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Message: 7075 Date: Wed, 23 Jul 2003 20:52:55 Subject: Re: Poor man's harmonic entropy? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Um, not clear how this could possibly work. Paul empirically > verified that p*q approximates the "width"

that's 1/sqrt(p*q), carl. i also verified that for triads, the "area" is approximately 1/cuberoot(p*q*r), for p:q:r in lowest terms . . . the total set of dyads or triads in each case is delimited by a very high (much higher than the p*q or p*q*r above) maximum value for the product of the terms. this ("tenney" as opposed to "farey") is the only rule that gives a "uniform" (trendless) density over dyad space or triad space.

> (and also the entropy?)

yes, if p and q are small enough, log(p*q) + c approximates the entropy. tenney's harmonic distance function is log(p*q).

> for x. Where I'm totally at a loss for how to define "width".

in harmonic entropy, you can use mediant-to-mediant width, or you can just use midpoint-to-midpoint width (which generalizes to voronoi cells for triads).

> 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...

yup, we've done that on the harmonic entropy list.

Message: 7076 Date: Wed, 23 Jul 2003 14:42:20 Subject: Re: Poor man's harmonic entropy? From: Carl Lumma Hooray!

>> Um, not clear how this could possibly work. Paul empirically >> verified that p*q approximates the "width"

> >that's 1/sqrt(p*q), carl.

Sure.

>i also verified that for triads, the "area" is approximately >1/cuberoot(p*q*r), for p:q:r in lowest terms . . .

I thought the rationals were infinitely dense. So how do you define "width"?

>the total set of dyads or triads in each case is delimited by a very >high (much higher than the p*q or p*q*r above) maximum value for the >product of the terms. this ("tenney" as opposed to "farey") is the >only rule that gives a "uniform" (trendless) density over dyad space >or triad space.

Thanks for the recap. But don't we want to eventually say the total set is infinite? Can we still define "width" then? -Carl

Message: 7078 Date: Wed, 23 Jul 2003 22:25:43 Subject: Re: Poor man's harmonic entropy? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Hooray! >

> >> Um, not clear how this could possibly work. Paul empirically > >> verified that p*q approximates the "width"

> > > >that's 1/sqrt(p*q), carl.

> > Sure. >

> >i also verified that for triads, the "area" is approximately > >1/cuberoot(p*q*r), for p:q:r in lowest terms . . .

> > I thought the rationals were infinitely dense.

sorry, i should have said *proportional to*, the width is *proportional* to 1/sqrt(p*q) (for p and q not too large), and the harmonic entropy is *proportional* to log(p*q) (for p and q quite small).

> But don't we want to eventually say the total set is infinite?

yes, we want to take the limit.

> Can we still define "width" then?

yes, the width will go to zero in the limit, but the whole entropy expression may still converge (or some function of it may), as you should know from calculus.

Message: 7079 Date: Thu, 24 Jul 2003 18:35:40 Subject: Re: Poor man's harmonic entropy? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >

> > i'm sure gene could tell us why we get 1/cuberoot(p*q*r) works.

> > I can? I don't even know what you mean.

where did you lose me? i thought the last few posts were quite clear, in particular the graphs that show the above to be true.

Message: 7080 Date: Thu, 24 Jul 2003 20:12:11 Subject: Re: Poor man's harmonic entropy? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote: Does that mean the original sum is over only those p/q in their lowest

> terms?

Of course.

Message: 7081 Date: Thu, 24 Jul 2003 00:05:34 Subject: Re: Poor man's harmonic entropy? From: Carl Lumma

>Carl, thanks for your reply, but it doesn't make a bit of >sense....thanks, shelly

Hi Shelly. Does Gene's original post make sense to you? How long have you been on this list? -Carl

Message: 7082 Date: Thu, 24 Jul 2003 00:06:44 Subject: Re: Poor man's harmonic entropy? From: Carl Lumma

>>>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.

Do you just see this stuff, or does your thought process involve multiple steps? -Carl

Message: 7083 Date: Thu, 24 Jul 2003 00:07:15 Subject: Re: Poor man's harmonic entropy? From: Carl Lumma

>> Can we still define "width" then?

> >yes, the width will go to zero in the limit, but the whole entropy >expression may still converge (or some function of it may), as you >should know from calculus.

Ok, good. So are there any open problems in harmonic entropy? Would it be accurate to say we've empirically verified the p*q*r stuff, but that we don't have a clean picture of why it works? -Carl

Message: 7084 Date: Thu, 24 Jul 2003 05:01:04 Subject: Re: Poor man's harmonic entropy? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:

> Do you just see this stuff, or does your thought process > involve multiple steps?

Experience tells me I could be completely out to lunch, working on a different problem than the one I think I am, seeing things as obvious which aren't, or mired in a host of other problems. Do you understand I'm working with an expression which, at least, involves all rational intervals and converges to a continuous function? It seems to me that has to be worth something.

Message: 7085 Date: Thu, 24 Jul 2003 06:16:14 Subject: Re: Poor man's harmonic entropy? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:

> >> Can we still define "width" then?

> > > >yes, the width will go to zero in the limit, but the whole entropy > >expression may still converge (or some function of it may), as you > >should know from calculus.

> > Ok, good. So are there any open problems in harmonic entropy?

yup -- summarized them for gene a while back.

> Would it be accurate to say we've empirically verified the p*q*r > stuff, but that we don't have a clean picture of why it works? > > -Carl

i'm sure gene could tell us why we get 1/cuberoot(p*q*r) works. as for the entropy for the very simplest triads being log(p*q*r)+c, that actually hasn't been verified yet. i'm hoping gene will help us do it analytically rather than numerically . . .

Message: 7086 Date: Thu, 24 Jul 2003 00:26:26 Subject: Re: Poor man's harmonic entropy? From: monz@xxxxxxxxx.xxx hi Gene,

> From: Gene Ward Smith [mailto:gwsmith@xxxxx.xxxx > Sent: Wednesday, July 23, 2003 10:01 PM > To: tuning-math@xxxxxxxxxxx.xxx > Subject: [tuning-math] Re: Poor man's harmonic entropy? > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote: >

> > Do you just see this stuff, or does your thought process > > involve multiple steps?

> > Experience tells me I could be completely out > to lunch, working on a different problem than > the one I think I am, seeing things as obvious > which aren't, or mired in a host of other problems. > Do you understand I'm working with an expression > which, at least, involves all rational intervals > and converges to a continuous function? It seems > to me that has to be worth something.

i really haven't been following this thread, or any developments in harmonic entropy theory since around early 2000, but my intuition has a sense that you're onto something good here. can you elaborate and then explain the math a little? PS to paul -- i'd like to update my sonic-arts pages on harmonic entropy so that they reflect some of the further developments of the theory. can you write an addendum to my current pages? i can just add it on at the bottom. -monz

Message: 7087 Date: Thu, 24 Jul 2003 00:54:39 Subject: Re: Poor man's harmonic entropy? From: Carl Lumma

>> Do you just see this stuff, or does your thought process >> involve multiple steps?

> >Experience tells me I could be completely out to lunch, working >on a different problem than the one I think I am, seeing things >as obvious which aren't, or mired in a host of other problems.

:)

>Do you understand I'm working with an expression which, at least, >involves all rational intervals and converges to a continuous >function?

Yes indeed. Though I don't see how you know it converges. Did you test a few values? No, you say it "isn't hard to see" that it does... maybe you can walk us through it.

>It seems to me that has to be worth something.

No argument here. -Carl

Message: 7088 Date: Thu, 24 Jul 2003 08:51:55 Subject: Re: Poor man's harmonic entropy? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx <monz@a...> wrote:

> PS to paul -- i'd like to update my sonic-arts > pages on harmonic entropy so that they reflect > some of the further developments of the theory. > can you write an addendum to my current pages? > i can just add it on at the bottom. > > > > -monz

this exchange from an e-mail from me to you dated apr. 14th: *******************************************************************

>if any text or links should be added to my harmonic entropy >definition, would you be so kind as to send them over? :)

could you add, right before "certain chords of three or more notes", the following: assuming an error distribution that follows an exponential decay on either side of the actual interval, instead of the usual default bell curve, yields harmonic entropy curves that have pointy, instead of round, local minima. accentuating the differences among the more dissonant intervals by taking the exponential of the entropy (in information theory, the exponential of the entropy is the so- called "alphabet size"), and assuming a tuning resolution representing the best human ears (s=0.6%), this "pointy" assumption lets us paint the most generous possible picture (within the harmonic entropy paradigm) for the ability to distinguish complex ratios by their relative consonance: {reproduce Yahoo groups: /harmonic_entropy/files/dyadic/m... * [with cont.] thanks, paul ******************************************************************** also, on any of the harmonic entropy pages, you might want to display Yahoo groups: /harmonic_entropy/files/trivoro.gif * [with cont.] the 2-dimensional space here is triad space, just like on the graphs on your eqtemp page. the center of each cell represents a triad a:b:c such that a*b*c is less than a million, and the cell itself contains all (and only) points that are closer to its center than to the center of any other cell. the largest cells are the ones with the smallest value of a*b*c, as shown more clearly in this closeup: Yahoo groups: /harmonic_entropy/files/Erlich/f... * [with cont.] the area of each cell as a function of the geometric mean of a, b, and c, in other words cuberoot(a*b*c), is shown at Yahoo groups: /harmonic_entropy/files/triadic.gif * [with cont.] showing that the areas are proportional to 1/cuberoot(a*b*c), at least for a*b*c not too large. finally, there's a bunch of stuff that can go on the "old ideas" page to connect them better to the "new ideas" assumed in the definition page. let me get back to you on that.

Message: 7089 Date: Thu, 24 Jul 2003 09:45:57 Subject: Re: Poor man's harmonic entropy? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> i'm sure gene could tell us why we get 1/cuberoot(p*q*r) works.

I can? I don't even know what you mean.

Message: 7090 Date: Thu, 24 Jul 2003 10:07:09 Subject: Re: Poor man's harmonic entropy? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Yes indeed. Though I don't see how you know it converges. Did > you test a few values? No, you say it "isn't hard to see" that > it does... maybe you can walk us through it.

Sorry! Assume d>1 and chop everything up into octaves. For the 1-2 octave, for a given denominator q of p/q, compare to the sum (p_i q)^(-d) (p_0/q + ... + p_phi(q)/q) where p_0 = q and we go through values p_i relatively prime to q. Our terms are all positive, and this expression is less than q^(-2d) (q/q + (q+1)/q + ... + (2*q-1)/q) < 2q^(-d) Since d>1, this converges by comparison to the Zeta function Dirichlet series 1+2^(-d) + 3^(-d) + ... Hence, the sum of the series over the octave is bounded by 2 Zeta(d), and in fact for each octave, the sum of the functions of x is bounded by 2 Zeta(d) M_O, where M_O is the maximum of Ps(x, p/q) over the ocatve O. The normal distribution tends rapidly to zero, so you have a uniformly convergent series of continuous functions summing over the octaves, leading to a continuous function. This is the sort of thing which was in my head; maybe it can be made clearer.

Message: 7091 Date: Thu, 24 Jul 2003 13:50:40 Subject: Re: Poor man's harmonic entropy? From: Graham Breed Gene Ward Smith wrote:

> Assume d>1 and chop everything up into octaves. For the 1-2 octave, > for a given denominator q of p/q, compare to the sum > > (p_i q)^(-d) (p_0/q + ... + p_phi(q)/q) > > where p_0 = q and we go through values p_i relatively prime to q. Our > terms are all positive, and this expression is less than > > q^(-2d) (q/q + (q+1)/q + ... + (2*q-1)/q) < 2q^(-d)

Does that mean the original sum is over only those p/q in their lowest terms?

> Since d>1, this converges by comparison to the Zeta function > Dirichlet series 1+2^(-d) + 3^(-d) + ... Hence, the sum of the series > over the octave is bounded by 2 Zeta(d), and in fact for each octave, > the sum of the functions of x is bounded by 2 Zeta(d) M_O, where M_O > is the maximum of Ps(x, p/q) over the ocatve O. The normal > distribution tends rapidly to zero, so you have a uniformly > convergent series of continuous functions summing over the octaves, > leading to a continuous function.

If c < 0. Graham

Message: 7093 Date: Fri, 25 Jul 2003 11:43:29 Subject: Re: Creating a Temperment /Comma From: monz@xxxxxxxxx.xxx

> From: paulhjelmstad [mailto:paul.hjelmstad@xx.xxx.xxxx > Sent: Friday, July 25, 2003 10:21 AM > To: tuning-math@xxxxxxxxxxx.xxx > Subject: [tuning-math] Creating a Temperment /Comma > > > I would like to know which temperment/comma connects > 12, 47, 35, and 23 ets on zoomr.gif. How could I > calculate these for myself? Thanks! If this is a > new one, could I name it? What is the 5-limit vector?

paul is referring to the "zoom:10" mouse-over link here Definitions of tuning terms: equal temperament... * [with cont.] (Wayb.) ... just trying to be helpful ... unfortunately, i'd be trying to figure out which comma it is empirically, somehow, but examining the cents errors of each temperament. Gene and paul erlich know how to do it easily with algebra. -monz

Message: 7094 Date: Fri, 25 Jul 2003 12:36:43 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>I would like to know which temperment/comma connects 12, 47, 35, and >23 ets on zoomr.gif. How could I calculate these for myself? Thanks! >If this is a new one, could I name it? What is the 5-limit vector?

The comma is 6561/6250, according to Gene's maple, if I did it right. This comma of 84 cents is quite large considering its large denominator, which according to Paul's heuristic makes the resulting temperaments "bad". Though the val I used for 12 is the standard meantone one... Scala does not show a name for this comma. -Carl

Message: 7098 Date: Fri, 25 Jul 2003 15:07:30 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>How about 12,25,37,49,61? Is this one any good?

You may not be aware that Gene has conducted searches of 5-limit comma space. He was able to look at everything within a region defined by max rms error, "complexity" and "badness". If this captures your idea of "good", he didn't miss anything. Try searching the archives or checking the database on the main list... Yahoo groups: /tuning/database/ * [with cont.] ...if you think you've found a temperament good temperament that isn't here, you're hearby encouraged to make a case for it! -Carl

Message: 7099 Date: Fri, 25 Jul 2003 23:03:54 Subject: Re: Creating a Temperment /Comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >I would like to know which temperment/comma connects 12, 47, 35, and > >23 ets on zoomr.gif. How could I calculate these for myself? Thanks! > >If this is a new one, could I name it? What is the 5-limit vector?

> > The comma is 6561/6250, according to Gene's maple, if I did it right.

I get the same. You get a temperament with a generator the size of a semitone, five of which give a fourth, and eight of which give a minor sixth, which obviously is compatible with 12-et.

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