[Smt-talk] Just intonation and tonality

[Marcel de Velde]

Hello List,

I'm writing this partly in reply to several references made to specific 
tuning systems in the discussion about the "governing tone".
I've researched just intonation full time for about 6 years now. A 
personal obsession of mine.
So I felt the urge to share my findings on the subject.

It indeed seems logical to me that armed with the knowledge of the 
interval ratios that make intervals "in perfect tune",  one can possibly 
find explanations for tonal behavior.
However. This field, called "just intonation", is anything but clear.
In this light, the references made earlier where intervals were called 
"harmonics" are I think largely unfounded. I will try to explain some of 
the problems of the tuning systems based on harmonics below.
There are too many systems that call themselves "just intonation" to 
mention. And none are currently unanimously accepted or proven to be 
correct.

Having said that. I belief I have found which system is most likely 
correct just intonation.
It may be a surprise (it certainly was to me), but I've actually found 
Pythagorean to be the true just intonation system.
The infinite chain of pure fifths and octaves.
Here a list of intervals for Pythagorean just intonation: 
http://www.justintonation.com/home/pythagorean-ji

In my opinion the main (very understandable) error that lead to all 
these centuries (millennia even) of confusion about just intonation is 
the major third.
It sounds very "synchronous" (and often better in isolation) when using 
the harmonic ratio of 5/4 instead of the Pythagorean ratio of 81/64 
(though this is timbre dependent).
But 5/4 for a major third and 3/2 for a perfect fifth will give 
mathematical impossibilities for even simple progressions. For instance 
circle progressions.
There are only 2 possible solutions:
- Comma shifts (where a tone, held or not, must shift by the interval of 
a Syntonic comma of 81/80).
- Drifting progressions. Where the circle progressions ascends in pitch 
by a Syntonic comma on every repetition.
The other solutions are either wolf fifths, or tempering each fifth to 
get 1/4 comma meantone.
Of all these solutions, only the 1/4 comma meantone is somewhat 
acceptable to the ear.
Problems like the above only get worse if one tries using even higher 
harmonics, like the 7th harmonic in a dominant 7th chord for instance.
And there are more problems. When intoning the major third as a 5/4, the 
tone looses some of it's individuality and timbre clarity in chords.
A Pythagorean 81/64 preserves the individuality of the tones and timbre 
in a clear way. (However, it is as such also much more critical of the 
timbre, there is no "floating" like in 12tone equal temperament to hide 
/ muddy things, and no "harmonic synchronicity" of 5/4 which imparts its 
own qualities upon the timbre. As a result, the Pythagorean 81/64 can 
sound less than good with certain timbres, though I feel one can only 
blame the timbre for this, not the tuning as it is perfect.)
Also, a 5/4 would logically be a perfect consonance, yet it should be an 
imperfect consonance according to the tried and tested rules of 
counterpoint.
Another objection is that to many people a 5/4 major thirds sound low. 
This can not be explained by habitual ear training due to 12tone equal 
temperament use in western music alone.
Furthermore studies have long shown that when singing solo melodies, 
people tend to sing Pythagorean.
Only when chords come into play a percentage of people (naturally or by 
training) start intoning for instance the major third as 5/4.
But does this make it correct? I think not, it is only an easy to find 
"synchronous" interval to sing or play, but not correct for true 
polyphony in my opinion.
There are more reasons why I think Pythagorean is the only correct just 
intonation system, but I've listed enough for now to make a case here I 
think.
I have not found a single "proof", but rather came to my conclusion by 
means of many listening tests (many of those only now practical thanks 
to computers), thought experiments and finding the unacceptable errors 
in all alternatives.
In the end, problems for just intonation systems that do not adhere to 
the chain of fifths are so large, that one will have to reject most of 
western music theory to make them theoretically possible.

Here are several audio files in 2 sets comparing Pythagorean just 
intonation to a few other tuning systems.
First set with exact synthesized waveforms, 2nd set with a sampled piano 
sound (perhaps a bit easier on the ear)
http://soundcloud.com/justintonation/sets/tuning-demonstration/
http://soundcloud.com/justintonation/sets/tuning-demonstration-2/

Soon I'll upload audio for another convincing argument for Pythagorean 
tuning.
I'm starting to successfully harmonize Arabic maqams, with full 
retention of the beautiful microtonal intervals. A first I belief, and I 
think only possible in Pythagorean just intonation.

Now for Pythagorean just intonation in relation to tonality.
The simplest scale is probably the pentatonic scale.
It is simply 5 tones connected by pure fifths in Pythagorean:
C (8/9) - G (4/3) - D (1/1) - A (3/2) - E (9/8)

For the major and minor scales add 2 tones:
F (32/27) - C (8/9) - G (4/3) - D (1/1) - A (3/2) - E (9/8) - B (27/16)

I've written D as (1/1) as it is the center of the chain of fifths in 
the above scales.
The tonic chords of the major and minor scales are symmetrical around D 
(1/1) and both use C (8/9) and E (9/8).
It's a very very clumsy way to describe it, please forgive me, but it 
appears to me like tonality is in some a ways a gravitational force 
towards the center of the chain of fifths, yet it somehow prefers to 
rest on 8/9 and 9/8. I'm not making much sense of this part yet as you 
can see. I think there is quite possibly great logic hiding in there, 
but it is escaping me so far.

I do think the Pythagorean tuning system is the correct system to base 
things like functional harmony on. (And this is mostly being done 
already as it corresponds 1:1 with western notation.)
Other than observing in Pythagorean what has already been described 
about tonality in normal textbooks, I've not yet found anything else 
myself (sensible enough to share here) other than applying it to Arabic 
maqams by seeing their microtonal intervals as (sometimes strongly) 
chromatic intervals.

I hope the info in this post is useful or interesting to some of you.
And if anybody had insights to share on the subject I'm very interested.


Kind regards,

Marcel de Velde
marcel at justintonation.com
Zwolle, Netherlands