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Message: 3750 - Contents - Hide Contents Date: Sat, 02 Feb 2002 01:42:26 Subject: Re: interval of equivalence, unison-vector, period From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Well I confused the two things, which is completely my fault, but was > not helped by Graham's opinion that the thing you declined to call a > temperament was in fact pajara.If you ever get around to trying 222223 with a period of 3/2 in the 22-et, tell us about it.
Message: 3751 - Contents - Hide Contents Date: Sat, 02 Feb 2002 10:11:29 Subject: Re: 43-edo (was: 171-EDO, Vogel (was: 7-limit MT reduced bases forets)) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> ------------ >>> From: monz <joemonz@y...> >> To: <tuning-math@y...> >> Sent: Saturday, February 02, 2002 1:50 AM >> Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reduced> bases for ets) >>>> from manuel's page: >> Stichting Huygens-Fokker: Logarithmic Interval... * [with cont.] (Wayb.) >>>>>> ... Sauveur ... found 43 to be optimal >>>> because 4 steps is almost exactly a 16/15 minor second >>>> and 7 steps almost exactly the geometric mean of >>>> three 9/8 and two 10/9 whole tones. The chromatic scale >>>> contained in 43-tET is virtually identical to 1/5-comma >>>> meantone tuning. > > > > > >> >> >>>> [-9 6 0] = 3 * [-3 2 0] (= 9:8 whole tone) >> + >> [ 2 -4 2] = 2 * [ 1 -2 1] (= 10:9 whole tone) >> ---------- >> [-7 2 2] (= 225:128 "augmented 6th") >> >> >> [7 2 2]^(1/2) = [-7/2 1 1] = ~488.2687147 cents >> >> >> but what significance does that have? i don't get it >> >> manuel? > > >> the only thing that i think i can see is some kind of > tritone-equivalence in action, because if you ignore > prime-factor 2 you get a mean for the 225:128 of 15:8, > which is 2^(1/2) higher than the above interval, and > which is the interval that is given exactly by 5 generators > of 1/5-comma meantone > > but i really don't understand what's going onyou have to take the weighted mean of three 9/8 and two 10/9 whole tones. that means three 9/8s "plus" two 10/9s "divided" by five, that is, ( (9/8)^3 * (10/9)^2 )^(1/5).
Message: 3752 - Contents - Hide Contents Date: Sat, 2 Feb 2002 02:52:10 Subject: Re: 43-edo (was: 171-EDO...) From: monz ----- Original Message ----- From: paulerlich <paul@xxxxxxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Saturday, February 02, 2002 2:11 AM Subject: [tuning-math] Re: 43-edo (was: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets))> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >> ------------ >>>>> From: monz <joemonz@y...> >>> To: <tuning-math@y...> >>> Sent: Saturday, February 02, 2002 1:50 AM >>> Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT > reduced>> bases for ets) >>>>>> from manuel's page: >>> Stichting Huygens-Fokker: Logarithmic Interval... * [with cont.] (Wayb.) >>>>>>>> ... Sauveur ... found 43 to be optimal >>>>> because 4 steps is almost exactly a 16/15 minor second >>>>> and 7 steps almost exactly the geometric mean of >>>>> three 9/8 and two 10/9 whole tones. The chromatic scale >>>>> contained in 43-tET is virtually identical to 1/5-comma >>>>> meantone tuning. >> >> >>>>>> [-9 6 0] = 3 * [-3 2 0] (= 9:8 whole tone) >>> + >>> [ 2 -4 2] = 2 * [ 1 -2 1] (= 10:9 whole tone) >>> ---------- >>> [-7 2 2] (= 225:128 "augmented 6th") >>> >>> >>> [7 2 2]^(1/2) = [-7/2 1 1] = ~488.2687147 cents >>> >>> >>> but what significance does that have? i don't get it >>> >>> manuel? >> >> >>>> the only thing that i think i can see is some kind of >> tritone-equivalence in action, because if you ignore >> prime-factor 2 you get a mean for the 225:128 of 15:8, >> which is 2^(1/2) higher than the above interval, and >> which is the interval that is given exactly by 5 generators >> of 1/5-comma meantone >> >> but i really don't understand what's going on >> you have to take the weighted mean of three 9/8 and two > 10/9 whole tones. that means three 9/8s "plus" two 10/9s > "divided" by five, that is, ( (9/8)^3 * (10/9)^2 )^(1/5).so that's what manuel means by "geometric mean"? i would have never understood it that way ok, i see it now ... but it took a little bit of work to comprehend what's going on there ... perhaps you'd like to reword that a bit, manuel? isn't there a better way to say that, with a quantifiable numeric? ... perhaps "almost exactly the geometric mean 1/5 part of the 5-tone interval composed of three 9/8 and two 10/9 whole tones"? anyway, 1/5-comma meantone whole-tone = ( (9/8)^3 * (10/9)^2 )^(1/5) = [2 3 5]^[-7/5 2/5 2/5] (did i write that correctly?) = ~195.3074859 cents 7 steps of 47-edo = 2^(7/43) = ~195.3488372 cents The difference between them is [2 3 5]^[336/215 -2/5 -2/5] ~0.041351317 cent = ~1/24 cent = ~1 jot and paul, there might be a hint of an answer (from the "new cylindrical meantones lattice" thread) as to how to draw spirals which compare a meantone-like edo with a fraction-of-a-comma meantone we need to include an axis for 2 ... but hmmm ... this pair of points would be rather far apart, since the exponents of 3 and 5 have their signs reversed and of course, what would including 2 mean when the lattice is wrapped into a cylinder? my spatial imagination can't handle this ... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3753 - Contents - Hide Contents Date: Sat, 02 Feb 2002 03:31:17 Subject: 7-limit MT reduced bases for ets From: genewardsmith 9: [21/20, 27/25, 128/125] 10: [25/24, 28/27, 49/48] 12: [36/35, 50/49, 64/63] 15: [28/27, 49/48, 126/125] 19: [49/48, 81/80, 126/125] 22: [50/49, 64/63, 245/243] 27: [64/63, 126/125, 245/243] 31: [81/80, 126/125, 1029/1024] 41: [225/224, 245/243, 1029/1024] 68: [245/243, 2048/2025, 2401/2400] 72: [225/224, 1029/1024, 4375/4374] 99: [2401/2400, 3136/3125, 4375/4374] 130: [2401/2400, 3136/3125, 19683/19600] 140: [2401/2400, 5120/5103, 15625/15552] For any prime limit, we could consider the most characteristic linear temperament of a particular et to be the one leaving off the last member of the MT reduced basis. It is interesting to note that the characteristic linear temperament of 99 and 130 is the same. Of course we can do the same for planar temperaments, etc.
Message: 3754 - Contents - Hide Contents Date: Sun, 03 Feb 2002 04:25:11 Subject: For Carl--5, 7, and 11-limit reduced bases for the 37-et From: genewardsmith 5: [250/243, 262144/253125] 7: [64/63, 250/243, 686/675] 11: [55/54, 64/63, 100/99, 686/675]
Message: 3756 - Contents - Hide Contents Date: Sun, 03 Feb 2002 04:38:36 Subject: Re: any ideas? From: paulerlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:> Here's something along the lines of what I'm looking for: > > All maximally even subsets favor palindromic symmetry. In fact, these > subsets are always the most palindromic or least skewed rotations or > modes--and if they're actual palindromes, then they're also unique, as > they have zero skew and therefore no inversion amongst their > rotations. > > This is the sort of a special condition that I had in mind... though > I'd want it to work with any given scale, not just scales with > Myhill's property. Any ideas?how about the rotation where the scale's center of gravity is closest to 600 cents?
Message: 3757 - Contents - Hide Contents Date: Sun, 3 Feb 2002 20:03:02 Subject: Re: Gene's notation formula: alternate duodene? From: monz my generalization of Gene's periodicity-block finding formula:> for a set of i rational unison-vectors {u1/v1, ... ui/vi}, > for any non-zero I can define a scale by calculating for 0<=n<d > > step[n] = (u1/v1)^round(7n/d) (u2/v2)^round(12n/d) > (u3/v3)^round(7n/d) (u4/v4)^round(-2n/d) (u5/v5)^round(5n/d) > From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 5:23 PM > Subject: [tuning-math] Re: Gene's notation formula: alternate duodene? > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> Gene, can you explain why your formula gave this >> result instead of what i expected? Is there a >> "correction factor" involved? >> My formula said to round to the nearest integer, but it > didn't say what to do when two integers are equally near, > which can happen when you have an even number of tones > in an octave. The different rounding choices then lead > to different blocks, which I think are equally correct.so then i guess i'm just getting the particular results i am because of the way Excel handles the rounding, yes?> The proposal for defining blocks a while back involved > defining a distance function designed to work with a > particular block problem in mind. In this case, it would give > > ||q|| = max(|h12(q)|, |12 h7(q) - 7 h12(q)|, |12 h3(q) - 3 h12(q)| > > If you take everything at a distance of less than six from > the unison using this measure, and transpose to the standard > octave (instead of the octave from 2^(-1/2) to 2^(1/2)) you > obtain the nine note scale > > 1--16/15--6/5--5/4--4/3--3/2--8/5--5/3--15/8 > > This is the core of the block, in every version of it. > If you now take everything at a distance of exactly six > from one, you get > {10/9, 9/8, 25/18, 45/32, 64/45, 36/25, 16/9, 9/5}. > To get a block, you add three of these to the core of the > block, in such a way that the diameter of the resulting > block is less than twelve: the diameter being the maximum > of all the distances between members of the block. > > You can therefore add 9/5, 9/8 and 45/32, getting what you > expected, or 64/45, 16/9, 10/9, which amounts to the same > thing. However, you could also add 36/25, 9/5 and 9/8, > which you didn't expect, or 16/9, 10/9, and 25/18.in fact, i just corrected an error in my spreadsheet and the last one is exactly what i got: i had my spreadsheet doing an incorrect calculation because i didn't divide the unimodular adjoint by the determinant first, as i should have. so, according to the way my Excel spreadsheet is handling the rounding in Gene's formula, here's one version of the 12-tone JI PB scale for Ellis's Duodene: kernel: 2 3 5 ratio ~cents unison-vector [-3 -1 2] = 25:24 70.6724269 chromatic [ 7 0 -3] = 128:125 41.0588584 commatic [-4 4 -1] = 81:80 21.5062896 commatic adjoint: [12 7 3] [19 11 5] [28 16 7] determinant = | 1 | JI periodicity-block: 2 / 1 1200 15 / 8 1088.268715 16 / 9 996.0899983 5 / 3 884.358713 8 / 5 813.6862861 3 / 2 701.9550009 25 / 18 568.717426 4 / 3 498.0449991 5 / 4 386.3137139 6 / 5 315.641287 10 / 9 182.4037121 16 / 15 111.7312853 1 / 1 0 triangular lattice: F# 25:18 / \ / \ / \ D A E B 10:9----5:3-----5:4-----15:8 / \ / \ / \ / / \ / \ / \ / / \ / \ / \ / Bb F C G 16:9----4:3-----1:1-----3:2 / \ / \ / / \ / \ / / \ / \ / Db Ab Eb 16:15----8:5-----6:5 -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3758 - Contents - Hide Contents Date: Sun, 03 Feb 2002 04:39:58 Subject: Re: For Carl--5, 7, and 11-limit reduced bases for the 37-et From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> 5: [250/243, 262144/253125] > 7: [64/63, 250/243, 686/675] > 11: [55/54, 64/63, 100/99, 686/675]given the notation you've been using for these, i thought you were talking about 5-equal, 7-equal, and 11-equal.
Message: 3759 - Contents - Hide Contents Date: Sun, 3 Feb 2002 20:17:58 Subject: Re: Gene's notation formula: alternate duodene? From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 5:23 PM > Subject: [tuning-math] Re: Gene's notation formula: alternate duodene? > > >>> The proposal for defining blocks a while back involved >> defining a distance function designed to work with a >> particular block problem in mind. In this case, it would give >> >> ||q|| = max(|h12(q)|, |12 h7(q) - 7 h12(q)|, |12 h3(q) - 3 h12(q)| >> >> If you take everything at a distance of less than six from >> the unison using this measure, and transpose to the standard >> octave (instead of the octave from 2^(-1/2) to 2^(1/2)) you >> obtain the nine note scale >> >> 1--16/15--6/5--5/4--4/3--3/2--8/5--5/3--15/8 >> >> This is the core of the block, in every version of it. >> If you now take everything at a distance of exactly six >> from one, you get >> {10/9, 9/8, 25/18, 45/32, 64/45, 36/25, 16/9, 9/5}. >> To get a block, you add three of these to the core of the >> block, in such a way that the diameter of the resulting >> block is less than twelve: the diameter being the maximum >> of all the distances between members of the block. >> >> You can therefore add 9/5, 9/8 and 45/32, getting what you >> expected, or 64/45, 16/9, 10/9, which amounts to the same >> thing. However, you could also add 36/25, 9/5 and 9/8, >> which you didn't expect, or 16/9, 10/9, and 25/18.i just realized that i mention exactly the same thing on my webpage "Ellis's Duodene and a "best-fit" meantone" Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)>> Note that this periodicity-block has three pitch-classes >> which fall right on the eastern boundary: (2,-1) = 9/5, >> (2,0) = 9/8 and (2,1) = 45/32. All three of these thus >> have alternates a comma lower -- in other words, by the >> -[4,-1] = 80:81 unison-vector--, and the alternate >> pitch-classes fall on the western boundary: (-2,0) = 16/9, >> (-2,1) = 10/9, and (-2,2) = 25/18, respectively. >> >> Also, since (-2,2) = 25/18 and (2,1) = 45/32 happen to >> fall right on the northwest and northeast *corners* of the >> boundary (respectively), they also have lower alternates at >> the distance of the *other* unison-vector -[0 3] = 64:125, >> which would place the alternates at (-2,-1) = 64/45 and >> (2,-2) = 36/25, respectively.you and i are talking about exactly the same structures here, even down to exactly the same pitches. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3760 - Contents - Hide Contents Date: Sun, 03 Feb 2002 06:18:00 Subject: 686/675 From: genewardsmith Has anyone noticed this one? I saw it turning up in the reduced bases of 37 and 55; it is a comma of 9,10,19,27,29,36,37,46,55,56, and is productive of a number of good temperaments.
Message: 3761 - Contents - Hide Contents Date: Sun, 03 Feb 2002 06:24:19 Subject: Re: 686/675 From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Has anyone noticed this one? I saw it turning up in the reduced bases of 37 and 55; it is a comma of 9,10,19,27,29,36,37,46,55,56, and is productive of a number of good temperaments.I should add it also turned up in the survey of 7-limit temperaments. It is (7/6)^3 / (5/4)^2.
Message: 3762 - Contents - Hide Contents Date: Sun, 3 Feb 2002 00:18:48 Subject: Uhrin's paper From: monz i would really appreciate a detailed explanation of this abstract: "Self-affine tiles and digit sets via the geometry of numbers" B. Uhrin Self-affine tiles and digit sets via the geome... * [with cont.] (Wayb.) in english thanks -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3763 - Contents - Hide Contents Date: Sun, 3 Feb 2002 23:32:28 Subject: Re: Gene's notation formula: alternate duodene? From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 10:12 PM > Subject: [tuning-math] Re: Gene's notation formula: alternate duodene? > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> but 25:24 only occurs between notes with the same letter-name >> and a change of accidental! in the JI duodene, the sharps >> and flats have distinct spellings, and 25:24 is indeed >> functioning as a chromatic change on the same letter-name, >> which is not what i would call a step. >> conventional diatonic notation is based on a 7-tone periodicity > block. the duodene mixes this notation with a 12-tone periodicity > block.hmmm ... that's interesting.> if an interval takes you from one pitch to another _within > the block_, that interval is *not* a unison vector of that > block. if it takes you between two adjacent, in pitch, notes > in the block, it is a step vector of the block.a h ! ! ! thanks, Paul, now it's clear as a mountain stream. the "within the block" bit is the key that unlocked that puzzle for me. but now i'm really curious -- why is it necessary to put a step-vector into the kernel to derive a notation. intuitively, it makes sense to me, even as i ask it, but i'd still like a good explanation of how and why it works. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3764 - Contents - Hide Contents Date: Sun, 03 Feb 2002 09:15:08 Subject: Re: Uhrin's paper From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > i would really appreciate a detailed explanation of this abstract: > > "Self-affine tiles and digit sets via the geometry of numbers" > B. Uhrin > Self-affine tiles and digit sets via the geome... * [with cont.] (Wayb.)Are you interested in this abstract in particular, or self-affine tilings, or what, exactly?
Message: 3765 - Contents - Hide Contents Date: Sun, 3 Feb 2002 20:41:12 Subject: Re: a notation for Schoenberg's rational implications From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 8:34 PM > Subject: [tuning-math] Re: a notation for Schoenberg's rational implications > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> The Schoenberg PBs i've been posting have been defined >> entirely by commatic unison-vectors. >> >> Paul also posted something about how i would need to include >> a *chromatic* unison-vector in order to arrive at a Smithian >> "notation" (... i've searched for that post but can't find it). >> Well, i was thinking about this and realized that here the >> 441/440 is a perfect candidate for a chromatic unison-vector! >> So i plugged it into my spreadsheet matrix in place of 2/1, >> using the unison-vectors i derived directly from _Harmonielehre_ >> (rather than Gene's Minkowski-reduced ones): >> >> >> kernel: >> >> 2 3 5 7 11 ratio ~cents >> >> [-3 2 -1 2 -1] = 441:440 3.93016 >> [-5 2 2 -1 0] = 225:224 7.71152 >> [-4 4 -1 0 0] = 81:80 21.50629 >> [ 6 -2 0 -1 0] = 64:63 27.26409 >> [-5 1 0 0 1] = 33:32 53.27294 >> And as Paul predicted, this time Gene's formula worked >> like a charm, >> can you remind me what you're referring to?i wish i could find it -- i've been searching like mad. i had posted a question about gene's formula, and he responded that i had to have a determinant of +/-1 in order to obtain a "notation". i asked about how to do this, and you posted something about needing to include a *chromatic* unison-vector in order to get the "notation". that's the post i can't find now. but as you can see, it does work.>> Paul, how does this scale compare with the PB you would >> find by your method using these criteria? >> what criteria? first of all, i have no idea what you did above, > as you have too many unison vectors to define a PB.i think if you replace the 441:440 with 2:1, you'll be able to derive the periodicity-block using your method. that was the one that i added in this time to get the "notation". -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3766 - Contents - Hide Contents Date: Sun, 03 Feb 2002 09:36:34 Subject: Re: Uhrin's paper From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > i would really appreciate a detailed explanation of this abstract: > > "Self-affine tiles and digit sets via the geometry of numbers" > B. Uhrin > Self-affine tiles and digit sets via the geome... * [with cont.] (Wayb.)R^n is real n-dimensional space, a compact set in such a space is a closed and bounded set, and an expanding matrix is one with all of its eigenvalues greater than one in absolute value. This talk comes from a curious generalization that Jeff Lagarias came up with, which generalizes the idea of a base b expansion to where the base is a matrix. What's the connection to music? I could probably get Jeff to send me a reprint of what he has done if we can't get it off the web.
Message: 3767 - Contents - Hide Contents Date: Sun, 3 Feb 2002 20:49:32 Subject: Re: Gene's notation formula: alternate duodene? From: paulerlich To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Sunday, February 03, 2002 8:37 PM Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >> >>>> for a set of rational unison-vectors {u1/v1, ... un/vn}, >> for any non-zero I can define a scale by calculating for 0<=n<d >> >> step[n] = (u1/v1)^round(7n/d) (u2/v2)^round(12n/d) >> (u3/v3)^round(7n/d) (u4/v4)^round(-2n/d) (u5/v5)^round(5n/d) >> you really think this is a correct generalization?no, i was responding to your previous post and realized that it's not. it still has numbers in it which come from a particular homomorphism. i'm working on a real generalization, and it will be in my other post. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3768 - Contents - Hide Contents Date: Sun, 3 Feb 2002 23:45:15 Subject: Re: Gene's PB formula, generalized From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 10:18 PM > Subject: [tuning-math] Re: Gene's PB formula, generalized > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> for a set of i rational unison-vectors {u1/v1, u2/v2,... ui/vi}, >> where {hx, hy, ...hq} is the top row of the unimodular adjoint >> of the kernel matrix of the unison-vectors, >> This should be h1, h2, ... hi to correspond to your unison vectors. absolutely. > Also, u1/v1 is a step vector, and the matrix therefore is not just > the kernel, but a set of generators for the kernel plus a step vector.right, got it.> "Unimodular adjoint" should just be "matrix inverse", and you should > note that since u1/v1 is a step vector, the matrix is unimodular, > and hence is invertible to an integral matrix.ok, thanks. how's this? : where M is the matrix composed of a set of i rational vectors {u1/v1, u2/v2,... ui/vi} in which u1/v1 is a step-vector and {u2/v2 ... ui/vi} are commatic unison-vector generators of the kernel, and where {h1, h2, ...hi} is the top row of M^-1, for any non-zero a scale can be defined by calculating for 0 <= n < d : step[n] = (u1/v1)^round(h1(2)*n/d) * (u2/v2)^round(h2(2)*n/d) * ... (ui/vi)^round(hi(2)*n/d) . (i think we should describe what n and d are.) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3769 - Contents - Hide Contents Date: Sun, 3 Feb 2002 09:45:08 Subject: Re: Uhrin's paper From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 1:36 AM > Subject: [tuning-math] Re: Uhrin's paper > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> i would really appreciate a detailed explanation of this abstract: >> >> "Self-affine tiles and digit sets via the geometry of numbers" >> B. Uhrin >> Self-affine tiles and digit sets via the geome... * [with cont.] (Wayb.) >> R^n is real n-dimensional space, a compact set in such a space > is a closed and bounded set, and an expanding matrix is one > with all of its eigenvalues greater than one in absolute value. > This talk comes from a curious generalization that Jeff Lagarias > came up with, which generalizes the idea of a base b expansion > to where the base is a matrix.thanks, gene ... but i still don't understand what's an eigenvalue?> What's the connection to music?ah ... that's for y o u to tell m e !!! i'm just hunting down stuff that discusses lattices, so that i can learn more about them BTW ... i took a look at the first couple of chapters from the Conway/Sloane book you recommended (_Sphere Packings..._), but don't understand much of that either :( -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3770 - Contents - Hide Contents Date: Sun, 3 Feb 2002 23:46:41 Subject: Re: Gene's notation formula: alternate duodene? From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 11:36 PM > Subject: [tuning-math] Re: Gene's notation formula: alternate duodene? > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> but now i'm really curious -- why is it necessary to put a >> step-vector into the kernel to derive a notation. >> because otherwise, you'd never get past the unison, to the second, > third, etc.!right, that makes sense ... but how does only one step-vector give you the whole scale? all the other steps can be derived from that and the commas, apparently. ? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3771 - Contents - Hide Contents Date: Sun, 03 Feb 2002 18:35:41 Subject: Re: For Carl--5, 7, and 11-limit reduced bases for the 37-et From: clumma> 5: [250/243, 262144/253125] > 7: [64/63, 250/243, 686/675] > 11: [55/54, 64/63, 100/99, 686/675] Thanks! -Carl
Message: 3772 - Contents - Hide Contents Date: Sun, 3 Feb 2002 21:03:55 Subject: Gene's PB formula, generalized (was: a notation for Schoenberg's...) From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 8:36 PM > Subject: [tuning-math] Re: a notation for Schoenberg's rational implications > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> And the JI periodicity-block scale derived from this >> again, i'm baffled. how do you get a PB when you have one too many > UVs?i'm using the formula Gene posted, which i generalized to:> for a set of i rational unison-vectors {u1/v1, ... ui/vi}, > for any non-zero I can define a scale by calculating for 0<=n<d > > step[n] = (u1/v1)^round(7n/d) (u2/v2)^round(12n/d) > (u3/v3)^round(7n/d) (u4/v4)^round(-2n/d) (u5/v5)^round(5n/d)but oops! ... i realize now that this is still not entirely generalized. all those numbers (7,12,7,-2,5) are from a particular set of homomorphisms (the first Schoenberg PB Gene calculated, back around Christmas), and need to be replaced by variables. i don't really know what to call them, so i'll just make this do: {hv, hw, hx, hy, hz}. it's the top row of numbers in the adjoint (or is it a unimodular inverse?) of the kernel. so the generalized formula really is: for a set of i rational unison-vectors {u1/v1, u2/v2,... ui/vi}, where {hx, hy, ...hq} is the top row of the unimodular adjoint of the kernel matrix of the unison-vectors, for any non-zero I can define a scale by calculating for 0 <= n < d : step[n] = (u1/v1)^round(hx*n/d) * (u2/v2)^round(hy*n/d) * ... (ui/vi)^round(hq*n/d) . -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3773 - Contents - Hide Contents Date: Sun, 3 Feb 2002 11:43:33 Subject: a notation for Schoenberg's rational implications From: monz I am here referring specifically to Schoenberg's 1911 theory as introduced in his _Harmonielehre_, and not to his later 1927/34 theory articulated in the paper "Problems of Harmony" (the latter was the basis for Partch's criticism). I apologize for the long quotes, but want to be complete for anyone who's interested in following this thread.> Message 2819 > From: monz <joemonz@y...> > Date: Sun Jan 20, 2002 4:09pm > Subject: Re: Re: lattices of Schoenberg's rational implications Yahoo groups: /tuning-math/messages/2819?expand=1 * [with cont.] > > > Help! > > I set up an Excel spreadsheet to calculate the notes of > a periodicity-block according to Gene's formula as expressed here: > > >> Message 2185>> From: "genewardsmith" <genewardsmith@j...> >> Date: Wed Dec 26, 2001 6:25 pm >> Subject: Re: Gene's notation & Schoenberg lattices >> <Yahoo groups: /tuning-math/message/2185 * [with cont.] > >> >> ... >> >> For any non-zero I can define a scale by calculating for 0<=n<d >> >> step[n] = (56/55)^round(7n/d) (33/32)^round(12n/d) >> (64/63)^round(7n/d) (81/80)^round(-2n/d) (45/44)^round(5n/d) > > >> It worked just fine for both of these examples <snipped>, > the 7-tone and 12-tone versions. > > > > But for the kernel I recently posted for Schoenberg ... > >> kernel >>>> 2 3 5 7 11 unison vectors ~cents >> >> [ 1 0 0 0 0 ] = 2:1 0 >> [-5 2 2 -1 0 ] = 225:224 7.711522991 >> [-4 4 -1 0 0 ] = 81:80 21.5062896 >> [ 6 -2 0 -1 0 ] = 64:63 27.2640918 >> [-5 1 0 0 1 ] = 33:32 53.27294323 >> >> adjoint >> >> [ 12 0 0 0 0 ] >> [ 19 1 2 -1 0 ] >> [ 28 4 -4 -4 0 ] >> [ 34 -2 -4 -10 0 ] >> [ 41 -1 -2 1 12 ] >> >> determinant = | 12 | > >> ... it doesn't work. All I get are powers of 2. > > Why? How can it be fixed? Do I need yet another > independent unison-vector instead of 2:1? > > ******** > > Message 2822 > From: genewardsmith <genewardsmith@j...> > Date: Sun Jan 20, 2002 5:08pm > Subject: Re: lattices of Schoenberg's rational implications Yahoo groups: /tuning-math/messages/2822?expand=1 * [with cont.] > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> determinant = | 12 | >>> ... it doesn't work. >> This determinant is why. In my example, the determinant > had an absolute value of 1, and so we get what I call a > "notation", meaning every 11-limit interval can be expressed > in terms of integral powers of the basis elements. You > have a determinant of 12, and therefore torsion. In fact, > you map to the cyclic group C12 of order 12, and the twelveth > power (or additively, twelve times) anything is the identity. >>> Why? How can it be fixed? Do I need yet another >> independent unison-vector instead of 2:1? >> If you want a notation, yes. One which makes the matrix > unimodular, ie with determinant +-1. > > > ****** > > > Message 2900 > From: monz <joemonz@y...> > Date: Tue Jan 22, 2002 3:34pm > Subject: Re: Re: Minkowski reduction > (was: ...Schoenberg's rational implications) Yahoo groups: /tuning-math/messages/2900?expand=1 * [with cont.] > >>> From: paulerlich <paul@s...> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Tuesday, January 22, 2002 4:32 AM >> Subject: [tuning-math] Re: Minkowski reduction >> (was: ...Schoenberg's rational implications) >> > >>> [monz]>>> With variant alternate pitches written on the same line >>> -- and thus with invariant ones on a line by themselves -- >>> these scales are combined into: >>> >>> 1/1 >>> 21/20 15/14 >>> 35/32 9/8 >>> 7/6 25/21 6/5 >>> 5/4 >>> 21/16 >>> 7/5 10/7 >>> 3/2 >>> 49/32 25/16 63/40 >>> 5/3 12/7 >>> 7/4 >>> 147/80 15/8 >>> >>> ... >>> >>> One thing I did notice in connection with this, is that >>> 147/80 is only a little less than 4 cents wider than 11/6, >>> which is one of the pitches implied in Schoenberg's overtone >>> diagram (p 23 of _Harmonielehre_) : >>> >>> vector ratio ~cents >>> >>> [ -4 1 -1 2 0 ] = 147/80 1053.2931 >>> - [ -1 -1 0 0 1 ] = 11/6 1049.362941 >>> -------------------- >>> [ -3 2 -1 2 -1 ] = 441/440 3.930158439 >>> >>> >>> So I know that 441/440 is tempered out. >>>> NO IT ISN'T! I believe it maps to 1 semitone given the set of unison >> vectors you've put forward. >>>>> But I don't see >>> how to get this as a combination of two of the other >>> unison-vectors. >> >> YOU CAN'T! > >> Oops... my bad. Thanks, Paul. I see it now. If "C" is Schoenberg's > 1/1, the 147/80 is mapped to "B" but 11/6 is mapped to "Bb". > This is precisely the note which was misprinted in the diagram in > the English edition ... guess I accepted it for so long that I > got confused.The Schoenberg PBs i've been posting have been defined entirely by commatic unison-vectors. Paul also posted something about how i would need to include a *chromatic* unison-vector in order to arrive at a Smithian "notation" (... i've searched for that post but can't find it). Well, i was thinking about this and realized that here the 441/440 is a perfect candidate for a chromatic unison-vector! So i plugged it into my spreadsheet matrix in place of 2/1, using the unison-vectors i derived directly from _Harmonielehre_ (rather than Gene's Minkowski-reduced ones): kernel: 2 3 5 7 11 ratio ~cents [-3 2 -1 2 -1] = 441:440 3.93016 [-5 2 2 -1 0] = 225:224 7.71152 [-4 4 -1 0 0] = 81:80 21.50629 [ 6 -2 0 -1 0] = 64:63 27.26409 [-5 1 0 0 1] = 33:32 53.27294 and got a unimodular adjoint (or is that unimodular inverse?): adjoint: [12 5 -2 19 12] [19 8 -3 30 19] [28 12 -5 44 28] [34 14 -6 53 34] [41 17 -7 65 42] Here i see two alternative mappings to 12, in which the only difference is h12(11)=41 or 42. The pentatonic mapping is in there, and now there's also one that goes to 19. But what to make of that third column? the -h2(2)=-2 means that some form of tritone is the period, correct? But how do i find the generator? Until i know what that is, the other numbers don't make any sense ... do they? And as Paul predicted, this time Gene's formula worked like a charm, and i got the following JI PB scale: degree ratio vector 2 3 5 7 11 ( 12 2/1 [ 1 0 0 0 0] ) 11 15/8 [-3 1 1 0 0] 10 16/9 [ 4 -2 0 0 0] 9 5/3 [ 0 -1 1 0 0] 8 8/5 [ 3 0 -1 0 0] 7 3/2 [-1 1 0 0 0] 6 10/7 [ 1 0 1 -1 0] 5 4/3 [ 2 -1 0 0 0] 4 5/4 [-2 0 1 0 0] 3 32/27 [ 5 -3 0 0 0] 2 9/8 [-3 2 0 0 0] 1 16/15 [ 4 -1 -1 0 0] 0 1/1 [ 0 0 0 0 0] triangular lattice: A E B 5:3.------.5:4-----15:8 / \ ` F# ' / \ / \ / \ 10:7 / \ / \ / \ | / \ / \ Eb Bb F C G D 32:27----16:9-----4:3-------1:1------3:2-----9:8 \ / \ / \ / \ / \ / \ / Db Ab 16:15----8:5 In my quest to find this notation, Paul has already suggested that i "forget it", since Schoenberg clearly meant for all of these unison-vectors to be tempered out of his system. But, more than once in _Harmonielehre_, Schoenberg did indeed allude to a rational basis which might underlie the compositions from his "free atonality" period, so i'm very interested in examining that rational basis. So, guys, am i on the right track with this one? Paul, how does this scale compare with the PB you would find by your method using these criteria? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3774 - Contents - Hide Contents Date: Sun, 3 Feb 2002 21:11:16 Subject: Re: a notation for Schoenberg's rational implications From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 8:57 PM > Subject: [tuning-math] Re: a notation for Schoenberg's rational implications > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> i think if you replace the 441:440 with 2:1, you'll be >> able to derive the periodicity-block using your method. >> well . . . i don't use 2:1 explicitly . . . >>> that was the one that i added in this time to get the >> "notation". >> so the PB comes _without_ using 441:440?well, no ... when i use 2:1 instead of 441:440, i can recover the set of homorphisms, but i only get "octaves" when i try to find the PB pitches. as i said, i have to use a chromatic unison-vector in addition to all the commatic ones, in order to get the full PB.> anyway, how strange to call that a chromatic unison vector. > are you sure you got that from me?n o ! i didn't get t h a t from you. i got the necissity of having a chromatic unison-vector from you. i got the idea to use 441:440 as a chromatic unison-vector when i derived it from the Minkowski-reduced version of the Schoenberg PB which Gene and you calculated, and you pointed out to me that it was not a commatic unison-vector because in fact the pitches separated by it did involve a change of accidental. so i replaced the 2:1 i had in my matrix with that, and _voilą_! -- out came the PB! but as you can see from my subsequent post, i believe there's more validity to Schoenberg's actual theory in using 45:44 as a chromatic unison-vector instead, and it does result in a scale which has one different note. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
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