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Message: 7050 Date: Thu, 17 Jul 2003 22:02:38 Subject: Re: Obvious things proven From: Carl Lumma

> Everything you already knew about maximally even scales: > > Maximal Evenness Proofs * [with cont.] (Wayb.)

Great stuff, indeed. That's everything I've ever suspected, and more. -Carl

Message: 7051 Date: Fri, 18 Jul 2003 14:21:15 Subject: Re: Obvious things proven From: Manuel Op de Coul Nice indeed. Was the formula p(n) = floor(na/b) proven by Clough and Douthett? Because I remember the definition being in terms of interval sizes. Manuel

Message: 7052 Date: Fri, 18 Jul 2003 14:46:13 Subject: Re: Obvious things proven From: Graham Breed Manuel Op de Coul wrote:

> Nice indeed. Was the formula p(n) = floor(na/b) proven by Clough and > Douthett? > Because I remember the definition being in terms of interval sizes.

I don't have that paper. I got it from equation 1.1 in Aytan Agmon's 1996 one. He says in Note 17 "In theorems 1.2 and 1.5 Clough and Douthett establish that 'being a J-set' (Definition 1.9) is equivalent to 'having the property that the spectrum of each dlen is either a single integer or two consecutive integers' (Definition 1.7). Thus Clough and Douthett's 'maximal evenness' ... and the present relation (1.1) are the same ..." Graham

Message: 7053 Date: Fri, 18 Jul 2003 16:44:30 Subject: Re: Obvious things proven From: Manuel Op de Coul Ah, so Agmon proved it then, good. Manuel

Message: 7057 Date: Sun, 20 Jul 2003 19:49:55 Subject: Poor man's harmonic entropy? From: Gene Ward Smith If x is a positive real number representing an interval, it's suggested that Pc(x) = exp((log(p/q)-x)^2/2c)) can model the probability that x is heard as p/q; here c is a parameter. If we take the sum sum q^(-d) Pc(x) over positive rationals p/q, then it isn't hard to see that this converges absolutely for high enough values of d--anything above 2, at any rate. A problem with this is that it mixes multiplicative and implicitly additive distance measures, since |log(p/q)-x| is multiplicative, while q does not depend on the octave and, like the Farey sequence, is implicitly additive.

Message: 7059 Date: Mon, 21 Jul 2003 11:09:38 Subject: Re: Poor man's harmonic entropy? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> If x is a positive real number representing an interval, it's > suggested that Pc(x) = exp((log(p/q)-x)^2/2c)) can model the > probability that x is heard as p/q; here c is a parameter. If we

take

> the sum > > sum q^(-d) Pc(x) > > over positive rationals p/q, then it isn't hard to see that this > converges absolutely for high enough values of d--anything above 2, > at any rate.

I think maybe the plan should be to get a continuous function the Tenney Height way, by sum_{p/q > 0} Pc(x, p/q)/(p*q)

Message: 7060 Date: Mon, 21 Jul 2003 18:38:20 Subject: Re: Interesting numerical coincidences (Combinatorics etc) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "paulhjelmstad"

> New finding: C{18,6} Reduces to 493 = 29 X 17 (in terms of unique > interval vectors). 17 is on meantone line, but 29 is on schismic.

So,

> not along the same line.

what do you mean? look again. the schismic line has a 17 on it as close to the ji center as the 17 on the meantone line!

Message: 7062 Date: Mon, 21 Jul 2003 21:44:30 Subject: Re: Interesting numerical coincidences (Combinatorics etc) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "paulhjelmstad" <paul.hjelmstad@u...> wrote:

> Hooray! Thanks for pointing the out. What a totally weird coincidence. > My brother is crunching C{30,6} as we speak. There are 19,811 sets > reduced to 10,133 Tn/TnI set types... Maybe C{4n,6} have factors on > meantone, and the remaining C{2n,6} are along schismic.

I don't see any weird coincidences; however, here is C(n, 6) for n from 7 to 60: 7 7 8 2^2 * 7 9 2^2 * 3 * 7 10 2 * 3 * 5 * 7 11 2 * 3 * 7 * 11 12 2^2 * 3 * 7 * 11 13 2^2 * 3 * 11 * 13 14 3 * 7 * 11 * 13 15 5 * 7 * 11 * 13 16 2^3 * 7 * 11 * 13 17 2^3 * 7 * 13 * 17 18 2^2 * 3 * 7 * 13 * 17 19 2^2 * 3 * 7 * 17 * 19 20 2^3 * 3 * 5 * 17 * 19 21 2^3 * 3 * 7 * 17 * 19 22 3 * 7 * 11 * 17 * 19 23 3 * 7 * 11 * 19 * 23 24 2^2 * 7 * 11 * 19 * 23 25 2^2 * 5^2 * 7 * 11 * 23 26 2 * 5 * 7 * 11 * 13 * 23 27 2 * 3^2 * 5 * 11 * 13 * 23 28 2^2 * 3^2 * 5 * 7 * 13 * 23 29 2^2 * 3^2 * 5 * 7 * 13 * 29 30 3^2 * 5^2 * 7 * 13 * 29 31 3^2 * 7 * 13 * 29 * 31 32 2^4 * 3^2 * 7 * 29 * 31 33 2^4 * 7 * 11 * 29 * 31 34 2^3 * 11 * 17 * 29 * 31 35 2^3 * 5 * 7 * 11 * 17 * 31 36 2^4 * 3 * 7 * 11 * 17 * 31 37 2^4 * 3 * 7 * 11 * 17 * 37 38 3 * 7 * 11 * 17 * 19 * 37 39 3 * 7 * 13 * 17 * 19 * 37 40 2^2 * 3 * 5 * 7 * 13 * 19 * 37 41 2^2 * 3 * 13 * 19 * 37 * 41 42 2 * 7 * 13 * 19 * 37 * 41 43 2 * 7 * 13 * 19 * 41 * 43 44 2^2 * 7 * 11 * 13 * 41 * 43 45 2^2 * 3 * 5 * 7 * 11 * 41 * 43 46 3 * 7 * 11 * 23 * 41 * 43 47 3 * 7 * 11 * 23 * 43 * 47 48 2^3 * 3 * 11 * 23 * 43 * 47 49 2^3 * 3 * 7^2 * 11 * 23 * 47 50 2^2 * 3 * 5^2 * 7^2 * 23 * 47 51 2^2 * 5 * 7^2 * 17 * 23 * 47 52 2^3 * 5 * 7^2 * 13 * 17 * 47 53 2^3 * 5 * 7^2 * 13 * 17 * 53 54 3^2 * 5 * 7^2 * 13 * 17 * 53 55 3^2 * 5^2 * 11 * 13 * 17 * 53 56 2^2 * 3^2 * 7 * 11 * 13 * 17 * 53 57 2^2 * 3^2 * 7 * 11 * 13 * 19 * 53 58 2 * 3^2 * 7 * 11 * 19 * 29 * 53 59 2 * 3^2 * 7 * 11 * 19 * 29 * 59 60 2^2 * 5 * 7 * 11 * 19 * 29 * 59

Message: 7066 Date: Mon, 21 Jul 2003 22:47:15 Subject: Re: Interesting numerical coincidences (Combinatorics etc) From: Carl Lumma

> This is 89 * 109. I found 89 along "Schismic" in > Zoomr.gif (hard to read, its really small, and > doesn't show up in Zooms.gif. I can't find 109. Help! > Can someone tell me where 109 would show up in a > taxicab diagram. Thanks!

The diagram appears at different scales here: Definitions of tuning terms: equal temperament... * [with cont.] (Wayb.) These are not "taxicab" diagrams. I'm don't know of any cute name for them. Maybe Paul E. would care to coin one. -Carl

Message: 7067 Date: Mon, 21 Jul 2003 23:09:09 Subject: Re: Interesting numerical coincidences (Combinatorics etc) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "paulhjelmstad" <paul.hjelmstad@u...> wrote:

> Thanks, Gene, but actually look at my first post. I am not just > calculating C{6n,6} I am looking at sets based on unique interval > vectors. I just called the sets C{18,6} and C{30,6} as a kind of > shorthand. Should have said "sets reduced from ..."

I've taught this stuff too many times to be able to see C(n, m) in more than one way. :) Evidently, you have another function you want computed.

Message: 7070 Date: Tue, 22 Jul 2003 06:45:49 Subject: Re: Interesting numerical coincidences (Combinatorics etc) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "paulhjelmstad"

> Well good news. C{30,6} reduces to 9,701 sets (based on unique > interval vector count). This is 89 * 109. I found 89 along

"Schismic"

> in Zoomr.gif (hard to read, its really small, and doesn't show up

in

> Zooms.gif. I can't find 109. Help! Can someone tell me where 109 > would show up in a taxicab diagram. Thanks!

a taxicab diagram? why do you call it that? you should be looking at Definitions of tuning terms: equal temperament... * [with cont.] (Wayb.) . . . zoom in, zoom out . . .

Message: 7071 Date: Wed, 23 Jul 2003 11:15:09 Subject: Re: Poor man's harmonic entropy? From: Carl Lumma Gene wrote...

>>If x is a positive real number representing an interval, it's >>suggested that Pc(x) = exp((log(p/q)-x)^2/2c)) can model the >>probability that x is heard as p/q; here c is a parameter. If >>we take the sum >> >>sum q^(-d) Pc(x) >> >>over positive rationals p/q, then it isn't hard to see that >>this converges absolutely for high enough values of d--anything >>above 2, at any rate.

Why are you summing? Presumably to get harmonic entropy for x, which is the entropy of the distribution of probabilities for all p/q. Does the q^(-d) term do that, and if so, how?

>>A problem with this is that it mixes multiplicative and >>implicitly additive distance measures, since |log(p/q)-x| is >>multiplicative, while q does not depend on the octave and, >>like the Farey sequence, is implicitly additive.

> >I think maybe the plan should be to get a continuous function >the Tenney Height way, by > >sum_{p/q > 0} Pc(x, p/q)/(p*q)

Um, not clear how this could possibly work. Paul empirically verified that p*q approximates the "width" (and also the entropy?) for x. Where I'm totally at a loss for how to define "width". Anyway, goal number 1 is to do extend things to triads and up. Where I think p*q*r is supposed tell you something about the space around triads on a 2-D plot... or something. It's been a long time... -Carl

Message: 7072 Date: Wed, 23 Jul 2003 03:59:22 Subject: Re: Interesting numerical coincidences (Combinatorics etc) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "paulhjelmstad"

> Another cool coincidence. C{19,6} reduces to a count of 735 unique > interval vectors. That's 49 X 15. 49 and 15 appear along "Kleismic" > in the zoom diagram.

what's the coincidence? any two ets will be collinear along some line, whether the particular line connecting them is special enough to merit inclusion on the chart or not -- and quite a few did. any two ets define a 5-limit linear temperament (represented by a line in the diagram).

Message: 7073 Date: Wed, 23 Jul 2003 20:13:02 Subject: Re: Poor man's harmonic entropy? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Gene wrote... >

> >>If x is a positive real number representing an interval, it's > >>suggested that Pc(x) = exp((log(p/q)-x)^2/2c)) can model the > >>probability that x is heard as p/q; here c is a parameter. If > >>we take the sum > >> > >>sum q^(-d) Pc(x) > >> > >>over positive rationals p/q, then it isn't hard to see that > >>this converges absolutely for high enough values of d--anything > >>above 2, at any rate.

> > Why are you summing? Presumably to get harmonic entropy for x, > which is the entropy of the distribution of probabilities for all > p/q. Does the q^(-d) term do that, and if so, how?

I've replace the q^(-d) with (pq)^(-d), and I think taking d=1 is fine. The weighting gives more weight to the better consonances, and it makes the series converge to a continuous function. What values of c have mostly been used? A graph would be nice.

> >I think maybe the plan should be to get a continuous function > >the Tenney Height way, by > > > >sum_{p/q > 0} Pc(x, p/q)/(p*q)

> > Um, not clear how this could possibly work. Paul empirically > verified that p*q approximates the "width" (and also the entropy?) > for x. Where I'm totally at a loss for how to define "width".

I'm not clear why it wouldn't work.

> Anyway, goal number 1 is to do extend things to triads and > up. Where I think p*q*r is supposed tell you something about the > space around triads on a 2-D plot... or something. It's been > a long time...

You could take a similar sum over all p:q:r: F(x) = sum_{p:q:r reduced} 1/(pqr) (Pc(x, p/q)+Pc(x, p/r)+Pc(x,r/q))

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