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Message: 6801 - Contents - Hide Contents Date: Mon, 05 May 2003 21:19:32 Subject: Re: Graphing chord connections in equal temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "paulhjelmstad" <paul.hjelmstad@u...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> I've put graphs of the triads in 7, 12, 19 and 22 equal, with an > edge>> drawn whenever the triads share an interval. Unfortunately Maple did >> not draw 19 and 22 in a way which shows the symmetry. The 7-et graph >> can also be thought of as a diatonic graph. >> >> I also put up a graph for tetrads in 12-et. Checking the >> characteristic polynomial of this, I find that there are 240 chord >> triangles for 12-et tetrads, where a chord triangle means three >> chords, each of which shares an interval with the other chords. >> >> The graphs can be found in the "chord connection graphs" album in > the>> "Photos" for this group. >> These are great! Since I can't read the labels on the nodes (too > small and faint) I need to ask you why there are 24 nodes apiece for > 12-et triads and 12-et tetrads. Thanks!!!seems obvious to me -- there are 24 5-limit triads, and 24 7-limit tetrads, approximated in 12-equal. the former are the conventional major and minor triads, the latter are the conventional dominant seventh and half-diminished seventh chords. none of these chords are of the "limited transposition" variety, so there are 12 of each type of triad (24 total) and 12 of each type of tetrad (24 total).

Message: 6803 - Contents - Hide Contents Date: Tue, 6 May 2003 16:10:07 Subject: Re: Doing 12-equal within 133-et From: Manuel Op de Coul Carl asked:

Message: 6805 - Contents - Hide Contents Date: Thu, 08 May 2003 09:42:12 Subject: Re: Doing 12-equal within 133-et From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:> > Carl asked:>> Does the .scl format support stretch/compression? >> IIRC the last pitch line is taken as the interval >> of equivalence, so instead of 2/1, we could give >> a cents value of 1197? > > Yes indeed!What would be really interesting would be allowing the cents values to occur in any order. By the way, I read somewhere that Scala can produce Csound score files. How is that done?

Message: 6806 - Contents - Hide Contents Date: Thu, 8 May 2003 11:50:51 Subject: Re: Doing 12-equal within 133-et From: Manuel Op de Coul Gene wrote:>What would be really interesting would be allowing the cents values to >occur in any order. Doesn't it? >By the way, I read somewhere that Scala can produce Csound score >files. How is that done?There's an example in cmd\cs-demo.cmd. Or maybe easier if you have a midi file that you want to convert to a tuned Csound score is to use midi2cs by Rüdiger Borrmann. There's a tip about it in tips.par. Manuel

Message: 6807 - Contents - Hide Contents Date: Mon, 12 May 2003 02:32:31 Subject: how come i never saw this before? From: wally paulrus %PDF-1.3 * [with cont.] (Wayb.) btw, what makes much more sense to me is to do multidimensional scaling based on a rationalization of the *interval matrix*, not the pitch matrix -- intervals are the objects of psychoacoustic consonance, not pitches. i've posted such multidimensional scaling results years ago. better yet would be to simply use the harmonic entropies of the intervals, which i think i did in a few posts as well. hmm, i'm getting an idea for a movie . . . --------------------------------- Do you Yahoo!? The New Yahoo! Search - Faster. Easier. Bingo. [This message contained attachments]

Message: 6808 - Contents - Hide Contents Date: Mon, 12 May 2003 14:12:32 Subject: Re: how come i never saw this before? From: Carl Lumma You're mentioned, with a link to sonic arts. You didn't know it?>btw, what makes much more sense to me is to do multidimensional >scaling based on a rationalization of the *interval matrix*, not >the pitch matrix Indeed. >i've posted such multidimensional scaling results years ago. You did? >better yet would be to simply use the harmonic entropies of the >intervals, which i think i did in a few posts as well.You're referring to the minimum pairwise entropy posts? Those were awesome. Harmonic entropy sort of makes the idea of rationalizing a scale irrelevant. On the other hand, starting with a *temperament*, it's useful to have a method for snapping it to the lattice in a simple way, as in TM reduction. There's sooo much publishable on tuning-math . . . . .

Message: 6809 - Contents - Hide Contents Date: Mon, 12 May 2003 21:36:34 Subject: Re: how come i never saw this before? From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> You're mentioned, with a link to sonic arts. You didn't know it? >>> btw, what makes much more sense to me is to do multidimensional >> scaling based on a rationalization of the *interval matrix*, not >> the pitch matrix > > Indeed. >>> i've posted such multidimensional scaling results years ago. > > You did?yeah, baby, yeah i sure did! back on the mills list, i believe . . .>>> better yet would be to simply use the harmonic entropies of the >> intervals, which i think i did in a few posts as well. >> You're referring to the minimum pairwise entropy posts?no, just multidimensional scaling solutions.> Harmonic entropy sort of makes the idea of rationalizing a scale > irrelevant. On the other hand, starting with a *temperament*, > it's useful to have a method for snapping it to the lattice in a > simple way, as in TM reduction.yeah but then you break a lot of the consonant connections. which makes this whole "rationalization" business look pretty unhealthy to me.

Message: 6810 - Contents - Hide Contents Date: Mon, 12 May 2003 14:52:07 Subject: Re: how come i never saw this before? From: Carl Lumma>>> >etter yet would be to simply use the harmonic entropies of the >>> intervals, which i think i did in a few posts as well. >>>> You're referring to the minimum pairwise entropy posts? >>no, just multidimensional scaling solutions.How should I go about finding those posts?>> Harmonic entropy sort of makes the idea of rationalizing a scale >> irrelevant. On the other hand, starting with a *temperament*, >> it's useful to have a method for snapping it to the lattice in a >> simple way, as in TM reduction. >

Message: 6811 - Contents - Hide Contents Date: Mon, 12 May 2003 23:15:18 Subject: Re: how come i never saw this before? From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> better yet would be to simply use the harmonic entropies of the >>>> intervals, which i think i did in a few posts as well. >>>>>> You're referring to the minimum pairwise entropy posts? >>>> no, just multidimensional scaling solutions. >> How should I go about finding those posts?search for "multidimensional scaling" on the tuning list here and in robert's mills tuning list archive.

Message: 6812 - Contents - Hide Contents Date: Tue, 13 May 2003 21:34:28 Subject: Efficicency and ambiguity From: Gene Ward Smith If I'm understanding Eytan Agmon's paper "Numbers and the Western Tone-System", he is interested in MOS in an equal temperament which are efficent and have at most one ambiguous interval. It seems to me that a 7 or 8 note MOS in Blackwood/15 (with 2/15 as a generator) or a 9 or 10 note MOS of Negri/19 (with a 2/19 generator) would qualify. Eytan claims to have a proof this is not so, but I don't see how he can in the face of such examples. Is there anyone following this business who thinks they could explain this? I suppose I should look the proof up before the library here closes for moving.

Message: 6813 - Contents - Hide Contents Date: Tue, 13 May 2003 23:09:53 Subject: Re: Efficicency and ambiguity From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> If I'm understanding Eytan Agmon's paper "Numbers and the Western > Tone-System", he is interested in MOS in an equal temperament which > are efficent and have at most one ambiguous interval. It seems to me that > a 7 or 8 note MOS in Blackwood/15 (with 2/15 as a generator) or a 9 or > 10 note MOS of Negri/19 (with a 2/19 generator) would qualify.Here are more examples: 15 or 16 notes with a 2/31 generator (Quartaminorthirds/31) 26 or 27 notes with a 2/53 generator 49 or 50 notes with a 2/99 generator But 2/odd is not the only possibility: 20 or 21 notes of Miracle/41 (4/41 generator.) 51 or 52 notes of Miracle/103 Then there are some nice ones fitting Eytan's conditions: 35 or 37 notes with 35/72 generator 41 or 43 notes with 41/84 generator

Message: 6814 - Contents - Hide Contents Date: Wed, 14 May 2003 19:39:19 Subject: Re: Efficicency and ambiguity From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:>> If I'm understanding Eytan Agmon's paper "Numbers and the Western >> Tone-System", he is interested in MOS in an equal temperament which >> are efficent and have at most one ambiguous interval. >> Not at most, exactly one. So your examples don't qualify. > > Manuelwhy on earth should there be exactly one ambiguous interval? it seems to me that musical academia has been staring for too long at its 7- out-of-12-equal navel.

Message: 6815 - Contents - Hide Contents Date: Wed, 14 May 2003 22:10:26 Subject: Re: Efficicency and ambiguity From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" > <manuel.op.de.coul@e...> wrote: > why on earth should there be exactly one ambiguous interval? it seems > to me that musical academia has been staring for too long at its 7- > out-of-12-equal navel.Good question. However, as we've seen, Eytan actually exterminates these with an extra assumption that the generator is to be an approximate fifth. If we solve 1/2+1/(4*n) = log2(3/2) for n, we get n = 2.94; the nearest interger solution and clearly by far the best is n = 3, leading to the 12-et. Needless to say, I am not convinced by all these assumptions, and wish Eytan had presented this as "here are some interesting conditions on scales, which lead to 12 as their solution".

Message: 6816 - Contents - Hide Contents Date: Wed, 14 May 2003 12:06:24 Subject: Re: Efficicency and ambiguity From: Manuel Op de Coul>If I'm understanding Eytan Agmon's paper "Numbers and the Western >Tone-System", he is interested in MOS in an equal temperament which >are efficent and have at most one ambiguous interval.Not at most, exactly one. So your examples don't qualify. Manuel

Message: 6817 - Contents - Hide Contents Date: Wed, 14 May 2003 22:28:01 Subject: Re: Efficicency and ambiguity From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" > <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" >>> why on earth should there be exactly one ambiguous interval? it seems >> to me that musical academia has been staring for too long at its 7- >> out-of-12-equal navel. > > Good question. However, as we've seen, Eytan actually exterminates > thesethese? what are these?

Message: 6818 - Contents - Hide Contents Date: Wed, 14 May 2003 23:13:03 Subject: Some Eytan candidates From: Gene Ward Smith By this I mean numbers 4n such that the standard vals for 2n-1 and 2n+1 add up to the standard val for 4n, and all three have badness scores below 2. All conditions except the requirement that generators be a fifth are satisfied, and if Paul can call something which isn't an approximate 5/4 a major third, it seems to me I can call something which isn't an approximate 3/2 a fifth. Fair is fair. Paul? Is this legitimate in your view, and if not, why not? 5-limit: 4, 8, 12 7-limit: 8, 12, 16, 32, 40, 56, 60, 72, 84, 224 11-limit: 16, 128, 176, 224, 320, 364, 460, 764, 1164 13-limit: 12, 16, 84, 176, 224, 320, 364, 624, 764, 1048 17-limit: 24, 224, 248, 296, 316, 320, 344, 364, 412, 460 19-limit: 24, 224, 248, 296, 316, 320, 344, 364, 412, 416, 460, 528, 768, 904, 1116

Message: 6819 - Contents - Hide Contents Date: Wed, 14 May 2003 23:03:07 Subject: Re: Efficicency and ambiguity From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> Good question. However, as we've seen, Eytan actually exterminates >> these >> these? what are these?Systems with an odd et n and 2/n as generator.

Message: 6820 - Contents - Hide Contents Date: Thu, 15 May 2003 00:30:28 Subject: Scala and Agmon diatonic systems From: Gene Ward Smith I loaded the 24-et, 13-note MOS with 13/24 generator into Scala to check whether it connected to any Arabic scales. I didn't find out, since when I tried to compare to the scales in my scl directory, which I got by donwloading and unzipping from the Scala site, it said that every one of the scales had a read error, then that it had encountered a bug and that I should send a bug report to Manuel. This is, among other things, the bug report. I did note, however, that it gave the following information about it: Scale is a Winograd/Gammer deep scale Scale is an Agmon diatonic system Rothenberg stability 77/78 Lumma stability 1/2

Message: 6821 - Contents - Hide Contents Date: Thu, 15 May 2003 00:34:15 Subject: Re: Efficicency and ambiguity From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> Good question. However, as we've seen, Eytan actually exterminates >> these >> these? what are these?MOS of size (n+1)/2 within an n-et, such that n is odd and 2/n is the generator.

Message: 6822 - Contents - Hide Contents Date: Thu, 15 May 2003 03:57:29 Subject: Re: Scala and Agmon diatonic systems From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> I loaded the 24-et, 13-note MOS with 13/24 generator into Scala to > check whether it connected to any Arabic scales. I didn't find out, > since when I tried to compare to the scales in my scl directory, which > I got by donwloading and unzipping from the Scala site, it said that > every one of the scales had a read error, then that it had encountered > a bug and that I should send a bug report to Manuel. This is, among > other things, the bug report.I tried to do the same under W98, and this time the zip file would not upzip correctly.

Message: 6823 - Contents - Hide Contents Date: Thu, 15 May 2003 12:26:27 Subject: Re: Efficicency and ambiguity From: Manuel Op de Coul>> >hy on earth should there be exactly one ambiguous interval? it seems >> to me that musical academia has been staring for too long at its 7- >> out-of-12-equal navel. >> these? what are these?>MOS of size (n+1)/2 within an n-et, such that n is odd and 2/n is the >generator.Yeah, those have no ambiguous interval, leaving only the even n with exactly one ambiguous interval. Manuel

Message: 6824 - Contents - Hide Contents Date: Sat, 17 May 2003 04:59:35 Subject: Efficient MOS From: Gene Ward Smith Those with good memories may recall the notation n+m;s I introduced to describe MOS on ets. Here the n+m part defines the generator and et; if we assume n>m and d = gcd(n,m) then if q/r is the next element of the nth row of the Farey sequence after m/n, so that if u = m/d and v = n/d then u*r-v*q=1, the generator Gen(n,m) = (q+r)/(u+v), the period is 1/d, and the equal temperament is n+m. Then n+m;s means the s-element scale in the n+m-et with generators Gen(n,m) and 1/d. In terms of this notation, Eytan is considering 2*n+1+2*n-1;2*n+1. This suggests how to put things in a wider context, by looking at n+m;n for various values of n-m togeter with n mod n-m. Difference of one We get scales n+1+n;n+1, with et 2*n+1 and generator 2/(2*n+1) Differences of two (1) Scales 2*n+2+2*n;2*n+2 with generators [1/2, 1/(2*n+1)] in the 4*n+2 et (2) Scales 2*n+1+2*n-1;2*n+1 with generator (2*n+1)/(4*n) in the 4*n et; if you add to this the very strange requirement that the generator maps from 3/2, you get Eytan's diatonic systems Differences of three (1) Scales 3*n+3+3*n;3*n+3 with generators [1/3, 1/(2*n+1)] in the 6*n+3 et (2) Scales 3*n+2+3*n-1;3*n+2 with generator (2*n+1)/(6*n+1) in the 6*n+1 et (3) Scales 3*n+1+3*n-1;3*n+1 with generator (4*n)/(6*n-1) in the 6*n-1 et And so forth...

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