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Message: 9250 - Contents - Hide Contents Date: Sun, 18 Jan 2004 15:47:42 Subject: Re: Annotated Dave Keenan file From: Carl Lumma>Single chain: No. generators in > Min Min interval >Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6 >(+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7 >------------------------------------------------------------- >125c 6 12.2c 17.9c -4 3 -7 -2 -5 2 >tertiathirdsWhy isn't this negri? By the way, anybody know names for these... ! Two pentatonic chains of 7:4's rooted a 5:4 apart, tuned in 31-tet. 10 ! 154.839 !.....4 232.258 !.....6 387.097 !....10 464.516 !....12 619.355 !....16 696.774 !....18 851.613 !....22 967.742 !....25 1083.871 !...28 2/1 !........31 ! ! Four 5-limit triads on 1-4-7, strictly proper. ! Two pentatonic chains of 3:2's rooted a 7:4 apart, tuned in 31-tet. 10 ! 154.839 !......4 193.548 !......5 387.097 !.....10 464.516 !.....12 658.065 !.....17 696.774 !.....18 890.323 !.....23 967.742 !.....25 1161.290 !....30 2/1 !.........31 ! ! Four 5-limit triads on 1-4-7, not proper. ? -Carl

Message: 9251 - Contents - Hide Contents Date: Sun, 18 Jan 2004 02:29:24 Subject: Re: A new graph for Paul? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >> wrote:>>> Dual to the 5-limit symmetrical lattice of intervals is a 5- limit >>> symmetrical lattice of vals whose first component is zero--which >>> includes the generators in the period-generator of linear >>> temperaments. >>>> Don't get it. >> What's the hang-up? Do you understand the part about pairs of integers > representing generators for 5-limit linear temperaments?No, it would seem you need 4 integers to specify the mapping. Oh, you're throwing out the period. Seems dirty to do that.>> What does that line mean? >> That's to be explored. We have, along a line, > > porcupine-->meantone-->tetracot-->amity > > In terms of generator mapping, but not the period part of the map, and > therefore not in terms of the generators in cents, we can transform > one to the next continuously. That's the sort of thing I think is > worth thiking about from a compositional point of view, for >starters.What can you do with it?

Message: 9252 - Contents - Hide Contents Date: Sun, 18 Jan 2004 05:33:43 Subject: Re: A new graph for Paul? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Let me get back to this after we're done talking about error > functions and the metrics of their duals.OK. Here's something to note when you do--these lines can go on for quite a ways including temperaments of reasonably low badness. Here is my line, each temperament of which differs from the next by meantone, meaning the successive differences are <1 4|: <0 -7| 2187/2048 <1 -3| 125/128 <2 1| 25/24 <3 5| 250/243 <4 9| 20000/19683 <5 13| 1600000/1594323 <6 17| 129140163/128000000

Message: 9253 - Contents - Hide Contents Date: Sun, 18 Jan 2004 05:41:19 Subject: Re: A new graph for Paul? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

Message: 9254 - Contents - Hide Contents Date: Sun, 18 Jan 2004 02:33:52 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Can you demonstrate how to get length log(9) out >>> of 9/5? >>>> 9/5 is a ratio of 9. >> I meant on the lattice.Yes, that's how this 'lattice' is defined, isn't it?>> OK, which part were we talking about? >> You were looking for Paul Hahn's algorithm, which is > like the 2nd or 3rd message in there. It isn't that > long in any case.OK -- that's the algorithm when each consonance in a given odd-limit is given a rung of length 1. So going back to the above, if the given odd-limit is less than 9, 9/5 will have to be constructed out of 3 and 3/5, thus has a length of 2, per Paul Hahn's lattice. No logs get involved there.

Message: 9255 - Contents - Hide Contents Date: Sun, 18 Jan 2004 16:52:54 Subject: Re: TOP on the web From: Carl Lumma>Monzos: |...> >Vals: <...| >Linear temperament wedgies: <<...|| >Planar temperament wedgies: <<<...|||Thank GOD you wrote this. Where's monz?>It makes sense to use <...| also for tuning maps, but I didn't above, >as I'm not sure if doing so would sow confusion.How could it sow confusion? I didn't know there was a difference between a map a some number of vals. -Carl

Message: 9256 - Contents - Hide Contents Date: Sun, 18 Jan 2004 02:38:57 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> The two obvious variations are rectangular odd-limit >>>> How can odd-limit be rectangular? Makes no sense to me. >> One can certainly have a rectangular lattice with a 9-axis.A 'lattice'-like thing, yes. But then it has nothing to do with odd- limit. And is there a 2-axis too?>>> and triangular octave-specific. >>>> Then the metric is not log(n*d) anymore. >> We actually haven't specified how to find the lengths of > rungs like 9:5...True, but if you use something different from what Tenney gives, you'll be hard pressed to get all the consonant intervals within a given range (say, 260-500 cents) in the correct order of consonance.

Message: 9257 - Contents - Hide Contents Date: Sun, 18 Jan 2004 06:52:02 Subject: Re: A new graph for Paul? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> <1 -3| 125/128 >> >> 135/128 >> Gene and all, > > What if, instead of issuing a correction post like this, > we were to post a full corrected version and delete the > original from the archives? Posterity may thank us...Not a bad plan; I do wonder how it works for people who read via email.

Message: 9258 - Contents - Hide Contents Date: Sun, 18 Jan 2004 06:54:16 Subject: Re: A new graph for Paul? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Let me get back to this after we're done talking about error > functions and the metrics of their duals.OK. Here's something to note when you do--these lines can go on for quite a ways including temperaments of reasonably low badness. Here is my line, each temperament of which differs from the next by meantone, meaning the successive differences are <1 4|: <0 -7| 2187/2048 <1 -3| 135/128 <2 1| 25/24 <3 5| 250/243 <4 9| 20000/19683 <5 13| 1600000/1594323 <6 17| 129140163/128000000

Message: 9259 - Contents - Hide Contents Date: Sun, 18 Jan 2004 19:02:28 Subject: Re: Annotated Dave Keenan file From: Carl Lumma>>> >ingle chain: No. generators in >>> Min Min interval >>> Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6 >>> (+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7 >>> ------------------------------------------------------------- >>> 125c 6 12.2c 17.9c -4 3 -7 -2 -5 2 >>> tertiathirds >>>> Why isn't this negri? >>Interesting question. This is the only 7-limit version of negri with >a badness score which is much good, so using the "reasonable tuning" >criterion perhaps it should be. // >However, <4 -3 -17 -14 -38 -31| is closer, <4 -3 21 -14 22 57| much >closer yet, and <4 -3 40 -14 52 101| has the identical TOP tuning.The TOP tuning of what?>What to do?I don't get it. Paul's temperament database doesn't list tertiathirds so I don't know what comma(s) tertiathirds was based on. If it's previously been a 5-limit linear temperament I don't see how it could have <-4 3] the same as negri. -Carl

Message: 9260 - Contents - Hide Contents Date: Sun, 18 Jan 2004 02:51:03 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> Can you demonstrate how to get length log(9) out >>>>> of 9/5? >>>>>>>> 9/5 is a ratio of 9. >>>>>> I meant on the lattice. >>>> Yes, that's how this 'lattice' is defined, isn't it? >> I was asking for any way it could be defined to make it > equal odd-limit, but this seems like cheating because > you require odd-limit infinity, and thus you're never > taking any multi-stop routes.OK -- but without 'cheating', how can one do in the octave-equivalent case what Tenney does in the octave-specific case?>>>> OK, which part were we talking about? >>>>>> You were looking for Paul Hahn's algorithm, which is >>> like the 2nd or 3rd message in there. It isn't that >>> long in any case. >>>> OK -- that's the algorithm when each consonance in a given >> odd-limit is given a rung of length 1. > > Right. >>> So going back to the above, if the given >> odd-limit is less than 9, 9/5 will have to be constructed out of 3 >> and 3/5, thus has a length of 2, per Paul Hahn's lattice. No logs >> get involved there. >> Right. It's easy. But it doesn't correspond to the "ratio-of" > the ratio.Right -- no logs, so no log(odd-limit) or log("ratio-of").> My point, if any, is that I think this will be impossible > with odd-limit < inf. on a triangular lattice.Well, that's exactly what this: lattice orientation * [with cont.] (Wayb.) was attempting to address, at least for a prime limit of 5.

Message: 9261 - Contents - Hide Contents Date: Sun, 18 Jan 2004 02:58:17 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> I'm fishing for something we can use to weed down the >> number of "lattices" we're interested in. Am I correct >> that you think log(odd-limit) is the best octave-equivalent >> concordance heuristic, >> That or something very similar to it, like perhaps > log(2*odd-limit - 1) > or > log(2*odd-limit + 1) > etc. >>> and that it constitutes a norm >> on the triangular odd-limit lattice with log weighting? >> Technically, it can't, because you don't have uniqueness, etc.Here's something you might try. Take every consonance in a given odd limit, expressed as a monzo. Multiply this by the log of the odd limit of that consonance. In this way, get a collection of points, and take the convex hull. This gives a convex solid containing the origin, and therefore defines a metric in the usual way, where the usual way is to call this the ball of radius one, and then find the norm of a point by scaling the ball so that the point is on the boundry; the scale factor is the norm.

Message: 9262 - Contents - Hide Contents Date: Sun, 18 Jan 2004 07:32:48 Subject: Warping along a line From: Gene Ward Smith Suppose I have a piece in four-part, mostly 5-limit harmony in porcupine, and another such piece in meantone. Suppose I take each part in the porcupine piece, calculate the generator, and add it to the generator of the meantone piece, then consider this to be a generator of something in tetracot, and find a suitable octave to go with the generator to get a note somewhere in the region of the average of my other two notes. What does this give us? If the two chords are major triads and the root, thirds and fifths correspond for the two pieces, then the result will be a major triad with the same arrangement of root, thirds and fifths. The same is also true if both chords are minor triads. If one chord is major and the other is minor, then the result will be the neutral thirds triad which is one of the features of tetracot. If the chords are in a different inversion, then results will vary, but an alternative approach would be to match root, third and fifth before doing the addition. This is an example of what sort of warping could be done; a mutant tetracot piece obtained by crossing a porcupine with a meantone piece. I think it has possibilities. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9263 - Contents - Hide Contents Date: Sun, 18 Jan 2004 03:07:12 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: >>>> I'm fishing for something we can use to weed down the >>> number of "lattices" we're interested in. Am I correct >>> that you think log(odd-limit) is the best octave-equivalent >>> concordance heuristic, >>>> That or something very similar to it, like perhaps >> log(2*odd-limit - 1) >> or >> log(2*odd-limit + 1) >> etc. >>>>> and that it constitutes a norm >>> on the triangular odd-limit lattice with log weighting? >>>> Technically, it can't, because you don't have uniqueness, etc. >> Here's something you might try. Take every consonance in a given odd > limit, expressed as a monzo. Multiply this by the log of the odd limit > of that consonance. In this way, get a collection of points, and take > the convex hull. This gives a convex solid containing the origin, and > therefore defines a metric in the usual way, where the usual way is to > call this the ball of radius one, and then find the norm of a point by > scaling the ball so that the point is on the boundry; the scale factor > is the norm.I'm interested in this approach. Also (NB Carl), these alternatives to log(odd-limit) don't work:>> log(2*odd-limit - 1) >> or >> log(2*odd-limit + 1)since the length for 1-limit ratios must be log(1)=0 on the lattice. But I still wonder whether anything else might make sense here.

Message: 9264 - Contents - Hide Contents Date: Sun, 18 Jan 2004 03:11:05 Subject: Re: Duals to ems optimization From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Now what if we apply 'odd-limit-weighting' to each of the intervals, > including 9:3 which is treated as having an odd-limit of 9? Try > using 'odd-limit' plus-or-minus 1 or 1/2 too.Is the weighting by multiplying or dividing by the log of the odd limit? Presumably mutliplying will make more sense. Do we square and then multiply, since we will be taking square roots?>> I think my idea of using >> the dual norm to my "geometric" norm makes more sense. >> Why is that?It's more or less reasonable to start with. We have ||3/2|| = log2(3), ||9/8|| = 2log2(3), ||5/4|| = log2(5), ||6/5|| = log2(5), ||7/6|| = ||7/5|| = ||7/4|| = log2(7), ||11/6|| = ||11/7|| = ||11/8|| = ||11/10|| = log2(11), ||9/5|| = sqrt(2log2(3)^2 + log2(5)^2) ||9/7|| = sqrt(2log2(3)^2 + log2(7)^2) ||11/9|| = sqrt(2log2(3)^2 + log2(11)^2) which isn't too bad.

Message: 9265 - Contents - Hide Contents Date: Sun, 18 Jan 2004 03:12:37 Subject: Re: A new graph for Paul? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> No, it would seem you need 4 integers to specify the mapping. Oh, > you're throwing out the period. Seems dirty to do that. Dirty??>>> What does that line mean? >>>> That's to be explored. We have, along a line, >> >> porcupine-->meantone-->tetracot-->amity >> >> In terms of generator mapping, but not the period part of the map, > and>> therefore not in terms of the generators in cents, we can transform >> one to the next continuously. That's the sort of thing I think is >> worth thiking about from a compositional point of view, for >> starters. >> What can you do with it?I just said.

Message: 9266 - Contents - Hide Contents Date: Sun, 18 Jan 2004 03:18:43 Subject: Re: Duals to ems optimization From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> Now what if we apply 'odd-limit-weighting' to each of the intervals, >> including 9:3 which is treated as having an odd-limit of 9? Try >> using 'odd-limit' plus-or-minus 1 or 1/2 too. >> Is the weighting by multiplying or dividing by the log of the odd > limit? Presumably mutliplying will make more sense.Divide. As in TOP, errors of more complex intervals are divided by larger numbers.> Do we square and > then multiply, since we will be taking square roots?No, we want to apply the weighting directly to the errors, before deciding how overall error is calculated from the individual weighted errors.>>> I think my idea of using >>> the dual norm to my "geometric" norm makes more sense. >>>> Why is that? >> It's more or less reasonable to start with. We have > > ||3/2|| = log2(3), ||9/8|| = 2log2(3), ||5/4|| = log2(5), > ||6/5|| = log2(5), ||7/6|| = ||7/5|| = ||7/4|| = log2(7), > ||11/6|| = ||11/7|| = ||11/8|| = ||11/10|| = log2(11), > ||9/5|| = sqrt(2log2(3)^2 + log2(5)^2) > ||9/7|| = sqrt(2log2(3)^2 + log2(7)^2) > ||11/9|| = sqrt(2log2(3)^2 + log2(11)^2) > > which isn't too bad.OK, but I think my proposal above will make even more sense than this. The two should agree for 7-odd-limit and below.

Message: 9267 - Contents - Hide Contents Date: Sun, 18 Jan 2004 00:11:23 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> The Thing I was referring to here was most certainly rectangular. >>> >>> -Carl >>>> Well then it's no Thing that I've ever thought about or talked about >> or heard of before! >> This was a different thing from our thread.You were talking about odd-limit thing: Yahoo groups: /tuning-math/message/8662 * [with cont.] When and where did you switch to a rectangular thing?

Message: 9268 - Contents - Hide Contents Date: Sun, 18 Jan 2004 03:23:43 Subject: Re: A new graph for Paul? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> No, it would seem you need 4 integers to specify the mapping. Oh, >> you're throwing out the period. Seems dirty to do that. > > Dirty??For one thing, the generator is not unique, and its multiplicity is proportional to periods per octave. For example, diaschismic can be understood as having, like meantone, a generator of a fourth, but its 'canonical' generator is sort of a minor second. Which do you use, and what's the rule to determine which?>>>> What does that line mean? >>>>>> That's to be explored. We have, along a line, >>> >>> porcupine-->meantone-->tetracot-->amity >>> >>> In terms of generator mapping, but not the period part of the map, >> and>>> therefore not in terms of the generators in cents, we can transform >>> one to the next continuously. That's the sort of thing I think is >>> worth thiking about from a compositional point of view, for >>> starters. >>>> What can you do with it? >> I just said.If you're not transforming continuously in terms of the generator in cents, what are you transforming continuously in terms of?

Message: 9269 - Contents - Hide Contents Date: Sun, 18 Jan 2004 00:46:15 Subject: Re: Question for Dave Keenan From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>>> What does "yes" mean here? >>>> the sound holds together as a single pitch. >> My guess is that it will be experienced as a single pitch, but one > that cannot be accurately determined. The pitch will be fuzzy or vague > in a similar way to that of a harmonic note of very short duration. Yahoo groups: /tuning_files/files/Erlich/dave.wav * [with cont.]

Message: 9270 - Contents - Hide Contents Date: Sun, 18 Jan 2004 01:17:55 Subject: A new graph for Paul? From: Gene Ward Smith Dual to the 5-limit symmetrical lattice of intervals is a 5-limit symmetrical lattice of vals whose first component is zero--which includes the generators in the period-generator of linear temperaments. The 3 axis and the 5 axis for intervals is 60 degrees apart; for a graph of the lattice of generators, |0 1> and |1 0> should be 120 degrees apart. There are interesting lines to draw on such a graph; the |-3 -5> of porcupine, |1 4> of meantone and |5 13> of amity lie along a line, for instance. Each generator can be graphed twice, by graphing +-|1 4>, etc. This would give us additional lines; the line between |-1 -4> and |-3 -5> includes pelogic at |1 -3>.

Message: 9271 - Contents - Hide Contents Date: Sun, 18 Jan 2004 03:37:49 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> The two obvious variations are rectangular odd-limit >>>>>>>> How can odd-limit be rectangular? Makes no sense to me. >>>>>> One can certainly have a rectangular lattice with a 9-axis. >>>> A 'lattice'-like thing, yes. >> If we're going to be going over to the mathematical definition > of lattice, we should come up with a term that means "anything > with rungs".A graph (as in graph-theory) but with lengths for each rung?>> But then it has nothing to do with odd-limit. And is there a >> 2-axis too? >> What would happen either way?If there is a 2-axis, a 9-axis in rectangular lattice seems superfluous (it doesn't change anything in terms of the taxicab distances you get, but adds an infinite number of copies of each pitch), unless you have a reason for treating '9' as different from '3*3' (and therefore '9/3' different than '3'), etc., such as a constraint to 768-equal partials. If there is no 2-axis, you get bad consonance evaluation, for the usual reasons.> So summing up, can we say that we're happy with our > octave-specific concordance heuristic and associated > lattice/metric, and that we have an octave-equivalent > concordance heuristic but *no* associated lattice/metric?I'd prefer not to say 'concordance heuristic', but yes.

Message: 9272 - Contents - Hide Contents Date: Sun, 18 Jan 2004 03:37:28 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> I'm interested in this approach.I just found out Maple's convex hull finder only works in two dimensions, which seems terribly lame. Maybe Mathematica or Matlab can do better?

Message: 9273 - Contents - Hide Contents Date: Sun, 18 Jan 2004 01:44:19 Subject: Re: Question for Dave Keenan From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > Thanks for that. Sounds to come.Thanks. I've listened to them. Definitely single pitches. Can you tell us the relative amplitides of all the partials. And can we hear a sustained note around middle C.>> So the waveform is essentially sinusoidal? Why not use sinusoidal >> waves for this thought experiment? >> They're not especially musical -- you'll have an easier time hearing > chords as sets of separate notes when the timbre is not a pure sine > wave.>>> The fact is that, when using inharmonic timbres of the sort I >>> described, Western music seems to retain all it meaning: certain >>> (dissonant) chords resolving to other (consonant) chords, etc., > all>>> sounds quite logical. My sense (and the opinion expressed in >>> Parncutt's book, for example) is that *harmony* is in fact very >>> closely related to the virtual pitch phenomenon. We already know, >>> from our listening tests on the harmonic entropy list, that the >>> sensory dissonance of a chord isn't a function of the sensory >>> dissonances of its constituent dyads. Furthermore, you seem to be >>> defining "something special" in a local sense as a function of >>> interval size, but in real music you don't get to evaluate each >>> sonority by detuning various intervals various amounts, which >>> this "specialness" would seem to require for its detection. >>> >>> The question I'm asking is, with what other tonal systems, > besides>>> the Western one, is this going to be possible in. >>>> If by "Western tonal systems", you mean any based on approximating >> small whole number ratios of frequency, >> No, I meant diatonic/meantone.OK. So is your question, "In what tonal systems other than diatonic/meantone is it going to be possible to have dissonant chords resolving to consonant chords?"? The obvious answer would seem to be systems in which there are consonant chords, i.e which approximate (or are) JI at least partially.>> What's your point? >> Did the above really not say anything to you?Certainly not until you clarified the above. And it might still be a good idea for you to spell out the conclusion you intend.

Message: 9274 - Contents - Hide Contents Date: Sun, 18 Jan 2004 03:40:34 Subject: Re: summary -- are these right? 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 interested in this approach. >> I just found out Maple's convex hull finder only works in two > dimensions, which seems terribly lame. Maybe Mathematica or Matlab can > do better? Yes:CONVHULLN N-D Convex hull. K = CONVHULLN(X) returns the indices K of the points in X that comprise the facets of the convex hull of X. X is an m-by-n array representing m points in n-D space. If the convex hull has p facets then K is p-by-n. [K,V] = CONVHULLN(X) also returns the volume of the convex hull in V. CONVHULLN is based on Qhull. See also CONVHULL, QHULL, DELAUNAYN, VORONOIN, TSEARCHN, DSEARCHN. QHULL Copyright information for Qhull. %%%% Qhull Copyright information copied from: %%%%%%%%%%%%%%%%%% %%%% Copyright notice for Geometry Center Software * [with cont.] (Wayb.) %%%% Copyright (c) 1993 The National Science and Technology Research Center for Computation and Visualization of Geometric Structures (The Geometry Center) University of Minnesota 400 Lind Hall 207 Church St. SE Minneapolis, MN 55454 USA email: software@xxxx.xxx.xxx The software distributed here is copyrighted as noted above. It is free software and may be obtained via anonymous ftp from ftp.geom.umn.edu. It may be freely copied, modified, and redistributed under the following conditions: 1. All copyright notices must remain intact in all files. 2. A copy of this file (COPYING) must be distributed along with any copies that you redistribute; this includes copies that you have modified, or copies of programs or other software products that include this software. 3. If you modify this software, you must include a notice giving the name of the person performing the modification, the date of modification, and the reason for such modification. 4. When distributing modified versions of this software, or other software products that include this software, you must provide notice that the original source code may be obtained as noted above. 5. There is no warranty or other guarantee of fitness for this software, it is provided solely "as is". Bug reports or fixes may be sent to the email address above; the authors may or may not act on them as they desire. If you use an image produced by this software in a publication or presentation, we request that you credit the Geometry Center with a notice such as the following: Figures 1, 2, and 5-300 were generated with software written at the Geometry Center, University of Minnesota. %%%% End of Qhull Copyright notice %%%%

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