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Message: 5925 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
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Message: 5928 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
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Message: 5930 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 * I asked a question about the math here: Yahoo groups: /tuning/message/15054 * Paul's explanation of the math, in answer to that question, is here: Yahoo groups: /tuning/message/15059 * Now I'm making a Tuning Dictionary entry for "Transformation", and using one of Regener's examples. Definitions of tuning terms: transformation, (c) 2001 by Joe Monzo * 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 + 2b But 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 Setup *
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Message: 5931 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
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Message: 5934 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.
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Message: 5935 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
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Message: 5937 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
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Message: 5938 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
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Message: 5940 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
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Message: 5941 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 5 which 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 4 which 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
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Message: 5945 Date: Thu, 03 Jan 2002 09:36:16 Subject: Re: Some 9-tone 72-et scales From: clumma >>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? 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
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Message: 5949 Date: Thu, 3 Jan 2002 13:38 +00 Subject: Re: Some 9-tone 72-et scales From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a1173r+ob05@xxxxxxx.xxx> gene wrote: > 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? I don't have a definitive answer. It, or something like it, may be important for modality. Especially for subsets of "comprehensible" ETs. The Pythagorean diatonic works fine despite being slightly improper, so you shouldn't be over-strict. For a scale with three step sizes to be proper shows that it has a certain level of cohesion. The extremely improper Magic subsets you gave composers, performers and listeners are likely to expect the large gaps to be filled in by more notes. This is a general problem with Magic scales of between 3 and 19 notes. The Decimal MOS has the opposite problem -- its step sizes are so closely equal that it doesn't have any shape. So it's great as a basis for notation, but doesn't have any sense of tonal center. Your top 10 note scale might solve this problem, because it's largest interval is almost twice the size of its smallest. And that single 5/72 step could be extremely important for leading to the tonic. 10 notes still seems like a lot for a mode. Perhaps the 6 note [12, 14, 9, 14, 9, 14] would work. How is it in terms of connectedness? Graham > > 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. > > > > 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 * > >
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