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Message: 2925 - Contents - Hide Contents Date: Tue, 01 Jan 2002 09:18:28 Subject: 7 and 10 note Magic scales From: genewardsmith These are 7-connected scales in the Magic temperament [5 1 12 25 -5 -10]. They are presented in terms of the 41-et, but are generic 7-limit Magic scales. It is striking that in both the 7 and 10 note cases, there is a non-MOS scale with the same connectivity as the MOS scale. 7 notes [0, 11, 13, 24, 26, 37, 39] [11, 2, 11, 2, 11, 2, 2] 2 [0, 11, 13, 15, 26, 37, 39] [11, 2, 2, 11, 11, 2, 2] 2 [0, 11, 13, 24, 26, 28, 30] [11, 2, 11, 2, 2, 2, 11] 1 10 notes [0, 9, 11, 13, 22, 24, 26, 35, 37, 39] [9, 2, 2, 9, 2, 2, 9, 2, 2, 2] 4 [0, 9, 11, 13, 15, 24, 26, 35, 37, 39] [9, 2, 2, 2, 9, 2, 9, 2, 2, 2] 4 [0, 9, 11, 13, 15, 24, 26, 28, 30, 32] [9, 2, 2, 2, 9, 2, 2, 2, 2, 9] 3 [0, 9, 11, 13, 22, 24, 26, 28, 30, 39] [9, 2, 2, 9, 2, 2, 2, 2, 9, 2] 3 [0, 9, 11, 13, 22, 24, 26, 28, 30, 32] [9, 2, 2, 9, 2, 2, 2, 2, 2, 9] 2 [0, 9, 11, 13, 15, 17, 19, 28, 30, 39] [9, 2, 2, 2, 2, 2, 9, 2, 9, 2] 2 [0, 9, 11, 13, 15, 17, 19, 21, 23, 32] [9, 2, 2, 2, 2, 2, 2, 2, 9, 9] 2 [0, 9, 11, 13, 15, 17, 19, 21, 30, 39] [9, 2, 2, 2, 2, 2, 2, 9, 9, 2] 1
Message: 2926 - Contents - Hide Contents Date: Tue, 01 Jan 2002 09:52:37 Subject: Nine tone Orwell scales From: genewardsmith Here are the 7-limit connectivities of the nine note Orwells--I wish I had known about these excellent non-MOS scales when I was doing my Orwell piece! So many scales, so little time ... The notation is the 53-et, where Orwell is 12/53. [0, 7, 12, 19, 24, 31, 36, 43, 48] [7, 5, 7, 5, 7, 5, 7, 5, 5] 3 [0, 7, 12, 19, 24, 29, 36, 43, 48] [7, 5, 7, 5, 5, 7, 7, 5, 5] 3 [0, 7, 12, 19, 24, 29, 36, 41, 46] [7, 5, 7, 5, 5, 7, 5, 5, 7] 3 [0, 7, 12, 17, 24, 31, 38, 43, 48] [7, 5, 5, 7, 7, 7, 5, 5, 5] 3 [0, 7, 12, 19, 24, 29, 34, 41, 46] [7, 5, 7, 5, 5, 5, 7, 5, 7] 2 [0, 7, 12, 19, 24, 29, 34, 39, 46] [7, 5, 7, 5, 5, 5, 5, 7, 7] 2 [0, 7, 12, 19, 26, 31, 36, 41, 46] [7, 5, 7, 7, 5, 5, 5, 5, 7] 2 [0, 7, 12, 17, 24, 31, 36, 41, 46] [7, 5, 5, 7, 7, 5, 5, 5, 7] 2 [0, 7, 12, 19, 24, 31, 36, 41, 46] [7, 5, 7, 5, 7, 5, 5, 5, 7] 2 [0, 7, 12, 17, 22, 27, 32, 39, 46] [7, 5, 5, 5, 5, 5, 7, 7, 7] 2
Message: 2927 - Contents - Hide Contents Date: Tue, 1 Jan 2002 18:17 +00 Subject: Re: Optimal 5-Limit Generators For Dave From: graham@xxxxxxxxxx.xx.xx genewardsmith@xxxx.xxx (genewardsmith) wrote:> You have an even and odd set of pitches, meaning an even or odd number > of generators to the pitch. You can't get from even to odd by way of > consonant 7-limit intervals, so basically we have two unrelated > meantone systems a half-fifth or half-fourth apart. You can always glue > together two unrelated systems and call it a temperament, and this > differs only because it does have a single generator.These are the [2 8] systems. There is some ambiguity, but if you mean the half-fifth system, isn't that Vicentino's enharmonic? That's 31&24 or [(1, 0), (1, 2), (0, 8)]. Two meantone scales, only 5-limit consonances recognize, but neutral intervals used in melody. It may not be a temperament, but does have a history of both theory and music, so don't write it off so lightly. The half-fifth system is 24&19 or [(1, 0), (2, -2), (4, -8)]. There's also a half-octave system, [(2, 0), (3, 1), (4, 4)]. That's the one my program would deduce from the octave-equivalent mapping [2 8]. If I had such a program. If anybody cares, is it possible to write one? Where torsion's present, we'll have to assume it means divisions of the octave for uniqueness. Gene said it isn't possible, but I'm not convinced. How could [1 4] be anything sensible but meantone? Perhaps the first step is to find an interval that's only one generator step, take the just value, period-reduce it and work everything else out from that. But there may be some cases where the optimal value should cross a period boundary. But if we could get the periodicity block in pitch-order, we could reconstruct an equal-tempered mapping and get all the information the wedge product gives us. Can we do that? Anybody? If you think it can't be done, show a counter-example: an octave-equivalent mapping without torsion that can lead to two different but equally good temperaments. Graham
Message: 2928 - Contents - Hide Contents Date: Tue, 01 Jan 2002 20:42:20 Subject: 12 note, 225/224 planar temperament scales From: genewardsmith These are, of course, expressed in terms of the 72-et. They have three step sizes, which are 2, 5, and 7 in the 72-et, so this is closely related to Miracle, as one would expect. They are sorted in terms of number of edges (consonant intervals) and then connectivity. [0, 5, 12, 19, 21, 28, 35, 42, 49, 51, 58, 65] [5, 7, 7, 2, 7, 7, 7, 7, 2, 7, 7, 7] edges 34 connectivity 4 [0, 5, 12, 19, 21, 28, 35, 42, 49, 56, 58, 65] [5, 7, 7, 2, 7, 7, 7, 7, 7, 2, 7, 7] edges 34 connectivity 4 [0, 5, 12, 14, 21, 28, 35, 42, 49, 51, 58, 65] [5, 7, 2, 7, 7, 7, 7, 7, 2, 7, 7, 7] edges 33 connectivity 4 [0, 5, 12, 19, 26, 28, 35, 42, 49, 51, 58, 65] [5, 7, 7, 7, 2, 7, 7, 7, 2, 7, 7, 7] edges 33 connectivity 3 [0, 5, 7, 14, 21, 28, 35, 42, 49, 56, 63, 65] [5, 2, 7, 7, 7, 7, 7, 7, 7, 7, 2, 7] edges 32 connectivity 5 [0, 5, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70] [5, 2, 7, 7, 7, 7, 7, 7, 7, 7, 7, 2] edges 32 connectivity 5 [0, 5, 7, 14, 21, 28, 35, 42, 44, 51, 58, 65] [5, 2, 7, 7, 7, 7, 7, 2, 7, 7, 7, 7] edges 32 connectivity 4 [0, 5, 7, 14, 21, 28, 35, 42, 49, 51, 58, 65] [5, 2, 7, 7, 7, 7, 7, 7, 2, 7, 7, 7] edges 32 connectivity 4 [0, 5, 7, 14, 21, 28, 35, 42, 49, 56, 58, 65] [5, 2, 7, 7, 7, 7, 7, 7, 7, 2, 7, 7] edges 32 connectivity 4 [0, 5, 12, 14, 21, 28, 35, 42, 44, 51, 58, 65] [5, 7, 2, 7, 7, 7, 7, 2, 7, 7, 7, 7] edges 32 connectivity 4 [0, 5, 12, 14, 21, 28, 35, 42, 49, 56, 58, 65] [5, 7, 2, 7, 7, 7, 7, 7, 7, 2, 7, 7] edges 32 connectivity 4 [0, 5, 12, 19, 21, 28, 35, 42, 44, 51, 58, 65] [5, 7, 7, 2, 7, 7, 7, 2, 7, 7, 7, 7] edges 32 connectivity 3 [0, 5, 12, 14, 21, 28, 35, 42, 49, 56, 63, 65] [5, 7, 2, 7, 7, 7, 7, 7, 7, 7, 2, 7] edges 31 connectivity 4 [0, 5, 7, 14, 21, 28, 35, 37, 44, 51, 58, 65] [5, 2, 7, 7, 7, 7, 2, 7, 7, 7, 7, 7] edges 31 connectivity 4 [0, 5, 12, 14, 21, 28, 35, 37, 44, 51, 58, 65] [5, 7, 2, 7, 7, 7, 2, 7, 7, 7, 7, 7] edges 30 connectivity 3 [0, 5, 7, 14, 21, 28, 30, 37, 44, 51, 58, 65] [5, 2, 7, 7, 7, 2, 7, 7, 7, 7, 7, 7] edges 30 connectivity 3 [0, 5, 12, 19, 26, 28, 35, 42, 44, 51, 58, 65] [5, 7, 7, 7, 2, 7, 7, 2, 7, 7, 7, 7] edges 29 connectivity 2 [0, 5, 7, 14, 21, 23, 30, 37, 44, 51, 58, 65] [5, 2, 7, 7, 2, 7, 7, 7, 7, 7, 7, 7] edges 29 connectivity 2 [0, 5, 12, 19, 21, 28, 35, 37, 44, 51, 58, 65] [5, 7, 7, 2, 7, 7, 2, 7, 7, 7, 7, 7] edges 28 connectivity 3 [0, 5, 7, 9, 16, 23, 30, 37, 44, 51, 58, 65] [5, 2, 2, 7, 7, 7, 7, 7, 7, 7, 7, 7] edges 28 connectivity 1 [0, 5, 7, 14, 16, 23, 30, 37, 44, 51, 58, 65] [5, 2, 7, 2, 7, 7, 7, 7, 7, 7, 7, 7] edges 28 connectivity 1 [0, 5, 12, 14, 21, 28, 30, 37, 44, 51, 58, 65] [5, 7, 2, 7, 7, 2, 7, 7, 7, 7, 7, 7] edges 27 connectivity 2 [0, 5, 12, 14, 16, 23, 30, 37, 44, 51, 58, 65] [5, 7, 2, 2, 7, 7, 7, 7, 7, 7, 7, 7] edges 24 connectivity 1 [0, 5, 12, 14, 21, 23, 30, 37, 44, 51, 58, 65] [5, 7, 2, 7, 2, 7, 7, 7, 7, 7, 7, 7] edges 24 connectivity 1 [0, 5, 12, 19, 21, 28, 30, 37, 44, 51, 58, 65] [5, 7, 7, 2, 7, 2, 7, 7, 7, 7, 7, 7] edges 23 connectivity 2 [0, 5, 12, 19, 26, 28, 35, 37, 44, 51, 58, 65] [5, 7, 7, 7, 2, 7, 2, 7, 7, 7, 7, 7] edges 23 connectivity 1 [0, 5, 12, 19, 26, 33, 35, 42, 44, 51, 58, 65] [5, 7, 7, 7, 7, 2, 7, 2, 7, 7, 7, 7] edges 23 connectivity 1 [0, 5, 12, 19, 21, 23, 30, 37, 44, 51, 58, 65] [5, 7, 7, 2, 2, 7, 7, 7, 7, 7, 7, 7] edges 21 connectivity 1 [0, 5, 12, 19, 26, 28, 30, 37, 44, 51, 58, 65] [5, 7, 7, 7, 2, 2, 7, 7, 7, 7, 7, 7] edges 19 connectivity 1 [0, 5, 12, 19, 26, 33, 35, 37, 44, 51, 58, 65] [5, 7, 7, 7, 7, 2, 2, 7, 7, 7, 7, 7] edges 18 connectivity 1
Message: 2929 - Contents - Hide Contents Date: Tue, 01 Jan 2002 21:11:14 Subject: Re: Optimal 5-Limit Generators For Dave From: genewardsmith --- In tuning-math@y..., graham@m... wrote: There is some ambiguity, but if you mean the> half-fifth system, isn't that Vicentino's enharmonic?I thought Vicentino was 31-et.>Gene said it isn't possible, but I'm not convinced. How > could [1 4] be anything sensible but meantone?Actually, I said normally there will be a clear best choice.
Message: 2930 - Contents - Hide Contents Date: Tue, 1 Jan 2002 21:45:47 Subject: Re: Optimal 5-Limit Generators For Dave From: Graham Breed Me:>> There is some ambiguity, but if you mean the >> half-fifth system, isn't that Vicentino's enharmonic? Gene:> I thought Vicentino was 31-et.He never actually says it's equally tempered. Only that the chromatic semitone is divided into 2 dieses, which follows from the perfect fifth being divided into two equal neutral thirds. Although he does say the usual diesis (the difference between a diatonic and chromatic semitone) can be treated equivalent to the other one, the tuning seems to be two meantone chains, corresponding to the two keyboards. The musical examples can all be understood as two meantone chains. He does obscure this by writing a Gb as F#, but each chord falls entirely on one keyboard. And they're all normal meantone chords. So the music is fully described by the meantone-with-neutral-thirds temperament. Although he mentions, briefly, that he considers neutral thirds as consonant and they may even be sung in contemporaneous music, he doesn't use them himself in chords. And he doesn't quite give the 11-limit interpretation. Graham
Message: 2931 - Contents - Hide Contents Date: Tue, 01 Jan 2002 22:06:34 Subject: Re: Optimal 5-Limit Generators For Dave From: genewardsmith --- In tuning-math@y..., Graham Breed <graham@m...> wrote: Although he> mentions, briefly, that he considers neutral thirds as consonant and they may > even be sung in contemporaneous music, he doesn't use them himself in chords. > And he doesn't quite give the 11-limit interpretation.If neutral thirds are consonant we are not talking about the 5-limit and the entire argument is moot.
Message: 2932 - Contents - Hide Contents Date: Tue, 1 Jan 2002 14:50:38 Subject: Transformation From: monz A while back (Tue Oct 24, 2000 1:34 am, to be exact), I posted to the tuning list a _Journal of Music Theory_ review of Eric Regener's 1973 book _Pitch notation and Equal Temperament: A Formal Study_: Yahoo groups: /tuning/message/14995 * [with cont.] I asked a question about the math here: Yahoo groups: /tuning/message/15054 * [with cont.] Paul's explanation of the math, in answer to that question, is here: Yahoo groups: /tuning/message/15059 * [with cont.] Now I'm making a Tuning Dictionary entry for "Transformation", and using one of Regener's examples. Definitions of tuning terms: transformation, (... * [with cont.] (Wayb.) This is the example which reinterprets coordinates (d,q) in interval space "I", in terms of (a,b) in interval space K_0. So the coordinates d(1,0) + q(0,1) define interval space "I", and the coordinates a(1,0) + b(0,1) define interval space "K_0". Their relationship is as follows:> The transformation equation from interval space 'I' > to interval space K_0, according to Chrisman, would be: > > d(1,0) + q(0,1) = a(3,-1) + b(1,2) so > (d,q) = (3a+b,-a+2b) and > d = 3a + b > q = -a + 2bBut isn't that wrong? I can see from a diagram that some of the signs should be reversed, and the answer should be: d = 3a - b q = a + 2b ???? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2933 - Contents - Hide Contents Date: Wed, 02 Jan 2002 02:36:03 Subject: Re: Some 10-tone, 72-et scales From: clumma Gene, Interested in calculating the 7-limit edge connectivity of Paul's decatonic scales in 22-tET? Just so I'm straight, this is the least number of connections, over every pitch in the scale, that the given pitch has with any other pitch in the scale, right? -Carl
Message: 2934 - Contents - Hide Contents Date: Wed, 02 Jan 2002 04:59:47 Subject: Some classes of 72-et scales From: genewardsmith Here is the result of a search for other possibilities for 72-et scales, which start from the premise that in the 7-limit they work as 225/224 planar temperament scales. I've already done the 10-note example and one of the 12-note examples. The first list is a list of scale step sizes in 72-et steps; below that is the multiplicity of each step size. The number of steps is of course the sum of these multiplicities. Note that in the 22-note case, there are two "diferent" steps of size 3/72; in the corresponding planar temperament these actually are a little different in the rms optimized tuning. 4 notes [7, 19, 23] [1, 1, 2] 5 notes [7, 12, 23] [2, 1, 2] 8 notes [5, 7, 16] [1, 5, 2] 9 notes [7, 12, 9] [6, 1, 2] 10 notes [7, 5, 9] [7, 1, 2] 11 notes [2, 5, 9] [1, 5, 5] 12 notes [6, 1, 11] [8, 2, 2] [7, 5, 2] [9, 1, 2] [6, 1, 11] [2, 5, 5] [7, 5, 6] [5, 5, 2] [6, 7, 5] [8, 2, 2] 15 notes [1, 7, 10] [8, 2, 5] [5, 2, 11] [8, 5, 2] [7, 4, 1] [9, 1, 5] [3, 7, 6] [8, 6, 1] 16 notes [2, 5, 7] [6, 5, 5] [6, 1, 9] [8, 6, 2] 17 notes [5, 2, 9] [8, 7, 2] [6, 1, 5] [7, 5, 5] 19 notes [5, 2, 7] [8, 9, 2] [3, 4, 5] [9, 5, 5] 20 notes [1, 7, 3] [8, 7, 5] 21 notes [3, 1, 6] [8, 6, 7] 22 notes [3, 3, 4] [8, 8, 6]
Message: 2935 - Contents - Hide Contents Date: Wed, 02 Jan 2002 05:08:44 Subject: Re: Some 10-tone, 72-et scales From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> Gene, > > Interested in calculating the 7-limit edge connectivity > of Paul's decatonic scales in 22-tET?That was the first example I did, but it might be interesting to do something with three step sizes in 22-et like the stuff I've been working out in 72-et.> Just so I'm straight, this is the least number of > connections, over every pitch in the scale, that the > given pitch has with any other pitch in the scale, right?It is how many edges (representing consonant intervals) would need to be removed in order to render the scale disconnected; very often this will be the same.
Message: 2936 - Contents - Hide Contents Date: Wed, 02 Jan 2002 05:36:43 Subject: Re: Some 10-tone, 72-et scales From: clumma>That was the first example I did,Found it. I don't see any other scales c=6 in the 7-limit, and only the 225:224 stuff has been up to c=5.>It is how many edges (representing consonant intervals) would >need to be removed in order to render the scale disconnected; >very often this will be the same. Cool. -C.
Message: 2937 - Contents - Hide Contents Date: Wed, 02 Jan 2002 06:29:40 Subject: tetrachordality From: clumma Paul, My current model works like this: pentachordal (0 109 218 382 491 600 709 873 982 1091) (1193 102 211 375 484 593 702 811 920 1084) 7 7 7 7 7 7 7 62 62 7 symmetrical (0 109 218 382 491 600 709 818 982 1091) (1193 102 211 320 484 593 702 811 920 1084) 7 7 7 62 7 7 7 7 62 7 So obviously, these two scales will come out the same. But you've view -- and I remember doing some listening experiments that back you up (the low efficiency of the symmetrical version was the other theory there) -- is that the symmetrical version is not tetrachordal. So what's going on here? Where's the error in tetrachordality = similarity at transposition by a 3:2? -Carl
Message: 2938 - Contents - Hide Contents Date: Wed, 02 Jan 2002 08:46:50 Subject: Some 9-tone 72-et scales From: genewardsmith The first three numbers on the third row are number of edges in the 5, 7, and 11 limits, and the second connectivity in the 5, 7, and 11 limits. [0, 7, 14, 21, 30, 37, 44, 56, 63] [7, 7, 7, 9, 7, 7, 12, 7, 9] 12 22 29 1 4 6 [0, 7, 14, 26, 33, 40, 49, 56, 63] [7, 7, 12, 7, 7, 9, 7, 7, 9] 12 21 29 1 3 6 [0, 7, 14, 21, 33, 40, 49, 56, 63] [7, 7, 7, 12, 7, 9, 7, 7, 9] 11 20 28 1 3 6 [0, 7, 14, 21, 33, 42, 49, 56, 63] [7, 7, 7, 12, 9, 7, 7, 7, 9] 10 20 28 1 3 5 [0, 7, 14, 21, 33, 40, 47, 56, 63] [7, 7, 7, 12, 7, 7, 9, 7, 9] 9 17 28 0 2 5 [0, 7, 14, 21, 28, 37, 44, 56, 63] [7, 7, 7, 7, 9, 7, 12, 7, 9] 10 21 27 0 3 5 [0, 7, 14, 21, 28, 37, 44, 51, 63] [7, 7, 7, 7, 9, 7, 7, 12, 9] 9 20 27 0 2 5 [0, 7, 14, 21, 28, 40, 49, 56, 63] [7, 7, 7, 7, 12, 9, 7, 7, 9] 8 18 26 0 2 5 [0, 7, 14, 21, 28, 40, 47, 56, 63] [7, 7, 7, 7, 12, 7, 9, 7, 9] 7 16 26 0 2 5 [0, 7, 14, 21, 33, 40, 47, 54, 63] [7, 7, 7, 12, 7, 7, 7, 9, 9] 6 12 26 0 1 5 [0, 7, 14, 21, 28, 35, 44, 51, 63] [7, 7, 7, 7, 7, 9, 7, 12, 9] 7 18 25 0 2 4
Message: 2939 - Contents - Hide Contents Date: Wed, 2 Jan 2002 10:11 +00 Subject: Re: Optimal 5-Limit Generators For Dave From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a0tbta+beck@xxxxxxx.xxx> Me:> Although he>> mentions, briefly, that he considers neutral thirds as consonant and >> they may even be sung in contemporaneous music, he doesn't use them >> himself in chords. And he doesn't quite give the 11-limit >> interpretation. Gene:> If neutral thirds are consonant we are not talking about the 5-limit > and the entire argument is moot.Neutral thirds are not consonant in Vicentino's enharmonic genus. If you wish to disagree, give specific references to the musical examples. On reviewing this, I think I should draw a distinction between the enharmonic system and the tuning of the archicembalo. The former has a minor diesis equal to half a chromatic semitone, and a major diesis equal to a chromatic semitone less a minor diesis. In the latter, the minor diesis is equal to the difference between a diatonic and chromatic semitone, and the major diesis is equal to the chromatic semitone. Vicentino starts off noting this difference, but doesn't always make it strict in the notation. The reference to neutral thirds being consonant is in the book on the archicembalo. The books on the diatonic, chromatic and enharmonic genera only recognize strict 5-limit vertical harmony. Also, although he does mention somewhere that the whole tone divides into five roughly equal parts, in the examples of the enharmonic genus he only divides it into four. So the system, but not always the notation, is fully consistent with a quartertone scale. Hence 24&31. In Book I of Music Practice, he's strict about this in the divisions of the whole tone and examples of the different dieses, but not when he introduces some of the derived intervals. It's here he says that the enharmonic dieses are "identical" to the extended meantone intervals on the archicembalo, and the one can stand in for the other for the sake of "compositional convenience". In Book III of Music Practice, he spells one of the enharmonic tetrachords such that the notation won't work in 24-equal, so must be ignoring the distinctions he made in Book I. Disclaimer: I'm writing this without the book to hand, but I did check the details last night. Graham
Message: 2940 - Contents - Hide Contents Date: Wed, 2 Jan 2002 10:20 +00 Subject: Re: Optimal 5-Limit Generators For Dave From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <memo.582096@xxx.xxxxxxxxx.xx.xx> Proof reading time> On reviewing this, I think I should draw a distinction between the > enharmonic system and the tuning of the archicembalo. The former has a > minor diesis equal to half a chromatic semitone, and a major diesis > equal to a chromatic semitone less a minor diesis. In the latter, the . ^^^^^^^^^That should be a *diatonic* semitone less a minor diesis. Graham
Message: 2941 - Contents - Hide Contents Date: Wed, 02 Jan 2002 11:43:42 Subject: Some 7-note, 7-limit scales From: genewardsmith These are the ones based on (16/15)^2 (15/14)^2 (9/8) (7/6)^2 = 2 [1, 16/15, 8/7, 4/3, 64/45, 32/21, 16/9] [16/15, 15/14, 7/6, 16/15, 15/14, 7/6, 9/8] edges 12 connectivity 2 [1, 16/15, 8/7, 4/3, 10/7, 32/21, 12/7] [16/15, 15/14, 7/6, 15/14, 16/15, 9/8, 7/6] edges 12 connectivity 2 [1, 16/15, 8/7, 4/3, 64/45, 8/5, 28/15] [16/15, 15/14, 7/6, 16/15, 9/8, 7/6, 15/14] edges 12 connectivity 1 [1, 16/15, 8/7, 4/3, 10/7, 5/3, 16/9] [16/15, 15/14, 7/6, 15/14, 7/6, 16/15, 9/8] edges 11 connectivity 2 [1, 16/15, 8/7, 4/3, 64/45, 8/5, 12/7] [16/15, 15/14, 7/6, 16/15, 9/8, 15/14, 7/6] edges 11 connectivity 1 [1, 16/15, 8/7, 4/3, 14/9, 5/3, 16/9] [16/15, 15/14, 7/6, 7/6, 15/14, 16/15, 9/8] edges 10 connectivity 2 [1, 16/15, 6/5, 32/25, 112/75, 8/5, 12/7] [16/15, 9/8, 16/15, 7/6, 15/14, 15/14, 7/6] edges 10 connectivity 2 [1, 16/15, 8/7, 9/7, 3/2, 8/5, 12/7] [16/15, 15/14, 9/8, 7/6, 16/15, 15/14, 7/6] edges 10 connectivity 1 [1, 16/15, 8/7, 128/105, 48/35, 8/5, 12/7] [16/15, 15/14, 16/15, 9/8, 7/6, 15/14, 7/6] edges 10 connectivity 1 [1, 16/15, 8/7, 9/7, 48/35, 8/5, 12/7] [16/15, 15/14, 9/8, 16/15, 7/6, 15/14, 7/6] edges 10 connectivity 1
Message: 2942 - Contents - Hide Contents Date: Wed, 2 Jan 2002 12:17 +00 Subject: Re: Some 10-tone, 72-et scales From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a0thh3+terr@xxxxxxx.xxx> Gene wrote:> I started out looking at these as 7-limit 225/224 planar temperament > scales, but decided it made more sense to check the 5 and 11 limits > also, and to take them as 72-et scales; if they are ever used that is > probably how they will be used. I think anyone interested in the 72-et > should take a look at the top three, which are all 5-connected, and the > top scale in particular, which is a clear winner. The "edges" number > counts edges (consonant intervals) in the 5, 7, and 11 limits, and the > connectivity is the edge-connectivity in the 5, 7 and 11 limits.Well, more than 72-equal, they're all Miracle consistent, aren't they? In which case, they're also all Blackjack subsets. But the decimal MOS isn't one of them. Interesting. Graham
Message: 2943 - Contents - Hide Contents Date: Wed, 2 Jan 2002 14:00 +00 Subject: Re: Some 9-tone 72-et scales From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a0uhdq+onbm@xxxxxxx.xxx> Can you get your program to check for propriety of these things? All these are strictly proper except for> [0, 7, 14, 21, 33, 42, 49, 56, 63] > [7, 7, 7, 12, 9, 7, 7, 7, 9] > 10 20 28 1 3 5 > [0, 7, 14, 21, 28, 37, 44, 56, 63] > [7, 7, 7, 7, 9, 7, 12, 7, 9] > 10 21 27 0 3 5 > [0, 7, 14, 21, 28, 37, 44, 51, 63] > [7, 7, 7, 7, 9, 7, 7, 12, 9] > 9 20 27 0 2 5 > [0, 7, 14, 21, 28, 40, 49, 56, 63] > [7, 7, 7, 7, 12, 9, 7, 7, 9] > 8 18 26 0 2 5 > [0, 7, 14, 21, 28, 40, 47, 56, 63] > [7, 7, 7, 7, 12, 7, 9, 7, 9] > 7 16 26 0 2 5which are proper but not strictly proper, and> [0, 7, 14, 21, 28, 35, 44, 51, 63] > [7, 7, 7, 7, 7, 9, 7, 12, 9] > 7 18 25 0 2 4which is improper. It's interesting that so many scales came out proper when that wasn't a criterion in the search. All the 10-note 72= scales are strictly proper. Graham
Message: 2944 - Contents - Hide Contents Date: Wed, 02 Jan 2002 23:08:21 Subject: Re: Some 10-tone, 72-et scales From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> Well, more than 72-equal, they're all Miracle consistent, aren't they? In > which case, they're also all Blackjack subsets. But the decimal MOS isn't > one of them. Interesting.The decimal MOS has two step sizes, and so belongs to a linear rather than a planar temperament, but perhaps I should run it through the evaluation process for the sake of comparison.
Message: 2945 - Contents - Hide Contents Date: Thu, 03 Jan 2002 09:05:55 Subject: Re: Some 10-tone, 72-et scales From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: Here by way of comparison is the 10-note Miracle MOS; it is relatively undistinguished in this company.> [0, 5, 12, 19, 28, 35, 42, 49, 58, 65] > [5, 7, 7, 9, 7, 7, 7, 9, 7, 7] > edges 15 27 35 connectivity 2 5 6 > > [0, 5, 12, 19, 28, 35, 42, 51, 58, 65] > [5, 7, 7, 9, 7, 7, 9, 7, 7, 7] > edges 14 25 35 connectivity 1 3 6 > > [0, 5, 12, 21, 28, 35, 42, 51, 58, 65] > [5, 7, 9, 7, 7, 7, 9, 7, 7, 7] > edges 13 25 35 connectivity 1 3 6 > > [0, 5, 12, 19, 26, 35, 42, 51, 58, 65] > [5, 7, 7, 7, 9, 7, 9, 7, 7, 7] > edges 11 21 35 connectivity 0 2 6 > > [0, 5, 12, 21, 28, 35, 42, 49, 58, 65] > [5, 7, 9, 7, 7, 7, 7, 9, 7, 7] > edges 12 25 33 connectivity 0 3 6 > > [0, 5, 14, 21, 28, 35, 42, 51, 58, 65] > [5, 9, 7, 7, 7, 7, 9, 7, 7, 7] > edges 10 24 33 connectivity 0 3 6 > > [0, 5, 14, 21, 28, 35, 44, 51, 58, 65] > [5, 9, 7, 7, 7, 9, 7, 7, 7, 7] > edges 10 23 33 connectivity 0 3 5 > > [0, 5, 12, 21, 28, 35, 44, 51, 58, 65] > [5, 7, 9, 7, 7, 9, 7, 7, 7, 7] > edges 11 22 33 connectivity 0 2 5 > > [0, 5, 12, 19, 28, 35, 44, 51, 58, 65] > [5, 7, 7, 9, 7, 9, 7, 7, 7, 7] > edges 10 20 33 connectivity 0 2 5 > > [0, 5, 12, 19, 26, 35, 44, 51, 58, 65] > [5, 7, 7, 7, 9, 9, 7, 7, 7, 7] > edges 7 15 32 connectivity 0 1 5 > > [0, 5, 14, 21, 28, 35, 42, 49, 58, 65] > [5, 9, 7, 7, 7, 7, 7, 9, 7, 7] > edges 9 23 31 connectivity 0 3 5 > > [0, 5, 12, 21, 28, 35, 42, 49, 56, 65] > [5, 7, 9, 7, 7, 7, 7, 7, 9, 7] > edges 9 22 31 connectivity 0 3 5 > > [0, 5, 14, 21, 28, 35, 42, 49, 56, 63] > [5, 9, 7, 7, 7, 7, 7, 7, 7, 9] > edges 7 22 31 connectivity 0 4 5 > > [0, 5, 14, 21, 28, 37, 44, 51, 58, 65] > [5, 9, 7, 7, 9, 7, 7, 7, 7, 7] > edges 8 21 31 connectivity 0 2 5 > > [0, 5, 14, 21, 28, 35, 42, 49, 56, 65] > [5, 9, 7, 7, 7, 7, 7, 7, 9, 7] > edges 7 21 31 connectivity 0 3 5 > > [0, 5, 14, 21, 30, 37, 44, 51, 58, 65] > [5, 9, 7, 9, 7, 7, 7, 7, 7, 7] > edges 6 19 31 connectivity 0 2 5 > > [0, 5, 12, 21, 28, 37, 44, 51, 58, 65] > [5, 7, 9, 7, 9, 7, 7, 7, 7, 7] > edges 7 18 31 connectivity 0 2 5[0, 7, 14, 21, 28, 35, 42, 49, 56, 63] [7, 7, 7, 7, 7, 7, 7, 7, 7, 9] edges 7 22 30 connectivity 0 3 5> [0, 5, 14, 23, 30, 37, 44, 51, 58, 65] > [5, 9, 9, 7, 7, 7, 7, 7, 7, 7] > edges 6 18 30 connectivity 0 1 5 > > [0, 5, 12, 19, 28, 37, 44, 51, 58, 65] > [5, 7, 7, 9, 9, 7, 7, 7, 7, 7] > edges 6 15 30 connectivity 0 2 5 > > [0, 5, 12, 21, 30, 37, 44, 51, 58, 65] > [5, 7, 9, 9, 7, 7, 7, 7, 7, 7] > edges 5 15 30 connectivity 0 1 5
Message: 2946 - Contents - Hide Contents Date: Thu, 03 Jan 2002 09:09:15 Subject: Re: Some 9-tone 72-et scales From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <a0uhdq+onbm@e...> > Can you get your program to check for propriety of these things?I'd need to write the code for it, and it isn't a graph property so I'm not going to start with any advantage from the Maple graph theory package. Paul did not think propriety was very important--what's your take on it?> It's interesting that so many scales came out proper > when that wasn't a criterion in the search. All the 10-note 72= scales > are strictly proper.It's also interesting that the best scores were all proper.
Message: 2947 - Contents - Hide Contents Date: Thu, 03 Jan 2002 09:36:16 Subject: Re: Some 9-tone 72-et scales From: clumma>> >an you get your program to check for propriety of these things? >>I'd need to write the code for it, and it isn't a graph property >so I'm not going to start with any advantage from the Maple graph >theory package. Paul did not think propriety was very important-- >what's your take on it?You might like to read Rothenberg's original papers on the subject. There's graph stuff in there that none of us have touched (propriety was just a starting point for Rothenberg), plus a fancy algorithm generating all the proper subsets of a scale. -Carl
Message: 2948 - Contents - Hide Contents Date: Thu, 03 Jan 2002 10:19:28 Subject: More 7-tone 7-limit scales From: genewardsmith These come from (21/20)^2 (10/9)^2 (9/8) (8/7)^2 = 2 [1, 21/20, 7/6, 21/16, 3/2, 5/3, 7/4] [21/20, 10/9, 9/8, 8/7, 10/9, 21/20, 8/7] edges 12 connectivity 2 [1, 21/20, 7/6, 21/16, 3/2, 63/40, 7/4] [21/20, 10/9, 9/8, 8/7, 21/20, 10/9, 8/7] edges 12 connectivity 2 [1, 21/20, 7/6, 49/40, 7/5, 14/9, 7/4] [21/20, 10/9, 21/20, 8/7, 10/9, 9/8, 8/7] edges 12 connectivity 1 [1, 21/20, 7/6, 21/16, 35/24, 5/3, 7/4] [21/20, 10/9, 9/8, 10/9, 8/7, 21/20, 8/7] edges 11 connectivity 2 [1, 21/20, 7/6, 4/3, 7/5, 63/40, 7/4] [21/20, 10/9, 8/7, 21/20, 9/8, 10/9, 8/7] edges 11 connectivity 1 [1, 21/20, 441/400, 63/50, 7/5, 63/40, 7/4] [21/20, 21/20, 8/7, 10/9, 9/8, 10/9, 8/7] edges 10 connectivity 2 [1, 21/20, 7/6, 21/16, 35/24, 49/32, 7/4] [21/20, 10/9, 9/8, 10/9, 21/20, 8/7, 8/7] edges 10 connectivity 2 [1, 21/20, 7/6, 21/16, 3/2, 63/40, 9/5] [21/20, 10/9, 9/8, 8/7, 21/20, 8/7, 10/9] edges 10 connectivity 1 [1, 21/20, 7/6, 4/3, 7/5, 8/5, 16/9] [21/20, 10/9, 8/7, 21/20, 8/7, 10/9, 9/8] edges 10 connectivity 1 [1, 21/20, 441/400, 49/40, 7/5, 63/40, 7/4] [21/20, 21/20, 10/9, 8/7, 9/8, 10/9, 8/7] edges 10 connectivity 1 [1, 21/20, 7/6, 4/3, 7/5, 14/9, 16/9] [21/20, 10/9, 8/7, 21/20, 10/9, 8/7, 9/8] edges 10 connectivity 1 [1, 21/20, 7/6, 49/40, 49/36, 14/9, 7/4] [21/20, 10/9, 21/20, 10/9, 8/7, 9/8, 8/7] edges 9 connectivity 2 [1, 21/20, 7/6, 21/16, 441/320, 63/40, 7/4] [21/20, 10/9, 9/8, 21/20, 8/7, 10/9, 8/7] edges 9 connectivity 1 [1, 21/20, 7/6, 49/40, 49/36, 49/32, 7/4] [21/20, 10/9, 21/20, 10/9, 9/8, 8/7, 8/7] edges 9 connectivity 1 [1, 21/20, 7/6, 4/3, 7/5, 63/40, 9/5] [21/20, 10/9, 8/7, 21/20, 9/8, 8/7, 10/9] edges 9 connectivity 1 [1, 21/20, 441/400, 49/40, 441/320, 63/40, 7/4] [21/20, 21/20, 10/9, 9/8, 8/7, 10/9, 8/7] edges 8 connectivity 1 [1, 21/20, 441/400, 49/40, 7/5, 63/40, 9/5] [21/20, 21/20, 10/9, 8/7, 9/8, 8/7, 10/9] edges 8 connectivity 1 [1, 21/20, 7/6, 49/40, 441/320, 63/40, 7/4] [21/20, 10/9, 21/20, 9/8, 8/7, 10/9, 8/7] edges 8 connectivity 1 [1, 21/20, 7/6, 35/27, 49/36, 14/9, 7/4] [21/20, 10/9, 10/9, 21/20, 8/7, 9/8, 8/7] edges 8 connectivity 1 [1, 21/20, 7/6, 35/27, 40/27, 14/9, 7/4] [21/20, 10/9, 10/9, 8/7, 21/20, 9/8, 8/7] edges 7 connectivity 1 [1, 21/20, 189/160, 3969/3200, 567/400, 63/40, 7/4] [21/20, 9/8, 21/20, 8/7, 10/9, 10/9, 8/7] edges 6 connectivity 1
Message: 2949 - Contents - Hide Contents Date: Thu, 03 Jan 2002 10:20:48 Subject: Re: Some 9-tone 72-et scales From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> You might like to read Rothenberg's original papers on the subject. > There's graph stuff in there that none of us have touched (propriety > was just a starting point for Rothenberg), plus a fancy algorithm > generating all the proper subsets of a scale.Where might Rothenberg's papers be found?
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