# Music Theory Approaches: 1. Tonality



## millionrainbows

This is the first of a series of threads devoted to various approaches to music theory and harmony.

*Tonality*
Rameau probably wrote the first book completely devoted to tonal harmony, in 1722. Since then not much has changed. There are some new innovations, though, which have appeared recently.

Leonhard Euler (1707-1783) proposed his "Euler plane" about the same time as Rameau was writing his treatise. He made a grid, and placed fifths on the horizontal axis, and major thirds on the vertical axis.

Guerino Mazzola used the Euler plane in 2002 to propose his "altered scales" of C major. He keeps C and G at the center, and adds any other five adjacent notes to create a 7-note scale. As you can see, the "C-G" at the start is what assures stability of the scale. A total of 32 scales can be generated this way.

It's also possible, on the Euler graph, to circle the notes of a Cmajor and D major scale. This way, one can visualize easily which notes are common to both, and which differ, in order to make modulations.

Mazzola also put the Euler grid on a torus (a doughnut shape). This replaces fifths with minor seconds, but now the intervals circle around the doughnut. Other intervals also circle around if followed at different angles.

There are other systems of graphing. Dmitri Tymoczko does this in his Geometry of Music (2011), and Franck Jedrzejewski (2006) also, by constructing a hexagonal grid (like interconnected stop signs), which graphs the 24 major and minor triads. This reveals connections of relative minors, parallel minors, and other "function" areas.

IRCAM has done similar research, and some have made grids using 24 notes per octave.


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## millionrainbows

The Euler grid, or variations on this, has shown up in other places as well; notably in Schoenberg's Structural Functions of Harmony, a text in which he explores modulation and moving within different areas of tonality. I've seen traces of it in Hindemith, and other newer theory texts.


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## millionrainbows

Traditional tonality is ostensibly based on the triad: the major third and the fifth. The "just" versions of the intervals are beatless, smooth phenomena of nature. Therefore, even in this age of equal temperament, tonality is a sensually-based system, derived from consonances, which the ear hears sensually.

Also, the 12-note scale was derived by "stacking" 3:2s or fifths. At the 12th note, the circle did not close, so the octaves were preserved by shrinking the interval. This gap is known as the Pythagoran comma.

So, even though tonally is based ostensibly on "perfect" consonances, the fact is, it is a compromise, and always has been. This is a very important principle to keep in mind.


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## Bettina

This is a fascinating series of posts! Are you familiar with Riemann's theories about transformation? The P transformation is a shift to the parallel minor, where only one note (the third of the chord) is lowered. The L transformation is when the root of the chord is lowered a half step. And so on...

In Riemann's opinion, most modulations could be explained in terms of such operations. I'd be interested in hearing your take on this approach.


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## millionrainbows

Yes, I've heard of Riemann's ideas, but not in detail. Hindemith had some interesting ideas as well, about consonance and dissonance, and a more modern way to deal with these.

Thus, we see tonality as a broader vista, greatly expanding the possibilities of "Tonality" (CP tonality) and its major/minor system.

The modern approach comes in when "tonality" is no longer dependent, or derived directly and sensually from the I-IV-V system. 
Besides, the perfect consonances which the CP Tonal system are based on never really existed as "perfect" anyway, because of the Pythagoran comma and the division of the octave into 12 notes. As chromaticism increases, so do these abstract principles and possibilities based on the "12"-ness of the octave.

The question then becomes a way of navigating through this 12-note chromatic landscape.

Grids are not the only way of charting these relationships; the circle of fifths, and the chromatic circle, also become valuable as maps.

John Coltrane became interested in the "magic triangle" which occurs when lines are drawn within the circle of fifths or a chromatic circle. Thus, geometry becomes another tool in music, harkening back to the Quadrivium of the Greeks: Music, geometry, number, and astronomy.


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## millionrainbows

In understanding tonality, we must always bear in mind Pythagoras and the division of the complete chromatic into 12 notes, derived from his principles.

This "12-ness" was arrived at by "stacking" or projecting fifths (3:2), like C-G-D-A-E-B-F#-C#-G#-D#-A#-E#, and cycling them back into one octave. See WIK, interval multiplication: https://en.wikipedia.org/wiki/Multiplication_(music)

So, we see from Pythagoras that this was based on the "perfect" fifth of 3:2.

But to preserve the octave, 2:1, perfect fifths are impossible within one octave. No number of "3:2s" can ever add up to "1" or "2". It's mathematically impossible. "1" is a prime number, divisible only by itself. "3" will not fit into it.

The only way this is remotely possible is by fitting "3" into "12" four times.

The Pythagoran comma is the "left over" of the near-coincidence of 12 stacked 3:2s, which almost coincide. It was close enough, so the difference was spread-out over 12 fifths. Thus, our ET fifth, which is 2 cents flat.

This "Pythagoran comma" and the search for better fifths and major thirds, was the reason for all the tempered versions of scales within the octave; mean-tone, Vallotti, etc.

Tonality is a compromise, and always has been. There are no "perfect" ratios in tonality, although the system itself is based on harmonic principles of "perfect" ratios. Bear this in mind, always: tonality is a compromised system based on unrealized "ideals."


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## BabyGiraffe

Western tonality practice is a system, a language.
Perfect harmony (shifting commas or drifting the tonic) is possible with instruments that can do adaptive retuning like the human voices or unfretted string instruments - or synthesizers... 
Of course, there are many other ways to tune and harmonize - there are many chinese instruments tuned in 4ths, so harmony in 4ths will sound better on them than harmony in 4ths using the western tuning systems.


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## millionrainbows

BabyGiraffe said:


> Western tonality practice is a system, a language.
> Perfect harmony (shifting commas or drifting the tonic) is possible with instruments that can do adaptive retuning like the human voices or unfretted string instruments - or synthesizers…


Just to ensure that we will not be missing the point of the Pythagoran comma, and why it was 'closed,' we must recognize that this was done to preserve the octave. Otherwise, the stacking of 3:2s would keep on spiraling outward.
Tonality is dependent on a tonic note. It is also desired that this tonic note is available in all octaves, hence the 'closing of the circle.'
Yes, it is possible to have music with perfect intervals; but as anyone who has experimented with 'just' intonation knows, this is possible with only one tonic, and key changes are impossible. Everything relates to that one tonic note.



> Of course, there are many other ways to tune and harmonize - there are many chinese instruments tuned in 4ths, so harmony in 4ths will sound better on them than harmony in 4ths using the western tuning systems.


Yes, this is a vast area of exploration; but my only goal in this thread is to show how the tonal system, ostensibly based on "perfect" intervals and harmonic principles, is essentially a compromise. Only by understanding that "12-ness" is ostensibly based on the Pythagoran principle of stacked 3:2s, with compromise, do we begin to see that "12 notes per octave" is essentially an intellectual construct which departs from harmonic principles, not only the minute Pythagoran comma, but the way other important intervals were compromised, namely the major third, which in our present ET tuning is 14 cents flat.

Our tonal system, being based on the projection of the fifth, favors that interval. Presently, in ET, our fifths are 'only' 2 cents flat each, to make up for the "comma" error; but our M3s are a full 14 cents flat. This was the reason for the mean-tone tunings of the past, which sought to find better thirds.

My goal is not to point out the imperfections of the tonal system, but to show that if we can grasp the above principles, the "geometry" of the 12-division octave emerges as a modernist principle, which allowed the expansion of tonality into new dimensions, and also allowed the flowering of a form of musical thought which was as dependent on harmonic factors as it was in geometric and arithmetical factors. Thus, modernism retained its harmonic, sensual characteristics, as well as having a new intellectual dimension based on 'geometries of sound.'


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## Bettina

millionrainbows said:


> Our tonal system, being based on the projection of the fifth, favors that interval. Presently, in ET, our fifths are 'only' 2 cents flat each, to make up for the "comma" error; but our M3s are a full 14 cents flat. This was the reason for the mean-tone tunings of the past, which sought to find better thirds.


Lots of thought-provoking issues here! I wonder what came first: the harmonic system based on the circle of fifths, or the tuning system that emphasized the accuracy of the fifth. Did the new musical styles in the Renaissance period lead to changes in the tuning system? Or did the new tuning system prompt composers to base their harmonic progressions on fifths rather than thirds? Such chicken-and-egg questions are often very difficult to answer, but I'd be interested in any thoughts on the topic!


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## millionrainbows

Bettina said:


> Lots of thought-provoking issues here! I wonder what came first: the harmonic system based on the circle of fifths, or the tuning system that emphasized the accuracy of the fifth.


The circle of fifths probably 'developed' out of the stacking (projection) principle, for the reason that not all 12 keys were used in earlier tonal music; the need to modulate was not yet a contingency. But the circle probably existed as a conceptual possibility, if not practical. Pythagoras did not have, or need, 12 notes, but his "school" extrapolated the stacking principle and followed it to its conclusion, the Pythagoran comma and the closing of the 12th step octave. (recovering Pythagorans take note).

Look at early keyboard music, and most of it stays in one key. Vallotti, mean-tone, and other early _tempering_ tunings attempted to 'adjust' the intervals within each octave (key) in search of better fifths and thirds (which are harmonic goals, purely sensual). These tunings were effective only within limited ranges, not all the way around the circle. Thus, a mean-tone tuning or variant would be good for C, G, A, and E (in the sharp direction), and C, F, Bb, and Eb in the "flat" direction (circle of fourths).

The complete circle of fifths, with its complete range of possibilities, probably planted the idea of 12 usable, good-sounding keys, which only became truly possible when Equal Temperament was achieved.

Bach's tuning for the WTC was such an attempt; not exactly equal, but "well-tempered," which meant all 12 key areas sounded passable. See the Bradley Lehman site, http://www.larips.com



> Did the new musical styles in the Renaissance period lead to changes in the tuning system? Or did the new tuning system prompt composers to base their harmonic progressions on fifths rather than thirds? Such chicken-and-egg questions are often very difficult to answer, but I'd be interested in any thoughts on the topic!


I think part of this has been answered.


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## Bettina

millionrainbows said:


> The circle of fifths probably 'developed' out of the stacking (projection) principle, for the reason that not all 12 keys were used in earlier tonal music; the need to modulate was not yet a contingency. But the circle probably existed as a conceptual possibility, if not practical. Pythagoras did not have, or need, 12 notes, but his "school" extrapolated the stacking principle and followed it to its conclusion, the Pythagoran comma and the closing of the 12th step octave. (recovering Pythagorans take note).
> 
> Look at early keyboard music, and most of it stays in one key. Vallotti, mean-tone, and other early _tempering_ tunings attempted to 'adjust' the intervals within each octave (key) in search of better fifths and thirds (which are harmonic goals, purely sensual). These tunings were effective only within limited ranges, not all the way around the circle. Thus, a mean-tone tuning or variant would be good for C, G, A, and E (in the sharp direction), and C, F, Bb, and Eb in the "flat" direction (circle of fourths).
> 
> The complete circle of fifths, with its complete range of possibilities, probably planted the idea of 12 usable, good-sounding keys, which only became truly possible when Equal Temperament was achieved.
> 
> Bach's tuning for the WTC was such an attempt; not exactly equal, but "well-tempered," which meant all 12 key areas sounded passable. See the Bradley Lehman site, Larips.com
> 
> I think part of this has been answered.


Thanks for a great--and very prompt--answer. That was fast!  You have an encylopedic knowledge of this stuff. I'll take a look at the website that you recommended.


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## millionrainbows

Bettina said:


> Thanks for a great--and very prompt--answer. That was fast!  You have an encylopedic knowledge of this stuff. I'll take a look at the website that you recommended.


You flatter me! Also, a bookshelf full of nothing but music theory books. I like to think that I have a good "overview" of theory, by reading repetitively. Sometimes the same basic information is stated in slightly different ways, which gives new perspective. This is why it is good to read books about what you think you already know.

I actually have some practical experience as well; I worked for over 10 yrs. in a tuned wind chime factory. The chimes (each pipe) were tuned using KORG digital tuners, to within 2 cents accuracy. The pipes were extruded aluminum, which is very uniform in thickness, and produces precise pitches which do not 'waver.'

I think this is why Dr. Bradley Lehman discovered the Bach tuning; he had practical experience tuning his own harpsichords. I urge all pianists and music students to try tuning things.

Harry Partch built his own instruments, using his own tunings. His book "Genesis of a Music" is a very valuable resource.


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## BabyGiraffe

Key changes are possible with some instruments, but you will hear some pitch drifting, comma pumps or wolf intervals.
After playing some retuned Kontakt libraries I hear now the normal 12tet major third as detuned and ugly... 
There are other things that sound bad on the normal piano - dom 7ths are way less dissonant, if 7/4 seventh is used - you can hear it in jazz and some classical recordings, and barbershop quartets; even different versions of the tritone sound more pleasant.
Playing Medieval, Renaissance and early Baroque music in 12tet is almost a sin. 
The funny thing about meantone temperament is that the different keys really have different flavour.

BTW, there are some dubious statements in "Genesis of music". I would stick to Wikipedia and the newer books.
Xenharmonics wiki has good information, but is pretty disorganized and it's hard to find the good articles.

The late baroque to early romanticism musical system is a set of rules for composing (usually mixing different modes like ionian, aeolian, harmonic and melodic minor). It's pretty sad that even now this is the default system taught in the schools.
For some reason people call this system the "western tonal system" when it's a modal mixture system and the other modes are just as tonal as the "western tonal system". There are many uneducated people (mainly jazz players) that insist on keeping the modes pure and similar nonsense. 
You can invent some kind of tonal system based on other modes that is just as valid (John Williams uses Lydian and Lydian variants, Danny Elfman - Lydian b3 and variants).
I learned more about harmony after I had to do Renaissance style counterpoint exercises in many different modes.
Harmony books are useless unless you are after some kind of cliches for specific historical styles with examples.


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## millionrainbows

One has to either remain in the harmonic hierarchy, which was becoming its own undoing, or depart into uncharted territories, and create a new hierarchical system of relations, as Schoenberg did. 

If one wishes not to admit that this change was historically inevitable (although things happen in sequence), then the numbers speak for themselves; 12 is more complex than 8, and produces more cross-relations. But one can choose not to recognize number in relation to music, and see it as purely art, with no dimension of number affecting it.

If one wishes (on artistic grounds) not to accept this inevitability as inevitable, then there is no point in arguing the point, as it becomes a matter of metaphysics or style.

Yet, the entire 12-note division of the octave is arbitrary and imperfect; but its hierarchy of vertical sonance relations and horizontal functions is based on natural harmonic principles; "1" is divided into fractional subservient parts.

Thus, as it has ever been, music represents the struggle between natural sensual factors and geometric constructs.


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## millionrainbows

BabyGiraffe said:


> ...BTW, *there are some dubious statements in "Genesis of music".* I would stick to Wikipedia and the newer books.
> 
> *...There are many uneducated people (mainly jazz players)* that insist on keeping the modes pure and similar nonsense.
> You can invent some kind of tonal system based on other modes that is just as valid (John Williams uses Lydian and Lydian variants, Danny Elfman - Lydian b3 and variants).
> 
> ...*Harmony books are useless* unless you are after some kind of cliches for specific historical styles with examples.
> 
> *...If you want a freakshow, you can go to the circus or listen to "avantgarde" orchestral. (Like I said before: this music is not classical, being art and orchestral doesn't make it "classical". *(from another thread)




Sorry, I can't concur with these statements, or the general emotional tone of these posts.


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## BabyGiraffe

millionrainbows said:


> Sorry, I can't concur with these statements, or the general emotional tone of these posts.


http://www.tonalsoft.com/sonic-arts/mclaren/partch/errors.htm


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## millionrainbows

I read the article. Note how many of the "contradictions" use sine tones. Sine tones are very artificial constructs, and can only be created with a frequency generator. Musically, they are useless, unless used in combination as Stockhausen did. The flute is the closest thing we have to a sine wave.

Sine tones have no harmonics at all. As such, they are more useful for scientists than by musicians.

I'm not bothered by this at all. Obviously, the guy has an agenda, because many of the points seem to be negative interpretations of Partch's statements from the get-go; statements removed from their context and picked-apart bit by bit.

It reminds me of the way the press works, picking apart every word, and putting a negative spin on it.

I found Partch's book to be a very useful practical guide when I was tuning chimes. The many tables and indexes were very useful.


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## BabyGiraffe

millionrainbows said:


> Sine tones have no harmonics at all. As such, they are more useful for scientists than by musicians.
> 
> I'm not bothered by this at all. Obviously, the guy has an agenda...


Well, every person has "an agenda" - (if we don't count Buddha and the rest of the enlightened saints - if they weren't charlatans).
I don't think that pure sine waves are useless - they are used all the time in electronic dance music for basslines, melodies and background chords.


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## millionrainbows

BabyGiraffe said:


> Well, every person has "an agenda" - (if we don't count Buddha and the rest of the enlightened saints - if they weren't charlatans).
> I don't think that pure sine waves are useless - they are used all the time in electronic dance music for basslines, melodies and background chords.


Well, I'm sure that if there is an exception in any of my posts, you'll find it.

And no, not everyone has an "agenda." Sometimes people just like to talk about things that interest them. Sometimes people respect other people. Sometimes people write books, and some people read them and appreciate them. Sometimes, people like to pick everything apart, and disagree with everything they read.

My only purpose in this thread is to present some information about tonality, not to push a Harry Partch agenda.

The Partch book is very good, and very useful. Like I said, I referred to it on many occasions when I worked in a chime factory, cutting pipe.

The guy who wrote the criticism has some sort of scholarly agenda, like he wrote this to please a professor who didn't like Partch, for his PHD dissertation. That's what I mean by "agenda."

I also find it very duplicitous the way he "praises" Partch's music, while at the same time dissing the book.

I'm pretty much immune to this sort of invalidation, unless it escalates into ugly talk.


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## BabyGiraffe

millionrainbows said:


> Well, I'm sure that if there is an exception in any of my posts, you'll find it.
> 
> And no, not everyone has an "agenda." Sometimes people just like to talk about things that interest them. Sometimes people respect other people. Sometimes people write books, and some people read them and appreciate them. Sometimes, people like to pick everything apart, and disagree with everything they read.
> 
> My only purpose in this thread is to present some information about tonality, not to push a Harry Partch agenda.
> 
> The Partch book is very good, and very useful. Like I said, I referred to it on many occasions when I worked in a chime factory, cutting pipe.
> 
> The guy who wrote the criticism has some sort of scholarly agenda, like he wrote this to please a professor who didn't like Partch, for his PHD dissertation. That's what I mean by "agenda."
> 
> I also find it very duplicitous the way he "praises" Partch's music, while at the same time dissing the book.
> 
> I'm pretty much immune to this sort of invalidation, unless it escalates into ugly talk.


Wow, the guy that wrote that criticism hosts a free encyclopedia on microtuning and music theory - updated with information that wasn't avaliable to Partch.
http://www.tonalsoft.com/
There is also a free software for composing microtonal music. It's not like he is doing it to please anyone.


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## millionrainbows

The article comes across to me like a dissertation, with a scholastic or intellectual agenda. I've seen articles like this used before in debates, and they range from opinions on tonality to other aspects of music. It's too academic for me.

I've got plenty of info on tunings, including The Just Intonation Primer, Sound and Light, Instrument making, and many others, and none of the writers felt compelled to dis Harry Partch. 

This writer needs to reconsider his criticism, and give respect where respect is due. I feel that Partch's views on certain intervals, tunings, and such, were informed by the nature of his instrument making, and by his art. I don't think you can separate his views from his art.

I already love Harry Partch, and have for years. I bought "Delusion" when it came out on vinyl. This is all lost on me. If you wish to discuss specifics, that could be arranged.


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## BabyGiraffe

millionrainbows said:


> The article comes across to me like a dissertation, with a scholastic or intellectual agenda. I've seen articles like this used before in debates, and they range from opinions on tonality to other aspects of music. It's too academic for me.
> 
> I've got plenty of info on tunings, including The Just Intonation Primer, Sound and Light, Instrument making, and many others, and none of the writers felt compelled to dis Harry Partch.
> 
> This writer needs to reconsider his criticism, and give respect where respect is due. I feel that Partch's views on certain intervals, tunings, and such, were informed by the nature of his instrument making, and by his art. I don't think you can separate his views from his art.
> 
> I already love Harry Partch, and have for years. I bought "Delusion" when it came out on vinyl. This is all lost on me. If you wish to discuss specifics, that could be arranged.


You can love whatever you want, it doesn't make it accurate.


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## lextune

I recommend Stuart Isacoff's book "Temperament" for an interesting read along this thread's lines.


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## millionrainbows

BabyGiraffe said:


> You can love whatever you want, it doesn't make it accurate.


Then let's get down to the details; I invite your discussion on another thread.


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## millionrainbows

lextune said:


> I recommend Stuart Isacoff's book "Temperament" for an interesting read along this thread's lines.


Well, it's diverging. This thread was about tonality, not microtonality and Harry Partch.

The book is good, I have it. Temperament is a relevant subject, relating to tonality.


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## millionrainbows

BabyGiraffe said:


> You can love whatever you want, it doesn't make it accurate.


So do you think I should read some music cognition and perception studies before just throwing out stuff like this?


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## Zellibrung

millionrainbows said:


> Musically, they are useless, unless used in combination as Stockhausen did. The flute is the closest thing we have to a sine wave.
> Sine tones have no harmonics at all. As such, they are more useful for scientists than by musicians.


I find it strange that someone who argues seems to argue strongly in favor of atonal music would dismiss sine waves as having no musical use simply because they aren't natural enough. How is it any different than the sound of a saxophone? as far as I know that exact sound could not be created either before the invention of a saxophone.

I have no opinion on the article in question though, and their sine wave based claims.


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## Vox Gabrieli

Zellibrung said:


> I find it strange that someone who argues seems to argue strongly in favor of atonal music would dismiss sine waves as having no musical use simply because they aren't natural enough. How is it any different than the sound of a saxophone? as far as I know that exact sound could not be created either before the invention of a saxophone.
> 
> I have no opinion on the article in question though, and their sine wave based claims.


Could you provide whatever source you derived the saxophone claim from?

In addition, I wouldn't exactly jump on the oppurtunity to hear a sine tone concerto. The closest thing would be a theremin, and only on it's contemporary relevance.

A cute little musical anecdote compliments of Dmitri Shostakovich.





I'm really not sure of my opinion here, and this is because of the general gravitation to the old, and neglecting the new.

This is a topic frequently brought up in the Music Theory sub forum.


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## Zellibrung

I was simply going for a respected instrument that had been invented fairly recently. Funny enough I thought of the theremin aftward. The general point being that many sounds seem to be "artificially constructed" by man in some way.

I said as far as I know, because I have no idea for certain if some bird in papua new guinea naturally sounds exactly like a saxophone or what have you.


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## Phil loves classical

millionrainbows said:


> I read the article. Note how many of the "contradictions" use sine tones. Sine tones are very artificial constructs, and can only be created with a frequency generator. Musically, they are useless, unless used in combination as Stockhausen did. The flute is the closest thing we have to a sine wave.
> 
> Sine tones have no harmonics at all. As such, they are more useful for scientists than by musicians.
> 
> I'm not bothered by this at all. Obviously, the guy has an agenda, because many of the points seem to be negative interpretations of Partch's statements from the get-go; statements removed from their context and picked-apart bit by bit.
> 
> It reminds me of the way the press works, picking apart every word, and putting a negative spin on it.
> 
> I found Partch's book to be a very useful practical guide when I was tuning chimes. The many tables and indexes were very useful.


Huh? I saw the waveform for a flute and it doesn't look close to a pure sine wave. Any wave can have higher harmonics.


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## millionrainbows

Phil loves classical said:


> Huh? I saw the waveform for a flute and it doesn't look close to a pure sine wave. Any wave can have higher harmonics.


Anyone familiar with the basics of sound synthesis knows that the flute is the closest thing to a sine wave, the violin the closest thing to a sawtooth, and the clarinet the closest thing to a square wave.


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## Phil loves classical

millionrainbows said:


> Anyone familiar with the basics of sound synthesis knows that the flute is the closest thing to a sine wave, the violin the closest thing to a sawtooth, and the clarinet the closest thing to a square wave.


Ok I saw some other pictures, and some look closer to the sine wave. Wonder why they have such a big discrepancy in the waveforms produced.


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