RE: [tuning] harmonic perception and Partch/Wilson

I wrote,

>> BTW, I discovered a detailed mathematical derivation of this fact
>> in 1993, which I could share with you.

Joe Pehrson wrote,

>Very cool... Thanks Paul for all your help. But, why don't you
>consider having Monz post this derivation as a continuation of your
>Entropy article... Even if some of us don't understand it, it probably
>will be fun to look at... (I'm being serious.)

Well, Monz is not accepting contributions to his website right now, but
here's the derivation:

You're already familiar with the Farey series -- for example, the Farey
series of order 6 is

1 6 5 4 3 5 2 5 3 4 5 6
-, -, -, -, -, -, -, -, -, -, -, -
1 5 4 3 2 3 1 2 1 1 1 1

This series has the property that any two consecutive fractions pi/qi and
pj/qj satisfy

pj*qi - pi*qj = 1 (formula 0)

(Three proofs of this are given in Hardy, G. H. and Wright, E. M. _An
Introduction to the Theory of Numbers_. Oxford University Press, London,
Chapter 3.)

If the next fraction after pj/qj is denoted by pk/qk, we find

1 = pj*qi - pi*qj = pk*qj - pj*qk
pj*qi + pj*qk = pk*qj + pi*qj

Thus

pj pk + pi
-- = ------- (formula 1)
qj qi + qk

Now assume that we are dealing with a Farey series of very high order N, say
at least 80, and that the fractions we're interested in examining are
relatively simple, so if we're looking at pj/qj, pj is small compared with
N. So

pk and pi are approximately equal to N, (formula 2)

because:

(a) membership in the Farey series requires that they be <= N;
(b) pi is > N - pj, since if pi were <= N - pj, the fraction (pi + pj)/(qi +
qj), which lies between pi/qi and pj/qj, would also belong to the Farey
series of order N, contradicting the assumption that pi/qi and pj/qj are
consecutive (same argument for pk).

Since they are consecutive, all three fractions pi/qi, pj/qj, and pk/qk are
similar in magnitude, thus formula 2 is equivalent to the assertion that

qk, qi, and N*qj/pj are similar in magnitude. (formula 2c)

By formula 1 we know that

c*pj = pk + pi, c*qj = qi + qk (formula 3)

for some c, so

pk = c*pj - pi, qi = c*qj - qk (formula 4)

whence

qk = c*qj - qi (formula 5)

From formula 4,

pk*qi = c*c*pj*qj - c*pi*qj - c*pj*qk + pi*qk

and using formula 5,

pk*qi - pi*qk = c*c*pj*qj - c*pi*qj - c*pj*(c*qj - qi)
pk*qi - pi*qk = -c*pi*qj + c*pj*qi
pk*qi - pi*qk = c (formula 6)

by virtue of formula 0. Now formulae 2 and 3 tell us that

c is approximately equal to 2*N/pj

so

pk*qi - pi*qk is approximately equal to 2*N/pj (formula 7)

While N is still finite, there is a range of intervals f1/f2 which could be
intepreted as pj/qj (i.e., they 'belong' to pj/qj's immediate realm on the
number line). A natural set of bounds for this range is defined by the
so-called mediants, the simplest fractions between pi/qi and pj/qj, and
between pj/qj and pk/qk:

pi + pj f1 pj + pk
leftbound = ------- < -- < ------- = rightbound
qi + qj f2 qj + qk

(See Mann, Chester D., 1990. _Analytic Study of Harmonic Intervals_. Tustin,
Calif., p. 163, for a justification for using mediants as "transition
points" coming from a consideration of beat rates.)

which unambiguously ascribes to any f1/f2 one and only one fraction from the
Farey series, namely pi/pj. Now the width of this range on a logarithmic
scale, say a scale of cents, is given by

rightbound (pi + pj)/(qi + qj)
Wj = log---------- = log-------------------
leftbound (pj + pk)/(qj + qk)

Since in our case pj and qj are relatively small,

pk*qi
Wj is approximately equal to log-----
pi*qk

and since Wj is a small number close to 0, we can use the approximation
log(x) = x - 1:

pk*qi
Wj is approximately equal to ----- - 1
pi*qk

or

pk*qi - pi*qk
Wj is approximately equal to -------------
pi*qk

Substituting formulae 7, 2, and 2c,

2*N/pj
Wj is approximately equal to ---------
N*N*qj/pj

or

2
Wj is approximately equal to ----
N*qj

So what we have found is, for any fraction much simpler than the Farey
limit, the size of the range it occupies is inversely proportional to the
denominator of the fraction.

It can be shown more precisely that for all fractions pj/qj in the Farey
limit, regardless of relative simplicity (in other words, dropping formula 2
and its consequents),

1 2
---- < Wj < ----
N*qj N*qj

I'll leave the proof of this as an exercise for the reader. In any case, to
within a fuzz factor of 2, the width of the range associated with each
fraction is inversely proportional to the fraction's denominator.

We've discussed all this on the list, and other related things, for example,
if one uses instead of a Farey series, what one might call a Tenney series,
where the product of numerator and denominator are less than a given limit,
one finds a less fuzzy expression relating width inversely to complexity.

>Also, the Utonal/Otonal midi chords should eventually be linked to the
>Monz site... or maybe Starrett... something central where people can
>find them. That is, if somebody actually does them. I'm anxious to
>hear them.

I'll make some .wavs for ya.

Joe Pehrson wrote,

>I would like to hear more about the Partch Utonal/Otonal vs. "major,
>minor" harmonic theory. If I'm understanding this correctly, the
>Utonal/Otonal is essentially ONE dimensional... actually one
>complementary set of ratios and while Partch may like to think of this
>as "Major-Minor," essentially it is totally separate from traditional
>"major minor" theory of traditional "functional" harmony. This is all
>very interesting... but I fear I should "brush up" on my Partch again...
>it's been a couple of years since I read Genesis for the first time (!!)

You got it right.

>I have tried virtually ALL the 19 tone scales in Scala, and this one
>seems the most particularly resonant. Of course, MANY of the ratios are
>quite small... this, of course, pertains only with reference to the
>fundamental, and other combinations of pitches would set up different
>complexities... but, on the whole, the entire thing seems uncannily
>resonant...

Let's lattice it out, as Kraig suggests:

25/24
/ \
/ \
/ \
/ \
5/3-------5/4------15/8
/|\`. ,'/|\ / \
/ | \10/7 / | \ / \
14/9-------7/6-------7/4 \ / \
,' `. /,' \`.\|/,'/ `.\ / \
16/9-------4/3-----\-1/1-/-----3/2-------9/8
\ / \ \/|\/ / `. .' `.
\ / \ /\|/\ / 9/7------27/14
\ / \ / 7/5 \ /
\ / \ /,' `.\ /
16/15------8/5-------6/5

I don't know if there's any underlying logic, but it looks like most notes
form a 7-limit consonance with at least 3 other notes, with 25/24, 9/7, and
27/14 the only exceptions, forming a 7-limit consonance with only 2 other
notes. Most conspicuous, though, is the absence of both 12/7 and 15/14 -- so
this is not a convex region in the 7-limit lattice (which indicates that
it's improvable in the resonance department, though those improvements might
have unwanted melodic consequences).