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Message: 2025 - Contents - Hide Contents Date: Tue, 20 Nov 2001 21:03:16 Subject: Re: LLL reduction pairs revised list From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> the LLL reductions > tend to give an interval of equivalence and a generator, which is > perfect.What do you mean, exactly? What do the LLL reductions give that an unreduced basis for a linear temperament don't, in the way of an interval of equivalence and a generator?>>> At least in the approach I envision in this paper. Another >> paper could more specifically address infinite temperaments. >> The scales will presumably be in one of three things: a temperament, > an et, or a p-limit.An ET is a temperament -- but, as I said, some might be in a linear temperament, some might be in a planar temperament, etc.> It seems to me you can't get away from > addressing one or more of these if you are going to work with scales.Right -- but that doesn't mean that I have to talk about ennealimmal temperament if it doesn't give me a scale with a reasonable number of notes (as I said, I have to delimit this project somewhere!)
Message: 2026 - Contents - Hide Contents Date: Tue, 20 Nov 2001 00:05:48 Subject: Re: "most compact" periodicity block From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Is it clear what I mean when I say "most compact" periodicity block >> for a particular equivalence class of bases? Won't it, in general, > be>> delimited by a hexagon in the 5-limit, and a rhombic dodecahedron > in >> the 7-limit? >> It seems to me that depends on how you define your distance.You know how I define distance.> Incidentally, I suspect that instead of using hexagonal blocks, or > rrhombic dodecahedra, ellipsoids would work.No, because then you'd have more than one note within some equivalence class, or no notes within some equivalence class.> I don't think you need a > tiling.You automatically get a tiling if you choose one and only one note from each equivalence class!> Could we use the resulting three or six unison vectors>> as a more complete characterization of a given system, rather than > an>> LLL or some such reduced basis, which has some element of >> arbitrariness because some "second best" reduction might be nearly > as >> good? >> What's the point? I thought you wanted to produce temperaments, not > PBs.Right, but I think any rule characterizing which unison vectors we should use, including relationships they may have with one another, should be evaluated with respect to some sort of non-arbitrary reduced basis, don't you think?
Message: 2027 - Contents - Hide Contents Date: Tue, 20 Nov 2001 21:08:46 Subject: Re: reduced basis for my decatonic From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Gene, what do you get as a reduced basis for my decatonic scale? >> I'm afraid I get <25/24, 28/27, 49/48>, which seems reasonable to me, > if not to you. Why did you expect 64/63 or even 225/224, given that > they clearly have a higher Tenney height?Sorry, Gene, I was thinking in terms of commatic vs. chromatic again, and you weren't. Of course, the reduced basis stuff we've been talking about has no provision for such distinctions. For me, my decatonic scale is MOS or altered-MOS by nature, so should be associated with two commatic UVs and only one chromatic one. The three you gave are all chromatic -- meaning none of them are tempered out in my decatonic scale.>> How about that reduced-basis-for-good-ETs request? >> I'm afraid I've forgotten you made one. Would the idea be to do for > other ets what I just did for 10? Do you only want them in the 7- > limit?That would be great.
Message: 2028 - Contents - Hide Contents Date: Tue, 20 Nov 2001 00:09:28 Subject: Re: LLL reduction pairs revised list From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Actually, we should be taking three at a time, not two. If all > three>> (in the Minkowski reduced basis, hopefully) are commatic UVs, we > get>> an ET. If one is chromatic, we get an MOS. If two are chromatic, a >> planar temperament. If three are chromatic, a JI block. >> This will prevent you from considering a lot of interesting > temperaments. Such as?>> I think the UVs in the reduced basis have to be in Kees's list or >> something like it -- if they aren't, then it seems likely that some >> step in the scale will be smaller than one of the commatic unison >> vectors -- which we shouldn't allow. >> Where is Kees's list? I was thinking of producing a list based on > some mathematical conditions, and using that, but I wondered if these > lists already represent such an effort.Start with Searching Small Intervals * [with cont.] (Wayb.) and follow the link to S2357 * [with cont.] (Wayb.). What Kees doesn't tell you (he told me) is that the unison vectors not in parentheses are the smallest (in octaves or cents) for their level of expressibility (i.e., for their length). The ones in parentheses are included so that you're guaranteed to have the three smallest for any level of expressibility. Perhaps Kees would like to chime in here?
Message: 2029 - Contents - Hide Contents Date: Tue, 20 Nov 2001 21:14:12 Subject: Re: LLL reduced 7-limit kernels of some good ets From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> Here's what I got: > > 34: [126/125, 49/48, 6272/6075]That depends on how you map the 7 in 34.> 53: [225/224, 1728/1715, 4000/3969]This is very heartening -- over on the tuning list, I wrote: "53-tone 7-limit Fokker periodicity block, which approximates the full 53-tET rather well, without redundancy, and with the smallest ratios of any of the 62 versions I tried . . . the unison vectors defining this FPB are 225:224, 1728:1715, and 4000:3969"> This calculation helps to answer the question of which commas we > should list--clearly, 4000/3969 and 2048/2025 are important commas > and belong there.Why is this so clear. Maybe the ETs with 4000/3969 don't satisfy the criteria I wish to employ to delimit the project.
Message: 2030 - Contents - Hide Contents Date: Tue, 20 Nov 2001 00:40:28 Subject: Re: "most compact" periodicity block From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> You know how I define distance.It doesn't work when you want to define blocks, though.>> Incidentally, I suspect that instead of using hexagonal blocks, or >> rrhombic dodecahedra, ellipsoids would work. >> No, because then you'd have more than one note within some > equivalence class, or no notes within some equivalence class.I was thinking of using the minimal diameter definition.>> I don't think you need a >> tiling. >> You automatically get a tiling if you choose one and only one note > from each equivalence class!You bet, which is exaclty why you don't need to require that the convex figures which result from taking everything less than or equal to a certain distance produce a tiling.>>> Could we use the resulting three or six unison vectors>>> as a more complete characterization of a given system, rather > than >> an>>> LLL or some such reduced basis, which has some element of >>> arbitrariness because some "second best" reduction might be > nearly >> as >>> good? >>>> What's the point? I thought you wanted to produce temperaments, not >> PBs. >> Right, but I think any rule characterizing which unison vectors we > should use, including relationships they may have with one another, > should be evaluated with respect to some sort of non-arbitrary > reduced basis, don't you think?
Message: 2031 - Contents - Hide Contents Date: Tue, 20 Nov 2001 22:32:15 Subject: Re: LLL reduction pairs revised list From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> My last list allowed in some duplicates; the following list of 81 > pairs is from an improved version of the program:I'm going to keep the ones where both UVs appear on my original list, until we have a firmer basis for making such a list. Commas like 4000/3969 seem like unlikely choices, since there are three or more commas that are both shorter vectors _and_ smaller musical intervals. Meanwhile, the superparticulars with smaller numbers than 50:49 lead to too much tempering for my taste.> {81/80, 126/125} > {64/63, 245/243} > {50/49, 81/80} > {64/63, 126/125} > {1728/1715, 3136/3125} > {81/80, 3136/3125} > {1029/1024, 4375/4374} > {3136/3125, 4375/4374} > {225/224, 1029/1024} > {50/49, 64/63} > {126/125, 245/243} > {50/49, 245/243} > {50/49, 6144/6125} > {245/243, 225/224} > {50/49, 4375/4374} > {126/125, 1029/1024} > {225/224, 4375/4374} > {81/80, 6144/6125} > {81/80, 1728/1715} > {64/63, 4375/4374} > {49/48, 225/224} > {81/80, 225/224} > {245/243, 1029/1024} > {245/243, 3136/3125} > {1029/1024, 3136/3125} > {2401/2400, 3136/3125} > {81/80, 2401/2400} > {2401/2400, 6144/6125} > {126/125, 1728/1715} > {225/224, 1728/1715} > {64/63, 225/224} > {64/63, 3136/3125} > {4375/4374, 6144/6125} > {2401/2400, 4375/4374}Still a whopping 34 possibilities! Who can be quickest to the draw to give the generator, period, and 7-limit complexity for all 34 linear temperaments? If Gene did this right, they should all be distinct . . . and this very well might be all the "interesting" ones for my present purposes.
Message: 2032 - Contents - Hide Contents Date: Tue, 20 Nov 2001 00:45:44 Subject: Re: "most compact" periodicity block From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> You know how I define distance. >> It doesn't work when you want to define blocks, though.What do you mean?>>>> Incidentally, I suspect that instead of using hexagonal blocks, > or>>> rrhombic dodecahedra, ellipsoids would work. >>>> No, because then you'd have more than one note within some >> equivalence class, or no notes within some equivalence class. >> I was thinking of using the minimal diameter definition.I could tell.>>>> I don't think you need a >>> tiling. >>>> You automatically get a tiling if you choose one and only one note >> from each equivalence class! >> You bet, which is exaclty why you don't need to require that the > convex figures which result from taking everything less than or equal > to a certain distance produce a tiling.??? I don't want to simply produce convex figures which result from taking everything less than or equal to certain distance! I want to define equivalence relations (i.e. a kernel), and _then_ use a block (possibly not unique up to reflections and such small changes) which, given that there's one and only one note from each equivalence class, is as compact as possible. Then, there should generally be three or six operative unison vectors, right?>>>>> Could we use the resulting three or six unison vectors>>>> as a more complete characterization of a given system, rather >> than >>> an>>>> LLL or some such reduced basis, which has some element of >>>> arbitrariness because some "second best" reduction might be >> nearly >>> as >>>> good? >>>>>> What's the point? I thought you wanted to produce temperaments, > not >>> PBs. >>>> Right, but I think any rule characterizing which unison vectors we >> should use, including relationships they may have with one another, >> should be evaluated with respect to some sort of non-arbitrary >> reduced basis, don't you think?You didn't answer.
Message: 2033 - Contents - Hide Contents Date: Tue, 20 Nov 2001 22:34:12 Subject: Re: LLL reduction pairs revised list From: Paul Erlich I wrote,> If Gene did this right, they should all be > distinct . . .Well, the least-squares generators should be distinct . . . some of the minimax generators might turn out to be the same (?)
Message: 2034 - Contents - Hide Contents Date: Tue, 20 Nov 2001 01:36:04 Subject: Re: LLL reduction pairs revised list From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., genewardsmith@j... wrote:>> This will prevent you from considering a lot of interesting >> temperaments. > Such as?If you take 2401/2400 and 4375/4374 by themselves, you get ennealimmal temperament. If you try to add 25/24, 28/27, 36/35 you get garbage. Besides, there's no point in it that I can see--why add anything when two commas are all you need to define a 7-limit linear temperament?
Message: 2035 - Contents - Hide Contents Date: Tue, 20 Nov 2001 01:46:12 Subject: Re: LLL reduction pairs revised list From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> --- In tuning-math@y..., genewardsmith@j... wrote: >>>> This will prevent you from considering a lot of interesting >>> temperaments. > >> Such as? >> If you take 2401/2400 and 4375/4374 by themselves, you get > ennealimmal temperament.I know you love that one.> If you try to add 25/24, 28/27, 36/35 you > get garbage.These are not the only canditates. You left out 49/48. Also, 50/49 and 64/63 can be used as _either_ commatic _or_ chromatic -- I don't know if I mentioned that but I think so. Other candidates may emerge once we have a solid foundation for all this. Anyway, I'd just like to set some reasonable bounds within which we can flesh out the possibilities. If someone wants to use 36/35 as a commatic unison vector, I'm all for that, but I'd like to start with a digestible array of possibilities just for the sake of presentation.> Besides, there's no point in it that I can see--why add > anything when two commas are all you need to define a 7-limit linear > temperament?The point is not so much infinite temperaments but rather finite scales. At least in the approach I envision in this paper. Another paper could more specifically address infinite temperaments.
Message: 2036 - Contents - Hide Contents Date: Tue, 20 Nov 2001 03:27:23 Subject: reduced basis for my decatonic From: Paul Erlich Gene, what do you get as a reduced basis for my decatonic scale? Looking at the most compact lattice arrangement, it appears that (64/63, 50/49, 49/48) should win, though 225/224 could replace 63/64 without too much damage, and 28/27 and 25/24, while both weak replacements for 49/48, are almost equally good in that role. How about that reduced-basis-for-good-ETs request?
Message: 2037 - Contents - Hide Contents Date: Tue, 20 Nov 2001 04:37:11 Subject: Re: LLL reduction pairs revised list From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> The point is not so much infinite temperaments but rather finite > scales.Given two generators, you are set to make scales; the LLL reductions tend to give an interval of equivalence and a generator, which is perfect. What I like about starting from two commas is your idea to use this as a canonical scheme for classification, but certainly one can go on to scales. At least in the approach I envision in this paper. Another> paper could more specifically address infinite temperaments.The scales will presumably be in one of three things: a temperament, an et, or a p-limit. It seems to me you can't get away from addressing one or more of these if you are going to work with scales. Scales are also less easy to classify than temperaments, because there are more reasonable possibilities. Moreover, if you are willing to restrict yourself to et scales, my a;n+m notation already does classify them.
Message: 2038 - Contents - Hide Contents Date: Tue, 20 Nov 2001 05:10:08 Subject: Re: reduced basis for my decatonic From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Gene, what do you get as a reduced basis for my decatonic scale?I'm afraid I get <25/24, 28/27, 49/48>, which seems reasonable to me, if not to you. Why did you expect 64/63 or even 225/224, given that they clearly have a higher Tenney height?> How about that reduced-basis-for-good-ETs request?I'm afraid I've forgotten you made one. Would the idea be to do for other ets what I just did for 10? Do you only want them in the 7- limit?
Message: 2039 - Contents - Hide Contents Date: Tue, 20 Nov 2001 08:19:16 Subject: LLL reduced 7-limit kernels of some good ets From: genewardsmith@xxxx.xxx Here's what I got: 10: [49/48, 28/27, 25/24] 12: [64/63, 50/49, 36/35] 15: [126/125, 49/48, 28/27] 19: [126/125, 81/80, 49/48] 22: [225/224, 245/243, 64/63] 27: [126/125, 245/243, 64/63] 31: [225/224, 1728/1715, 81/80] 34: [126/125, 49/48, 6272/6075] 41: [225/224, 4000/3969, 245/243] 46: [1029/1024, 126/125, 245/243] 53: [225/224, 1728/1715, 4000/3969] 58: [1728/1715, 126/125, 2048/2025] 68: [4000/3969, 245/243, 2048/2025] 72: [4375/4374, 225/224, 1029/1024] 99: [4375/4374, 6144/6125, 3136/3125] 171: [4375/4374, 2401/2400, 32805/32768] This calculation helps to answer the question of which commas we should list--clearly, 4000/3969 and 2048/2025 are important commas and belong there.
Message: 2040 - Contents - Hide Contents Date: Tue, 20 Nov 2001 08:48:32 Subject: Re: LLL reduced 7-limit kernels of some good ets From: genewardsmith@xxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote: Some of these temperaments derive from a comma shared by two reduced bases: 10&15: [49/48, 28/27] 15&34: [49/48, 126/125] 22&27: [64/63, 245/243] 41&53: [225/224, 4000/3969] 41&68: [245/243, 4000/3969] This might be one place to start.
Message: 2041 - Contents - Hide Contents Date: Tue, 20 Nov 2001 12:45 +0 Subject: Re: LLL definitions From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9t7fj7+odag@xxxxxxx.xxx> Gene wrote:> Starting from an ordered lattice basis [b_1, b_2, ..., b_m] we can > obtain another basis [g_1, g_2, ..., g_m] by the Gram-Schmidt > orthogonalization process. This will span the same linear subspace, > but *not*, normally, the same lattice. We also obtain Gram-Schmidt > coefficients:What's an ordered basis?> c[i,j] = <b_i, g_j>/<g_j, g_j>I've implemented this in Python code: import operator def dotprod(x,y): return reduce(operator.add, map(operator.mul, x, y)) def makec(b,g): c = [] for i in range(len(g)): row = [] for j in range(len(g)): row.append(float(dotprod(b[i],g[j])) / dotprod(g[j], g[j])) c.append(row) return c Please say if I'm going wrong anywhere. One thing that worries me is that I'm getting floating point results where you stay with integers.> We do this by the standard Gram-Schmidt recursion from linear algebra: > > g_1 = b_1 > g_i = b_i - sum_{j = 1 to i - 1} c[i,j] g_jAgain, I've implemented that def LLL(b, g): c = makec(b,g) newg = [] for i in range(len(g)): row = [] for k in range(len(g)): thing = b[i][k] for j in range(i): thing -= c[i][j] * j[j][k] row.append(thing) newg.append(row) return newg I'm assuming that repeatedly applying this will give the right result, but it clearly doesn't. For one thing, the first ratio can never change. Also, you haven't said what seed value to use for g. I'm using the initial value of b. If you can supply an alternative algorithm in some kind of procedural code, that would be cood too. I've found LiDIA, and it comes with the source code but an imperfect license. I'll have a look at it sometimt. Graham
Message: 2042 - Contents - Hide Contents Date: Wed, 21 Nov 2001 19:42:39 Subject: Re: LLL reduction pairs revised list From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> I'm going to keep the ones where both UVs appear on my original > list,>> until we have a firmer basis for making such a list. Commas like >> 4000/3969 seem like unlikely choices, since there are three or more >> commas that are both shorter vectors _and_ smaller musical > intervals. >> However, 4000/3969 shows up as part of the reduced basis for 41, 53 > and 68, which are all important ets.Maybe not for the conditions I have in mind.> I conclude therefore it's > significant, and in any case this gives us a way of picking them. > > The alternative approach, of course, is to start from pairs of ets; I > think perhaps we should do it both ways as a check. >>> Meanwhile, the superparticulars with smaller numbers than 50:49 > lead>> to too much tempering for my taste. >> I think drawing the line between 50/49 and 49/48 is a little absurd; > why not between 49/48 and 36/35?I wouldn't have a big problem with that -- I'm trying to keep the set of outcomes as small as possible, without introducing a larger number of conditions than is reasonable.
Message: 2043 - Contents - Hide Contents Date: Wed, 21 Nov 2001 19:46:04 Subject: Re: Start of survey From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> I did the first three pairs on my list, and got the following. (All > turned out to be Minkowski reduced according to Tenney height.) > > <1728/1715, 2048/2025> > > ets: 14, 22, 58, 80 > > LLL reduced map: > > [ 0 2] > [-3 4] > [ 6 3] > [-5 7] > > Generators: a = 0.1376381046 = 11.01104837 / 80; b = 1/2 > > Appromimately 58+22 in the 80-et. > > Errors: > > 3: 2.55 > 5: 4.68 > 7: 5.35 > > Extension of map to the 11-limit: > > [ 0 2] > [-3 4] > [ 6 3] > [-5 7] > [ 7 5] > > <225/224, 49/48> > > ets: 9, 10, 19, 29 > > LLL-reduced map: > > [-1 1] > [-2 -2] > [-2 5] > [-3 1] > > Adjusted map: > > [ 0 1] > [-4 2] > [ 3 2] > [-2 3] > > Generator a = 0.1045573299 = 1.986589268 / 19 > > This system is closely related to 10+9 in the 19-et, and also related > to 19+10. > > Errors: > > 3: -3.83 > 5: -9.91 > 7: -19.76 > > <245/243, 50/49> > > Map: > > [-2 -2] > [-1 5] > [-1 9] > [-2 8] > > Adjusted map: > > [0 2] > [3 1] > [5 1] > [5 2] > > Generator: 0.3629853525 = 7.985677755 / 22 > > Errors: > > 3: 4.79 > 5: -8.40 > 7: 9.09 > > This one may as well be taken as the generator 8/22 in the 22-et; > this is a supermajor third (9/7), and we have two parallel chains > separated by sqrt(2).Then shouldn't you have said a = 0.3629853525 = 7.985677755 / 22, b = 1/2 above, similar to what you did for the first example?
Message: 2044 - Contents - Hide Contents Date: Wed, 21 Nov 2001 19:50:10 Subject: Re: Start of survey From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Then shouldn't you have said > > a = 0.3629853525 = 7.985677755 / 22, b = 1/2 > > above, similar to what you did for the first example?I got lazy, but I suppose I'd better do it systematically. I also left out b=1 when that was a generator.
Message: 2045 - Contents - Hide Contents Date: Wed, 21 Nov 2001 23:11:00 Subject: Re: LLL definitions From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:> Oh dear. I wanted something like this: > > (2, -1, 2, -1, 0) > (0, -3, 1, 1, 1) > (-3, 1, -1, -1, 1) > (-3, 0, 0, 1, -1)I can certainly LLL reduce this, after adding in the 2s; I get <196/195, 364/363, 441/440, 1575/1573>> So it looks like this off the shelf LLL algorithm isn't what I want at > all. Do you know of any way of doing that kind of reduction? Or, more > specifically, of getting a simple set of unison vectors for the consistent > 29+58 temperament?The above does it; however you did most of the work, so I presume you must have some idea how to proceed. One can brute force it by first getting a 13-limit notation with a basis of about the right size, dual to a set of ets containing 29 and 58, and then searching for elements of the kernel of 29&58--one should not need exponents beyond +-2, so a search would be feasible. If we find something of rank 4 which can be extended to make a basis for the kernel of both 29 and 58, we are ready to LLL reduce it. Since you did the above calculation, however, perhaps you have another idea.
Message: 2046 - Contents - Hide Contents Date: Wed, 21 Nov 2001 00:08:11 Subject: Two versions of 12&34 in the 11-limit From: genewardsmith@xxxx.xxx From 12 and 34, we can obtain a unique generator/period for the 46=12+34 et, which I denote by "34+12", by the following procedure: (1) Find the penultimate convergent to 12/34, obtaining 1/3 (2) Take the mediant of 1/12 and 3/34, obtaining 4/46 = 2/23 (3) Find the mapping to primes of 46, obtaining [46, 73, 107, 129, 159]. Note that this does *not* require us to even look at mappings for 12 or 34, much less worry about validity! (4) Taking our period of 23 steps and our generator of 4/46, we calculate generator steps: 73/4 = 1 mod 23 107/4 = -2 mod 23 129/4 = -8 mod 23 159/4 = 11 mod 23 (5) Our other generator is 1/2, and we find the corresponding number of steps for it: 73/46 - 2/23 = 3/2 107/46 + 2(2/23) = 5/2 129/46 + 8(2/23) = 7/2 159/46 - 11(2/23) = 9/2 (6) Since the mapping to number of generator steps from an interval is a val, and we represent vals by column vectors, we can put our results together in a 5x2 matrix: [ 0 2] [ 1 3] [-2 5] [-8 7] [11 5] (7) We now have all we need so far as the 46-et goes; however we may also detemper using the above map and linear programming or least squares to find an optimal tuning. Using least squares in the 11- limit gives a generator a1 = .08700594368 = 4.002273409 / 46; this is not much different from the 46-et and gives similar tuning errors. However, the map to primes was only unique mod 23, and we might have used instead -12 = 11 mod 23 for the number of steps we mapped 11 to. We obtain instead the map: [ 0 2] [ 1 3] [ -2 5] [ -8 7] [-12 9] Since the other maps to primes have a negative tendency, this seems like it is probably the best plan. If we adopt it, we get instead a2 = .08648628 = 3.97836888 / 46 as our generator, which is quite a bit farther from 46-et than the other system. A comparison of tunings shows: 3: 2.39 2.45 1.83 5: 4.99 4.87 6.12 7: -3.61 -4.08 0.91 11: -3.49 -2.84 3.28 Here the first column is the 46-et, the second our first detempering, and the third the alternative--which looks pretty good!
Message: 2047 - Contents - Hide Contents Date: Wed, 21 Nov 2001 05:04:25 Subject: Re: LLL reduction pairs revised list From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> I'm going to keep the ones where both UVs appear on my original list, > until we have a firmer basis for making such a list. Commas like > 4000/3969 seem like unlikely choices, since there are three or more > commas that are both shorter vectors _and_ smaller musical intervals.However, 4000/3969 shows up as part of the reduced basis for 41, 53 and 68, which are all important ets. I conclude therefore it's significant, and in any case this gives us a way of picking them. The alternative approach, of course, is to start from pairs of ets; I think perhaps we should do it both ways as a check.> Meanwhile, the superparticulars with smaller numbers than 50:49 lead > to too much tempering for my taste.I think drawing the line between 50/49 and 49/48 is a little absurd; why not between 49/48 and 36/35? The schisma showed up in the reduced basis for 171; perhaps we should include that, then look at 130 and 140 and call it a day?
Message: 2048 - Contents - Hide Contents Date: Wed, 21 Nov 2001 05:25:03 Subject: Re: LLL reduction pairs revised list From: genewardsmith@xxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote:> I think drawing the line between 50/49 and 49/48 is a little absurd; > why not between 49/48 and 36/35? The schisma showed up in the reduced > basis for 171; perhaps we should include that, then look at 130 and > 140 and call it a day? I get130: <2401/2400, 3136/3125, 19683/19600> 140: <2401/2400, 5120/5103, 15625/15552>
Message: 2049 - Contents - Hide Contents Date: Wed, 21 Nov 2001 06:25:17 Subject: Yet another revised list From: genewardsmith@xxxx.xxx This started from the commas 49/48, 50/49, 64/63, 81/80, 2048/2025, 245/243, 126/125, 4000/3969, 1728/1715, 1029/1024, 225/224, 3136/3125, 5120/5103, 6144/6125, 2401/2400, 4375/4374 I obtained the following 72 reduced pairs; the number following the pair is the ratio between the largest and the smallest comma. I think some bound needs to be placed on this, as the [4375/4374, 50/49] system is obviously a little absurd. [1728/1715, 2048/2025] 1.495580025 [225/224, 49/48] 4.629021956 [245/243, 50/49] 2.464716366 [5103/5000, 49/48] 1.011210863 [2401/2400, 3136/3125] 8.434916361 [49/48, 28/27] 1.763768279 [5120/5103, 1728/1715] 2.270583084 [64/63, 50/49] 1.282845376 [4000/3969, 245/243] 1.053543673 [3136/3125, 245/243] 2.332723121 [126/125, 49/48] 2.587706812 [3645/3584, 50/49] 1.197064798 [6144/6125, 81/80] 4.010835974 [245/243, 64/63] 1.921288732 [225/224, 1728/1715] 1.695329007 [126/125, 245/243] 1.028688881 [126/125, 81/80] 1.559018011 [2401/2400, 2048/2025] 27.11124152 [4375/4374, 6144/6125] 13.54885592 [1029/1024, 686/675] 3.318654575 [3136/3125, 49/48] 5.868055561 [225/224, 4000/3969] 1.746649139 [81/80, 128/125] 1.909155841 [4375/4374, 225/224] 19.48552996 [1728/1715, 81/80] 1.645020488 [64/63, 686/675] 1.026452250 [1029/1024, 245/243] 1.682792874 [4375/4374, 2401/2400] 1.822327650 [50/49, 525/512] 1.241102862 [875/864, 50/49] 1.596910924 [4000/3969, 2048/2025] 1.451636818 [2048/2025, 50/49] 1.788798893 [4375/4374, 3136/3125] 15.37118131 [3136/3125, 64/63] 4.481834648 [4375/4374, 2048/2025] 49.40556506 [225/224, 64/63] 3.535500094 [225/224, 1029/1024] 1.093522071 [2401/2400, 5120/5103] 7.983665859 [2401/2400, 81/80] 29.82023498 [2401/2400, 6144/6125] 7.434917601 [1029/1024, 126/125] 1.635861828 [875/864, 64/63] 1.244819488 [49/48, 2240/2187] 1.161297413 [4375/4374, 64/63] 68.89109299 [81/80, 50/49] 1.626297004 [81/80, 875/864] 1.018401828 [3136/3125, 1029/1024] 1.386221179 [245/243, 2048/2025] 1.377861075 [6144/6125, 5120/5103] 1.073806905 [4375/4374, 50/49] 88.37662008 [126/125, 64/63] 1.976408356 [49/48, 250/243] 1.377325721 [49/48, 25/24] 1.979796637 [225/224, 245/243] 1.840171149 [1728/1715, 126/125] 1.055164518 [6144/6125, 4000/3969] 2.511974761 [49/48, 6272/6075] 1.547739961 [3136/3125, 81/80] 3.535332623 [49/48, 392/375] 2.150210807 [4375/4374, 1029/1024] 21.30785708 [64/63, 36/35] 1.788813730 [4000/3969, 875/864] 1.626068363 [225/224, 81/80] 2.788850951 [50/49, 128/125] 1.173928155 [81/80, 49/48] 1.659831249 [1728/1715, 4000/3969] 1.030271488 [126/125, 2048/2025] 1.417390368 [36/35, 21/20] 1.731936266 [50/49, 36/35] 1.394411021 [3136/3125, 1728/1715] 2.149111606 [6144/6125, 50/49] 6.522810530 [5120/5103, 3136/3125] 1.056521717
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