math, just do it.
(If we play them along the major thirds of 12 equal, of course, in a triadic style - you know, like in every pop song).

Btw, I have never complained about 12 equal fifths.

You can never have good fifths in meantone (that's not full with other distorted intervals). Or good fifths and thirds in combination. Why - it's easy to do the math - harmonic and arithmetic mean, you know how ancient Greeks came with their musical scales - 10/9 is the harmonic mean between a major third and the root; 9/8 is the arithmetic mean; there is no way to compromise this relation without detuning lots of stuff in our musical system - 9/8:10/9 = 81/80 - the interval that doesn't exist officially in European music theory. (Btw, the differences between major and minor thirds and between fifths and fourths have the same relations, but noone is dumb enough to try and temper them to "improve" the complexity of the music system.)
Close to pure sounds require something like schismic or diaschismic temperament. And these are pain in the *** - too complex, I guess - you have to deal with a more than one comma when modulating and also when playing simple chord progressions.


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## millionrainbows

BabyGiraffe said:


> They could... if the fifth was 720 cents, not around 702. It's simple math, just do it.
> (If we play them along the major thirds of 12 equal, of course, in a triadic style - you know, like in every pop song).
> 
> Btw, I have never complained about 12 equal fifths.


Then quit complaining about ET so much. At least it has good fifths. 720 cents is a wolf fifth. Why don't you explain some of what you're talking about?

Oh, you must mean *IF* a fifth was 720 cents. 

Ah ha ha ha ha ha.


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## fluteman

BabyGiraffe said:


> I don't know, it's hard to generalize, even back then we had plenty of diversity. It's just that few, selected composers are considered representative of the musical styles.


Well, no, that's not an accurate historical description of western classical music before Stravinsky. Charles Rosen discusses this in The Classical Style, where he demonstrates that Beethoven's piano sonatas, for example, were almost certainly written with the equal-tempered scale in mind. (Yes, Rosen considers only a few select composers, e.g., Haydn, Mozart and Beethoven, but in the long term these were by far the most influential of the classical period.) This is a significant evolution from the days of J.S. Bach. Many scholars argue that the Well-tempered Clavier was not composed with equal temperament in mind. Some suggest Bach intended the use of Werckmeister III, where the Pythagorian comma is equally distributed over four neighboring fifths (rooted on C, G, D and B).

But what clinched the near-universal adoption of equal temperament was the increasingly dominant role of the piano in western music. Inventions such as Erard's double escapement action and Steinway's overstrung cast iron frame made it powerful and flexible, and before the phonograph and radio came along, the piano was the home music system of the middle class. (I estimate the golden age of the piano as from about 1820 to about 1920.) It could be tuned differently, but the piano is designed to play equal-tempered, diatonic scales, most easily C major and nearby keys. Wind and brass instruments were redesigned to more easily accommodate the 12-tone equal tempered scale. You are naive, and certainly not a musician yourself, if you don't understand the enormous impact of these developments on nearly all western music.

Yes, a cappella choral singers deviate from equal temperament (been there, sung that), but a cappella singing has long been a relatively small niche in western music, for better or worse.


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## BabyGiraffe

fluteman said:


> Charles Rosen discusses this in The Classical Style, where he demonstrates that Beethoven's piano sonatas, for example, were almost certainly written with the equal-tempered scale in mind. (Yes, Rosen considers only a few select composers, e.g., Haydn, Mozart and Beethoven, but in the long term these were by far the most influential of the classical period.) This is a significant evolution from the days of J.S. Bach. Many scholars argue that the Well-tempered Clavier was not composed with equal temperament in mind.
> 
> Wind and brass instruments were redesigned to more easily accommodate the 12-tone equal tempered scale. You are naive, and certainly not a musician yourself, if you don't understand the enormous impact of these developments on nearly all western music.
> 
> Yes, a cappella choral singers deviate from equal temperament (been there, sung that), but a cappella singing has long been a relatively small niche in western music, for better or worse.


Dude, string and brass instruments rely on harmonics, you are not a musician, if you don't understand how they work. You have to manually adjust the intonation all the time. It's not that easy compared to a fretted guitar for example.

Unequal temperaments, that some composers like Bach used, is basically a mixture between pythagorean and meantone. There are probably infinite ways to create such systems mathematically. We get some out of tune (in 5-limit) intervals as result, but no wolves. (Mapping was 12 equal's val. I have a friend who works as a tuner that created a special 19 keys unequal temperament for my needs.)

Beethoven may have used unequal temperament.

Rosen argues that some of his enharmonic modulations sound awful in just intonation, but we can't know, if he used equal or unequal temperament. His early music is most certainly in meantone. Considering how often he relies on augmented sixths as effect, I am not sure that he used 12-equal temperament at all.


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## millionrainbows

fluteman said:


> Many scholars argue that the Well-tempered Clavier was not composed with equal temperament in mind. Some suggest Bach intended the use of Werckmeister III, where the Pythagorian comma is equally distributed over four neighboring fifths (rooted on C, G, D and B).


I'm going with the ideas of Dr. Bradley Lehman (larips.com) who deciphered Bach's well-tempered tuning from a decorative flourish on the cover page of the original manuscript. It was basically an attempt at ET, in which all keys sounded good.



> ...but the piano is designed to play equal-tempered, diatonic scales, most easily C major and nearby keys.


What you say is true, especially in light of key signatures, but Chopin later used distant keys like A flat because of ergonomics: the hand sits naturally on the piano when the middle fingers are covering black keys, and the thumb and pinkies cover white keys. He said that C major was the most difficult scale to play because of the thumb cross-under, and started his students out with other scales like A flat. See book "Natural Fingering."


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## millionrainbows

BabyGiraffe said:


> Beethoven may have used unequal temperament. Rosen argues that some of his enharmonic modulations sound awful in just intonation, but we can't know, if he used equal or unequal temperament. His early music is most certainly in meantone. Considering how often he relies on augmented sixths as effect, I am not sure that he used 12-equal temperament at all.


Other sources indicate that Beethoven probably used Thomas Young tuning. Nobody had ET back then, it wasn't achieved until 1919. Many tunings were approaching ET, and that's what things were moving towards, but none achieved it. Nevertheless, Bach's tuning (Lehman/Bach) and other tunings were approximations, by ear & stopwatch, of ET.

View attachment 125615


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## BabyGiraffe

millionrainbows said:


> It was basically an attempt at ET, in which all keys sounded good.
> 
> What you say is true, especially in light of key signatures, but Chopin later used distant keys like A flat because of ergonomics: the hand sits naturally on the piano when the middle fingers are covering black keys, and the thumb and pinkies cover white keys. He said that C major was the most difficult scale to play because of the thumb cross-under, and started his students out with other scales like A flat. See book "Natural Fingering."


You still get horrible intervals, there is no such thing as system with just 12 keys that sounds good. What you don't get - unlike any meantone, there is no mistuned fifth (but enjoy other mistuned intervals). The closest to sounding good with just 12 keys per octave is 12 equal, wow, which sounds "bad" (to purists). We need minimum 43 equal for meantone that sounds close to 5-limit just intonation. 31 keys sounds also ok and was used in theory - it is known as 1/4 comma meantone.
https://en.wikipedia.org/wiki/Quarter-comma_meantone

The guy that posts in the historical performance thread has quotes from Chopin letters about unequal temperament or it was meantone?... What was the name of the thread - I remember it was in general discussions.


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## millionrainbows

I'd like to hear serial music played in 43-tone tuning.


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## fluteman

BabyGiraffe said:


> Dude, string and brass instruments rely on harmonics, you are not a musician, if you don't understand how they work. You have to manually adjust the intonation all the time. It's not that easy compared to a fretted guitar for example.
> 
> Unequal temperaments, that some composers like Bach used, is basically a mixture between pythagorean and meantone. There are probably infinite ways to create such systems mathematically. We get some out of tune (in 5-limit) intervals as result, but no wolves. (Mapping was 12 equal's val. I have a friend who works as a tuner that created a special 19 keys unequal temperament for my needs.)
> 
> Beethoven may have used unequal temperament.
> 
> Rosen argues that some of his enharmonic modulations sound awful in just intonation, but we can't know, if he used equal or unequal temperament. His early music is most certainly in meantone. Considering how often he relies on augmented sixths as effect, I am not sure that he used 12-equal temperament at all.


After playing the flute for 50 years, I do have some idea about how it works, including its use of harmonics. In the case of the flute, there was a dramatic change based on the radical invention of Theobald Boehm in 1847 (which also influenced the development of the other winds). The old system flute consisted of a wood tube with tone holes where the fingers could reach them and sized so the fingers could cover them, with a few keys for additional holes for the chromatic scale, or for the lowest notes where the fingers weren't long enough to reach. Boehm made the flute with tone holes optimally located and sized for the 12-tone equal tempered scale, with optimum venting, regardless of what the fingers could reach or cover, with the holes covered by a system of keys connected by rods, clutches and springs.

The second octave notes on the Boehm flute are still essentially overblown second harmonics, and the third octave overblown third harmonics, but with adjustments to keep them in equal-tempered tune. It's a highly sophisticated yet practical acoustic system with a few minor compromises, especially in the venting holes, so all 12 tones in 3 octaves can be played without too complicated and heavy a mechanism. As a result, 2 or 3 "problem" notes can be slightly harder to play in tune. But in more recent years, very slight and sophisticated modifications have been made to the scale (tone hole spacing and size), often with the help of computer programs, to get a more uniformly satisfactory result.

The end result is an acoustic instrument that can play three octaves plus of the 12-tone equal tempered scale, with a surprising degree of intonation accuracy and consistency. This is true even of the older Boehm flutes, made before modern computer-assisted improvements and as far back as the mid-19th century, though alas many of those are tuned to a=435 or a=438, or even as high as a=452, rather than a=440, which didn't become the international standard until 1938.

Unfretted string instruments are obviously another story, and didn't need to be redesigned for equal temperament. But as I said above, the development and success of the modern piano is probably the biggest single factor in making equal temperament the universal standard of western music. It was ideal for playing diatonic scales and triads in any key with relative ease.


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## BabyGiraffe

Many developers for Kontakt (and similar engines) absolutely always autotune the samples, because many customers (check vi-control forum for many such threads from the last decade) complain about unplayable keys. And many of these libraries record the best orchestral musicians available (Vienna, London, L.A.).

Something funny:

"



"


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## fluteman

millionrainbows said:


> I'm going with the ideas of Dr. Bradley Lehman (larips.com) who deciphered Bach's well-tempered tuning from a decorative flourish on the cover page of the original manuscript. It was basically an attempt at ET, in which all keys sounded good.
> 
> What you say is true, especially in light of key signatures, but Chopin later used distant keys like A flat because of ergonomics: the hand sits naturally on the piano when the middle fingers are covering black keys, and the thumb and pinkies cover white keys. He said that C major was the most difficult scale to play because of the thumb cross-under, and started his students out with other scales like A flat. See book "Natural Fingering."


Yes, I read about that analysis of the cover page of the WTC manuscript. Very interesting. I was really only making a general statement that while equal temperament was around by the early 18th century, by Beethoven's time it was well on its way to becoming standard, and the dominance of the piano from the mid-19th century on helped seal the deal. And all this isn't some amazing new insight by me. Historians know all about it.

It's appropriate that you mention Chopin. He was a pioneer in expanding piano technique, and that included playing ferocious virtuoso passages and entire pieces in any key. And in fact, one of the most important, if not the most important, features of equal temperament is that every interval is identical in every key. So any or all 12 keys can be used, in the same piece even, with no odd sounding intervals or chords. Again, no special insight by me. I would think it's obvious how this has had a profound impact on western music and how we hear it.


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## millionrainbows

fluteman said:


> Yes, I read about that analysis of the cover page of the WTC manuscript. Very interesting. I was really only making a general statement that while equal temperament was around by the early 18th century, by Beethoven's time it was well on its way to becoming standard, and the dominance of the piano from the mid-19th century on helped seal the deal. And all this isn't some amazing new insight by me. Historians know all about it.
> 
> It's appropriate that you mention Chopin. He was a pioneer in expanding piano technique, and that included playing ferocious virtuoso passages and entire pieces in any key. And in fact, one of the most important, if not the most important, features of equal temperament is that every interval is identical in every key. So any or all 12 keys can be used, in the same piece even, with no odd sounding intervals or chords. Again, no special insight by me. I would think it's obvious how this has had a profound impact on western music and how we hear it.


Well, I'm so glad that you've been pleased with what I've said here. BabyGiraffe, as usual, always has some fascinating insights as well.

Is everybody happy?


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## fluteman

millionrainbows said:


> Well, I'm so glad that you've been pleased with what I've said here. BabyGiraffe, as usual, always has some fascinating insights as well.
> 
> Is everybody happy?


I'm happy so long as I can get out of bed in the morning and walk down the street (so far, so good). But I'd be even happier if you discontinued those pre-enlightenment, anti-empiricist philosophy posts. There is no theory that explains what makes good music good, and that includes harmonic theory. Music is a fundamentally practical, empirical art, and imo, the best explanation is that of Duke Ellington: If it sounds good, it _is good. ;-)_


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## millionrainbows

fluteman said:


> I'm happy so long as I can get out of bed in the morning and walk down the street (so far, so good). But I'd be even happier if you discontinued those pre-enlightenment, anti-empiricist philosophy posts. There is no theory that explains what makes good music good, and that includes harmonic theory. Music is a fundamentally practical, empirical art, and imo, the best explanation is that of Duke Ellington: If it sounds good, it _is good. ;-)_


_

I'm so glad that you have identified "the enemy," and you seem to feel as if you are in control again. Just remember that "tonality is God" and things will continue as is. Remember, if you get scared, there's always a big trouser-leg you can hide behind._


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## millionrainbows

BabyGiraffe said:


> Many developers for Kontakt (and similar engines) absolutely always autotune the samples, because many customers (check vi-control forum for many such threads from the last decade) complain about unplayable keys. And many of these libraries record the best orchestral musicians available (Vienna, London, L.A.).


That's fascinating info about auto-tune.


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## fluteman

millionrainbows said:


> I'm so glad that you have identified "the enemy," and you seem to feel as if you are in control again. Just remember that "tonality is God" and things will continue as is. Remember, if you get scared, there's always a big trouser-leg you can hide behind.


Especially as I get older, three things (among others) become ever clearer: I am not in control (nor is anyone else), things will not continue as is, and there is nowhere to hide. What scares me is the seemingly increasing number of people worldwide who fail to understand one or more of those things.


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## BabyGiraffe

fluteman said:


> There is no theory that explains what makes good music good, and that includes harmonic theory. Music is a fundamentally practical, empirical art, and imo, the best explanation is that of Duke Ellington: If it sounds good, it _is good. ;-)_


_

Well, there are many theories, but they are just theories... right.
Still, trying to explain certain styles (even Duke Ellington's use of diminished "tonality") with simple meantone and 5-limit simplifications is like trying to explain quantum physics with classical physics' theories. It just doesn't work well.

Hindemith noted that what was considered dissonant in the past, became a new consonance (or something in this vein). For example - in 3-limit Pythagorean tuning we have only 3 in-tune intervals (9/8, 4/3 and 3/2)) that don't have simpler and better sounding variant in 5 limit (pythagorean major third 81/64 and countless others differ to their 5-limit versions by syntonic comma, diaschisma or schisma and probably other exotic commas in way bigger tonal systems). 
In 5-limit we get major and minor thirds and sixths as consonances. Dissonances in 5-limit (like augmented sixth, tritones, diminished thirds and fourths etc) can be interpreted as simpler 7 or 13 limit intervals. (That's why I suggested 43 as good meantone - you can't mistake some "arabic" out of tune intervals as consonances unlike more 7-limit flavoured meantone, which can sound somewhat "bluesy".) 
Dissonances in 7-limit are basically 11-limit intervals. (I haven't played much with higher just intonation that that, but these harmonics in acoustic instruments are pretty quiet (unless we synthesize some special electronic timbres with additive synthesis) and these ratios - more discordant, so these higher prime limits are useless as consonant harmony; still, they work perfectly fine for melody.)_


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## millionrainbows

fluteman said:


> Especially as I get older, three things (among others) become ever clearer: I am not in control (nor is anyone else), things will not continue as is, and there is nowhere to hide. What scares me is the seemingly increasing number of people worldwide who fail to understand one or more of those things.


That's just fine. Why don't you make a report about this? I'm sure it would be fascinating.


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## millionrainbows

BabyGiraffe said:


> Well, there are many theories, but they are just theories... right.
> Still, trying to explain certain styles (even Duke Ellington's use of diminished "tonality") with simple meantone and 5-limit simplifications is like trying to explain quantum physics with classical physics' theories. It just doesn't work well.
> 
> Hindemith noted that what was considered dissonant in the past, became a new consonance (or something in this vein). For example - in 3-limit Pythagorean tuning we have only 3 in-tune intervals (9/8, 4/3 and 3/2)) that don't have simpler and better sounding variant in 5 limit (pythagorean major third 81/64 and countless others differ to their 5-limit versions by syntonic comma, diaschisma or schisma and probably other exotic commas in way bigger tonal systems).
> In 5-limit we get major and minor thirds and sixths as consonances. Dissonances in 5-limit (like augmented sixth, tritones, diminished thirds and fourths etc) can be interpreted as simpler 7 or 13 limit intervals. (That's why I suggested 43 as good meantone - you can't mistake some "arabic" out of tune intervals as consonances unlike more 7-limit flavoured meantone, which can sound somewhat "bluesy".)
> Dissonances in 7-limit are basically 11-limit intervals. (I haven't played much with higher just intonation that that, but these harmonics in acoustic instruments are pretty quiet (unless we synthesize some special electronic timbres with additive synthesis) and these ratios - more discordant, so these higher prime limits are useless as consonant harmony; still, they work perfectly fine for melody.)


I suspect that when you "rank" intervals as good, better than, etc, that you are simply basing this on numbers, unless you can explain how you are doing this. 
What actual listening have you done, and how do you accomplish the actual sounds? These ratios have to be converted into numbers in order to be read by any tuner.

I know that it's possible to modulate the keyboard voltage on some synthesizers to get different divisions of the octave, by simply tuning the octave to different locations on the keyboard.


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## BabyGiraffe

millionrainbows said:


> I suspect that when you "rank" intervals as good, better than, etc, that you are simply basing this on numbers, unless you can explain how you are doing this.
> What actual listening have you done, and how do you accomplish the actual sounds? These ratios have to be converted into numbers in order to be read by any tuner.


For dyadic chords (= intervals), modern psycho-acoustic theories agree with Partch and similar just intonation theorists (at least for most ratios - of course, octave reduced 17th harmonic (17/16 can be thought as 12 equal's semitone) would be dissonant, but use an additive synthesizer, generate a saw wave (or violin sound etc), then mute selected harmonics - you will notice the difference in timbre; I haven't really tried 17-limit harmony/ratios, but it's dissonant, at least anywhere near the root of your voicings. It should be some sort of cluster harmony, I guess, if that makes sense).
You can join the facebook microtonal group and ask for references - some guys there work at universities (and sometimes give free links to such articles) and/or have published such academic papers on related topics. There are also many graphs using different models of human hearing on the most consonant theoretical dyads and triads.

These ratios actually have very deep connection to physics and dynamical systems in mathematics.

If you have not read Parch's book, this one is a good introduction to his method:
https://en.wikipedia.org/wiki/Tonality_diamond

(You can mess with a stereo plugin and a simple vectorscope to generate pretty interesting visualizations of the patterns of interferences with different ratios. It's very easy to visualize how much better are simple ratios. I have seen neo-pythagorean "sacred geometry" charlatans to actually use such visualizations in their youtube "educational" videos without really explaining what is going on.)

"... how do you accomplish the actual sounds? "

Use any DAW and a free microtonal synth, and retuning software (like the free Scala) for generating scripts or write your own scale files in "tun" or "scala" formats, if you are interested in such stuff.


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## fluteman

BabyGiraffe said:


> Well, there are many theories, but they are just theories... right.
> Still, trying to explain certain styles (even Duke Ellington's use of diminished "tonality") with simple meantone and 5-limit simplifications is like trying to explain quantum physics with classical physics' theories. It just doesn't work well.
> 
> Hindemith noted that what was considered dissonant in the past, became a new consonance (or something in this vein). For example - in 3-limit Pythagorean tuning we have only 3 in-tune intervals (9/8, 4/3 and 3/2)) that don't have simpler and better sounding variant in 5 limit (pythagorean major third 81/64 and countless others differ to their 5-limit versions by syntonic comma, diaschisma or schisma and probably other exotic commas in way bigger tonal systems).
> In 5-limit we get major and minor thirds and sixths as consonances. Dissonances in 5-limit (like augmented sixth, tritones, diminished thirds and fourths etc) can be interpreted as simpler 7 or 13 limit intervals. (That's why I suggested 43 as good meantone - you can't mistake some "arabic" out of tune intervals as consonances unlike more 7-limit flavoured meantone, which can sound somewhat "bluesy".)
> Dissonances in 7-limit are basically 11-limit intervals. (I haven't played much with higher just intonation that that, but these harmonics in acoustic instruments are pretty quiet (unless we synthesize some special electronic timbres with additive synthesis) and these ratios - more discordant, so these higher prime limits are useless as consonant harmony; still, they work perfectly fine for melody.)


Thanks, that's interesting. You seem to actually have significant experience working with all these different scales, which is nice. All of that experimentation is now possible with electronic 'instruments', if that word is still useful, or synthesizers, which I assume you are experienced in working with. One thing your post demonstrates in some detail is how every scale is a compromise, as some intervals can be 'right' but others will be off, like the Dutch boy with not enough thumbs to plug all the leaks in the ****. I tried to make a similar point with my discussion of wind instrument scales, though avoiding technical specifics.

In equal temperament, unlike just temperament, every interval is bad, some very bad, though fifths are very close to good. The crucial advantage being that each interval is identical in every key. That advantage has immense importance in traditional, pre-early 20th century western music. But with your electronic synthesizers, you like many others seem to have moved in a new direction. Good for you. Which only goes to show, music is not a mathematical puzzle that can be solved.



millionrainbows said:


> Other sources indicate that Beethoven probably used Thomas Young tuning. Nobody had ET back then, it wasn't achieved until 1919. Many tunings were approaching ET, and that's what things were moving towards, but none achieved it. Nevertheless, Bach's tuning (Lehman/Bach) and other tunings were approximations, by ear & stopwatch, of ET.


Yes, but as you say or imply, musicians estimated equal temperament long before mathematicians figured out how to calculate it exactly. And btw, exactly what BabyGiraffe means by "ranking" good sounding intervals is a good question, as you suggest. I assume he means closer to the Pythagorean "ideal", but surely there could be other definitions of "good sounding".


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## BabyGiraffe

fluteman said:


> In equal temperament, unlike just temperament, every interval is bad, some very bad, though fifths are very close to good. The crucial advantage being that each interval is identical in every key. That advantage has immense importance in traditional, pre-early 20th century western music.
> 
> music is not a mathematical puzzle that can be solved.


Well, well, 12 equal is great for its size for 5-limit music.
5, 7 and 9 equal are actually great for their sizes (and see usage in non-Western music; or at least close approximations to them).
If you want to play "oriental" music with neutral thirds, 17 equal works (but it would be the same as playing Western music in 7 equal).
About the theory - modern tuning/mathematical theory is generalized abstract theory that works for any style. It doesn't matter whether it's Indonesian, Western or Arabic, or African etc music...

You can do pretty cool pre-compositional tricks with mathematical theory- let's say you want a diatonic scale that sound more "arabic".
Diatonic in 12 equal is generated by stacking fifths. "Diatonic" in 10 equal (which sounds vaguely like something between harmonic major/double harmonic/harmonic minor or some kind of authentic arabic scale) is generated by stacking sharp neutral thirds.
Cartesian product of these two is... the familiar "just intonation" diatonic scale (but only structurally - created by three step intervals, so syntonic comma exists, tuning wise, it's somewhat off). Try it in 22 or 29 equal (the pattern is the same as "just intonation diatonic").

Or create a "tonal" pentatonic - (or whatever size you choose - each equal division of the octave can be retuned in several "tonal" framework using similar methods). 
5 equal resembles 3 and 7 limit intervals.
So there can exist a chord like 1-4/3-7/4 or 1-21/16-7/4 - these two are inversions of each other like major and minor (21/16 is the 21th octave reduced harmonic and a close sounding interval to a perfect fourth; 7/4 is the octave reduced 7th harmonic ). Using the diatonic analogy in 12 equal, we can tune it to a scale with 4 consonant chords (2 "major", 2 "minor") and one characteristic dissonance - stacking two 21/16 gives us the tempered version of 12/7 (like two minor thirds in 12 equal give a false fifth, this will give false 7th harmonic).
Try it in 31 equal. It sounds melodically somewhat like black keys pentatonic or 5 equal. (Pattern: 6 6 7 6 6)


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## millionrainbows

fluteman said:


> In equal temperament, unlike just temperament, *every interval is bad,* some very bad, though fifths are very close to good.


You should think about this sentence. How can a 2 cent difference be "bad" if the human ear has a hard time hearing 4 cents? All the intervals generated by ANY equal-divison are going to be off.


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## millionrainbows

BabyGiraffe said:


> Or create a "tonal" pentatonic - (or whatever size you choose - each equal division of the octave can be retuned in several "tonal" framework using similar methods).
> 5 equal resembles 3 and 7 limit intervals.


Why do we need to, when you can stack fifths, stop at five, and have our ET pentatonic? Redundant, needless complex.


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## BabyGiraffe

millionrainbows said:


> Why do we need to, when you can stack fifths, stop at five, and have our ET pentatonic? Redundant, needless complex.


Because you need an approximation of 7/4, not 9/5. 
It may be useless for Arabic or European music, but it's a typical folk music interval in various African regions (I have heard some Indian sitar players using it, but it has nothing to do with Indian theory, so they probably played it by ear, not following some traditions). 
There are plenty of different septimal pentatonic, hepta- and hexa-tonic collections used in African music.

Anyway, Pythagorean minor seventh is a dissonance that needs resolution, compared to other two ratios -
7/4 968.825906 harmonic seventh
16/9 996.089998 Pythagorean minor seventh
9/5 1017.596288 just minor seventh, BP seventh

(Having a consonant relation to other 3 and 5 limit ratios doesn't change that - you can have relation by pure fifths or major/minor thirds in a "musical system" even with dissonant intervals - 24 equal is a good example).


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## fluteman

millionrainbows said:


> You should think about this sentence. How can a 2 cent difference be "bad" if the human ear has a hard time hearing 4 cents? All the intervals generated by ANY equal-divison are going to be off.


OK, both reasonable points. So, in your terminology, the equal-tempered fifth is "very close to good", or good and very close to perfect, but not quite. BabyGiraffe would probably go along with that. FWIW, testing my own hearing, after resting my ears with a long period of silence, I could distinguish between tones as close as .75 cents apart, putting me well in the upper 50th, but not the 90th, percentile (maybe 70th?) according to the online testing site. I assume younger people generally do better in these hearing tests. But after listening to a lot of out of tune noise, I could do no better than 2 cents.



BabyGiraffe said:


> Well, well, 12 equal is great for its size for 5-limit music.
> 5, 7 and 9 equal are actually great for their sizes (and see usage in non-Western music; or at least close approximations to them).
> If you want to play "oriental" music with neutral thirds, 17 equal works (but it would be the same as playing Western music in 7 equal).
> About the theory - modern tuning/mathematical theory is generalized abstract theory that works for any style. It doesn't matter whether it's Indonesian, Western or Arabic, or African etc music...


Makes sense. As I said above, Blackwood's Microtonal Etudes are written for 13 to 24 tone equal tempered scales, and mostly in early Baroque, Classical or Romantic/Impressionist styles. To me, there is a bit of 'aliens from another planet', futuristic, sci-fi feel to them, but they don't sound bad. And 5-, 7- and 9-equal scales you mention seem to be useful options, too, at least for you electronic synthesizer mavens who can easily choose between any scales.

Still, the advantages of 12-tone ET, both practical and sonic, with its nearly perfect fifths, understandably brought it to the forefront, of acoustic western music, anyway, before the electric and electronic era. Now musicians who think like you are beginning to take over, e.g., those who have a broader view of harmony, and/or look more extensively beyond harmony. I notice you still have a very practical view of these things, and readily adapt your now very wide range of options depending on your goals. I think that will be the new tradition.


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## BabyGiraffe

fluteman said:


> FWIW, testing my own hearing, after resting my ears with a long period of silence, I could distinguish between tones as close as .75 cents apart, putting me well in the upper 50th, but not the 90th, percentile (maybe 70th?) according to the online testing site. I assume younger people generally do better in these hearing tests.


Most people even with very good pitch fail around 5 cents, so you have exceptional pitch recognition.
I have tested my unmusical friends and family with short excerpts of music with out of tune fifths and thirds - they can't notice anything wrong. (For example the out of tune chord in just intonation diatonic that lead to meantone and 12 equal is 32/27 as minor third = 21 cents flatter than pure and 40 /27 as fifth - again the same amount of detuning, resulting in a wolf fifth; tempering this gives a series of equal temperaments with bad major and minor thirds or bad fifths, or bad - both of them (12 is one of these with bad thirds). This means that in non-meantone system, we have to modulate not only a chromatic semitone, but also a syntonic comma.)

If most people are just as tone deaf as my friends, all this tuning business is as useful as number theory... Still, there is always a slight noticeable aural difference between performances in 12 equal, pythagorean and various meantone systems.


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## fluteman

BabyGiraffe said:


> Most people even with very good pitch fail around 5 cents, so you have exceptional pitch recognition.
> I have tested my unmusical friends and family with short excerpts of music with out of tune fifths and thirds - they can't notice anything wrong. (For example the out of tune chord in just intonation diatonic that lead to meantone and 12 equal is 32/27 as minor third = 21 cents flatter than pure and 40 /27 as fifth - again the same amount of detuning, resulting in a wolf fifth; tempering this gives a series of equal temperaments with bad major and minor thirds or bad fifths, or bad - both of them (12 is one of these with bad thirds). This means that in non-meantone system, we have to modulate not only a chromatic semitone, but also a syntonic comma.)
> 
> If most people are just as tone deaf as my friends, all this tuning business is as useful as number theory... Still, there is always a slight noticeable aural difference between performances in 12 equal, pythagorean and various meantone systems.


Again, interesting, but you flatter me. My hearing is pretty good, but not exceptional. I've heard selections from the Well-tempered Clavier and the 5th Brandenburg Concerto of Bach played in Werckmeister III (which Millionrainbows tells us is not the tuning Bach intended, but I'm not going to worry about that debate) and it did sound very slightly odd in the way Blackwood's Microtonal Etudes do, but not as much. If you can, please find some music played in that scale and let me know what you think.

I have an antique flute made in 1908 and tuned to a=438 (equal temperament, of course), a common compromise with the French "diapason normale" of a=435 before a=440 became universal. I was once foolish enough to bring it to a rehearsal of the Poulenc sextet for piano and winds with some very good musicians, and we tuned perhaps slightly above a=440 but not by much, due to the piano. I thought I could adjust. The oboist / leader of the group condescendingly informed me I was painfully out of tune, and should carefully retune. I said, that won't work, I need to bring another flute.

Listening to myself on tape (actually WAV digital file) playing up to a=440, I could only cringe. As you know, equal tempered scales are logarithmic, not linear, so they can't be adjusted on wind instruments without distorting many of the notes. It turns out the difference between a=438 and a=440 is indeed painfully obvious. Again, I'd be most interested if you programmed your fancy equipment and tested your own hearing in this regard.

I should correct the above, in that I could detect the difference between two pitches within .75 Hz, not .75 cents. A4, which was likely the base tone in the test I took, is 24.7 Hz above G#4 and 26.2 Hz below A#4. According to the Michigan Tech physics department, one should "easily" be able to tell the difference between sustained tones within 1 Hz. They also provide some hearing tests:

https://pages.mtu.edu/~suits/scales.html


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## BabyGiraffe

438 to 440 hz is approximately 7.88 cents.

Maybe you will find this interesting:

"Music production applications

In music production, a single change in a property of sound which is below the JND does not affect perception of the sound. For amplitude, the JND for humans is around 1 dB (Middlebrooks & Green, 1991; Mills, 1960).

The just-noticeable difference (JND) (the threshold at which a change is perceived) depends on the tone's frequency content. Below 500 Hz, the JND is about 3 Hz for sine waves, and 1 Hz for complex tones; above 1000 Hz, the JND for sine waves is about 0.6% (about 10 cents).[3] The JND is typically tested by playing two tones in quick succession with the listener asked if there was a difference in their pitches.[4] The JND becomes smaller if the two tones are played simultaneously as the listener is then able to discern beat frequencies. The total number of perceptible pitch steps in the range of human hearing is about 1,400; the total number of notes in the equal-tempered scale, from 16 to 16,000 Hz, is 120"

https://en.wikipedia.org/wiki/Just-noticeable_difference

Still, all this is irrelevant to harmony discussions. Such small intervals are considered useless in actual music compositions, that's why equal and unequal temperaments are used.


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## fluteman

BabyGiraffe said:


> 438 to 440 hz is approximately 7.88 cents.
> 
> [ .... ]
> 
> Still, all this is irrelevant to harmony discussions. Such small intervals are considered useless in actual music compositions, that's why equal and unequal temperaments are used.


Thanks, that was a very interesting and useful post. However, these "small intervals" aren't entirely irrelevant once we start playing around with these other scales, especially those that aren't equal tempered, in actual musical compositions, right? That's why I was interested in your opinion of the Well-tempered Clavier played in a non-equal tempered tuning like Werckmeister III.


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## Luchesi

millionrainbows said:


> You should think about this sentence. How can a 2 cent difference be "bad" if the human ear has a hard time hearing 4 cents? All the intervals generated by ANY equal-divison are going to be off.


Yes, when I tune a piano my goal is 2 to 3 cents, depending upon the instrument and its condition. Customers don't want to pay you to come right back 2 weeks later because the piano has inevitably 'fallen' and you hadn't anticipated it.


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## fluteman

Luchesi said:


> Yes, when I tune a piano my goal is 2 to 3 cents, depending upon the instrument and its condition. Customers don't want to pay you to come right back 2 weeks later because the piano has inevitably 'fallen' and you hadn't anticipated it.


Yes, 2 cents is very good, so you are welcome to tune my piano. If you saw my correction to my original post, you saw that I meant 2 Hz, not two cents. BabyGiraffe gave a good basic summary of human ability to hear differences in pitch above, although I'm sure he'd agree that it was only a summary of a much larger subject.


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## millionrainbows

I was tested, with another guy, and we both could hear a difference of 2 cents, on fairly high-pitched wind chimes, probably A=880. The difference was identifiable, but it really wasn't pitch I was hearing; it was that the higher note sounded ever-so-slightly "brighter."

I got to do all kinds of experiments when I was working at the chime factory. I cut a 17-note per octave Arabic chime, a 7-note ET chime (which is Thai tuning), A "harmonic" bass tuning which had a harmonic seventh, just fifth and third and some kind of second, a solfeggio tuning, a septimal whole tone scale based on a septimal second, all sorts of stuff. 
It was valuable, because it showed me that sound is linked to physics, arithmetic, and actual materials.


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## BabyGiraffe

fluteman said:


> Thanks, that was a very interesting and useful post. However, these "small intervals" aren't entirely irrelevant once we start playing around with these other scales, especially those that aren't equal tempered, in actual musical compositions, right? That's why I was interested in your opinion of the Well-tempered Clavier played in a non-equal tempered tuning like Werckmeister III.


There is a theory that there are so many sequences, arpeggios and ostinatos in a typical Baroque music work, because they were sounding somewhat different, because there are more unique intervals in irregular temperaments (known as well-temperaments). This won't happen in equal or just intonation (where you shouldn't really play the out of tune notes, that's the whole point, at least in theory).
About Werckmeister 3 - Scala gives this data, comparing it to 12 equal - I compared the most even mode, it's the closest one to 12 equal. 
Standard deviation: 3.7435 c. Maximum deviation: 7.8200 c.

If we compare 12 equal to:

55 equal meantone - SD: 6.3420 c. M:-10.9091 c. (Some of the modes of Werkmeister are more uneven than that)

43 meantone, we get: SD: 8.1118 c. M:-13.9535 c.

50 equal meantone SP: 7.1181 c. M:-12.0000 c.

31 equal meantone: SD: 11.2519 c. M:-19.3548 c.

19 equal meantone: SD: 18.3583 c. M:-31.5789 c.

Equal 19 meantone is potentially useful for chromatic 12 tone music, because it is not really that close to it, so it can be considered as melodic and harmonic scale on its own - we have small and large semitone ( 63.15789 cents and 126.31579 cents).
Other useful scales for 12 equal based chromatic tonality are 22 equal (semitone is split as 54.54545 cents and 109.09091 cents, but it's not meantone, so there is different logic in many chord progressions), 17 equal (70, 140, but 17 equal clearly won't work for good triadic harmony on most timbres - some slightly inharmonic instruments like piano can do it, but not flutes or strings, still counterpoint is possible in a medieval style - using fifths, fourths and quartal/quintal stacks.)

" However, these "small intervals" aren't entirely irrelevant once we start playing around with these other scales, especially those that aren't equal tempered, in actual musical compositions, right?"

Not really, 99 % of the listeners won't notice them or the musician will adjust their melodic or harmonic intonation or occasionaly modulate a comma up/down, or just drift away from the initial tonic as the piece progresses, if there are many distant modulations. 
It's more of a compositional and notational problem.
If you want to compose or perform in a close to optimal 5-limit tuning, you have to basically deal with 53 equal - where the difference of pythagorean and syntonic comma, called schisma, is the only significant interval tempered.
53 equal is the modern theoretical Turkish system. I was very surprised when read in modern Turkish dissertation that "Rast" - their diatonic is basically the Western major scale. But that is because in 53 equal Pythagorean and just intonation major are different scales. (Another layer of confusion was the naming - "Rast" in Arabic countries is the neutral, not the major tetrachord; but other Arabic tetrachord names are also swapped in their Turkish variants).


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## millionrainbows

One advantage of creating other ET octave divisions on a normal keyboard (by modulating/"stretching" the keyboard voltage) is that you can see some connections with normal tuning. In 19-tone, tune the octave past the "C to C" 12-note ET octave to "C-G", the G past the octave C. If you compose melodies in 19, then convert them back to 12, it gives a diminished tonality.

!7 note ET is octave on F; 22 tone ET is C-Bb (flat seven).

You begin to see that these other ETs are based on _overtones_ of the 12 ET, and that is why certain ET tunings are favored, such as 17 (C-F fourth), 19 (C-G fifth), 21 (C-A relative minor), and 22 (C-Bb flat seven).


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## fluteman

Well, BabyGiraffe, I guess I and a lot of people I know have exceptional hearing, as we can easily tell the difference between equal temperament and various forms of just intonation in the right contexts. In his Well-tempered Clavier, Bach wrote a prelude and fugue for all 24 major and minor keys of the 12-tone scale. As only equal temperament gives you identical intervals and chords in every key, the differences there are especially clear. Have you listened yourself?

Or take a famous traditional Japanese melody such as Sakura (cherry blossoms). This uses a pentatonic scale roughly similar to the Phrygian mode. Compare a recording made with traditional Japanese instruments such as the koto with a westernized recording. This should be very easy to do. The famous French flutist Jean-Pierre Rampal made a series of recordings of traditional Japanese melodies, including Sakura, all in westernized arrangements with equal-tempered tuning. I think in most the flute is accompanied by a western harp, although the koto and other instruments are also used. Listen and compare:


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## millionrainbows

fluteman said:


> ...In his Well-tempered Clavier, Bach wrote a prelude and fugue for all 24 major and minor keys of the 12-tone scale. As only equal temperament gives you identical intervals and chords in every key, the differences there are especially clear. Have you listened yourself?


In the case of Bach, who used his own "ET"-ish tuning, there may be slight differences in fifths and thirds which influenced the way he _actually composed each prelude/fugue for the WTC_. For example, if the key had a really good sounding major third, he might emphasize that note more. If it had a bad minor third, he might not linger on that note, etc.


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## fluteman

millionrainbows said:


> In the case of Bach, who used his own "ET"-ish tuning, there may be slight differences in fifths and thirds which influenced the way he _actually composed each prelude/fugue for the WTC_. For example, if the key had a really good sounding major third, he might emphasize that note more. If it had a bad minor third, he might not linger on that note, etc.


Yes, excellent point, I've read work by scholars on the subject who have said something similar. I guess there will always be some debate on that topic. After all, musicologists need to get published.


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## Luchesi

fluteman said:


> Yes, excellent point, I've read work by scholars on the subject who have said something similar. I guess there will always be some debate on that topic. After all, musicologists need to get published.


I don't see that. We could ask one of the composers in this forum whether they ever think about that. But I expect that composers will begin composing in the same way that performers will first play a piece from which patterns they see standing out as intriguing and interesting. This is difficult to put into words …but I don't think the exact sound of a major third etc. influences them. I play many pianos that are even somewhat out of tune and your hearing adapts very quickly! because other notes in the constellation you grab (and the accompaniment) wash out the very slight deviations you could pinpoint acoustically.

I always remember from the man who taught me how to tune pianos, he was legally blind. He was an elderly man of few words and he said to me, "We're not going to tune musically, we're going to tune acoustically. ..And then we go back to make it musical.". I initially only got a sense of what he meant, but now I know it's very important to keep in mind.


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## fluteman

Luchesi said:


> I don't see that. We could ask one of the composers in this forum whether they ever think about that. But I expect that composers will begin composing in the same way that performers will first play a piece from which patterns they see standing out as intriguing and interesting. This is difficult to put into words …but I don't think the exact sound of a major third etc. influences them. I play many pianos that are even somewhat out of tune and your hearing adapts very quickly! because other notes in the constellation you grab (and the accompaniment) wash out the very slight deviations you could pinpoint acoustically.
> 
> I always remember from the man who taught me how to tune pianos, he was legally blind. He was an elderly man of few words and he said to me, "We're not going to tune musically, we're going to tune acoustically. ..And then we go back to make it musical.". I initially only got a sense of what he meant, but now I know it's very important to keep in mind.


Hey, don't shoot the messenger, Luchesi. Musicologists like to argue about exactly what Bach meant by "well-tempered" when he wrote The Well-tempered Clavier. And I wholeheartedly agree with your comment that our hearing adapts to 'slightly' out of tune notes. I made that point above, and my own experience confirms it, as does what I have been told by conductors, etc. But have you ever been to a Chinese or Indian restaurant where they are piping in traditional music? (Less common these days, as many of those restaurants try to become more hip for a younger crowd and pipe in westernized music.) Can't you hear that it sounds a little 'off'?


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## JeffD

millionrainbows said:


> Is everybody happy?


What, exactly and completely in all its senses, does the term "happy" mean?


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## millionrainbows

JeffD said:


> What, exactly and completely in all its senses, does the term "happy" mean?


It's what they used to say before they said "Is everybody ready to rock and roll?"


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## millionrainbows

This tome should clear up some questions about the Pythagorean method and scales, and clarify things a bit.

View attachment 126142


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## fluteman

millionrainbows said:


> This tome should clear up some questions about the Pythagorean method and scales, and clarify things a bit.
> 
> View attachment 126142


Or this one I just found, which is written in a more non-technical, non-mathematical style, and really only summarizes and simplifies what musicologists have written about with great precision and in great detail, but looks like a fun read for those who don't want to get too technical. He discusses what I was trying to talk about, i.e., the imperfections of equal temperament and why it isn't necessarily the best-sounding scale in all circumstances, but he also gives more general information. I do not go along with the idea that equal temperament "ruined" harmony, and I don't think the author does either. He's just trying to sell books by using a provocative title.


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## Luchesi

fluteman said:


> Or this one I just found, which is written in a more non-technical, non-mathematical style, and really only summarizes and simplifies what musicologists have written about with great precision and in great detail, but looks like a fun read for those who don't want to get too technical. He discusses what I was trying to talk about, i.e., the imperfections of equal temperament and why it isn't necessarily the best-sounding scale in all circumstances, but he also gives more general information. I do not go along with the idea that equal temperament "ruined" harmony, and I don't think the author does either. He's just trying to sell books by using a provocative title.
> 
> View attachment 126361


What I always conclude is it depends upon what we want from music. Do we want sweet sounds alone, or do we want the intellectual achievement? Can we have both? Bach decided that we needed something close to ET.


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## fluteman

Luchesi said:


> What I always conclude is it depends upon what we want from music. Do we want sweet sounds alone, or do we want the intellectual achievement? Can we have both? Bach decided that we needed something close to ET.


I think the point this author is making is that in the right (or wrong) circumstances, the sounds can be as sour with equal temperament as with any other tuning (they are all compromises). And of course, a lot of music, including western music from over 150 or 200 years ago and certainly music from other cultures, was written with other tuning systems in mind. As to what tuning Bach intended, there is a wide range of opinions, even from musicologists who spend most of their time examining such issues.


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## Luchesi

fluteman said:


> Hey, don't shoot the messenger, Luchesi. Musicologists like to argue about exactly what Bach meant by "well-tempered" when he wrote The Well-tempered Clavier. And I wholeheartedly agree with your comment that our hearing adapts to 'slightly' out of tune notes. I made that point above, and my own experience confirms it, as does what I have been told by conductors, etc. But have you ever been to a Chinese or Indian restaurant where they are piping in traditional music? (Less common these days, as many of those restaurants try to become more hip for a younger crowd and pipe in westernized music.) Can't you hear that it sounds a little 'off'?


"Can't you hear that it sounds a little 'off'?"

I do hear them here and there, but they're not irritating. As humans, we say it all adds to the charm, if it isn't overdone (and the composers don't want to get ambitious, beyond their tradition).

I've read that the philosophy about music in the Far East is that it should be full of metaphors for the patterns and symmetries and realities of Nature. Therefore an artificial system won't do it for them..


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## fluteman

Luchesi said:


> "Can't you hear that it sounds a little 'off'?"
> 
> I do hear them here and there, but they're not irritating. As humans, we say it all adds to the charm, if it isn't overdone (and the composers don't want to get ambitious, beyond their tradition).
> 
> I've read that the philosophy about music in the Far East is that it should be full of metaphors for the patterns and symmetries and realities of Nature. Therefore an artificial system won't do it for them..


What an interesting comment. I don't think we need to say that composers of traditional, non-western music couldn't be too ambitious because of the limitations of their tradition. That could be seen as condescending, and after all, all composers of any tradition work within limitations, none more strict and elaborate than those of western classical music. Yet, even there, you make a good point, as equal temperament, whatever its compromises and "artificial" intervals, alone makes every interval identical in all 12 keys, and thereby removes a significant limitation of nearly all other tuning systems, admittedly at a price.


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## Guest

You gain and you loose. In equal temperament you can use any key and so a broader range of modulation is available, but you lose the unique "sound" of each key. Probably we should use whatever tuning the composer was familiar with, if only we knew. Wouldn't it be wonderful if Bach had told someone how he liked his harpsichord tuned?


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## millionrainbows

Baron Scarpia said:


> Wouldn't it be wonderful if Bach had told someone how he liked his harpsichord tuned?


Bach's tuning has already been revealed by Dr. Bradley Lehman. See larips.com


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## Guest

millionrainbows said:


> Bach's tuning has already been revealed by Dr. Bradley Lehman. See larips.com


Yes, I've seen that, and other versions which interpret it differently. When I followed the arguments there were inconsistencies and ambiguities. I'm wishing Bach could have left something more explicit than some squiggles to which different people assign different meanings. And since he put "well tempered" in the title, why not specify what that means in the published work?


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## BabyGiraffe

Baron Scarpia said:


> Yes, I've seen that, and other versions which interpret it differently. When I followed the arguments there were inconsistencies and ambiguities. I'm wishing Bach could have left something more explicit than some squiggles to which different people assign different meanings. And since he put "well tempered" in the title, why not specify what that means in the published work?


It was well known what "well tempered" means in his own time.

https://en.wikipedia.org/wiki/Well_temperament

Free book - by J. Barbour
Tuning and temperament : a historical survey

https://archive.org/details/tuningtemperamen00barb


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## BabyGiraffe

A personal note - it's impossible to have "great" tuning in less than... let's say 31 keys (31 is the smallest equal system that starts to somewhat sound like just intonation). So, while it's probably possible to create 31 "well temperament", the idea that with only 12 keys you can sound very good in all keys is idiotic - some of the "modes" in these historical well temperaments are worse than 12 equal or various meantone selections (but some are better).
Something that I have said before:-every meantone (like 12,19,31,43,50,55 equal) system is flawed, because diatonic scale is symmetrical around two different "D" pitches - one is 10/9, another - 9/8. Good luck playing with that - "bending" traditions like Indian music and Blues may give you an idea what you have to do when dealing with such small intervals.
And lastly - for Pythagorean fanatics - it is decent around 41 notes per octave and good only at 53 equal.
Diaschismic temperaments - like 34 and 46 equal are better than meantone, but worse than Pythagorean (in sound; for practical chord progressions it's meantone>diaschismic>extended pythagorean systems - I say extended, because traditional pythagorean with only 12 keys is obviously bad - worse than 12 equal - that's how much 2 cents difference makes. There exist also other theoretical systems created by chains of various thirds, but they are also more complex that meantone )


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## Luchesi

BabyGiraffe said:


> It was well known what "well tempered" means in his own time.
> 
> https://en.wikipedia.org/wiki/Well_temperament
> 
> Free book - by J. Barbour
> Tuning and temperament : a historical survey
> 
> https://archive.org/details/tuningtemperamen00barb


Paul Barton has finished his WTC video. Wow!






He says he has to wear the arm band because other youtube uploaders stole some of his earlier videos and then reported him.


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## fluteman

BabyGiraffe said:


> A personal note - it's impossible to have "great" tuning in less than... let's say 31 keys (31 is the smallest equal system that starts to somewhat sound like just intonation). So, while it's probably possible to create 31 "well temperament", the idea that with only 12 keys you can sound very good in all keys is idiotic - some of the "modes" in these historical well temperaments are worse than 12 equal or various meantone selections (but some are better).
> Something that I have said before:-every meantone (like 12,19,31,43,50,55 equal) system is flawed, because diatonic scale is symmetrical around two different "D" pitches - one is 10/9, another - 9/8. Good luck playing with that - "bending" traditions like Indian music and Blues may give you an idea what you have to do when dealing with such small intervals.
> And lastly - for Pythagorean fanatics - it is decent around 41 notes per octave and good only at 53 equal.
> Diaschismic temperaments - like 34 and 46 equal are better than meantone, but worse than Pythagorean (in sound; for practical chord progressions it's meantone>diaschismic>extended pythagorean systems - I say extended, because traditional pythagorean with only 12 keys is obviously bad - worse than 12 equal - that's how much 2 cents difference makes. There exist also other theoretical systems created by chains of various thirds, but they are also more complex that meantone )


Well, you electronic computer program music mavens can talk about 31 or 56 note scales, but it doesn't seem very practical for traditional acoustic instruments. There are top players of some traditional instruments who can play quarter-tone scales very well, and in some cases the instruments have been modified to make that even easier to do, so those are 24 note scales. But 31 or 56? That would be tough.


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## BabyGiraffe

fluteman said:


> Well, you electronic computer program music mavens can talk about 31 or 56 note scales, but it doesn't seem very practical for traditional acoustic instruments. There are top players of some traditional instruments who can play quarter-tone scales very well, and in some cases the instruments have been modified to make that even easier to do, so those are 24 note scales. But 31 or 56? That would be tough.


It is possible to design such instruments, but they wouldn't be that easy to be played.

But the most practical methodology (remember, there was no 12 equal back in the day!!!, many of the characteristics of most old compositions are based on these limitations) is tuning/designing around specific subset/scale of big equal temperament (historical "well temperaments" or "meantone" temperaments) are exactly this (the main difference is that meantone is achieved by stacking a single interval modulo octave - slightly flat fifth, so it's easier to understand how to construct it; well temperament will need each fifth having a different size).

Btw, if we look at equal temperaments, we can notice that all the good melodic and harmonic scales somewhat heavily distort the sizes of equal intervals.
Compare pentatonic and 5 equal or diatonic and 7 equal. 
The trivial tuning of 5 equal is "African"/blues pentatonic, because it's very close to it; the trivial tuning of 7 equal is "Arabic" style tuning with neutral intervals. 
The trivial tuning of 12 equal is something close to 5-limit (Western music), the nontrivial tuning should be something with septimal intervals - that's why something like 31 or 22 or 27 equal maybe are good for 12 tone gamuts - 31 will temper the syntonic comma, but won't temper the septimal comma; 22 will temper the septimal comma, but won't temper syntonic comma, 27 will temper the septimal comma and fraction of syntonic comma.
Basically, there is a structure on just intonation lattice, defined by 3 interval differences - 126/125, 64/63, 225/224. 
126/125 x 225/224 = 81/80 - that's the syntonic comma.
64/63 is the difference between pythagorean 16/9 "minor seventh" and septimal "minor seventh". 81/80 is the difference between pythagorean minor seventh and 5-limit "pure" minor seventh.

There is a whole branch of mathematics behind this theory, linking geometry and number theory (it's usually used only in computer graphics, I think).
https://en.wikipedia.org/wiki/Geometry_of_numbers

Here is picture, demonstrating 12 equal in terms of fifths/fourths and major and minor sixths and thirds in just intonation and 12 equal.
https://en.wikipedia.org/wiki/Fokke...media/File:Fokker_periodicity_block_12TET.svg
For septimal tuning, we need a picture of parallelepiped, because it will be in 3d space.


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## fluteman

BabyGiraffe said:


> It is possible to design such instruments, but they wouldn't be that easy to be played.
> 
> But the most practical methodology (remember, there was no 12 equal back in the day!!!, many of the characteristics of most old compositions are based on these limitations) is tuning/designing around specific subset/scale of big equal temperament (historical "well temperaments" or "meantone" temperaments) are exactly this (the main difference is that meantone is achieved by stacking a single interval modulo octave - slightly flat fifth, so it's easier to understand how to construct it; well temperament will need each fifth having a different size).
> 
> Btw, if we look at equal temperaments, we can notice that all the good melodic and harmonic scales somewhat heavily distort the sizes of equal intervals.
> Compare pentatonic and 5 equal or diatonic and 7 equal.
> The trivial tuning of 5 equal is "African"/blues pentatonic, because it's very close to it; the trivial tuning of 7 equal is "Arabic" style tuning with neutral intervals.
> The trivial tuning of 12 equal is something close to 5-limit (Western music), the nontrivial tuning should be something with septimal intervals - that's why something like 31 or 22 or 27 equal maybe are good for 12 tone gamuts - 31 will temper the syntonic comma, but won't temper the septimal comma; 22 will temper the septimal comma, but won't temper syntonic comma, 27 will temper the septimal comma and fraction of syntonic comma.
> Basically, there is a structure on just intonation lattice, defined by 3 interval differences - 126/125, 64/63, 225/224.
> 126/125 x 225/224 = 81/80 - that's the syntonic comma.
> 64/63 is the difference between pythagorean 16/9 "minor seventh" and septimal "minor seventh". 81/80 is the difference between pythagorean minor seventh and 5-limit "pure" minor seventh.
> 
> There is a whole branch of mathematics behind this theory, linking geometry and number theory (it's usually used only in computer graphics, I think).
> https://en.wikipedia.org/wiki/Geometry_of_numbers
> 
> Here is picture, demonstrating 12 equal in terms of fifths/fourths and major and minor sixths and thirds in just intonation and 12 equal.
> https://en.wikipedia.org/wiki/Fokke...media/File:Fokker_periodicity_block_12TET.svg
> For septimal tuning, we need a picture of parallelepiped, because it will be in 3d space.


Yes, all that is obvious. When I was six, my music teacher told my mother to buy a piano with nontrivial tuning and septimal intervals. But I had set fire to the garage that day, and as a punishment, I was forced to play a 12-tone equal-tempered piano and simultaneously sing while tempering both the syntonic and septimal commas.

Last night I was at a get-together with some musicians, one of whom plays the Persian ney (significantly different from the Arabic ney), had one with him, and demonstrated it for me in the kitchen, near a Breville espresso machine. (Everything in that last sentence is true, most remarkably that he could get a sound out of that thing.) Since we were in a kitchen, he used the just intonation lattice, with goat cheese, chopped walnuts and a very good raspberry vinaigrette. I don't want to adopt a mean tone, but I though the whole thing was off by 81/80.


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## Guest

BabyGiraffe said:


> It was well known what "well tempered" means in his own time.
> 
> https://en.wikipedia.org/wiki/Well_temperament
> 
> Free book - by J. Barbour
> Tuning and temperament : a historical survey
> 
> https://archive.org/details/tuningtemperamen00barb


Well yes, but there was more than one system in use. I remember reading somewhere that Bach's preference was probaly the Vallotti tuning or something closely related. I don't find the sound of a harpsichord very pleasant so I'm almost always listening to recordings of modern piano, which means I am accustomed to equal temperament. It would be a hoot if someone performed on modern piano with Vallotti tuning. 

Which make me wonder, can I assume that all harpsichord recordings use a historically appropriate tuning, or do people sometimes tune harpsichords with equal temperament?


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## Guest

millionrainbows said:


> What, exactly and completely in all its senses, does the term "harmonic" mean?


I've never understood the term to be honest. Noise sounds "harmonic" to me. For a few months now, I've been falling asleep using apps that supply a tanpura drone. A tanpura is an Indian instrument that gives these really AMSR-type drones. I can change the note values of each tanpura app to anything I want. So, I tune one to C, for example, and the other to D# which should produce dissonance but it doesn't. In fact, I find it very relaxing. But, at the same time, I'm not tone deaf. I know if someone is playing out of tune. But another part of me doesn't hear dissonance, even welcomes it. I mean, if we are talking a harmonic series, I know that that is and we need it because I like notes to have that purity but harmonizing notes doesn't mean a lot me. I don't care if chords have wrong notes in them. I don't hear them that way. I use them anyway--especially in jazz. Used to drive my instructor crazy.


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## millionrainbows

Well, they say a C to D is a dissonance, but I hear it as a harmonic. I have that harmonic series memorized in my head, from working with synth filters: 1-5-1-2-3-5-b7-1, you know what I mean? You can hear it on any string.


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## Guest

millionrainbows said:


> exactly and completely in all its senses


Rereading this in Victor Redseal's last post, I'm struck by the oxymoron. If a word has more than one sense, or can be used in several different contexts, then it's probably not possible to list and define them all "exactly and completely" - not here at any rate. By all means collate findings from dictionary and encyclopedia research and post them here. Or invite people's general instinctive responses. But "exact and complete" is unlikely to be achieved.

I wonder what prompted the question, million? Some dissatisfaction with the way it's being used by someone somewhere?


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## BabyGiraffe

Baron Scarpia said:


> I don't find the sound of a harpsichord very pleasant so I'm almost always listening to recordings of modern piano,
> 
> Which make me wonder, can I assume that all harpsichord recordings use a historically appropriate tuning, or do people sometimes tune harpsichords with equal temperament?


Well, there exists such thing as https://en.wikipedia.org/wiki/Historically_informed_performance
About piano - pianos are excellent for accompaniment, not because they are very harmonious, but for the opposite reason. It's because of the sharp, metallic attack that cuts though other instruments.
I think harpsichords and similar have different envelope profiles than piano - https://en.wikipedia.org/wiki/Envelope_(music)
They also have more clearly audible harmonics than pianos. I am not sure, if the characteristic "buzz" in their timbre comes from some kind of string inharmonicity or pianos would sound the same, if they had the same sustain and release envelopes.

Millionrainbows:

"

Well, they say a C to D is a dissonance, but I hear it as a harmonic. I have that harmonic series memorized in my head, from working with synth filters: 1-5-1-2-3-5-b7-1, you know what I mean? You can hear it on any string.

"

C->D should be a dissonance, if they are in the same octave, but C->D (in higher octaves) can be a combination tone that can be heard even if it is not played, if the music is very loud or we apply some kind of guitar amplification/distortion.


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## Larkenfield

I can’t remember the last time I heard a baroque keyboard recording where it sounded like anything other than the equal temperament was being used, though the mean-tone temperament was certainly available and would work in certain pieces. But recordings do exist and when the mean-tone temperament is used one can hear it in the purity of the scale and intervals. It’s a wonderful sound, as if the intervals of those pure tones and intervals were intended to be used in everything because they’re so pleasing. But again, I can’t remember the last time I heard a performance with any kind of a mean-tone temperament being used, perhaps because it limits the modulation to certain keys to avoid the unpleasant, out of tune “wolf” tones that the equal temperament doesn’t have. That’s the problem with the mean-tone temperament because it can require more inconvenient retuning of the instrument to get those pure intervals.


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## millionrainbows

MacLeod said:


> Rereading this in Victor Redseal's last post, I'm struck by the oxymoron. If a word has more than one sense, or can be used in several different contexts, then it's probably not possible to list and define them all "exactly and completely" - not here at any rate. By all means collate findings from dictionary and encyclopedia research and post them here. Or invite people's general instinctive responses. But "exact and complete" is unlikely to be achieved.
> 
> I wonder what prompted the question, million? Some dissatisfaction with the way it's being used by someone somewhere?


Yes, someone confused "harmonic" to mean 'derived exclusively from the natural harmonic series,' in order to derail a thread.

Yes, it is possible to have an exact an complete definition of the word 'harmonic,' and this is made even easier by the fact that it can be used both as a noun and an adjective.


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## millionrainbows

BabyGiraffe said:


> Well, there exists such thing as https://en.wikipedia.org/wiki/Historically_informed_performance
> About piano - pianos are excellent for accompaniment, not because they are very harmonious, but for the opposite reason. It's because of the sharp, metallic attack that cuts though other instruments.
> I think harpsichords and similar have different envelope profiles than piano - https://en.wikipedia.org/wiki/Envelope_(music)
> They also have more clearly audible harmonics than pianos. I am not sure, if the characteristic "buzz" in their timbre comes from some kind of string inharmonicity or pianos would sound the same, if they had the same sustain and release envelopes.
> 
> Millionrainbows:
> 
> "
> 
> Well, they say a C to D is a dissonance, but I hear it as a harmonic. I have that harmonic series memorized in my head, from working with synth filters: 1-5-1-2-3-5-b7-1, you know what I mean? You can hear it on any string.
> 
> "
> 
> C->D should be a dissonance, if they are in the same octave, but C->D (in higher octaves) can be a combination tone that can be heard even if it is not played, if the music is very loud or we apply some kind of guitar amplification/distortion.


The "c-d" I speak of does occur in the same octave, right next to each other as a major second. Haven't you ever heard this harmonic when a note is filtered? I can't believe you would question this.
No difference tone, no amplification, I hear C-D as non-dissomant. In chords, it is commonly used in pop songs as a second, right next to the triadic third, distinguished from a ninth. It's used to create a very smooth, rich, harmonic sound.

If you are insisting that this is a dissonance, then you are more academic than I realized, and I am very disappointed. I suggest that you avoid 'diaphonic' music, like the Bulgarian Women's Choir.

In fact, I'm disappointed in the whole lot of you critics. Why doesn't everyone stop focussing on contradicting me, and try to get at some truth? Just like all the other traditionalist right-fighters, you are only interested in being "right" and proving every assertion I make as "wrong" in some irrelevant detail.


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## Guest

millionrainbows said:


> Well, they say a C to D is a dissonance, but I hear it as a harmonic. I have that harmonic series memorized in my head, from working with synth filters: 1-5-1-2-3-5-b7-1, you know what I mean? You can hear it on any string.


What it is is that once you get used to it, you hear it differently. First encountered, you wince but prolonged subjection causes your brain to adjust to it. Try C and C# and see how that works. I find the smaller the interval, the more dissonant. In a C scale, a D could serve as a passing note. We use those quite a lot in double bass because we tend to arpeggiate the chords and sometimes to get from one chord to the next if they share no notes is to transition through a passing note--it's in the scale but not in the chord. So it can be harmonic in that sense, I suppose. But shrink the interval to a half-step and it's a different story but then jazz uses that too. I'll try it with the apps tonight and see how it goes.


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## millionrainbows

Victor Redseal said:


> What it is is that once you get used to it, you hear it differently. First encountered, you wince but prolonged subjection causes your brain to adjust to it. Try C and C# and see how that works. I find the smaller the interval, the more dissonant.


A m2 is dissonant, but as the interval gets smaller, towards C, it becomes a unison.

In a C scale, a D could serve as a passing note. We use those quite a lot in double bass because we tend to arpeggiate the chords and sometimes to get from one chord to the next if they share no notes is to transition through a passing note--it's in the scale but not in the chord. So it can be harmonic in that sense, I suppose. But shrink the interval to a half-step and it's a different story but then jazz uses that too. I'll try it with the apps tonight and see how it goes.[/QUOTE]

Also, a major second (C-D) suggests a stacking of fifths, as in C-G-D. In this sense, it is not dissonant at all, especially since G is the most prominent overtone of C.

If the minor second is in a seventh chord, as C-E-G-A-Bb, as jazz players use it, it is also not as dissonant, as the "A" tends to pull the ear down from Bb towards a more "harmonic" seventh which does not need "resolving."


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## fluteman

BabyGiraffe said:


> Well, there exists such thing as https://en.wikipedia.org/wiki/Historically_informed_performance
> About piano - pianos are excellent for accompaniment, not because they are very harmonious, but for the opposite reason. It's because of the sharp, metallic attack that cuts though other instruments.
> I think harpsichords and similar have different envelope profiles than piano - https://en.wikipedia.org/wiki/Envelope_(music)


That's partly right but not the whole story. In the 19th century, not only the piano, but many of the other standard instruments of the orchestra, were redesigned mainly so they could be played louder. There are a number of possible explanations for this, but in my opinion a key reason is the much larger concert halls that began to be built in the mid- to late-19th century, in response to the industrial revolution and the expanding middle class. Yes, the piano had to be louder to cut through the din of all the other instruments, but it also needed to be louder as a solo instrument. Even its name, pianoforte, is a direct reference to its dynamic range, in comparison to the harpsichord, for example, which with its plucked strings can only be played at a single dynamic level.

But one of the great advantages of the piano, whether played alone or with other instruments or voices, is the ease with which chords can be played, of four or even five notes, even with a single hand. And though it is a percussion instrument, with developments of the mechanism and further help of the sustaining pedal, notes and chords can be sustained to a great extent for several seconds. This is a major advantage for the piano when it accompanies other instruments as well as when it is heard alone.


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## Luchesi

fluteman said:


> That's partly right but not the whole story. In the 19th century, not only the piano, but many of the other standard instruments of the orchestra, were redesigned mainly so they could be played louder. There are a number of possible explanations for this, but in my opinion a key reason is the much larger concert halls that began to be built in the mid- to late-19th century, in response to the industrial revolution and the expanding middle class. Yes, the piano had to be louder to cut through the din of all the other instruments, but it also needed to be louder as a solo instrument. Even its name, pianoforte, is a direct reference to its dynamic range, in comparison to the harpsichord, for example, which with its plucked strings can only be played at a single dynamic level.
> 
> But one of the great advantages of the piano, whether played alone or with other instruments or voices, is the ease with which chords can be played, of four or even five notes, even with a single hand. And though it is a percussion instrument, with developments of the mechanism and further help of the sustaining pedal, notes and chords can be sustained to a great extent for several seconds. This is a major advantage for the piano when it accompanies other instruments as well as when it is heard alone.


Glenn Gould was a kook, but here's his discovery;

"Somehow, I cannot help thinking of something that happened to me when I was thirteen or fourteen... I happened to be practicing at the piano one day - I clearly recall, not that it matters, that it was a fugue by Mozart, K. 394, for those of you who play it too - and suddenly a vacuum cleaner started up just beside the instrument. Well, the result was that in the louder passages, this luminously diatonic music in which Mozart deliberately imitates the technique of Sebastian Bach became surrounded with a halo of vibrato, rather the effect that you might get if you sang in the bathtub with both ears full of water and shook your head from side to side all at once. And in the softer passages I couldn't hear any sound that I was making at all. I could feel, of course - I could sense the tactile relation with the keyboard, which is replete with its own kind of acoustical associations, and I could imagine what I was doing, but I couldn't actually hear it. But the strange thing was that all of it suddenly sounded better than it had without the vacuum cleaner, and those parts which I couldn't actually hear sounded best of all. Well, for years thereafter, and still today, if I am in a great hurry to acquire an imprint of some new score on my mind, I simulate the effect of the vacuum cleaner by placing some totally contrary noises as close to the instrument as I can. It doesn't matter what noise, really - TV Westerns, Beatles records; anything loud will suffice - because what I managed to learn through the accidental coming together of Mozart and the vacuum cleaner was that the inner ear of the imagination is very much more powerful a stimulant than is any amount of outward observation".

He said he didn't like the sound of a piano. He couldn't understand how people could go to recitals and listen to an hour of it. He would never do that..


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## fluteman

Interesting as always, Luchesi. As for his observations about not listening to the piano for over an hour, Bach's Goldberg Variations, played with all the indicated repeats, usually clocks in at well over an hour. But in his famous 1955 recording, Gould skips most of the repeats and reaches the finish line in only 38:23, conveniently just short enough to fit on the typical single vinyl LP of the day. His more stately 1981 version, just before the dawn of the CD era (some call it the CD error) is only 51:20, again, conveniently just short enough to fit on a single vinyl LP of the day, as the industry had become adept at jamming more music onto a single record, possibly at the cost of sound quality but at the gain of profit.

Gould was an expert showman, and one of the many amusing and entertaining facets of his essays is how often his opinions, expressed in terms of righteous artistic intellectual honesty, happen to coincide with the commercial realities of the music business. In addition to being just outrageous enough to gain him maximum notoriety, but not so outrageous as to cause him to be altogether shunned. A very clever man.


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## Luchesi

fluteman said:


> Interesting as always, Luchesi. As for his observations about not listening to the piano for over an hour, Bach's Goldberg Variations, played with all the indicated repeats, usually clocks in at well over an hour. But in his famous 1955 recording, Gould skips most of the repeats and reaches the finish line in only 38:23, conveniently just short enough to fit on the typical single vinyl LP of the day. His more stately 1981 version, just before the dawn of the CD era (some call it the CD error) is only 51:20, again, conveniently just short enough to fit on a single vinyl LP of the day, as the industry had become adept at jamming more music onto a single record, possibly at the cost of sound quality but at the gain of profit.
> 
> Gould was an expert showman, and one of the many amusing and entertaining facets of his essays is how often his opinions, expressed in terms of righteous artistic intellectual honesty, happen to coincide with the commercial realities of the music business. In addition to being just outrageous enough to gain him maximum notoriety, but not so outrageous as to cause him to be altogether shunned. A very clever man.


In this clip they depict how Glenn might've wanted to make money in the stock market. He was clever in many things.


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## fluteman

Luchesi said:


> In this clip they depict how Glenn might've wanted to make money in the stock market. He was clever in many things.


That was a wonderful movie, one of the best classical music-themed ones ever, imho. One aspect of his life that it skips over is that despite a probably deserved reputation as a weirdo, loner and recluse, he was more than a little successful as a ladies' man. Unfortunately, Cornelia Foss, apparently the love of his life (though there were others), left him to return to her husband, the American composer Lucas Foss, and he never entirely recovered. Here is the couple in happier times:







We may never know how much of his eccentricity was genuine and how much was an act.


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## Luchesi

fluteman said:


> That was a wonderful movie, one of the best classical music-themed ones ever, imho. One aspect of his life that it skips over is that despite a probably deserved reputation as a weirdo, loner and recluse, he was more than a little successful as a ladies' man. Unfortunately, Cornelia Foss, apparently the love of his life (though there were others), left him to return to her husband, the American composer Lucas Foss, and he never entirely recovered. Here is the couple in happier times:
> View attachment 126563
> 
> We may never know how much of his eccentricity was genuine and how much was an act.


From the interview with her, I gather she left her husband for normal reasons (he was too busy and/or he was distant). She couldn't stay with Gould because of his abnormal need to often be alone, and his unhealthy obsessions about wellness.
Yes, he was acting part after part in his own life. He lived his whole life like that. Those pressures and the pills for moods and the pills for pain did him in at a young age. I think his cousin said he never ate anything which we would consider nutritious (fruits and vegetables) and essential today.


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## JeffD

millionrainbows said:


> In fact, I'm disappointed in the whole lot of you critics. Why doesn't everyone stop focussing on contradicting me, and try to get at some truth? Just like all the other traditionalist right-fighters, you are only interested in being "right" and proving every assertion I make as "wrong" in some irrelevant detail.


OK we give up. What is does "Harmonic" mean?

This incessant "wrong, guess again" is like click bate, it keeps the thread moving, but simultaneously going nowhere.


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## millionrainbows

JeffD said:


> OK we give up. What is does "Harmonic" mean?
> 
> This incessant "wrong, guess again" is like click bate, it keeps the thread moving, but simultaneously going nowhere.


I asked the thread question "what is harmonic" in hopes that pro-active thinkers would come forward with answers, instead of viewing the thread question in this negative way. Like an old manager of mine once said, "Come to me with solutions, not problems."

If you read the thread thoroughly, you'll see that "harmonic" has meanings both as a noun and as an adjective. You will also see that I reject a one-dimensional definition.

Now, let's not interrupt floot-man and Lucheezy's discussion of Glenn Gould.


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## Luchesi

millionrainbows said:


> I asked the thread question "what is harmonic" in hopes that pro-active thinkers would come forward with answers, instead of viewing the thread question in this negative way. Like an old manager of mine once said, "Come to me with solutions, not problems."
> 
> If you read the thread thoroughly, you'll see that "harmonic" has meanings both as a noun and as an adjective. You will also see that I reject a one-dimensional definition.
> 
> Now, let's not interrupt floot-man and Lucheezy's discussion of Glenn Gould.


When I got here in 2013 TC gave me Luchesi. I wanted Woodgoose or Duckandcover or maybe the more cryptic Wouldduck.


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## millionrainbows

I chose 'millionrainbows' over 'millionbrainsow.'


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## JeffD

millionrainbows said:


> I asked the thread question "what is harmonic" in hopes that pro-active thinkers would come forward with answers, instead of viewing the thread question in this negative way. Like an old manager of mine once said, "Come to me with solutions, not problems."
> 
> If you read the thread thoroughly, you'll see that "harmonic" has meanings both as a noun and as an adjective. You will also see that I reject a one-dimensional definition.


Stirring the pot for the intellectual amusement of it all. Gotcha.


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## MAXSWAGGER

Harmonic can mean:

1.) The overtones of a given pitch.

2.) The maj7 in any minor scale - giving the kind of "cheapo hollywood oriental cliché" feeling.

3.) As an a describing character of a feeling. Harmonic music feels full, warm and bright, while dissonant music only only spreads fear, anxiety and terror. So harmonic music is the music of life and dissonant music is the music of death. Harmonious music is love. Dissonant music is hate.


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