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Message: 10376 - Contents - Hide Contents Date: Wed, 25 Feb 2004 19:40:59 Subject: Re: non-1200: Tenney/heuristic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:>>> That's not correct. For one thing, it's not equal! >> Yes, sorry, I forgot again how to use my own program! > The code is correct, but I gave it the wrong parameters. > It should have been (1200.6171 1900.9770 2801.4398). > Step is 100.051421. > > ManuelIncorrect, I'm afraid.

Message: 10377 - Contents - Hide Contents Date: Wed, 25 Feb 2004 19:51:51 Subject: Re: non-1200: Tenney/heuristic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" > <manuel.op.de.coul@e...> wrote: >>>> 31-equal TOP is this, identical for 5-limit and 7-limit: >> (1201.6366 1899.3611 2790.8979 3372.3350) >> I get 1201.4675Yes, to two places I get [1201.47 1899.09 2790.51 3371.86].

Message: 10378 - Contents - Hide Contents Date: Wed, 25 Feb 2004 20:01:45 Subject: Re: non-1200: Tenney/heuristic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:> > Gene wrote:>>> 31-equal TOP is this, identical for 5-limit and 7-limit: >>> (1201.6366 1899.3611 2790.8979 3372.3350) >>> I get 1201.4675 >> I don't see how that can be correct. Your twelfth will be > 1899.094. Then (1901.955 - 1899.094) / (1201.4675 - 1200.0) = > 1.95 which is not log2(3)/log2(2) = 1.585. > > ManuelBut Manuel, your tuning gives (2790.8979 - 2786.3137)/(1201.4675 - 1200.0) = 2.80 which is not log(5)/log(2) = 2.32. So I'm not sure what your point is.

Message: 10379 - Contents - Hide Contents Date: Wed, 25 Feb 2004 20:04:12 Subject: Re: non-1200: Tenney/heuristic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:> > Gene wrote:>>> 31-equal TOP is this, identical for 5-limit and 7-limit: >>> (1201.6366 1899.3611 2790.8979 3372.3350) >>> I get 1201.4675 >> I don't see how that can be correct. Your twelfth will be > 1899.094. Then (1901.955 - 1899.094) / (1201.4675 - 1200.0) = > 1.95 which is not log2(3)/log2(2) = 1.585. > > ManuelBut Manuel, your tuning gives (2790.8979 - 2786.3137)/(1201.6366 - 1200.0) = 2.80 which is not log(5)/log(2) = 2.32. So I'm not sure what your point is.

Message: 10382 - Contents - Hide Contents Date: Thu, 26 Feb 2004 17:23:26 Subject: Re: non-1200: Tenney/heuristic meantone temperament From: Manuel Op de Coul Paul wrote:>But Manuel, your tuning gives (2790.8979 - 2786.3137)/(1201.4675 - >1200.0) = 2.80 which is not log(5)/log(2) = 2.32. So I'm not sure >what your point is.Yes, never mind, I shouldn't be posting when in a hurry. Today I found your post where you explain the equal tempered case, so I understand it now. So the method in Scala is only compatible with TOP when tempering out a single comma. I could implement it for more than one comma sometime, but it's more complicated than the existing procedure. Manuel

Message: 10383 - Contents - Hide Contents Date: Thu, 26 Feb 2004 22:41:36 Subject: Re: non-1200: Tenney/heuristic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:> > Paul wrote:>> But Manuel, your tuning gives (2790.8979 - 2786.3137)/(1201.4675 - >> 1200.0) = 2.80 which is not log(5)/log(2) = 2.32. So I'm not sure >> what your point is. >> Yes, never mind, I shouldn't be posting when in a hurry. > Today I found your post where you explain the equal tempered case, > so I understand it now.Not so fast -- the single-comma case is the only one where the TOP tempering is motivated geometrically, so I'd be interested if there are other methods of tempering you'd come up with for multiple commas.> So the method in Scala is only compatible > with TOP when tempering out a single comma. > I could implement it for more than one comma sometime, but it's > more complicated than the existing procedure.If you have a method that doesn't require searching 2^n corners or whatever, I'd be most interested in learning it.

Message: 10384 - Contents - Hide Contents Date: Thu, 26 Feb 2004 22:59:49 Subject: Re: TOP and Tenney space webpage From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" > <paul.hjelmstad@u...> wrote: >> Gene, >>>> Did you mean (2^n)-1 instead of 2^(n-1)? ("Since we have 2^n > corners>> to a ball there are 2^(n-1) lines.." etc.) >> The 2^n corners come in 2^(n-1) pairs of opposite corners, and lines > between them pass through the center of the ball, which is the JIP. > > What's your take on the acronym JIP? I think it makes sense, since it > is the point corresponding to just intonation, but Paul objects for > reasons not entirely clear to me.First of all, it measures pitch, something you fail to note at all on your website, but would be the most comprehensible thing about this whole business. Secondly, who discovered the duality between points and linear functionals? Poincare? That's very recent in the history of mathematics, and far more advanced than what most musicians could ever, let alone should be expected to already, comprehend.

Message: 10385 - Contents - Hide Contents Date: Thu, 26 Feb 2004 23:58:54 Subject: Re: non-1200: Tenney/heuristic meantone temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Not so fast -- the single-comma case is the only one where the TOP > tempering is motivated geometrically...Did you read my TOP tuning page? Clearly, this is false.

Message: 10386 - Contents - Hide Contents Date: Thu, 26 Feb 2004 08:33:52 Subject: Re: TOP and Tenney space webpage From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> Gene, > > Did you mean (2^n)-1 instead of 2^(n-1)? ("Since we have 2^n corners > to a ball there are 2^(n-1) lines.." etc.)The 2^n corners come in 2^(n-1) pairs of opposite corners, and lines between them pass through the center of the ball, which is the JIP. What's your take on the acronym JIP? I think it makes sense, since it is the point corresponding to just intonation, but Paul objects for reasons not entirely clear to me. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10387 - Contents - Hide Contents Date: Fri, 27 Feb 2004 17:53:50 Subject: Re: non-1200: Tenney/heuristic meantone temperament From: Manuel Op de Coul Paul wrote:>Not so fast -- the single-comma case is the only one where the TOP >tempering is motivated geometrically, so I'd be interested if there >are other methods of tempering you'd come up with for multiple commas.What I have is not so good as TOP, it needs experimenting with the prime weights to get a good result for more than one comma.>If you have a method that doesn't require searching 2^n corners or >whatever, I'd be most interested in learning it.I had indeed the brute force approach in mind. I thought it was a linear programming problem, and Gene's TOP and Tenney space page confirmed it. Manuel

Message: 10388 - Contents - Hide Contents Date: Fri, 27 Feb 2004 20:30:10 Subject: Re: non-1200: Tenney/heuristic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Not so fast -- the single-comma case is the only one where the TOP >> tempering is motivated geometrically... >> Did you read my TOP tuning page?Yes, several times.

Message: 10389 - Contents - Hide Contents Date: Fri, 27 Feb 2004 20:33:37 Subject: Re: TOP and Tenney space webpage From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>>> What's your take on the acronym JIP? I think it makes sense, > since >> it>>> is the point corresponding to just intonation, but Paul objects > for>>> reasons not entirely clear to me. >>>> First of all, it measures pitch, something you fail to note at all > on>> your website, but would be the most comprehensible thing about this >> whole business. >> It doesn't measure anything. It is the just intonation mapping, just > as other points represent other tunings. Hence, "J I Point".I have no idea how to reconcile this with Yahoo groups: /tuning-math/message/9797 * [with cont.] and with the fact that it's obvious that this linear operator, when acting on a monzo, is the only one returns its pitch (or interval size) in cents.

Message: 10390 - Contents - Hide Contents Date: Fri, 27 Feb 2004 20:34:34 Subject: Re: non-1200: Tenney/heuristic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:> > Paul wrote:>> Not so fast -- the single-comma case is the only one where the TOP >> tempering is motivated geometrically, so I'd be interested if there >> are other methods of tempering you'd come up with for multiple commas. >> What I have is not so good as TOP, it needs experimenting with the > prime weights to get a good result for more than one comma. >>> If you have a method that doesn't require searching 2^n corners or >> whatever, I'd be most interested in learning it. >> I had indeed the brute force approach in mind. I thought it was a > linear programming problem, and Gene's TOP and Tenney space page > confirmed it.Amazingly, Gene's page would have you believe you need to search even in the codimension 1 case!

Message: 10391 - Contents - Hide Contents Date: Fri, 27 Feb 2004 00:01:17 Subject: Re: TOP and Tenney space webpage From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> What's your take on the acronym JIP? I think it makes sense, since > it>> is the point corresponding to just intonation, but Paul objects for >> reasons not entirely clear to me. >> First of all, it measures pitch, something you fail to note at all on > your website, but would be the most comprehensible thing about this > whole business.It doesn't measure anything. It is the just intonation mapping, just as other points represent other tunings. Hence, "J I Point". ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10392 - Contents - Hide Contents Date: Sat, 28 Feb 2004 04:28:44 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma>Two 7-limit notes less than 2 apart in the symmetric lattice can be >harmonized by two tetrads sharing at least one common note, and notes >2 or more apart cannot be so harmonized. Hence, a scale consisting >only of such intervals has the property that stepwise progressions can >be harmonized by tetrads with common notes--though not, of course, >necessarily tetrads all of whose notes belong to the scale. > >If list all notes reduced to an octave which are at a distance of less >than <correction>two</correction> from the unison and smaller than 200 >cents in size, we obtain this: > >{28/25, 10/9, 35/32, 15/14, 16/15, 21/20, 25/24, 36/35, 49/48} > >The possible types of JI scale with the above property, in terms of >the intervals and their multiplicities, with the above restriction on >step size are given below. We can obtain tempered versions of these by >temperaments which equate steps; we have (28/25)/(10/9) = 126/125, >(10/9)/(35/32) = 64/63, (35/32)/(15/14) = 49/48, (15/14)/(16/15) = >225/224, (16/15)/(21/20) = 64/63, (21/20)/(25/24) = 126/125, >(25/24)/(36/35) = 875/864, (36/35)/(49/48) = 1728/1715. Meantone, >magic, orwell, pajara, porcupine, blackwood, superpythagorean, >tripletone, kleismic or nonkleismic would all be reasonable linear >temperaments to try--or beep if you think that is reasonable. Rock! >[28/25, 10/9, 35/32, 36/35] [1, 3, 2, 3] >[28/25, 10/9, 25/24, 36/35] [3, 1, 4, 3] >[28/25, 15/14, 16/15, 49/48] [1, 5, 3, 2] >[28/25, 16/15, 25/24, 36/35] [3, 1, 5, 3] > >[10/9, 35/32, 16/15, 36/35] [2, 3, 2, 3] >[10/9, 35/32, 36/35, 49/48] [4, 1, 5, 2] >[10/9, 35/32, 21/20, 36/35] [4, 1, 2, 3] >[10/9, 15/14, 16/15, 21/20] [2, 3, 2, 3] >[10/9, 15/14, 21/20, 36/35] [4, 1, 3, 2] >[10/9, 15/14, 36/35, 49/48] [4, 1, 5, 3] >[10/9, 21/20, 25/24, 36/35] [4, 3, 1, 3] >[10/9, 25/24, 36/35, 49/48] [4, 1, 6, 3] > >[35/32, 15/14, 16/15, 36/35] [3, 2, 4, 1] >[35/32, 16/15, 25/24, 36/35] [3, 4, 2, 3] > >[15/14, 16/15, 21/20, 25/24] [3, 4, 3, 2] >[15/14, 16/15, 21/20, 49/48] [5, 4, 1, 2] >[15/14, 16/15, 36/35, 49/48] [5, 4, 1, 3] > >[16/15, 21/20, 25/24, 36/35] [4, 3, 5, 3] >[16/15, 25/24, 36/35, 49/48] [4, 5, 6, 3] > >[28/25, 10/9, 15/14, 25/24] [3, 1, 3, 1] >[28/25, 15/14, 16/15, 25/24] [3, 3, 1, 2] >[28/25, 35/32, 15/14, 16/15] [1, 2, 3, 3] >[28/25, 10/9, 15/14, 21/20] [2, 2, 3, 1] > >[35/32, 15/14, 16/15, 21/20] [2, 3, 4, 1]Rad. No 6- or 7-toners, eh? I wonder about the "9-limit". The 9-limit has the further advantage that you can hit more fifths, and thus improve omnitetrachordality. Also, could we screen based on which of the above combinations, and which orderings of those, produce the most low-numbered ratios in the scale? Or does such an approach fail on the grounds that it ignores temperament (aka TOLERANCE)? -Carl

Message: 10393 - Contents - Hide Contents Date: Sat, 28 Feb 2004 21:06:34 Subject: Re: Harmonized melody in the 7-limit From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Rad. No 6- or 7-toners, eh? I wonder about the "9-limit". > The 9-limit has the further advantage that you can hit more > fifths, and thus improve omnitetrachordality.The 9-limit would be different, for sure. The simple symmetrical lattice criterion wouldn't work, but it would be easy enough to find what does.> Also, could we screen based on which of the above > combinations, and which orderings of those, produce the > most low-numbered ratios in the scale? Or does such > an approach fail on the grounds that it ignores temperament > (aka TOLERANCE)?The main problem I see with that is that it is a huge computational job, since in each case you need to find the optimal version of the scale in question. I wouldn't want to try it using Maple.

Message: 10394 - Contents - Hide Contents Date: Sat, 28 Feb 2004 21:43:27 Subject: Re: Harmonized melody in the 7-limit From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Also, could we screen based on which of the above > combinations, and which orderings of those, produce the > most low-numbered ratios in the scale? Or does such > an approach fail on the grounds that it ignores temperament > (aka TOLERANCE)?One approach would be to use tempering to simplify the problem. If we pick linear temperaments which reduce the four sizes of scale step to two, we also automatically enforce Myhill's property. By the way, it seems to me the more general property of a scale with a set of steps (generating the temperament) with the number of different sizes of step equal to the number of generators, or rank, or dimension + 1 or whatever you want to call it for a regular temperament might be worth exploring; it generalizes Myhill's property for linear temperaments.

Message: 10395 - Contents - Hide Contents Date: Sat, 28 Feb 2004 14:37:14 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma>> >ad. No 6- or 7-toners, eh? I wonder about the "9-limit". >> The 9-limit has the further advantage that you can hit more >> fifths, and thus improve omnitetrachordality. >>The 9-limit would be different, for sure. The simple symmetrical >lattice criterion wouldn't work, but it would be easy enough to >find what does.Nobody ever answered me if symmetrical is synonymous with unweighted.>> Also, could we screen based on which of the above >> combinations, and which orderings of those, produce the >> most low-numbered ratios in the scale? Or does such >> an approach fail on the grounds that it ignores temperament >> (aka TOLERANCE)? >>The main problem I see with that is that it is a huge computational >job, since in each case you need to find the optimal version of the >scale in question. I wouldn't want to try it using Maple.The thing is to only store one permutation in memory at a time. Alas, I haven't come up with an easy way to code these kinds of evaluations in scheme. They're very natural in C, I think. The present problem may still be hard on account of CPU cycles, tho. -Carl

Message: 10396 - Contents - Hide Contents Date: Sat, 28 Feb 2004 04:00:52 Subject: Re: TOP and Tenney space webpage From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I have no idea how to reconcile this with > > Yahoo groups: /tuning-math/message/9797 * [with cont.] > > and with the fact that it's obvious that this linear operator, when > acting on a monzo, is the only one returns its pitch (or interval > size) in cents.I'm being contrary; it seems to me "measure" isn't really how we want to look at it, since just intonation is now being viewed as one of an infinite set of possible tuning maps. It is the just *tuning* point, but is that a measurement point?

Message: 10397 - Contents - Hide Contents Date: Sat, 28 Feb 2004 14:40:26 Subject: Re: Harmonized melody in the 7-limit From: Carl Lumma>By the way, it seems to me the more general property of a scale with >a set of steps (generating the temperament) with the number of >different sizes of step equal to the number of generators, or rank, >or dimension + 1 or whatever you want to call it for a regular >temperament might be worth exploring; it generalizes Myhill's >property for linear temperaments.I've always wanted to understand what's going on geometrically (in terms of block selection on the lattice) with Myhill scales. -Carl

Message: 10398 - Contents - Hide Contents Date: Sat, 28 Feb 2004 04:02:06 Subject: Re: non-1200: Tenney/heuristic meantone temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Amazingly, Gene's page would have you believe you need to search even > in the codimension 1 case!You want I should derive the codimension 1 formula instead?

Message: 10399 - Contents - Hide Contents Date: Sat, 28 Feb 2004 23:09:50 Subject: DE scales with the stepwise harmonization property From: Gene Ward Smith Here is what I get by the most plausible temperings of the corresponding JI scales. The DE scales for these are now unique up to transposition, so all the headaches about correctly ordering the steps vanish. Diminished[8] [28/25, 10/9, 15/14, 21/20] [2, 2, 3, 1] [28/25, 10/9, 15/14, 25/24] [3, 1, 3, 1] (28/25)/(10/9) = 126/125 (15/14)/(21/20) = 50/49 (15/14)/(25/24) = 36/35 Augmented[9] [28/25, 35/32, 15/14, 16/15] [1, 2, 3, 3] (28/25)/(35/32) = 128/125 (15/14)/(16/15) = 225/224 Blackwood[10] [10/9, 35/32, 16/15, 36/35] [2, 3, 2, 3] [10/9, 35/32, 21/20, 36/35] [4, 1, 2, 3] [10/9, 15/14, 16/15, 21/20] [2, 3, 2, 3] [35/32, 15/14, 16/15, 21/20] [2, 3, 4, 1, [35/32, 15/14, 16/15, 36/35] [3, 2, 4, 1] (10/9)/(35/32) = 64/63 (16/15)/(36/35) = 28/27 (21/20)/(36/35) = 49/48 (10/9)/(15/14) = 28/27 (16/15)/(21/20) = 64/63 Kleismic[11] [28/25, 10/9, 25/24, 36/35] [3, 1, 4, 3] (28/25)/(10/9) = 126/125 (25/24)/(36/35) = 876/864 Superpythagorean[12] [10/9, 35/32, 36/35, 49/48] [4, 1, 5, 2] (10/9)/(35/32) = 64/63 (36/35)/(49/48) = 1728/1715 Meantone[12] [15/14, 16/15, 21/20, 25/24] [3, 4, 3, 2] (15/14)/(16/15) = 225/224 (21/20)/(25/24) = 126/125 Augmented[12] [15/14, 16/15, 21/20, 49/48] [5, 4, 1, 2] (15/14)/(16/15) = 225/224 (21/20)/(49/48) = 36/35 Orwell[13] [15/14, 16/15, 36/35, 49/48] [5, 4, 1, 3] (15/14)/(16/15) = 225/224 (36/35)/(49/48) = 1728/1715 Porcupine[15] [16/15, 21/20, 25/24, 36/35] [4, 3, 5, 3] (16/15)/(21/20) = 64/63 (25/24)/(36/35) = 875/864

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