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Message: 5278 Date: Thu, 10 Oct 2002 01:34:32 Subject: Re: mathematical model of torsion-block symmetry? From: monz hi Hans, thanks very much for your replies to this, but i'm afraid some of the math language is over my head. i defer to Gene, paul, Graham, et al. for comment. -monz "all roads lead to n^0" ----- Original Message ----- From: "Hans Straub" <straub@xxxxxxxx.xx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Wednesday, October 09, 2002 2:47 PM Subject: [tuning-math] Re: mathematical model of torsion-block symmetry? > From: "monz" <monz@a...>: > > > >Is there some way to mathematically model > >the symmetry in a torsion-block? > > > >see the graphic and its related text in my > >Tuning Dictionary definition of "torsion" > >-- i've uploaded it to here: > >Yahoo groups: /monz/files/dict/torsion.htm * > > > > Well, they are translation symmetries in the quotient group of the full lattice > and the subgroup generated by the unison vectors. The symmetries in the > example are pairs because the element has order 2 in the quotient group, > but there are other elements such as (0,1) with order 6 or (0,2), (1,1) with > order 3. Something like this? > > > BTW, I think the definition of torsion can be made simpler. You do not need > the condition that some power of the interval is in the unison vector group, > because this is always the case (at least when the periodicity block is finite). > Do I see this correctly? > > Hans Straub
Message: 5279 Date: Thu, 10 Oct 2002 17:20:27 Subject: Re: Piano tuning and "BODE'S LAW EXPLAINED" II From: manuel.op.de.coul@xxxxxxxxxxx.xxx You can add another note to your solar system scale now. Perhaps it's also an escaped moon from Neptune? Manuel
Message: 5280 Date: Thu, 10 Oct 2002 15:10:26 Subject: EDO superset containing approximation of Werckmeister III? From: monz could someone please explain how to find an EDO superset that gives a good approximation of the 12 pitches in Werckmeister III, with the scale data given here? Yahoo groups: /monz/files/dict/werckmeister.htm * -monz
Message: 5281 Date: Thu, 10 Oct 2002 14:43:36 Subject: Re: Piano tuning and "BODE'S LAW EXPLAINED" II From: monz ----- Original Message ----- From: <manuel.op.de.coul@xxxxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Thursday, October 10, 2002 8:20 AM Subject: Re: [tuning-math] Re: Piano tuning and "BODE'S LAW EXPLAINED" II > You can add another note to your solar system scale now. > Perhaps it's also an escaped moon from Neptune? thanks -- john chalmers and david beardsley wrote me about this already a few days ago. unofortunately, even Pluto is already beyond the audible range in my sonic mapping, and so since it's more distant than Pluto, this planet won't sound like much either! ;-) -monz
Message: 5282 Date: Thu, 10 Oct 2002 16:05:18 Subject: 7-limit signatures From: Gene W Smith Recall that cubic lattice coordinates for 7-limit tetrads associate the 3-tuple of integers [a,b,c] with the major triad with root 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+b-c)/2) if a+b+c is even, and the minor tetrad with root 3^((-a+b+c-1)/2) 5^((a-b+c+1)/2) 7^((a+b-c+1)/2) if a+b+c is odd. This means that [2,0,0], [0,2,0], [0,0,2] represent the major tetrads with roots 5*7/3, 3*7/5, 3*5/7 respectively; when octave reduced these are 35/24, 21/20, and 15/14. If L is a wedgie for a 7-limit linear temperament, we may define the *signature* of L as S = [-L[1]+L[2]+L[3], L[1]-L[2]+L[3], L[1]+L[2]-L[3]]. This is a 3-tuple representing the number of generator steps in the octave plus generator formulation of the temperament for 35/24, 21/20, 15/14 respectively, weighted by the number of periods to the octave. In the case where the octave is the period, it uniquely defines the tetrad in terms of steps by sending the tetrad [a,b,c] to S[1]*a + S[2]*b + S[3]*c steps. For example, taking the meantone wedgie of [1,4,10,12,-13,4] gives us a signature of [13,-7,5], so the minor tonic tetrad [-1,0,0] is sent to -13 steps, the dominant major tetrad [0,1,1] to -2 steps, and so forth; for major tetrads these steps are twice the number of generator steps for the root of the tetrad, while the minor tetrads fill in the gaps in ways which depend on the temperament--for instance, here we get [-1,1,-1] ~ [1,-2,0] at -1 step, equivalent under 126/125. Just as temperaments with a generator which is a consonant interval are of particular interest, temperaments where one of the signature values is +-1 are of interest, with miracle, whose signature is [-15,11,1] an example. In this case the ordering of tetrads by steps corresponds to a chain of adjacent tetrads in the lattice, so the step ordering is of particular interest. Miracle now relates [-1,0,0] not just to 15 steps, but to the tetrad [0,0,15], and [0,1,1] to [0,0,12], and so forth. This helps to keep track of the connectivity of the tetrads when using miracle. Moreover, we may define miracle MOS in terms of tetrads--Blackjack for instance can be described as a chain of sixteen consecutive [0,0,n] tetrads, where n starts from an even number (representing a major tetrad) and runs up to an odd number (minor tetrad.) For example, the chain from [0,0,0] (major tonic) to [0,0,15] (minor tonic.) Here is a list of temperaments with this unital signature property: [[1, 1, 3, 3], [0, 6, -7, -2]] [6, -7, -2, 15, 20, -25] Miracle generators [1200., 116.5729472] signatures [-15, 11, 1] rms 1.637405196 comp 24.92662917 bad 1017.380173 ets [10, 21, 31, 41, 72, 103] [[1, 0, -4, 6], [0, 1, 4, -2]] [1, 4, -2, -16, 6, 4] Dominant seventh generators [1200., 1902.225977] signatures [1, -5, 7] rms 20.16328150 comp 9.836559603 bad 1950.956872 ets [5, 7, 12] [[1, 1, 2, 3], [0, 9, 5, -3]] [9, 5, -3, -21, 30, -13] Quartaminorthirds generators [1200., 77.70708739] signatures [-7, 1, 17] rms 3.065961726 comp 27.04575317 bad 2242.667500 ets [15, 16, 31, 46] [[1, 1, 1, 2], [0, 8, 18, 11]] [8, 18, 11, -25, 5, 10] Octafifths generators [1200., 88.14540671] signatures [21, 1, 15] rms 2.064339812 comp 34.23414357 bad 2419.357925 ets [27, 41, 68] [[1, 2, 2, 3], [0, 4, -3, 2]] [4, -3, 2, 13, 8, -14] Tertiathirds generators [1200., -125.4687958] signatures [-5, 9, -1] rms 12.18857055 comp 14.72969740 bad 2644.480844 ets [1, 9, 10, 19, 29] [[1, 0, 7, -5], [0, 1, -3, 5]] [1, -3, 5, 20, -5, -7] Hexadecimal generators [1200., 1873.109081] signatures [1, 9, -7] rms 18.58450012 comp 12.33750942 bad 2828.823679 ets [7, 9, 16] [[1, 25, -31, -8], [0, 26, -37, -12]] [26, -37, -12, 76, 92, -119] generators [1200., -1080.705187] signatures [-75, 51, 1] rms .2219838332 comp 118.1864167 bad 3100.676640 ets [10, 171, 513] [[1, 3, 6, 5], [0, 20, 52, 31]] [20, 52, 31, -74, 7, 36] generators [1200., -84.87642563] signatures [63, -1, 41] rms .3454637898 comp 96.52895120 bad 3218.975773 ets [99, 212, 311, 410] [[1, 2, 2, 2], [0, 5, -4, -10]] [5, -4, -10, -12, 30, -18] generators [1200., -97.68344522] signatures [-19, -1, 11] rms 6.041345016 comp 24.27272426 bad 3559.349900 ets [12, 37] [[1, 3, 0, 2], [0, 14, -23, -8]] [14, -23, -8, 46, 52, -69] generators [1200., -121.1940013] signatures [-45, 29, -1] rms .8353054234 comp 68.53846955 bad 3923.865443 ets [10, 99] [[1, 12, 15, 1], [0, 23, 28, -4]] [23, 28, -4, -88, 71, -9] generators [1200., -543.2692838] signatures [1, -9, 55] rms .7218691130 comp 78.22290415 bad 4416.989140 ets [53] [[1, 2, 3, 4], [0, 5, 8, 14]] [5, 8, 14, 10, -8, 1] generators [1200., -102.3994286] signatures [17, 11, -1] rms 8.609470174 comp 22.70605087 bad 4438.739304 ets [12] [[1, 2, 1, 1], [0, 6, -19, -26]] [6, -19, -26, -7, 58, -44] generators [1200., -83.37933102] signatures [-51, -1, 13] rms 1.487254275 comp 55.50097036 bad 4581.275174 ets [29, 72] [[1, 43, -58, -17], [0, 46, -67, -22]] [46, -67, -22, 137, 164, -213] generators [1200., -1080.392876] signatures [-135, 91, 1] rms .1267147296 comp 211.5126443 bad 5668.912722 ets [10, 301, 311, 612] [[1, 2, 3, 3], [0, 6, 10, 3]] [6, 10, 3, -21, 12, 2] generators [1200., -82.00647655] signatures [7, -1, 13] rms 12.62928610 comp 21.39334917 bad 5780.113425 ets [15, 29] [[1, 2, 1, 2], [0, 4, -13, -8]] [4, -13, -8, 18, 24, -30] generators [1200., -122.3321832] signatures [-25, 9, -1] rms 6.403982242 comp 31.21994593 bad 6241.865585 ets [10] [[1, 1, 2, 2], [0, 4, 2, 5]] [4, 2, 5, 6, 3, -6] generators [1200., 187.6316444] signatures [3, 7, 1] rms 47.68000484 comp 11.69073209 bad 6516.579639 ets [6] [[1, 0, -3, 6], [0, 3, 10, -6]] [3, 10, -6, -42, 18, 9] generators [1200., 638.4642643] signatures [1, -13, 19] rms 9.885351494 comp 25.98120378 bad 6672.839126 ets [15] [[1, 2, 3, 3], [0, 5, 8, 2]] [5, 8, 2, -18, 11, 1] generators [1200., -100.0317906] signatures [5, -1, 11] rms 21.64417648 comp 17.58481613 bad 6692.936885 ets [12] [[1, 2, 5, 6], [0, 4, 26, 31]] [4, 26, 31, -1, -38, 32] generators [1200., -123.5352658] signatures [53, 9, -1] rms 2.267858844 comp 56.46645397 bad 7230.978171 ets [29, 68] [[1, 2, 2, 3], [0, 5, -4, 2]] [5, -4, 2, 16, 11, -18] generators [1200., -99.19646785] signatures [-7, 11, -1] rms 21.21541236 comp 18.58251802 bad 7325.893533 ets [1, 12] [[1, 3, 2, 4], [0, 13, -3, 11]] [13, -3, 11, 34, 19, -35] generators [1200., -130.2049690] signatures [-5, 27, -1] rms 4.481233722 comp 41.46170034 bad 7703.566083 ets [9, 37, 46] [[1, 12, 10, 5], [0, 19, 14, 4]] [19, 14, 4, -30, 47, -22] generators [1200., -657.8863907] signatures [-1, 9, 29] rms 3.032624788 comp 52.44877824 bad 8342.369709 ets [31] [[1, 23, -56, 83], [0, 47, -128, 176]] [47, -128, 176, 768, -147, -312] generators [1200., -546.7680257] signatures [1, 351, -257] rms .3610890892e-1 comp 481.2637469 bad 8363.357505 ets [1578] [[1, 13, 17, -1], [0, 21, 27, -7]] [21, 27, -7, -92, 70, -6] generators [1200., -652.3887024] signatures [-1, -13, 55] rms 1.469925034 comp 75.92946624 bad 8474.535049 ets [46, 57, 103] [[1, 2, 4, 5], [0, 4, 16, 21]] [4, 16, 21, 4, -22, 16] generators [1200., -125.5372720] signatures [33, 9, -1] rms 6.562501740 comp 35.99263747 bad 8501.523814 ets [19] [[1, 1, 3, 4], [0, 7, -8, -14]] [7, -8, -14, -10, 42, -29] generators [1200., 101.5775171] signatures [-29, 1, 13] rms 7.012328960 comp 35.52454740 bad 8849.513343 ets [12] [[1, 2, -1, -1], [0, 6, -48, -55]] [6, -48, -55, 7, 104, -90] generators [1200., -83.05774075] signatures [-109, -1, 13] rms .6644554968 comp 115.7156146 bad 8897.127847 ets [29, 130] [[1, 2, 3, 3], [0, 7, 11, 3]] [7, 11, 3, -24, 15, 1] generators [1200., -73.16557361] signatures [7, -1, 15] rms 16.40779159 comp 24.26315309 bad 9659.276719 ets [16] [[1, 2, 3, 3], [0, 8, 13, 4]] [8, 13, 4, -27, 16, 2] generators [1200., -63.00613990] signatures [9, -1, 17] rms 12.64637740 comp 28.07029990 bad 9964.608569 ets [19] It might be remarked that the signatures with the middle-sized (in absolute value) components relatively small are an interesting subclass of these unital signature temperaments; they are associated with certain planar temperaments of a kind not usually considered. Examples are [-5,27,-1], covered by 46, [-109,-1,13], covered by 130, and [1,-9,55], covered (though not very well) by 53.
Message: 5283 Date: Fri, 11 Oct 2002 08:13:06 Subject: Werckmeister as subset of 612edo From: monz hi Gene, i've just done a comprehensive analysis of Werckmeister III as a subset of 612edo: Yahoo groups: /monz/files/dict/werckmeister.htm * i've put an entry for this into the EDO historical table: Yahoo groups: /monz/files/dict/eqtemp.htm * have you analyzed Werckmeister III like this before? has anyone else? the only reference i've found to 612edo besides your posts is a mention by Bosanquet in his book, referring to Captain Herschel's advocacy of this tuning. -monz
Message: 5286 Date: Fri, 11 Oct 2002 14:23:25 Subject: Re: EDO superset containing approximation of Werckmeister III? From: monz > From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, October 11, 2002 12:37 PM > Subject: [tuning-math] Re: EDO superset containing approximation of Werckmeister III? > > > --- In tuning-math@y..., "monz" <monz@a...> wrote: > > > awesome!! i was hoping you'd give some details as to how > > you found out that 612edo was the best approximation. > > I ran a search and 612 came out the best, well, OK, but ... AARRRGGH! -- *how* did you do that search? since i'm math-challenged, the only way i know how to do it is to set up an Excel spreadsheet with the EDO-cardinality as a variable, but then i have to manually enter each cardinality and look at the graphs of deviation to see which EDOs are best. > but other strange-looking possibilities are out there, > such as 200 and 412 (200+412=612, of course.) ah, now that's useful! i was hoping to find something smaller than 612edo which could describe Werckmeister III, and 200 does the trick nicely. unfortunately, however, neither 200 nor 412 give integer-divisions for 12edo, so they're not as useful for comparing Werckmeister III to 12edo as 612edo is. please, Gene, more info on how your search method works. do you know how to set it up in an Excel spreadsheet? if not, then do you have some code that i could run on my PC? i have Mathematica -- just don't know a lot about how to use it. -monz "all roads lead to n^0"
Message: 5289 Date: Fri, 11 Oct 2002 00:57:35 Subject: Re: EDO superset containing approximation of Werckmeister III? From: monz hi Gene, > From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, October 10, 2002 10:08 PM > Subject: [tuning-math] Re: EDO superset containing approximation of Werckmeister III? > > > --- In tuning-math@y..., "monz" <monz@a...> wrote: > > > could someone please explain how to find an EDO superset > > that gives a good approximation of the 12 pitches in > > Werckmeister III, with the scale data given here? > > > > Yahoo groups: /monz/files/dict/werckmeister.htm * > > I used Manual's scale data rather than trying to figure out > where the data was on your page. there's a table showing the tunings as a chain of generators. anyway, i tried it and came up with the same results you did. > It turns out that Werckmeister III can be expressed with > extreme accuracy in terms of what I call "schismas", steps > of the 612 et. In 612-et terms, it is > > 0, 46, 98, 150, 199, 254, 300, 355, 404, 453, 508, 557 awesome!! i was hoping you'd give some details as to how you found out that 612edo was the best approximation. -monz "all roads lead to n^0"
Message: 5293 Date: Fri, 11 Oct 2002 00:57:35 Subject: Re: EDO superset containing approximation of Werckmeister III? From: monz hi Gene, > From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, October 10, 2002 10:08 PM > Subject: [tuning-math] Re: EDO superset containing approximation of Werckmeister III? > > > --- In tuning-math@y..., "monz" <monz@a...> wrote: > > > could someone please explain how to find an EDO superset > > that gives a good approximation of the 12 pitches in > > Werckmeister III, with the scale data given here? > > > > Yahoo groups: /monz/files/dict/werckmeister.htm * > > I used Manual's scale data rather than trying to figure out > where the data was on your page. there's a table showing the tunings as a chain of generators. anyway, i tried it and came up with the same results you did. > It turns out that Werckmeister III can be expressed with > extreme accuracy in terms of what I call "schismas", steps > of the 612 et. In 612-et terms, it is > > 0, 46, 98, 150, 199, 254, 300, 355, 404, 453, 508, 557 awesome!! i was hoping you'd give some details as to how you found out that 612edo was the best approximation. -monz "all roads lead to n^0"
Message: 5295 Date: Fri, 11 Oct 2002 22:52:04 Subject: Re: Historical well-temeraments, 612, and 412 From: monz ----- Original Message ----- From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Friday, October 11, 2002 5:31 PM Subject: [tuning-math] Historical well-temeraments, 612, and 412 > It seems that Werckmeister III is not the only well-temperament > to be nailed by 612. Here are some others, using data taken from > Manual's list of scales: > <snip> wow, Gene, thanks for these!!! they'll eventually all become Tuning Dictionary webpages. my guess is that the reason 612 works so well has something to do with the fact that these temperaments temper out the Pythagorean comma. wanna look into that more? -monz
Message: 5297 Date: Sat, 12 Oct 2002 13:37:39 Subject: Re: EDO superset containing approximation of Werckmeister III? From: manuel.op.de.coul@xxxxxxxxxxx.xxx Joe and Gene, I must have told this before but in Scala it's very easy to do too: load werck3 fit/mode This show successively better approximations and stops at some point. To go beyond that, and show all divisions, use a negative number: fit/mode -612 With a positive parameter it only shows that division. Manuel
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