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Message: 5275 - Contents - Hide Contents Date: Fri, 04 Oct 2002 17:07:58 Subject: Re: A common notation for JI and ETs From: gdsecor (This is a continuation of my message #4664.) --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote (#4662):> At 06:19 PM 17/09/2002 -0700, George Secor wrote:>> From: George Secor (9/17/02, #4626) >> Subject: A common notation for JI and ETs >> >> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: >>> ...>>> Here are some others for your consideration: >>> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >> 15>>> 282: )| ~| ~)| |~ /| |) )|) (| (|( //| /|) (|~ /|\ (|) >>> |( ~|( /|~ ~|\ |~) >>> >>> 11deg282 is the difficult one. /|) is only correct as the 5- comma + >>> 7-comma, not the 13-comma, and |~) is a two-flags-on-the-same- side symbol >>> I'm proposing to stand for the 13:19-comma (and possibly the 5:13-comma). >>> But if you'd rather, I'll just accept that 282-ET and 294-ETare not notatable. I find 282 a little difficult, but still notatable. If we don't use |\, then we can't have both matching symbols and ||\ as RC of /|. With that constraint I would do 282 this way with rational complementation: 282a: |( ~| ~)| |~ /| |) )|) (| (|( //| |~) (|~ /|\ (|) ||( )||( ~|| ~||( )||~ )/|| ||) ||\ ~||) ~||\ //|| /||) /||\ (RC) The )/|| symbol is the double-shaft version of the one that I am proposing below for 306 and 494; here it is the proposed rational complement of )|). But if we use |\ with matching symbols, then I get this: 282b: |( ~| ~)| |~ /| |) )|) |\ (|( //| |~) (|~ /|\ (|) ||( ~|| ~)|| ||~ /|| ||) )||) ||\ (||( //|| ||~) (||~ /||\ (MS) But this shifts symbols such as ||) into the wrong positions and makes them almost meaningless, besides not having ||\. So I prefer 282a.>> Yes, I think that there are too many problems. >>>>>> However, 306-ET _is_ notatable without using any two-flags-on- the-same-side >>> symbols. Alternatives for some degrees are given on the line below. >>> >>> 306: )| |( )|( ~|( /| ~|~ |) (| |\ //| ~|\ /|)(|~ /|\ (|)>>> ~| ~)| |~ )|) ~|) (|( |~) >>>> (|( is a better choice than //| for the comma roles it fulfills. >> I guess so. Since //| only works as 5+5 comma and (|( works in all its > possible roles. > >> (|~>> and ~|~ look like they may be a little shaky in the flag arithmetic for >> |~. (A wavy flag becomes a shaky flag?) >> I hadn't noticed that, thanks. But in cases like this (where the only > alternative is incomplete notation, I don't think we should let flag > arithmetic stop us.>>> 318 is notatable if you accept (/| (the 31' comma) for 15 steps. >>>> Neither 306 nor 318 are 7-limit consistent, so I don't see much point >> in doing these, other than they may have presented an interesting >> challenge. >> Good point. Forget 318-ET, but 306-ET is of interest for being strictly > Pythagorean. The fifth is so close to 2:3 that even god can barely tell the > difference. ;-)What's making me hesitate about 306 is a 5 factor 49 percent of a degree false. But I tried it anyway without looking at what you have and came up with the following, which surprised me with how well it works. It eliminates the shaky flag with a new symbol )/|, which I will explain below when I discuss 494: 306: )| |( )|( ~|( /| )/| |) )|) |\ (|( |~) /|) (|~ /|\ (|) )|| (|\ )||( ~||( /|| )/| ||) )||) ||\ (||( ||~) /||) (||~ /||\ (RC & MS)> If we can accept fuzzy arithmetic with the right wavy flag, and the > addition of the 13:19 comma symbol |~) then the 31-limit-consistent 388-ET > can be notated (but surprisingly, not 311-ET). > > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 > 388: )| |( ~| ~)| ~|( |~ /| ~|~ |) |\ (| ~|) ~|\ //| > > 15 16 17 18 19 20 21 22 > |~) /|) /|\ (/| |\) (|) (|\ ||( ... (MS) > > The symbols (/| and |\) are of course the 31-comma symbols we agreed on > long ago.Yes, and they work quite well here, as well as in 494, below. Rational complementation doesn't work very well when /| and |\ are 3 degrees apart, so I will go along with the matching symbols, even if they don't really mean much of anything; 388 is therefore agreed! I was wondering why you said that we can't do 311. Is it because (/| is not the proper number of degrees for the 31 comma? But neither is |~ as 6deg388, the 23 comma, nor is )|~ as 8deg494 valid as the 19' comma, but you have proposed these here. And I agree with your decision, because there is no alternative. So I would do 311 thus: 311: |( )|( ~)| ~|( |~ /| |) |\ (| (|( ~|\ //| /|) /|\ (/| (|) (|\ ~|| ~)|| ~|( )|~ /|| ||) ||\ ~||) (||( ~||\ ||~) /||) /||\ (RC) I have selected the best single-shaft symbols and used their rational complements. The symbols are not matched in the half-apotomes.> Here's another one I think should be on the list, 494-ET, if only because > of the fineness of the division, and because it shows all our rational > complements*. It is 17-limit consistent. Somewhat surprisingly, it is fully > notatable with the addition of the 13:19 comma symbol |~). It has the same > problem as 306 and 388, with right-wavy being fuzzy, taking onvalues 6, 7> and 8 here. > > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 > 494: )| |( )|( ~| ~)| ~|( |~ )|~ /| ~|~ |) )|) |\ (| ~|) > > 16 17 18 19 20 21 22 23 24 25 26 27 28 > (|( ~|\ //| |~) /|) (|~ /|\ (/| |\) (|) )|| (|\ )||( ... > (RC* & MS) > > * It agrees with all our rational complements so far, except that we'd need > to accept > ~|~ <---> )|) [where the |~ flag corresponds to 6 steps of 494] > instead of > ~|~ <---> /|( > which might become an alternative complement. > > and we'd need to add > > )|( <---> |~) [where the |~ flag corresponds to 8 steps of 494] > > In all other symbols above, the |~ flag corresponds to 7 steps of 494. > > My interpretations are > ~|~ 5:19 comma > )|) 7:19 comma > )|( 19 comma + 5:7 comma > |~) 13:19 comma > > Obviously these symbols should be the last to be chosen for any purpose. > > So we see that the addition of that one new symbol |~) for the 13:19 comma > and the acceptance of a fuzzy right wavy flag, lets the maximum notatable > ET leap from 217 to 494, more than double! > > So who cares about notating 282, 388 and 494? I dunno, but here's a funny > thing: The difference between them is 106. 176 is the next one down.And (surprise!) 600 is the next one up (but 7 and 17 are really bad). All I can say about 106 is that it's twice 53. I first found 494 in the 1970s when I was looking for a division with a low-error 17 limit. I noticed that two excellent 7-limit divisions, 99 and 171, have their 11 errors in opposite directions, so in their sum, 270, they cancel out (reckoned as fractions of a degree). For the 13 limit both 224 and 270 are good, but their 17 errors are in opposite directions, so in their sum, 494, they also cancel out. (Also note their difference of 46, which is also quite good for the 17 limit.) But I digress. I have a problem changing ~|~ to represent 10deg494 in that it must be given a different complement to make this work. The proposed complement, )||), has an offset of -2.64 cents, large enough that it would be invalid in most other larger divisions. This would also make the complementation we previously had for ~|~ <--> /||( and /|( <--> ~||~ (offset of 0.49 cents) unavailable for other divisions such as 342 and 388 (except as an alternate complement). Instead of ~|~ I propose )/| for 10deg494 (and 6deg306 above), which is the correct number of degrees and has the actual flags for the 5:19 comma (hence is easy to remember; besides, the symbol that I made for this looks pretty good). This also makes a consistent complement to )||) in 282, 306 and 494 (the three places where I have found a use for it); the offset of -2.25 cents is still rather large, but not as much as before. (It makes a nice alternate complement with /|( with an offset of 0.88 cents.) It also restricts the fuzzy arithmetic to only one symbol, |~), which has its two flags on the same side. This would put the total number of single-shaft symbols at 30, and the only symbols that would be left without rational complements are )|\ and /|~. I don't object to the fuzzy |~) arithmetic for 19deg494, because this makes it consistent with its proposed complement )||(, which has an offset of only 0.09 cents (and would probably be valid a lot of other places). The symbol does somewhat resemble |\), but I believe that the two are sufficiently different in size that this shouldn't cause any problem. So I get: 494: )| |( )|( ~| ~)| ~|( |~ )|~ /| )/| |) )|) |\ (| ~|) (|( ~|\ //| |~) /|) (|~ /|\ (/| |\) (|) )|| (|\ )||( ~|| ~)|| ~||( ||~ )||~ /|| )/|| ||) )||) ||\ (|| ~||) (||( ~||\ //|| ||~) /||) (||~ /||\ (RC & MS) The only irregularities with this are the fuzzy symbol arithmetic with |~) and ||~) and the fact that )|~ is not valid as the 19' comma. Considering that 19 is not well represented in 494 and that the 19' comma will be the much less used of the two 19 commas, I think that this is inconsequential. I tried messing around with some 3-flag symbols as alternatives to |~), which would eliminate the remaining fuzzy symbol arithmetic. Since )/| looked so good, I tried ~|\( for the 37 comma for 19deg494, which seems pretty easy to distinguish from everything else. As a u- d complement to )|( it has an offset of -2.60 cents, rather large, so it's not valid in a lot of other places. I eventually decided that it wasn't worth it, especially since the symbol would have 3 flags, so I would stick with your proposal for |~). However, I am intrigued by the idea of )|)), the 19+7^2 diesis, as being very close to half an apotome (and thus its own rational complement); this would be very useful in a lot of places, e.g., 270, 311, and 400. We may have to explore this a bit more, or at least leave open the possibility of future expansion, i.e., more flag combinations. I figure that the more bells and whistles we have, the less likely it is that anybody is ever going to use all of them.> Here's another big one we can notate this way. Only 11-limit consistent, > but its relative accuracy at that limit is extremely good. 342 = 2*3*3*19. > > 342: > )| |( )|( ~|( )|~ /| ~|~ |) |\ ~|) (|( //||~) /|) /|\ (/| (|) (|\ Agreed! I spoke about 224 and 270 above, but we don't have a notation for them. How about this: 224: |( )|( ~|( /| |) |\ (|( //| /|) /|\ (|) (|\ ~|| ~||( /|| ||) ||\ (||( ~||\ /||) /||\ (RC) 270: |( ~| ~)| )|~ /| |) |\ (| (|( //| /|) /|\ (/| (|) (|\ ~|| ~||( )||~ /|| ||) ||\ (|| ~||\ //|| /||) /||\ (RC) --George

Message: 5276 - Contents - Hide Contents Date: Wed, 9 Oct 2002 23:47:34 Subject: Re: mathematical model of torsion-block symmetry? From: monz" : >>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 * [with cont.] >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: 5278 - Contents - Hide Contents 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 * [with cont.] >> >> 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 * [with cont.] -monz

Message: 5281 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 * [with cont.] i've put an entry for this into the EDO historical table: Yahoo groups: /monz/files/dict/eqtemp.htm * [with cont.] 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: 5284 - Contents - Hide Contents Date: Fri, 11 Oct 2002 19:37:16 Subject: Re: EDO superset containing approximation of Werckmeister III? From: Gene Ward Smith --- 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, but other strange-looking possibilities are out there, such as 200 and 412 (200+412=612, of course.)

Message: 5286 - Contents - Hide Contents 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: 5287 - Contents - Hide Contents Date: Fri, 11 Oct 2002 22:46:56 Subject: Re: EDO superset containing approximation of Werckmeister III? From: Gene Ward Smith --- In tuning-math@y..., "monz" <monz@a...> wrote:> well, OK, but ... AARRRGGH! -- *how* did you do that search?Brute force, much like a search for good ets. I totaled up the relative error for each n from 1 to 1000 by running a simple Maple routine, and insisted they be at least as good as 12-et.> 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.Mathematica is very similar to Maple, but you need to learn how to use it.

Message: 5288 - Contents - Hide Contents Date: Fri, 11 Oct 2002 05:08:28 Subject: Re: EDO superset containing approximation of Werckmeister III? From: Gene Ward Smith --- 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 * [with cont.]I used Manual's scale data rather than trying to figure out where the data was on your page. 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

Message: 5289 - Contents - Hide Contents 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 * [with cont.] >> 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, 557awesome!! 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: 5291 - Contents - Hide Contents Date: Fri, 11 Oct 2002 05:08:28 Subject: Re: EDO superset containing approximation of Werckmeister III? From: Gene Ward Smith --- 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? >=20 > Yahoo groups: /monz/files/dict/werckmeister.htm * [with cont.]I used Manual's scale data rather than trying to figure out where the data = was on your page. It turns out that Werckmeister III can be expressed with = extreme accuracy in terms of what I call "schismas", steps of the 612 et. I= n 612-et terms, it is 0, 46, 98, 150, 199, 254, 300, 355, 404, 453, 508, 557

Message: 5293 - Contents - Hide Contents 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 * [with cont.] >> 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, 557awesome!! 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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

Message: 5298 - Contents - Hide Contents Date: Sat, 12 Oct 2002 22:25:30 Subject: Re: EDO superset containing approximation of Werckmeister III? From: Gene Ward Smith --- In tuning-math@y..., manuel.op.de.coul@e... wrote:> > 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 -612Nice! Is there a way to go beyond the stop point, and *not* show all divisions? I notice 612 popping up a lot with temperaments I hadn't looked at yet, but the stop point is set too low to easily see it.

Message: 5299 - Contents - Hide Contents Date: Sat, 12 Oct 2002 22:34:00 Subject: Re: mathematical model of torsion-block symmetry? From: Gene Ward Smith --- In tuning-math@y..., "hs" <straub@d...> wrote:> The base lattice (Z^2) is a Z-module (like a vector space but only integers as > coefficients and for scalar multiplication), and so is the quotient (the > elements of which are simply equivalence classes of intervals with respect to > the unison vectors). A periodicity block, BTW, is nothing else but a set of > adjacent representants of the quotient module (one representant for each > equivalence class).I've mentioned this before, but readers used to the "abelian group" terminology should keep in mind that abelian group and Z-module mean the same thing.> Now, the quotient module being finite...Whups--you are sticking "2" into the mix when you conclude this. The math is more straightforward if you treat 2 as just another prime number.

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