# Forget the piano player - Shoot the piano.



## NickNotabene (Jun 13, 2013)

In 1917 an extraordinary non-event occurred in the music world. A 19-year old kid named Ernst Bacon published what most agree was the first complete list of set classes contained in the 12-tone chromatic. His article, "Our Musical Idiom," was rejected by the _Journal of Musicology_ as being inappropriate (read "over their heads") for that journal's readership, and so it was published in the non-music journal _The Monist_ - and then ignored and promptly forgotten by all but a very few. Anyone who has ever tried to make The List from scratch knows this wasn't an easy feat. Even though The List is manna for post-tonal theory & composition, it wasn't until 1960 that Howard Hanson published his version, followed later by the versions of Forte & Carter. None of these later lists acknowledged Bacon's, and it is still unclear if any of them knew of Bacon's work when they came up with their own.

If you follow the link above to Bacon's article, the list per se starts on p.35, but leading up to it Bacon discusses the difficulties presented by a notational system that assumes the diatonic scale as its basis. E.g., when you start to look at the entire SC universe you suddenly realize the absurdity of trying to write any SC as a stack of thirds. This realization gradually resulted in today's situation where neotonalists (attempting to establish a new common practice by fiat, IMO) are forced to do something that would have been a pointless exercise to Mozart: to justify the diatonic triad as a basic building block. I wouldn't state it as a fact, but, to me, the creation of the first set class list, laying out the entire harmonic material of the 12tet universe, was the point where Western music lost its innocence.

But still, we cling to that innocence in a highly intimidating way that no one I know of has previously remarked upon. The embodiment of diatonicism is omnipresent. Whether on a piano, a synthesizer, or a laptop, it's very difficult to remain blind to the traditional keyboard. To get at all those other sonorities and patterns waiting for you out there, you still have to work through or around that damn white-key-black-key trap: 2212221/23223.

I'm just wondering if anyone else shares my love-hate relationship with the keyboard specifically & the diatonic generally as standing in the way of real "musical exploration"? You love your Beethoven, but you hate that you have no choice but to use his piano if you want to make your own music.


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## Mahlerian (Nov 27, 2012)

Schoenberg once created a new system of notation for writing chromatic music, which did not require sharp or flat symbols to modify notes. Of course, even though if learned, it could be a useful tool, old habits die hard, and the notation that's stayed the same for the last few centuries wasn't going to be replaced so easily.

He details this in _Style and Idea_.


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## Marisol (May 25, 2013)

NickNotabene said:


> E.g., when you start to look at the entire SC universe you suddenly realize the absurdity of trying to write any SC as a stack of thirds.


Western music is not built on stacks of thirds, it is built on a combination of octaves, fifths and thirds.


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## NickNotabene (Jun 13, 2013)

Marisol said:


> Western music is not built on stacks of thirds, it is built on a combination of octaves, fifths and thirds.


OK. I'm perfectly willing to give you octaves and fifths into the mix. Now: Express the chord C-Db-D-Eb-F-Ab as a combination of octaves, fifths and thirds. Yes, you _can_ do it, of course (though the octave is redundant). But you either have to admit that the result of generating/expressing this sonority in this way is either absurd, or you have to say the chord is "unacceptable" as a proper chord - which is what the academic gate keepers were doing in the beginning of the twentieth century (e.g., Verklaerte Nacht). So either you set yourself up as the arbiter of what is & is not a proper chord & have to defend why - or you get tangled in a notational nightmare.


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## Marisol (May 25, 2013)

NickNotabene said:


> OK. I'm perfectly willing to give you octaves and fifths into the mix. Now: Express the chord C-Db-D-Eb-F-Ab as a combination of octaves, fifths and thirds. Yes, you _can_ do it, of course (though the octave is redundant). But you either have to admit that the result of generating/expressing this sonority in this way is either absurd, or you have to say the chord is "unacceptable" as a proper chord - which is what the academic gate keepers were doing in the beginning of the twentieth century (e.g., Verklaerte Nacht). So either you set yourself up as the arbiter of what is & is not a proper chord & have to defend why - or you get tangled in a notational nightmare.


I am at a loss as to what you are talking about.

The C major chord for instance is a C + a third + a fifth
The C minor chord for instance is a C + a third below the fifth + a fifth.

Both are a combination of thirds and fifths not stacked thirds.

A chord based on stacked thirds would for instance be: C - E - G#
A chord based on stacked fourths would for instance be: G - C - F
A chord based on stacked fifths would for instance be: C - G - D

The western music notation is inadequate for describing more complex relationships, for instance how would be able to describe a chord based on stacked fourths starting with C?

We would have C - F and then what? It is not Bb, because Bb relates to C as the third below two stacked fifths. There is simply no way to describe it in western music notation.

Another example, say we have a simple song in the key of C, at one point we have a D major chord, the A of his chord cannot be represented by western notation. The D - F# are fine but the A is the third above the fourth not the Pythagorean A there is no way to express that unless we temporarily modulate to the key of G.


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## NickNotabene (Jun 13, 2013)

Marisol said:


> I am at a loss at to what you are talking about.


This could go on a very long time and go nowhere. Why don't you try reading the Bacon article I cited. Maybe that will help. Again, it can be downloaded from this page:

https://urresearch.rochester.edu/in...458451B04691231BAB2?institutionalItemId=26947


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## Marisol (May 25, 2013)

NickNotabene said:


> This could go on a very long time and go nowhere. Why don't you try reading the Bacon article I cited. Maybe that will help. Again, it can be downloaded from this page:
> 
> https://urresearch.rochester.edu/in...458451B04691231BAB2?institutionalItemId=26947


The premise of the article is that harmony is formed by "superimposed thirds", I think that is simply incorrect and I gave some examples that counters this notion.


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## Mahlerian (Nov 27, 2012)

Marisol said:


> I am at a loss as to what you are talking about.
> 
> The C major chord for instance is a C + a third + a fifth
> The C minor chord for instance is a C + a third below the fifth + a fifth.
> ...


No, they are stacked thirds, alternating major and minor. Most chords considered normal can be built in this way.



Marisol said:


> A chord based on stacked thirds would for instance be: C - E - G#
> A chord based on stacked fourths would for instance be: G - C - F
> A chord based on stacked fifths would for instance be: C - G - D


And all of these can be found in the literature pre-1910. The first is an augmented chord (you left out the diminished seventh, a chord that stacks three minor thirds), found in Impressionist and Russian music in particular. The second is found in Mahler, Schoenberg, and Scriabin, and the third is found as far back as the beginning of the finale of Beethoven's Pastoral symphony.



Marisol said:


> The western music notation is inadequate for describing these relationships, for instance how would be describe a chord based on stacked fourths starting with C?
> 
> We would have C - F and then what? It is not Bb, because Bb relates to C as the third below two stacked fifths. There is simply no way to describe it in western music notation.


Well, quartal chords have a dominant-like function, so you can describe it in terms of how it resolves.

Take the beginning of Schoenberg's First Chamber Symphony, for instance:








The first chord here is a 6-note quartal: G-C-F-Bf-Ef-Af, doubled on the top and the bottom at the octave. It resolves to a chord that acts as a dominant of F minor, and the final chord is a minor chord with a suspension on the fourth where both minor third and fourth resolve to the major third.


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## Marisol (May 25, 2013)

Mahlerian said:


> No, they are stacked thirds, alternating major and minor. Most chords considered normal can be built in this way.


Give me your harmonic definition of a minor third. Really there is no such thing as a minor third harmonically, it is a misnomer. It does not make any sense as the 'third' in a third has to do with ratios between frequencies. A 'minor third' is simply a third below the fifth above.



Mahlerian said:


> Take the beginning of Schoenberg's First Chamber Symphony, for instance:
> View attachment 19618
> 
> 
> The first chord here is a 6-note quartal: G-C-F-Bf-Ef-Af, doubled on the top and the bottom at the octave.


It is not, at least not the way it is written here. You simply cannot write it down in the common notation. That the current key is E makes this chord harmonically completely out of place.

Did Schoenberg intended stacked fourths here? I do not know the piece.
Perhaps you can point me to a recording where it is played in just intonation with stacked fourths?

I think you misinterpret Schoenberg completely if you attribute harmonic structures that he did not intend to be played at all, for him it is enharmonics all the way.

If you want to hear the differences try a keyboard with microtuning abilities and play G-C-F-Bb-Eb-Ab first in equal temperament, then in pure intonation and finally as stacked fourths by tuning the keyboard in 'reversed' Pythagorean tuning so that you get true stacked fourths.


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## Mahlerian (Nov 27, 2012)

Marisol said:


> It is not, at least not the way it is written here. You simply cannot write it down in the common notation. Furthermore since the current key is E this chord would is harmonically completely out of place.


I transcribed it from the full score, available here.

The music converges on E for the first theme, which opens up with a fanfare in rising fourths.



Marisol said:


> Did Schoenberg intended stacked fourths here? I do not know the piece.
> Perhaps you can point me to a recording where it is played in just intonation with stacked fourths?
> 
> I think you misinterpret Schoenberg completely if you attribute harmonic structures that he did not intend to be played at all, for him it is enharmonics all the way.


Just intonation? Western music theory depends on the equal tempered scale, and much of the repertoire (from Wagner on) would be nonsense if played on a non-tempered scale.

He uses stacked tempered fourths, and I would have thought that was obvious. Furthermore, he did clearly intend them, because the relationship of the fourth is melodically and harmonically the backbone of the piece.


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## Marisol (May 25, 2013)

Mahlerian said:


> Western music theory depends on the equal tempered scale...




By the way Wagner could certainly be played in just intonation but it would be very difficult not only because of the frequent modulations but also because often the harmonic centers shift in relation to the leading melody tone instead of diatonic or chromatic offsets of the current harmonic center. But in principle it is possible.

Below is an attempt to identify the shifts in the harmonic center and chords in the Prelude of Wagner's Tristan and Isolde:


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## Mahlerian (Nov 27, 2012)

The idea of the 12-note chromatic octave, in which C-flat and B natural are either the same or functionally equivalent became the ideal in the 18th century, at the same time as the codification of tonal, key-based harmony and the rise of modulation in the Classical era. The bulk of the harmonic theory taught today derives from the practices of that period, only supplemented by what followed and what preceded it.


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## PetrB (Feb 28, 2012)

That "keyboard trap" has everything to do with a player's acquired habits, the keyboard itself not at all "tyrannical" to triads or anything else. The keyboard is literally "Handy = user / hands friendly" and nothing more. (On a slightly narrower keyed instrument of Bach's era, the modern pianist would initially be hitting a lot of ninths where motor habits intended an octave 

Play a bunch of triads the majority of your life, maybe not have the most fluent of full piano technique, and one might assume there is a trap having to do with the keyboard: the real trap is a heavily habituated / conditioned mind connected, of course, to the ear.

The keyboard is not a problem, players motor and listening habits are the problem: the problem is conditioned, so re-conditioning should be effective if an individual wishes to do so.


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## NickNotabene (Jun 13, 2013)

Thanks for trying, Mahlerian!  I for one have learned that I don't have to respond to everything. It's too exhausting. But please know I agree with everything you have said here.


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## PetrB (Feb 28, 2012)

NickNotabene said:


> OK. I'm perfectly willing to give you octaves and fifths into the mix. Now: Express the chord C-Db-D-Eb-F-Ab as a combination of octaves, fifths and thirds. Yes, you _can_ do it, of course (though the octave is redundant). But you either have to admit that the result of generating/expressing this sonority in this way is either absurd, or you have to say the chord is "unacceptable" as a proper chord - which is what the academic gate keepers were doing in the beginning of the twentieth century (e.g., Verklaerte Nacht). So either you set yourself up as the arbiter of what is & is not a proper chord & have to defend why - or you get tangled in a notational nightmare.


It is not 1900 anymore, and ever since Hindemith, the modern - contemporary maxim is thus:
_*Any three discrete pitches constitute a chord.*_


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## NickNotabene (Jun 13, 2013)

Marisol said:


> Did Schoenberg intended stacked fourths here? I do not know the piece.
> ...
> I think you misinterpret Schoenberg completely if you attribute harmonic structures that he did not intend to be played at all, for him it is enharmonics all the way.


Just a final word or two...
One: You don't do your case - _whatever_ it is - any good by saying you don't know a piece and then claiming someone else's view of it must be wrong. Hello? You don't have standing to say anything about the piece.
Two: Schoenberg devoted an entire chapter to quartal harmony in his _Harmonielehre_, so there is hardly any way, on any grounds, of arguing away a clear example of quartal harmony found in one of his scores.


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## NickNotabene (Jun 13, 2013)

No, that's not the premise of the article. Read it again.


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## NickNotabene (Jun 13, 2013)

((Ignore this -- Sorry! -- I'm trying to get used to this format which is not the most user friendly I've experienced recently.))


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## NickNotabene (Jun 13, 2013)

Marisol said:


> The premise of the article is that harmony is formed by "superimposed thirds", I think that is simply incorrect and I gave some examples that counters this notion.


(Second try 
No, that is not the premise of the article. Seriously: READ it.


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## PetrB (Feb 28, 2012)

Marisol said:


> I am at a loss as to what you are talking about.
> 
> The C major chord for instance is a C + a third + a fifth
> The C minor chord for instance is a C + a third below the fifth + a fifth.
> ...


There is only ONE FIFTH in your C chord example (above) in case you have not noticed.

Within the compass of that fifth are two thirds, for the major chord a major with a minor on top, for the minor chord those qualities in opposite positions.

I continually seem to miss whatever actual point you are trying to make about these basics... imagining always some further explanation or argument to follow, but it never does.

As to the rest, as I said elsewhere in this thread, in the 20th century a chord is any three discrete pitches.

That stack of fourths is a chord, and called "Quartal", the fifths a chord and "Quintal."

You seem very interested in understanding and staying within very old common practice harmony, chord functions as per that mindset, and tonality as thought of during that period.... while that is all good and proper for common practice repertoire (and it can be argued it is best to know it even while studying how to compose in more contemporary ways), we are now in the 21st century.


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## NickNotabene (Jun 13, 2013)

PetrB said:


> It is not 1900 anymore, and ever since Hindemith, the modern - contemporary maxim is thus:
> _*Any three discrete pitches constitute a chord.*_


THANK YOU!!! That's the point!! But you realize we are - unbelievably - still in the minority.


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## PetrB (Feb 28, 2012)

NickNotabene said:


> THANK YOU!!! That's the point!! But you realize we are - unbelievably - still in the minority.


The majority of those who seem to be consuming art do not seem to realize that for the most part, *artists have not and do not set out to make people "Comfortable."

Art, *even the most "beautiful" *is meant*, imho, _*primarily to disturb.*_


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## KenOC (Mar 7, 2011)

PetrB said:


> *Art, *even the most "beautiful" *is meant*, imho, _*primarily to disturb.*_


I am listening to Mozart's Great C minor Mass as I read your message. Yes, it's disturbing to think anybody could write music this good! But not in any other way that I can find... 

Perhaps it disturbed people in Mozart's time, but I doubt it. Or perhaps it's neither "art" nor "beautiful"... ?


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## PetrB (Feb 28, 2012)

KenOC said:


> I am listening to Mozart's Great C minor Mass as I read your message. Yes, it's disturbing to think anybody could write music this good! But not in any other way that I can find...
> 
> Perhaps it disturbed people in Mozart's time, but I doubt it. Or perhaps it's neither "art" nor "beautiful"... ?


Since when does true beauty not disturb? I believe you've attached a quality, and only one at that, to "disturb."

If we are normally in Stasis, and something disturbs = Ecstasis = Ecstasy. 
Which direction out of stasis is another matter, not qualified in my post.


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## Marisol (May 25, 2013)

Looks like this topic is about ramblings instead of a serious discussion about harmony.


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## KenOC (Mar 7, 2011)

PetrB said:


> Since when does true beauty not disturb? I believe you've attached a quality, and only one at that, to "disturb."


If that's your definition of "disturbing" (some might find it a stretch I think) then I agree that Mozart's Mass is quite disturbing!


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## PetrB (Feb 28, 2012)

Marisol said:


> Looks like this topic is about ramblings instead of a serious discussion about harmony.


Well then how about some serious posts vs. a list of prime frontispiece factoids about common practice viewpoints and Pythagorean observations on acoustics?

Me, I hate discussing "Harmony" -- and for that matter, much chat about theory seems useless to me. Learn it, use it to make something, I say. I.e. put it into practice instead of diddling about with it.


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## Mahlerian (Nov 27, 2012)

Marisol said:


> By the way Wagner could certainly be played in just intonation but it would be very difficult not only because of the frequent modulations but also because often the harmonic centers shift in relation to the leading melody tone instead of diatonic or chromatic offsets of the current harmonic center. But in principle it is possible.


You mean changing the tuning for every key center? That's not only unfeasible for most instruments and players, it makes nonsense out of the music by making the relationships between key centers even less clear than before.



Marisol said:


> Below is an attempt to identify the shifts in the harmonic center and chords in the Prelude of Wagner's Tristan and Isolde:


I see diminished chords in there. Those have neither major thirds nor fifths in them, so how do you justify them?

On a more serious note, I haven't bothered to watch the video, but I see little point in analyzing something as if every secondary dominant is a change in key. It's laborious, time-consuming, and tells the listener almost nothing about the way the music actually works, which is on a larger time-scale than that.


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## Marisol (May 25, 2013)

Mahlerian said:


> You mean changing the tuning for every key center? That's not only unfeasible for most instruments and players, it makes nonsense out of the music by making the relationships between key centers even less clear than before.


How do you know, did you ever try it? 
I compose and play in just intonation and modulate all the time, both to a key on the current scale and also by recalculating a new harmonic center based on the leading melody tone.



Mahlerian said:


> I see diminished chords in there. Those have neither major thirds nor fifths in them, so how do you justify them?
> 
> On a more serious note, I haven't bothered to watch the video, but I see little point in analyzing something as if every secondary dominant is a change in key. It's laborious, time-consuming, and tells the listener almost nothing about the way the music actually works, which is on a larger time-scale than that.


Nobody forces you to watch the video, apparently it is wasted on you.

And apparently the way I am attacked here it is wasted on a lot of people.


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## PetrB (Feb 28, 2012)

Mahlerian said:


> You mean changing the tuning for every key center? That's not only unfeasible for most instruments and players, it makes nonsense out of the music by making the relationships between key centers even less clear than before.
> 
> I see diminished chords in there. Those have neither major thirds nor fifths in them, so how do you justify them?
> 
> On a more serious note, I haven't bothered to watch the video, but I see little point in analyzing something as if every secondary dominant is a change in key. It's laborious, time-consuming, and tells the listener almost nothing about the way the music actually works, which is on a larger time-scale than that.


Typical Freshman / Soph misstep -- once all those secondary dominants and German sixths, etc. are learned, "They must be everywhere," and the tendency is to over-mark / minutia label every vertical along the way -- often without any real notion of context of function. I.e. as you say, comes a time, rather soon, when the big picture does not include a beat by beat vertical labeling, which is no longer the thing to analyze, and has little or nothing to say about the overall breadth of how a piece works.

Stewing in the details -- drowning in, without seeing -- the big picture. (the brightest of super detail oriented folk have a hard time pulling away from that impulse


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## NickNotabene (Jun 13, 2013)

PetrB said:


> That "keyboard trap" has everything to do with a player's acquired habits, the keyboard itself not at all "tyrannical" to triads or anything else. The keyboard is literally "Handy = user / hands friendly" and nothing more. (On a slightly narrower keyed instrument of Bach's era, the modern pianist would initially be hitting a lot of ninths where motor habits intended an octave
> 
> Play a bunch of triads the majority of your life, maybe not have the most fluent of full piano technique, and one might assume there is a trap having to do with the keyboard: the real trap is a heavily habituated / conditioned mind connected, of course, to the ear.
> 
> The keyboard is not a problem, players motor and listening habits are the problem: the problem is conditioned, so re-conditioning should be effective if an individual wishes to do so.


The keyboard is a _symbol_ of the problem. One response I've heard too often, for example, is that a preference for the sound of the perfect fifth or the traditional diatonic triad (in just, equal, whatever) is "hard-wired" into our brains. My response is that, no, it is not hard wired, but it _is_ "soft wired" in that our society/culture makes it next to impossible to _not_ "prefer" it. The pattern of keys on the keyboard then becomes a cultural symbol of what sorts of musics are most acceptable/usable/expressive/etc.

Following is not a real suggestion, just a thought experiment:

I understand what you are saying about the keyboard being "hands friendly," but it could be just as hands friendly using a pattern other than the usual diatonic that would get many of the same results. Let's say you want to keep complementary maximal evenness for a different pattern that would fit the hands just as nicely as the diatonic (after a little practice). The easiest solution would be keyboard pattern based on the octatonic scale. So the pattern of white & black keys (now 8 instead of 7 to the octave) might be: WBWWBWWBWWBW with a semitone between each key, so you're still in 12tet. Using only the white keys, you get 4 major and 4 minor triads, 4 "dominant 7th" chords, 2 diminished 7th chords, etc. etc. - all well known stuff to composers who have used the octatonic in the past. You also gain an interesting atonality feature: pick any 4 W notes at random, and the other 4 contain the same interval classes (a special case of the Babbitt hexachord theorem). What you lose, of course, is that the white keys don't express any unique tonic or dominant function, so if you want to make this scale "work like" the usual diatonic (tonics, key relationships, modulations, etc.) you have to work around the octatonic's fundamental atonal-ness.

In a sense, this is the mirror situation to what the atonalist/post-tonalist has to do to work around the biases built into the ubiquitous diatonic.

I admit that "shoot the piano" was purposely meant to be provocative. But there is still some truth about the keyboard & the traditional notation system it reflects - both in reality and as a cultural symbol - standing as a wall that has to be scaled to get at all 4,096 pitch-class sets.


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## Mahlerian (Nov 27, 2012)

Marisol said:


> How do you know, did you ever try it?
> I compose and play in just intonation and modulate all the time, both to a key on the current scale and also by recalculating a new harmonic center based on the leading melody tone.


Then you are using false relationships, by modulating to tones that were not even harmonically present before, either as overtones of the current tonic or as scale tones. Can't you see or hear a problem with that?


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## millionrainbows (Jun 23, 2012)

I think the point that Marisol is making, about harmony being triadic in nature, and how she sees a triad as a M3+5, is definitely tonal, because it considers the relation to "1" as being more important. Read Rameau. The "stacked major and minor thirds" is more of a non-hierarchical way of looking at it, because it emphasizes "note-to-note" relations more. But let us remember, if we are going to use "tonally loaded" terms like *triad* that triads started out as a harmonic/tonal phenomenon, based (ideally) on a 5/4 and a 3/2. See Rameau.

To assert that "any 3 notes are a triad" robs the harmonic basis of our original triad, and its reason for being. Peter Schat's Tone Clock system is based on this.

The fifth is primal in our 12-note system, because this particular octave division was formed by stacking fifths, or perfect, 'just' 3:2s, which as Mahlerian said, do not exist in equal temperament;

...but Mahlerian is missing or not acknowledging the point that the whole 12-note division is a compromise to begin with, as is any division; no ratios will ever "stack out" to an even octave, because they are fractional divisions of "1", and that's prime.

For example, if I wished to preserve the integrity of perfectly 'just' major thirds, by stacking, what would my octave look like, at its closest coincidence to the elusive "recursive octave?"

Here's an old post of mine which demonstrates an octave-division based on major thirds, rather than fifths, as Pythagoras did:

I'm going to fly by the seat of my pants & discuss fifths, cycles, and the "12 limit." As you may have guessed, when intervals other than the fifth are projected, they tend to "poop out" in a short time. The next interval Hanson discusses is the major third; it yields a cycle of only three notes before re-connecting with itself at the octave. There are four adjacent versions of this, hence the four augmented chords (3x4=12).

But we forget, don't we? that the "12 cycle" we take for granted is itself imperfect, the stacked circle of 3/2 fifths not really perfectly "reconnecting" with itself after 12 notes have been generated. Thus, Pythagorus coined the term "close enough for rock & roll."

What if we did the same thing with the major third, and used a stack of 'just' major thirds (5/4) to generate our cycle? Where will it stop, this Pythagoran Wheel of Fortune?

I decided to do it with cents. We know that an octave is 1200 cents, and that a 'just' major third is 14 cents flat of our equal-tempered third, which is 400 cents, making the 'just' M3rd about 386 cents (386.3 is closer).

So, how many times do we have to "stack" this 5/4 major third before it (apparently) comes back to its starting point? We're shooting for some multiple of 1200 cents, since that's the octave.

Okay, 1200 + 1200 is 2400, + 1200 is 3600, 4800, 6000, 7200, 9600, 10,800, 12,000, 13,200, 14,400, 15600, 16,800, 18,000, 19,200, 20,400, 21,600, 22,800, 24,000, 25200, 26400, 27,600, 28,800, 30,000, 31,200, 32,400, 33,600, 34,800, 36000,

Now, we start adding up the 5/4s, or 386.3s. That gives us 386.3, 772.6, 1158.9, 1545.2, 1931.5, 2317.8, 2704.1, 3090.4, 3476.7, 3863.0, 4249.3, 4635.6, 5021.9, 5408.2, 5794.5, 6180.8, 6567.1, 6953.4, 7339.7, 7726.0, 8112.3, 8498.6, 8884.9, 9271.2, 9657.5, 10043.8, 10430.1, 10816.4, 11202.7, 11589.0, 11975.3, 12361.6, 12747.9, 13134.2, 13520.5, 13906.8, 14293.1, 14679.4, 15065.7, 15452.0, 15838.3, 16224.6, 16610.9, 16997.2, 17383.5, 17769.98, 18156.1, 18542.4, 18928.7, 19315.0, 19701.3, 20087.6, 20473.9, 20860.2, 21246.5, 21632.8, 22019.1, 22405.4, 22791.7, 23178.0, 23564.3, 23950.6, 24336.9, 24723.2, 25109.5, 25495.8, 25882.1, 26268.4, 26654.7, 27041.0, 27427.3, 27813.6, 28199.9, 28586.2, 28972.5, 29358.8, 29745.1, 30131.4, 30517.7, 30904.0, 31290.3, 31676.6, 32062.9, 32449.2, 32835.5, 33221.8, 33608.1, *Wait! Bingo! 33,608.1 is almost equal to 33, 660: only 8.1 cents off!* I say, let's go for it, before our calculators run out of battery power.
So how many cycles of 5/4s is that? Let's see...33,608.1 divided by 386.3 is 87!

So our Pythagoran "major third" scale is 87 notes per octave!


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## Marisol (May 25, 2013)

Mahlerian said:


> Then you are using false relationships, by modulating to tones that were not even harmonically present before, either as overtones of the current tonic or as scale tones. Can't you see or hear a problem with that?


False relationships? 
So now a person who apparently has a problem with just intonation is going to tell me what I can do or not. 
And hearing? Frankly if you only listen to music in equal temperament I seriously doubt if you could even hear the difference.

When you want to change the harmonic center from a leading melody tone you either must recalculate a new harmonic center based on the new position of that tone in the scale or you must slightly change the pitch of the tone to map it onto the new center if this center is based on a prior scale.


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## aleazk (Sep 30, 2011)

Having written some very chromatic music, I can say yes, the diatonic-centered musical notation is a headache. The score ends being a mess of sharps and flats. It's even counterproductive if you want to sight read the piece.
I guess a solution that only requires a minimal change in the actual conventions would be to say that a given line or space in the staff represents a given note in the chromatic scale. So we wouldn't need sharps and flats, since all the 12 notes of the chromatic scale have their own line or space. The pay is that a staff will only be useful for representing a single octave, so more lines are needed, maybe, e.g., a staff with ten lines. Although I'm sure a more efficient system can be constructed if we still maintain some sharp and flats.
Anyway, it's certainly a consideration of some interest. But, "But there is still some truth about the keyboard & the traditional notation system it reflects - both in reality and as a cultural symbol"... maybe in the 1920's... by now, that wall has been demolished a couple of times... there are plenty of composers using (or who used) microtonal scales, chromatic way of thinking, etc. The disposition of the keys is just a historical curiosity now...


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## PetrB (Feb 28, 2012)

NickNotabene said:


> The keyboard is a _symbol_ of the problem.


Stop paying so much attention to "symbols" (nay, I'd say more like worshipping symbols) as thought icons and instead just get your hands on the keyboard.

You also need to meet more people who have little or no trouble reading much of anything contemporary, with sharps, flats, cancellations and all the rest for that sort of musician _(there are plenty enough)_ for whom it is just not a problem, and for whom the keyboard itself, being _*JUST A KEYBOARD AND NOT A SYMBOL OF ANYTHING,*_ presents little or no problems in the physical negotiation of its operation.


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## PetrB (Feb 28, 2012)

millionrainbows said:


> To assert that "any 3 notes are a triad" robs the harmonic basis of our original triad, and its reason for being. Peter Schat's Tone Clock system is based on this.


zOMG, we were ROBBED! Call the Politzbureau!



millionrainbows said:


> Now, we start adding up the 5/4s, or 386.3s. That gives us 386.3, 772.6, 1158.9, 1545.2, 1931.5, 2317.8, 2704.1, 3090.4, 3476.7, 3863.0, 4249.3, 4635.6, 5021.9, 5408.2, 5794.5, 6180.8, 6567.1, 6953.4, 7339.7, 7726.0, 8112.3, 8498.6, 8884.9, 9271.2, 9657.5, 10043.8, 10430.1, 10816.4, 11202.7, 11589.0, 11975.3, 12361.6, 12747.9, 13134.2, 13520.5, 13906.8, 14293.1, 14679.4, 15065.7, 15452.0, 15838.3, 16224.6, 16610.9, 16997.2, 17383.5, 17769.98, 18156.1, 18542.4, 18928.7, 19315.0, 19701.3, 20087.6, 20473.9, 20860.2, 21246.5, 21632.8, 22019.1, 22405.4, 22791.7, 23178.0, 23564.3, 23950.6, 24336.9, 24723.2, 25109.5, 25495.8, 25882.1, 26268.4, 26654.7, 27041.0, 27427.3, 27813.6, 28199.9, 28586.2, 28972.5, 29358.8, 29745.1, 30131.4, 30517.7, 30904.0, 31290.3, 31676.6, 32062.9, 32449.2, 32835.5, 33221.8, 33608.1, [/SIZE]


Thanks for the random assortment of lottery numbers to play, none, thank the Lord, related to anybody's Grandma's / granddaughter's birthday -- uh, oh, wait a minute....


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## aleazk (Sep 30, 2011)

millionrainbows said:


> Okay, 1200 + 1200 is 2400, + 1200 is 3600, 4800, 6000, 7200, 9600, 10,800, 12,000, 13,200, 14,400, 15600, 16,800, 18,000, 19,200, 20,400, 21,600, 22,800, 24,000, 25200, 26400, 27,600, 28,800, 30,000, 31,200, 32,400, 33,600, 34,800, 36000,
> 
> Now, we start adding up the 5/4s, or 386.3s. That gives us 386.3, 772.6, 1158.9, 1545.2, 1931.5, 2317.8, 2704.1, 3090.4, 3476.7, 3863.0, 4249.3, 4635.6, 5021.9, 5408.2, 5794.5, 6180.8, 6567.1, 6953.4, 7339.7, 7726.0, 8112.3, 8498.6, 8884.9, 9271.2, 9657.5, 10043.8, 10430.1, 10816.4, 11202.7, 11589.0, 11975.3, 12361.6, 12747.9, 13134.2, 13520.5, 13906.8, 14293.1, 14679.4, 15065.7, 15452.0, 15838.3, 16224.6, 16610.9, 16997.2, 17383.5, 17769.98, 18156.1, 18542.4, 18928.7, 19315.0, 19701.3, 20087.6, 20473.9, 20860.2, 21246.5, 21632.8, 22019.1, 22405.4, 22791.7, 23178.0, 23564.3, 23950.6, 24336.9, 24723.2, 25109.5, 25495.8, 25882.1, 26268.4, 26654.7, 27041.0, 27427.3, 27813.6, 28199.9, 28586.2, 28972.5, 29358.8, 29745.1, 30131.4, 30517.7, 30904.0, 31290.3, 31676.6, 32062.9, 32449.2, 32835.5, 33221.8, 33608.1, *Wait! Bingo! 33,608.1 is almost equal to 33, 660: only 8.1 cents off!* I say, let's go for it, before our calculators run out of battery power.
> So how many cycles of 5/4s is that? Let's see...33,608.1 divided by 386.3 is 87!
> ...


This is very awkward to read, I blame the decimal system and its symbolic meaning for this!. Those arabs knew that MR would use it and that we wouldn't like to read it. Ergo, I don't know how, but the arabs hated serialism, MR, and us!.


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## PetrB (Feb 28, 2012)

aleazk said:


> This is very awkward to read, I blame the decimal system and its symbolic meaning for this!. Those arabs knew that MR would use it and that we wouldn't like to read it. Ergo, I don't know how, but the arabs hated serialism, MR, and us!.


and the number 12 -- don't forget the number 12.


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## PetrB (Feb 28, 2012)

Marisol said:


> And hearing? Frankly if you only listen to music in equal temperament I seriously doubt if you could even hear the difference.


The ability to discern fine differences in pitch is not rare. No one need be vain about it, as if it is something special.

"Even a duck can hear." ~ Stravinsky


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## aleazk (Sep 30, 2011)

PetrB said:


> and the number 12 -- don't forget the number 12.


But, of course!: 12 -> 1+2=3 -> take the 1 in 12, combined with the fact that 1+2=3, then we have 111, i.e., three 1's -> take again the fact that 1+2=3, then 111x3=333 -> _symmetrically_ (the key point, all must be symmetric, that's the way the universe wants!), now take the 2 in 12, so we have 333x2=666!. QED


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## PetrB (Feb 28, 2012)

aleazk said:


> But, of course!: 12 -> 1+2=3 -> take the 1 in 12, combined with the fact that 1+2=3, then we have 111, i.e., three 1's -> take again the fact that 1+2=3, then 111x3=333 -> _symmetrically_ (the key point, all must be symmetric, that's the way the universe wants!), now take the 2 in 12, so we have 333x2=666!. QED


My head aches after all this. I think I'm just going to listen to some music, or perhaps continue working on that simple piano piece which is still incomplete.


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## Marisol (May 25, 2013)

This is just awful, people here cannot even find the decency of staying respectful in topics they either know nothing about or are not interested in.


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## millionrainbows (Jun 23, 2012)

aleazk said:


> This is very awkward to read, I blame the decimal system and its symbolic meaning for this!. Those arabs knew that MR would use it and that we wouldn't like to read it. Ergo, I don't know how, but the arabs hated serialism, MR, and us!.


That's it, aleazk; blame the Arabs! Incidentally, they outdid Pythagoras, kept going around the circle, and stopped at 17.

Not so they could have 17 chromatic notes (which sounds ghastly), but so they could derive different monotonic tunings which have different "pure" steps, depending on the tuning.

Take heart, Marisol; I do believe that I see your position.

This underscores for me that, sooner or later, if they are to have a sufficiently deep grasp of music theory, these "music scholars" are going to have to engage in the tedious task of "re-inventing the wheel" instead of assuming that everything in music revolves around a "12-division" of the octave. Yes, Mahlerian, that means you need to get the Harry Partch book and live with that bum for about a year or two.


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## millionrainbows (Jun 23, 2012)

PetrB said:


> and the number 12 -- don't forget the number 12.


Oh, how can we ever forget the number 12?


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## millionrainbows (Jun 23, 2012)

PetrB said:


> The ability to discern fine differences in pitch is not rare. No one need be vain about it, as if it is something special.
> 
> "Even a duck can hear." ~ Stravinsky


That takes a lot for granted, PetrB, and it dismisses entire tuning systems. I think Marisol's point is the one you conveniently used earlier, when it suited your argument: we tend to get accustomed to ways of hearing things.

"Any jackass can swing." ~ Dave Brubeck


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## millionrainbows (Jun 23, 2012)

aleazk said:


> But, of course!: 12 -> 1+2=3 -> take the 1 in 12, combined with the fact that 1+2=3, then we have 111, i.e., three 1's -> take again the fact that 1+2=3, then 111x3=333 -> _symmetrically_ (the key point, all must be symmetric, that's the way the universe wants!), now take the 2 in 12, so we have 333x2=666!. QED


That's very entertaining, aleazk. Who are you imitating? It must be someone who doesn't stop to think about what they are saying.

The fact is, our Western 12-note division of the octave is based on the fifths of Pythagoras. If you had ever stopped to try to think about why this is, you would be capable of contributing something less destructive and more useful to this farce of a discussion. And I'm doing you a favor by telling you this. :lol:


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## millionrainbows (Jun 23, 2012)

PetrB said:


> My head aches after all this. I think I'm just going to listen to some music, or perhaps continue working on that simple piano piece which is still incomplete.


I suggest a nice, warm bubble-bath as well. You need to rest, after all this high-pressure discussion. Have a nice day!


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## millionrainbows (Jun 23, 2012)

Marisol said:


> This is just awful, people here cannot even find the decency of staying respectful in topics they either know nothing about or are not interested in.


I agree completely, Marisol, and you have my empathy. I think you bring some interesting ideas to the fore, and apparently I'm the only one who is equipped to see both sides of the issue. The smug academic modernists have all gone home, and are listening to Schoenberg in the bath.


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## PetrB (Feb 28, 2012)

millionrainbows said:


> I agree completely, Marisol, and you have my empathy. I think you bring some interesting ideas to the fore, and apparently I'm the only one who is equipped to see both sides of the issue. The smug academic modernists have all gone home, and are listening to Schoenberg in the bath.


Very lonely at that imagined top, ain't it?


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## millionrainbows (Jun 23, 2012)

PetrB said:


> Very lonely at that imagined top, ain't it?


Not with you to constantly remind me that I am, indeed, human, PetrB. Just keep skirting the ice-fields of northern Adhominea, and this thread will stay perpetually within the bounds of whatever brilliantly humorous confines you deem it should conform to. 
Oh, by the way, you might want to mention Marisol in your next reply, or at least include her in the list.

Don't let the polar-bears getcha!


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