# Complementation



## millionrainbows (Jun 23, 2012)

An interval's complement is that interval which, when added to said interval, yields an octave (2:1). Every "just" interval less than an octave has a complement.

For example, the complement of a perfect fifth (3:2) is 4:3 (a perfect fourth), since 3:2 + 4:3 = 12:6, or 2:1 (octave).

In a slightly different way of explaining this, tuning a particular interval upward from 1:1 will generate the same tone as tuning its complement. Thus, a 3:2 above C and a 4:3 below C both yield the same tone, G.

So from this, we can see how tonality references everything to 1:1, and how intervals have inversions which reference to 1:1 and are the same note name or pitch identity (albeit in a different octave).

Modern music, and composers such as Bartok, using the equal tempered scale, often divide tempered intervals into some number of equal parts. For example, the octave can be divided symmetrically into two tempered tritones, three tempered major thirds, four tempered minor thirds, and six tempered whole tones. These divisions correspond numerically to 6, 4, 3, and 2, all of which are factors of 12.

This is what I call geometric, or numeric division, as opposed to "just" intervals which are small-number ratios corresponding to acoustic sonance, having an acoustic, harmonic basis rather than a numeric or geometric basis.

In using "just" intervals, no_ simple _interval is divisible into any number of _equal_ parts. These just ratios can only be divided _un_equally.

What is the relevance of all this?

As music proceeded out of limited key ranges, and tunings changed by moving gradually towards equal temperament (to accommodate more key areas), then the 12-note scale became more about chromatic division rather than acoustic factors such as 3:2 and 4:3, and began to _numerically _divide the octave into symmetrical parts based on numerical and geometric principles rather than acoustic or harmonic principles.

Remember, our 12-note scale was never perfect, being based on Pythagoras' stacking of 3:2 fifths, which would never completely close the circle. Additionally, other intervals suffered greatly, the most obvious casualty being the 6:5 major third, which was attempted to be reclaimed using various mean-tone tunings, which were only usable in a limited range of keys.

This shows how tonality, never perfect from its 12-note inception, was based on harmonic and acoustic principles, and how more modern, chromatic music began to base its principles on symmetrical divisions of 12, and was numeric and geometric by nature.

This should provide the best possible argument for those listeners who claim that modernism is "unnatural" compared to tonality, that is, provided they can understand it. :lol:

Then again, they must be reminded that tonality never really was a perfect system to begin with, ever since Pythagoras favored the fifth and octave, and sacrificed the other perfect consonances.


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## PetrB (Feb 28, 2012)

"Then again, they must be reminded that tonality never really was a perfect system to begin with, ever since Pythagoras favored the fifth and octave, and sacrificed the other perfect consonances." LOL. you might have led with that....

So after 23498 words on math, ratios, acoustics, etc. you end up with what it all actually means. _Nothing._

imo, a massive waste of your time, the reader's time, and virtual column space.


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## Huilunsoittaja (Apr 6, 2010)

Huh, I thought you guys liked each other.


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## PetrB (Feb 28, 2012)

Huilunsoittaja said:


> Huh, I thought you guys liked each other.


We do. We also (at least I do, often enough) _live to disagree _


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## MissLemko (May 11, 2014)

The equal temperament/ just temeperament/ EDO have almost nothing to do with the concept of tonality. I just don't see the point of concluding that tonality as a concept is imperfect because the equal temperament is imperfect. Extreme chromaticism does not solve the "problem" of equal temperament, nor does atonality, microtonality etc...


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## millionrainbows (Jun 23, 2012)

This all started with my investigation into Bartok, and his use of two 'systems' in composing, one based on the 12-note chromatic scale, divided into various symmetries (tritone, M3, m3, M2, m2), which is 'geometric' and numerical, since it is not based on acoustic principles of tonality, namely the consonant interval 3:2 and its counterpart, 4:3.

This is a very important concept to get, since these were the types of modern ideas that were 'in the air' around the beginning of the 20th century, influencing Bartok, Debussy, Stravinsky, and all modernist music thinkers. This was a 'sea change' in musical thinking, which permeated all modern musical thought, and ushered-in the way for serial thinking.

As far as tonality being an 'imperfect' system, I say this in the context that our 12-note scale was based on the projection (by Pythagoras) of the 3:2 interval, a 'perfect' consonance (nomenclature which still persists). For additional evidence of the struggle for tonality to 'achieve perfection,' witness the historical attempts at using various versions of mean-tone tunings, which sought to produce a more consonant major third (ideally 6:5). Our present-day major third is 14 cents sharp! That is very audible, and most people have never heard a true, consonant major third. Additionally, our fifths are all 2 cents sharp.

P. S. what is "EDO"?


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## millionrainbows (Jun 23, 2012)

PetrB said:


> "Then again, they must be reminded that tonality never really was a perfect system to begin with, ever since Pythagoras favored the fifth and octave, and sacrificed the other perfect consonances." LOL. you might have led with that....
> 
> So after 23498 words on math, ratios, acoustics, etc. you end up with what it all actually means. _Nothing._
> 
> imo, a massive waste of your time, the reader's time, and virtual column space.


I'm not so sure where this over-reaction is coming from, or what it means in terms of your 'position' on this subject (ahhh, the internet).

:lol:

I assume that you agree with me that tonality is not a perfect system, but you seem prepared to throw out all the rest of my statement, which goes a long way towards explaining the differences in tonal and modernist musical thinking, including specifically Bartok's approach. I thought you were a sympathetic supporter of modern music.


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## Aramis (Mar 1, 2009)

millionrainbows said:


> P. S. what is "EDO"?


A former capital of Japan.


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## MissLemko (May 11, 2014)

Equal division of the octave; basically the same as equal temeperament, only used with certain microtonal connotations. The divisions are generally from 5 to 24 (I never heard of a 30EDO).
I am an enthusiastic supporter of modern music, but I don't like fallacies of logic. No. Impliying that A is true because B is true and not making a functional explenation for the statement is not really logical. Sorry.


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## Vaneyes (May 11, 2010)

For sanity sake, it's all illusion. Just smell the flowers, and mow the grass.


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## millionrainbows (Jun 23, 2012)

MissLemko said:


> Equal division of the octave; basically the same as equal temeperament, only used with certain microtonal connotations. The divisions are generally from 5 to 24 (I never heard of a 30EDO).


Oh, I see. An equal tempered scale is any division of the octave into equal parts.

The Thai use a 7-note ET, and there are uses for 17 and 19-tone ET scales. Usually, the higher-division scales, such as 43 and 53-tone, are used to approximate 'just' intervals (as H*arry Partch u*sed them).

On a synthesizer keyboard with varying keyboard control voltages (such as an Ensoniq ESQ-1), these other odd divisions can be seen to make more sense. To tune a 17-note ET scale, tune your middle C to the "F" above the octave C until you get a good octave. Viola, you now have a 17-note octave of equal division. And you can see that this is based on going up a fourth to "F" from the normal octave.

Similarly, the 19-tone ET scale simply goes up to the following "G" for its octave, and we can see that this is based on going a fifth above the normal 12-ET octave.



MissLemko said:


> I am an enthusiastic supporter of modern music, but I don't like fallacies of logic. No. Impliying that A is true because B is true and not making a functional explenation for the statement is not really logical. Sorry.


Everybody's going to have to go into more detail if they wish to make a credible point; otherwise, this will look like any other internet shooting match I've seen, with short, pithy retorts, and no reasons given.


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## millionrainbows (Jun 23, 2012)

Vaneyes said:


> For sanity sake, it's all illusion. Just smell the flowers, and mow the grass.


Just so you can play golf? No way!


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## millionrainbows (Jun 23, 2012)

*Equal temperament - Theory

*You can have equal tunings with any number of divisions per octave. Each produces its own array of intervals, and each has its own characteristic sound. Usually these are used to approximate 'just' scales. Our standard 12-tone ET is somewhat imperfect in this regard, matching the fifths quite well, but falling way short in regard to major and minor thirds, and flat-sevens. But twelve has the great advantage that it is the smallest ET number that does a decent job.

By definition, each step in an equal temperament must be higher than the previous by a constant interval. This means that the frequency for each step must be multiplied by the same factor to get the frequency for the next successive step. This factor works just like the frequency ratios in just intonation, but it is expressed as a decimal for reason having to do with its *mathematical derivation.
*
So, you see, *equal temperaments are mathematical constructs, *totally artificial in comparison to 'just' interval scales which are based on small-number ratios which correspond to vibrating acoustic phenomena.

As music proceeded out of limited key ranges, and tunings changed by moving gradually towards equal temperament (to accommodate more key areas), then the 12-note scale became more about numerical, chromatic division rather than acoustic factors such as 3:2 and 4:3, and began to _numerically _divide the octave into symmetrical parts based on numerical and geometric principles rather than acoustic or harmonic principles.

Remember, our 12-note scale was never perfect, being based on Pythagoras' stacking of 3:2 fifths, which would never completely close the circle. Additionally, other intervals suffered greatly, the most obvious casualty being the 6:5 major third, which was attempted to be reclaimed using various mean-tone tunings, which were only usable in a limited range of keys.

*This shows how tonality, never perfect from its 12-note inception, was based on a striving for harmonic and acoustic principles, and how more modern, chromatic music began to base its principles on symmetrical divisions of 12, and was numeric and geometric by nature.

*Tonality as we knew it was also gradually degraded, left in the dust, a casualty of 12-note ET of the modern age of literal quantity over relationship, or ratio.

This could provide the best possible argument for those listeners who claim that modernism is "unnatural" compared to tonality, that is, provided they can understand it.









With one caveat: the argument 'for' tonality is also weakened, because 'tonality' as we now know it is a mere shadow of its former self. Hence, the proliferation of HIP performances and tunings which actually enhance to acoustic, sonant qualities of older music, rather than water it down with ET and drag it through a chromatic wasteland.


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## Matsps (Jan 13, 2014)

Does anyone here know how many hertz need to separate two tones for a typical brain to recognize them as separate tones?


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