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Message: 2150 - Contents - Hide Contents Date: Thu, 29 Nov 2001 22:14:49 Subject: Re: Survey IX From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:>> Graham, where does this one come in on your rankings? >> 11 in the 7-limit,Cool! I knew it had to be comparable to the ones in your top 10!
Message: 2151 - Contents - Hide Contents
Date: Thu, 29 Nov 2001 22:44:39
Subject: reduced basis for MIRACLE temperament?
From: Paul Erlich
Is the TM-reduced basis for the MIRACLE linear temperament in the 11-
limit
{225/224, 243/242, 384/385}??
Message: 2152 - Contents - Hide Contents Date: Thu, 29 Nov 2001 23:54:19 Subject: Re: Graham's Top Ten From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:> It might be interesting to try other methods, but the problem I'd like you > to set your mind to is going from the map to the simplest set of unison > vectors. That's something that's not working currently. I can go from > unison vectors to maps no problem.One way to do this would be to treat the unison vector problem in the exact same way as the generator map problem. We can find the wedge invariant from the map by taking the wedge product of the column vectors, and directly from the wedge invariant we can read off a 5-limit unison vector. We then can find the exponent of 7 for the other unison vector via the gcd of the exponents of the first, and the rest using the requirement that the wedge invariant is invariant. Once we have this pair of vectors we should have something which we can readily reduce further via LLL and Minkowski reduction. How well would this work? Why don't you give a list of hard examples and we'll see.
Message: 2153 - Contents - Hide Contents Date: Fri, 30 Nov 2001 19:11:39 Subject: Re: reduced basis for MIRACLE temperament? From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Can you think of a reason anyone would want to look at LLL rather > than Minkowski?Not in this case; in higher prime limits Minkowski will become increasingly difficult to compute.
Message: 2154 - Contents - Hide Contents Date: Fri, 30 Nov 2001 20:27:03 Subject: Hey Gene From: Paul Erlich May I anticipate any attempts to answer the other questions I've asked you lately?
Message: 2155 - Contents - Hide Contents Date: Fri, 30 Nov 2001 22:54:46 Subject: Re: Hey Gene From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> May I anticipate any attempts to answer the other questions I've > asked you lately?What were they? I somtimes lose track of outstanding questions, and sometimes decide I don't want to pursue something at the moment, and sometimes think I'll get around to it eventually anyway in the course of the survey or whatever.
Message: 2156 - Contents - Hide Contents Date: Fri, 30 Nov 2001 23:04:32 Subject: Re: Hey Gene From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> May I anticipate any attempts to answer the other questions I've >> asked you lately? >> What were they? Yahoo groups: /tuning-math/message/1570 * [with cont.] for instance.
Message: 2157 - Contents - Hide Contents Date: Fri, 30 Nov 2001 00:29:22 Subject: Re: reduced basis for MIRACLE temperament? From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Is the TM-reduced basis for the MIRACLE linear temperament in the 11- > limit > > {225/224, 243/242, 384/385}??That's what I get, as well as {225/224, 441/440 and 540/539} as an LLL reduced basis.
Message: 2158 - Contents - Hide Contents Date: Fri, 30 Nov 2001 18:13:05 Subject: Re: reduced basis for MIRACLE temperament? From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Is the TM-reduced basis for the MIRACLE linear temperament in the > 11- >> limit >> >> {225/224, 243/242, 384/385}?? >> That's what I get, as well as {225/224, 441/440 and 540/539} as an > LLL reduced basis.Can you think of a reason anyone would want to look at LLL rather than Minkowski?
Message: 2159 - Contents - Hide Contents Date: Sat, 01 Dec 2001 20:28:38 Subject: Re: Survey X From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> When I mentioned it before, as two interlaces diatonic scales, each > completing the other's triads into tetrads, you seemed interested and > said you'd print it out.My printer ran out of ink, and I haven't replaced it. I needed to print out your 22-et paper mostly, and I presume this isn't in there. What would I print out here?
Message: 2160 - Contents - Hide Contents Date: Sat, 01 Dec 2001 20:51:11 Subject: Re: RMS mean generators? From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Why not the root-mean-square of the deviations from zero?Good point--the list was supposed to cover the reciprocals also, so that we got the entire vertex figure, and the mean would be zero. I'd better go change things.
Message: 2161 - Contents - Hide Contents Date: Sat, 01 Dec 2001 00:50:24 Subject: Re: Graham's Top Ten From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., genewardsmith@j... wrote:>> It's simply a way of trying to interpret what the area defined by > the>> paralleogram of two generators or two vals--which is an invariant > of>> the temperament--means in some rough sense. Since area is > assoicated>> to counts of lattice points in the coordiante projections of this >> paralleogram, it seemed a reasonable way to look at it.> Confused. Can you give a low-limit example?Let's try 50/49^64/63 = [-2,4,4,-2,-12,11] We have, in the above order: 5-7: -2 [2 -2]^(-1) [1/2 -1] [0 -1] = [ 0 -1] The determinant is -2, corresponding to the first entry, so we have two classes in the block of 5-7 equivalence classes. Using the inverse matrix columns, we can calculate a block based on 0<=u<1, 0<=v<1 where u = i/2 and v = -i-j. The block turns out to be <1, 5/7> 7-3: 4 [ 0 2]^(-1) [ 1/2 0] [-1 -2] = [-1/4 0] Block: <1,7,1/3,7/3> 3-5: 4 Block <1,5,1/3,5/3> 2-3: -2 Block <1,3> 2-5: -12 Block: <1,2,4,8,16,32,10,20,40,80,160,320> 2-7: 11 Block: <1,2/7,4/7,8/7,16/7,32/7,4/49,8/49,16/49,32/49,64/49> Fodder for some kind of product set here, perhaps.>> I count the number of generator steps of the non-octave generator to >> {3,5,7,5/3,7/3,7/5}, take the mean and then the root-mean-square of >> the deviations from the mean, and finally multiply >> Multiply what? The mean or the root-mean-square of the deviations > from the mean?RMS of deviations from mean.
Message: 2162 - Contents - Hide Contents Date: Sat, 01 Dec 2001 08:30:09 Subject: Survey X From: genewardsmith@xxxx.xxx Not a very exciting list this time. <64/63,875/864> Minkowski reduction: <64/63,250/243> Wedge invariant: [-3,-5,6,28,-18,-1] length = 34.3366 Ets: 7,8,15,22,37,59 Map: [ 0 1] [-3 2] [-5 3] [ 6 2] Generators: a = .1353148451 = 5.006649269 / 37; b = 1 Errors and 37-et: 3: 10.91 11.56 5: 1.80 2.88 7: 5.44 4.15 Measures: 42.461 sc, 264.793 s2c Pretty much 22+15=37 if someone wants to try sharp fifths. <64/63, 4375/4374> Wedge invariant: [4,9,-8,-44,24,5] length = 51.9423 Ets: 7, 27 Map: [ 0 1] [ 4 1] [ 9 1] [-8 4] Generators: a = .14788728 = 3.99295636 / 27; b = 1 Errors and 27-et: 3: 7.90 9.16 5: 10.87 13.69 7: 11.46 8.95 Measures: 71.962 sc; 694.499 s2c This would be excluded by Paul's criterion, and it doesn't seem like a big loss. <50/49, 81/80> Wedge invariant: [2,8,8,-4,-7,8] length = 16.1555 Ets: 12, 26, 38 Map: [0 2] [1 2] [4 0] [4 1] Generators: a = .578042519 (~3/2) = 15.02910557 / 26; b = 1/2 Errors and 26 et: 3: -8.30 -9.65 5: -11.71 -17.08 7: 5.78 0.40 Measures: 33.657 sc; 100.970 s2c Paul has apparently played with this oddball temperament, and Graham has it on his catalog page as "Double Negative". <81/80, 875/864> Minkowski reduction: <81/80, 525/512> Wedge invariant: [1,4,-9,-32,17,4] length = 37.7757 Ets: 7,19,26,45 Map: [ 0 1] [ 1 1] [ 4 0] [-9 8] Generators: a = .577880065 (~3/2) = 26.00460293 / 45; b = 1 Errors and 45-et: 3: -8.50 -8.62 5: -12.49 -12.98 7: -9.93 -8.83 Measures: 52.276 sc; 357.112 s2c An alternative septimal meantone for those who like flat systems.
Message: 2163 - Contents - Hide Contents Date: Sat, 01 Dec 2001 09:51:35 Subject: Orwell = Miracle + Meantone From: genewardsmith@xxxx.xxx The heading is slightly deceptive, because it refers to adding wedge invariants, which are only determined up to sign, but it gives the idea. The wedge invariant represents each 7-limit linear temperament by a pair of 6-dimensional lattice points, of opposite sign. Hence, we can add and subtract wedge invariants, or in general take linear combinations, and produce new linear temperaments. I tried this with Miracle and Meantone with the above result. In particular, Meantone = [1,4,10,12,-13,4] and Miracle = [6,-7,-2,14,20,-25]. The sum gives us [7,-3,8,27,7,-21] and the difference [-5,11,12,-3,-33,29]. The sum is Orwell, but the difference was not on my list; by finding a reduced set of commas from the wedge invariant, I get that it is <225/224,50421/50000>, where the second comma has a Tenney height too high to make the cut. It seems like a pretty good system anyway; it is basically the 29+2 system (also 31+29) with a generator close to 7/5.
Message: 2164 - Contents - Hide Contents Date: Sat, 01 Dec 2001 12:03:00 Subject: Miracle + Ennealimmal From: genewardsmith@xxxx.xxx I thought I'd try another one of these. The two systems I obtained in this way are: <2401/2400, 19683/19600> Wedge invariant: [12,34,20,-49,2,26] length = 69.1448 Ets: 58,72,130 Map: [ 0 2] [ 6 7] [17 7] [10 4] Generators: a = .0693055608 = 9.00972229 / 130; b = 1/2 Errors and 130-et: 3: -.955 -.417 5: -.147 1.376 7: -.493 .405 Measures: 8.9154 sc; 132.4036 s2c <2401/2400,15625/15552> Wedge invariant: [24,20,16,-19,42,-24] length = 62.7136 Map: [ 0 4] [-6 18] [-5 19] [-4 19] Generators: a = .4857631578 (~7/5) = 68.00684209 / 140; b = 1/4 Measures: 11.4125 sc; 147.7268 s2c
Message: 2165 - Contents - Hide Contents Date: Sat, 01 Dec 2001 19:33:31 Subject: RMS mean generators? From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:>>> I count the number of generator steps of the non-octave generator > to>>> {3,5,7,5/3,7/3,7/5}, take the mean and then the root-mean- square > of>>> the deviations from the mean, and finally multiply >>>> Multiply what? The mean or the root-mean-square of the deviations >> from the mean? >> RMS of deviations from mean.Why not the root-mean-square of the deviations from zero?
Message: 2166 - Contents - Hide Contents Date: Sat, 01 Dec 2001 19:36:35 Subject: Re: Survey X From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> Ets: 12, 26, 38 > > Map: > > [0 2] > [1 2] > [4 0] > [4 1] > > Generators: a = .578042519 (~3/2) = 15.02910557 / 26; b = 1/2 > > Errors and 26 et: > > 3: -8.30 -9.65 > 5: -11.71 -17.08 > 7: 5.78 0.40 > > Measures: 33.657 sc; 100.970 s2c > > Paul has apparently played with this oddball temperament, and Graham > has it on his catalog page as "Double Negative".When I mentioned it before, as two interlaces diatonic scales, each completing the other's triads into tetrads, you seemed interested and said you'd print it out.
Message: 2167 - Contents - Hide Contents Date: Sun, 02 Dec 2001 20:41:35 Subject: Re: List cut-off point From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Can you come up with a goodness measure for linear temperaments just > as you did for ETs?Just what I was thinking of--my stuff on goodness of ets connecting to goodness of generator steps should allow for an estimate, and hence a more rational measure.
Message: 2168 - Contents - Hide Contents Date: Sun, 02 Dec 2001 04:25:05 Subject: List cut-off point From: genewardsmith@xxxx.xxx I looked at the step-cents of the 7-limit temperaments on my list, and they range from 2.09 for ennealimmal to 99.5 for the system [0 12] [0 19] [1 28] [1 34] One possibility on the low end is therefore to set 100 sc as the cut-off; which would leave us needing something else on the high end.
Message: 2169 - Contents - Hide Contents Date: Sun, 02 Dec 2001 06:08:21 Subject: double-diatonic system (in 26-tET) From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> When I mentioned it before, as two interlaces diatonic scales, each >> completing the other's triads into tetrads, you seemed interested > and>> said you'd print it out. >> My printer ran out of ink, and I haven't replaced it. I needed to > print out your 22-et paper mostly, and I presume this isn't in there.It is mentioned.> What would I print out here? >Date: Fri, 18 Dec 1998 17:30:43 -0500 >From: "Paul H. Erlich" <PErlich@A...> >To: "'tuning@e...'" <tuning@e...> >Subject: Assymmetrical 14-tone modes in 26-tET >Message-ID: <85B74BA01678D211ACDE00805FBE3C050B6587@M...> > >I decided to look at the modes of the assymmetrical scale 0 2 4 6 8 10 >12 14 15 17 19 21 23 25 in 26-tET -- because all octave species of this >scale is constructed of two identical heptachords, each spanning a ~4/3, >separated by either a ~4/3 or a ~3/2; and because the scale contains 10 >consonant 7-limit tetrads constructed by the generic scale pattern 1, 5, >9, 12 (5 are otonal and 5 are utonal) with a maximum tuning error of 17 >cents. > >In my paper, I introduce the notion of a charateristic dissonance. This >is a dissonant interval which is the same generic size (same number of >scale steps) as a consonant interval. Allowing the 7-limit to define >consonance and allowing errors up to 17 cents, the 14-out-of-26 scale >has three characteristic dissonances (plus their octave inversions and >extensions). Two are "sevenths" of 554 cents instead of the usual 508 >cents, and one is an "eighth" of 554 cents instead of the usual 600 >cents. > >None of the modes of this scale satisfy all the properties for a >strongly tonal mode according to my paper. But a few come close. The >mode > >0 2 4 6 8 10 11 13 15 17 19 21 23 25 > >or in cents, > >0 92 185 277 369 462 508 600 692 785 877 969 1062 1154 > >has all characteristic dissonances disjoint from the tonic tetrad (0 8 >15 21), which is major. The only other mode with this property is the >minor equivalent: > >0 2 4 5 7 9 11 13 15 16 18 20 22 24 > >or in cents > >0 92 185 231 323 415 508 600 692 738 831 923 1015 1108. > >The following modes have one characteristic dissonance which shares a >note with the tonic tetrad, but it approximates the 1 identity and the >11 identity when played along with the tetrad. Therefore the interval >does not disturb the stability of the tonic too much, and the mode can >be considered tonal: > >major: 0 2 4 6 8 10 12 13 15 17 19 21 23 24 > >or in cents > >0 92 185 277 369 462 554 600 692 785 877 969 1062 1108 > >and > >minor: 0 2 3 5 7 9 11 13 15 17 18 20 22 24 > >or in cents > >0 92 138 231 323 415 508 600 692 785 831 923 1015 1108.
Message: 2170 - Contents - Hide Contents Date: Sun, 02 Dec 2001 06:12:33 Subject: Re: List cut-off point From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> I looked at the step-cents of the 7-limit temperaments on my list, > and they range from 2.09 for ennealimmal to 99.5 for the system > > [0 12] > [0 19] > [1 28] > [1 34] > > One possibility on the low end is therefore to set 100 sc as the > cut-off; which would leave us needing something else on the high end.s means scale size and c means error? Did you have a problem with the (s^2)*c or whatever rule that graham used? Would you consider (2^s)*c? Or s*(2^s)*c? The latter should _not_ need a high-end cutoff, as the number of possibilities will be finite (if my brain is working right) . . .
Message: 2172 - Contents - Hide Contents Date: Sun, 02 Dec 2001 06:28:24 Subject: Re: Graham's Top Ten From: Paul Erlich I wrote,>> Confused. Can you give a low-limit example? Gene wrote,> Let's try 50/49^64/63 = [-2,4,4,-2,-12,11]Oh dear . . . how about a couple of 5-limit examples?
Message: 2173 - Contents - Hide Contents Date: Sun, 02 Dec 2001 06:59:29 Subject: Re: List cut-off point From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> s means scale size and c means error?s means steps and c means cents, so sc are units of step-cents.> Did you have a problem with the (s^2)*c or whatever rule that graham > used?I had no problem with it beyond failing to interpret the units; however it weighs the smaller systems more heavily, which might be a good idea from the point of view of practicality. Would you consider (2^s)*c? Or s*(2^s)*c? The latter should> _not_ need a high-end cutoff, as the number of possibilities will be > finite (if my brain is working right) . . .I'd consider anything, but 2^s sounds ferocious!
Message: 2174 - Contents - Hide Contents Date: Sun, 02 Dec 2001 07:05:45 Subject: Re: List cut-off point From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> > I'd consider anything, but 2^s sounds ferocious!Fokker evaluated equal temperaments using 2^n and found 31-tET best!
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