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Message: 5525 - Contents - Hide Contents Date: Fri, 08 Nov 2002 20:19:07 Subject: Re: from the realms of private correspondence From: Carl Lumma> Tenney Harmonic Distance. Note that it is only defined for > dyads. I attempted to extend it to triads. Paul claims that > in so doing, I removed its metric status.I also don't see how Tenney HD meets property 4... take dyads 33:26 and 39:22 (Paul's example). -Carl

Message: 5526 - Contents - Hide Contents Date: Fri, 08 Nov 2002 21:42:02 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> [Paul wrote...]>>>>> You're wondering how to define the taxicab distance between >>>>> three points? The shortest path that connects the subgraph... >>>>> wouldn't punish all the dyads, and a fully-connecting path >>>>> would be equiv. to the sum of the dyadic Tenney HD... maybe >>>>> the area of the enclosed polygon... for a chord like 4:5:25, >>>>> this leads to the problem that you can only compare it to other >>>>> 5-limit-only chords. Well, I give up. >>>>>>>> actually, i figured this out once. it's proportional to the >>>> total edge length of the hyper-rectangle defined by the points. >>>> but that doesn't make it a metric! >>>>>> 1. Can you give an example for 4:5:25? >>>> ok, in that case the hyper-rectangle collapses down to two >> dimensions, but the total edge length is unaffected. so no real >> problem there. the vertices of the hyper-rectangle are the >> pitches comprising the (non-octave reduced) euler genus whose >> factors are the notes in your chord. >> Come again? What points in the lattice does the rectangle > intersect?its vertices are 1 4 5 20 25 100 125 500 and it also intersects 5 (again), 20 (again), 25 (again), 100 (again), 2, and 10.>>>> do you know what "metric" means? >>>>>> According to mathworld, it's a function that satisfies the >>> following: >>> >>> 1. g(x,y) + g(y,z) >= g(x,z) >>> triangle inequality >>> >>> 2. g(x,y) = g(y,x) >>> symmetric >>> >>> 3. g(x,x) = 0 >>> >>> 4. g(x,y) = 0 implies x = y. >>> >>> 1. Makes no sense to me. does Tenney HD satisfy it for dyads? >>>> once again, carl -- tenney HD is a metric for *pitches*, not >> dyads (notwithstanding your "cop-out"). >> When did you ever say that? What does it mean? What meaning > do pitches have in terms of concordance?pitches are what you start with when you want to calculate concordance. the fact that you can transpose them (together) without affecting the concordance is one of the essential features in making this a metric at all.>>> 4. This would depend on the lack of chords sharing the same >>> set of factors. Without it, mathworld says we have a >>> "pseudometric". >>>> Tenney's HD is a metric, not a pseudometric. property 4 implies >> that any pitch has a zero harmonic distance from itself -- >> that's all. >> No, that's property 3. Property 4 says any time you see zero > distance you measuring the distance from a pitch to itself. true enough! > Also, still don't get how anything sensible could satisfy > property 1,of course any sensible metric satisfies property 1. euclidean, taxicab, you name it!> or how my suggestion is a cop-out.because it deprives "metric" of all meaning.

Message: 5528 - Contents - Hide Contents Date: Fri, 08 Nov 2002 21:46:52 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> My feeling is that we want a measure that works when comparing > dyads in the same pitch range, and that product limit always > was such a measure.if you're in the same pitch range, you can use product, you can use denominator, you can use sum, you can use numerator, and they'll all give you the same ranking. it's when comparing dyads in *different* pitch ranges that these diverge. product is at least preferable to the other options, in that case, because it leaves the curve with no overall trend -- it's asymptotically flat.

Message: 5530 - Contents - Hide Contents Date: Fri, 08 Nov 2002 21:50:29 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>> I don't understand how the term "limit" got into your question, >> or is this what others have called it? >> It comes from that it can be used as an alternative to odd limit. > Interestingly, IIRC Paul showed odd-limit is as close to the > product thing as we can get in an octave-equivalent measure. Is > that right, Paul?in a certain sense dealing with harmonic entropy, yes.

Message: 5532 - Contents - Hide Contents Date: Fri, 08 Nov 2002 21:51:31 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote: >>> I got it from Denny Genovese, who was using it at least since >> the mid 80's, and maybe since the mid 70's. When did Tenney >> come up with his HD? >> I think a lot of people used it. I used it, along with more >complicated things aimed at scale creation. I don't know why Euler >didn't use it, and have no idea who first did.probably it was benedetti. in the renaissance. he had very interesting reasons for choosing it -- margo has posted on this on the tuning list . . .

Message: 5534 - Contents - Hide Contents Date: Fri, 08 Nov 2002 21:52:58 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>> It says the distance from A to B, plus the distance from B to C >> (two legs of a triangle) is at least as big as the distance from >> A to C (the third leg.) > > Mint! >> Of course. Sorry. > > In which case, I'd think property 4 is the only one my cop-out > doesn't meet, and Tenney HD also doesn't meet it.what do you mean? of course tenney HD meets property 4, as i demonstrated in my e-mail which was your jumping-off point for all this . . .

Message: 5535 - Contents - Hide Contents Date: Fri, 08 Nov 2002 21:55:33 Subject: Re: from the realms of private correspondence From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>> it is a vector space norm and automatically induces a metric >> by d(X, Y) = ||X - Y|| >I wonder what Eric Weisstein would say if someone told him we used his mathematics dictionary a whole lot more than Treasure Trove of Music around here?

Message: 5536 - Contents - Hide Contents Date: Fri, 08 Nov 2002 04:36:31 Subject: Fwd: Re: from the realms of private correspondence From: Carl Lumma>> >our hypothesis is that because psychoacoustic dissonance >> varies from listener to listener, music had to develop a >> grammar to make the perception of dissonance more... >> uniform? >>I don't recall saying that it "has to", or even that >to whatever extent it "ought to", that variation in >perception of dissonance would be the only reason.Not sure how this answers my question... -Carl

Message: 5537 - Contents - Hide Contents Date: Fri, 08 Nov 2002 21:56:32 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote: >>> In which case, I'd think property 4 is the only one my cop-out >> doesn't meet, and Tenney HD also doesn't meet it. If this >> really isn't true, I'm hoping someone will refute it. >> What's Tenney HD? If you mean > > || 3^a 5^b 7^c || = |a/log(3)| + |b/log(5)| + |c/log(7)| no, it's|| 2^z 3^a 5^b 7^c . . . || = |z*log(2)| + |a*log(3)| + |b*log(5)| + |c*log(7)| . . . which ends up being the log of (numerator times denominator)> > or something like that, it is a vector space norm and automaticallyinduces a metric by> > d(X, Y) = ||X - Y||where "-" is really an arithmetic division, not an arithmetic subtraction . . .

Message: 5538 - Contents - Hide Contents Date: Fri, 08 Nov 2002 04:37:50 Subject: Re: from the realms of private correspondence From: Carl Lumma>> > must not understand property 1. If I have colinear >> points a b c, how is the sum of AB and BC >= AC? >> AB + BC = AC > >The two parts of the line segment AC add up to the whole in >length.Yeah, I must be reading the notation here... Metric -- from MathWorld * [with cont.] ...wrong then? -Carl

Message: 5539 - Contents - Hide Contents Date: Fri, 08 Nov 2002 21:59:02 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>> Tenney Harmonic Distance. Note that it is only defined for >> dyads. I attempted to extend it to triads. Paul claims that >> in so doing, I removed its metric status. >> I also don't see how Tenney HD meets property 4... take > dyads 33:26 and 39:22 (Paul's example). > > -Carljust because two pairs of points have the same distance from one another, or (more directly) just because two different points have the same distance from the origin, doesn't mean that they have to be the same point, or anything like that. property 4 only says that if the distance between two points is *zero*, they have to be the same point.

Message: 5540 - Contents - Hide Contents Date: Fri, 08 Nov 2002 04:43:10 Subject: Re: from the realms of private correspondence From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote: It says the distance from A to B, plus the distance from B to C (two legs of a triangle) is at least as big as the distance from A to C (the third leg.)

Message: 5541 - Contents - Hide Contents Date: Fri, 08 Nov 2002 22:06:29 Subject: Re: from the realms of private correspondence From: Carl Lumma>>>>> >ctually, i figured this out once. it's proportional to the >>>>> total edge length of the hyper-rectangle defined by the points. >>>>> but that doesn't make it a metric! >>>>>>>> 1. Can you give an example for 4:5:25? >>>>>> ok, in that case the hyper-rectangle collapses down to two >>> dimensions, but the total edge length is unaffected. so no real >>> problem there. the vertices of the hyper-rectangle are the >>> pitches comprising the (non-octave reduced) euler genus whose >>> factors are the notes in your chord. >>>> Come again? What points in the lattice does the rectangle >> intersect? >>its vertices are > >1 4 > >5 20 > >25 100 > >125 500 > >and it also intersects 5 (again), 20 (again), 25 (again), 100 >(again), 2, and 10.My chord is: 25 | 5 | 1 Here are your verticies on the lattice: 125 - x - 500 | | | 25 - x - 100 | | | 5 - x - 20 | | | 1 - x - 4 Obviously, you intend some of these to be on extra dimensions. Why? How did you figure out that the perimeter of these structures would be a consistent taxicab distance for three points?>>> once again, carl -- tenney HD is a metric for *pitches*, not >>> dyads (notwithstanding your "cop-out"). >>>> When did you ever say that? What does it mean? What meaning >> do pitches have in terms of concordance? >>pitches are what you start with when you want to calculate >concordance. the fact that you can transpose them (together) >without affecting the concordance is one of the essential >features in making this a metric at all.I don't understand how a pitch can have concordance. -Carl

Message: 5542 - Contents - Hide Contents Date: Fri, 08 Nov 2002 05:38:28 Subject: Re: from the realms of private correspondence From: Carl Lumma>It says the distance from A to B, plus the distance from B to C >(two legs of a triangle) is at least as big as the distance from >A to C (the third leg.) Mint!Of course. Sorry. In which case, I'd think property 4 is the only one my cop-out doesn't meet, and Tenney HD also doesn't meet it. If this really isn't true, I'm hoping someone will refute it. -Carl

Message: 5544 - Contents - Hide Contents Date: Fri, 08 Nov 2002 22:34:46 Subject: Re: from the realms of private correspondence From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> I don't understand how a pitch can have concordance.They have it with respect to "1". Because of the group structure, this automatically gives you concordance for dyads, which is why thinking aobut it in terms of norms rather than metrics makes the most sense.

Message: 5545 - Contents - Hide Contents Date: Fri, 08 Nov 2002 22:42:01 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>>>>> actually, i figured this out once. it's proportional to the >>>>>> total edge length of the hyper-rectangle defined by the points. >>>>>> but that doesn't make it a metric! >>>>>>>>>> 1. Can you give an example for 4:5:25? >>>>>>>> ok, in that case the hyper-rectangle collapses down to two >>>> dimensions, but the total edge length is unaffected. so no real >>>> problem there. the vertices of the hyper-rectangle are the >>>> pitches comprising the (non-octave reduced) euler genus whose >>>> factors are the notes in your chord. >>>>>> Come again? What points in the lattice does the rectangle >>> intersect? >>>> its vertices are >> >> 1 4 >> >> 5 20 >> >> 25 100 >> >> 125 500 >> >> and it also intersects 5 (again), 20 (again), 25 (again), 100 >> (again), 2, and 10. >> My chord is: > > 25 > | > 5 > | > 1you said 4:5:25!> Here are your verticies on the lattice: > > 125 - x - 500 > | | | > 25 - x - 100 > | | | > 5 - x - 20 > | | | > 1 - x - 4i don't know if you read my diagram right. it was meant to represent the eight vertices of (in concept) a cube.> Obviously, you intend some of these to be on > extra dimensions. Why?they're not on extra dimensions, but the edges overlap, since it's squashed down from> How did you figure > out that the perimeter of these structures > would be a consistent taxicab distance for three > points?it's easy. there are 12 edges. the three representing each of the pitches' distances from 1/1 (when they are expressed as simply as possible as harmonics thereof) are each present four times. so you can divide through by four, and you simply have log(a) + log(b) + log (c), which equals log(a*b*c). get it?>>>> once again, carl -- tenney HD is a metric for *pitches*, not >>>> dyads (notwithstanding your "cop-out"). >>>>>> When did you ever say that? What does it mean? What meaning >>> do pitches have in terms of concordance? >>>> pitches are what you start with when you want to calculate >> concordance. the fact that you can transpose them (together) >> without affecting the concordance is one of the essential >> features in making this a metric at all. >> I don't understand how a pitch can have concordance.it doesn't! that's why we have a concordance *metric*!

Message: 5546 - Contents - Hide Contents Date: Fri, 08 Nov 2002 22:42:21 Subject: Fwd: Re: from the realms of private correspondence From: wallyesterpaulrus 2:1 --- In tuning-math@y..., <Josh@o...> wrote:> How are we defining "octave"? > > ---- Original message ---->> Date: Fri, 08 Nov 2002 21:50:29 -0000 >> From: "wallyesterpaulrus" <wallyesterpaulrus@y...> >> Subject: [tuning-math] Re: from the realms of private > correspondence >> To: tuning-math@y... >> >> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>>> I don't understand how the term "limit" got into your > question,>>>> or is this what others have called it? >>>>>> It comes from that it can be used as an alternative to > odd limit.>>> Interestingly, IIRC Paul showed odd-limit is as close to > the>>> product thing as we can get in an octave-equivalent > measure. Is>>> that right, Paul? >>>> in a certain sense dealing with harmonic entropy, yes. >> >> >> To unsubscribe from this group, send an email to: >> tuning-math-unsubscribe@y... >> >> >> >> Your use of Yahoo! Groups is subject to > Yahoo! Terms of Service * [with cont.] (Wayb.) >> >>

Message: 5548 - Contents - Hide Contents Date: Fri, 08 Nov 2002 23:08:30 Subject: Re: from the realms of private correspondence From: Carl Lumma>> > don't understand how a pitch can have concordance. >>They have it with respect to "1".Dyad. So what you have is a function that assigns a value to pitches, and then you subtract them. The function assigns zero to 1/1 but you're still subtracting. I have a function that assigns a value to triads, and I can subtract them. What's the difference?>Because of the group structure, this automatically gives you >concordance for dyads, which is why thinking aobut it in terms >of norms rather than metrics makes the most sense.If you know what norms are and how to work with them. I'm still struggling with metrics. But do tell. Maybe Paul will follow. -Carl

Message: 5549 - Contents - Hide Contents Date: Fri, 08 Nov 2002 23:09:53 Subject: Fwd: Re: from the realms of private correspondence From: Carl Lumma --- In tuning-math@y..., <Josh@o...> wrote:> Try it with clarinet timbre, piano timbre and oboe timbre, > and I can practically guarantee that you'll each get an > inconsistent result across timbres.I have, and I don't. Have you? -Carl

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