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Message: 3775 - Contents - Hide Contents Date: Sun, 3 Feb 2002 12:10:31 Subject: Re: a notation for Schoenberg's rational implications From: monz Take 2 ...> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 11:43 AM > Subject: [tuning-math] a notation for Schoenberg's rational implications > > > I am here referring specifically to Schoenberg's 1911 theory > as introduced in his _Harmonielehre_, > <... etc> > > > 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): > > <snipped matrix details> > > > 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----8:5Here's another JI PB for comparison, derived from 45:44, which i had originally interpreted as a commatic unison-vector -- thanks to the misprint i discovered in the English translation (B instead of Bb for 11th/F), and which led me down the wrong path for years -- but which can now be used as a chromatic unison-vector. kernel 2 3 5 7 11 ratio ~cents [-2 2 1 0 -1] = 45:44 38.9057732 [-5 2 2 -1 0] = 225:224 7.7115230 [-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 53.2729432 adjoint [12 -7 -2 7 12] [19 -11 -3 11 19] [28 -16 -5 16 28] [34 -20 -6 19 34] [41 -24 -7 24 42] determinant = | 1 | We still have the same two mappings to 12, as well as the one to -2 (which i don't understand). But this time, the mappings to 5 and 19 disappear and are replaced by two "diatonic" mappings to 7, which differ only in their mapping of prime-factor 7: the first one sends it to the 20th degree, which is a "7th", and the second sends it to the 19th degree, which is a "6th". Am i right about this? And the JI periodicity-block scale derived from this has only one note different from the one in my last post, and that is the "tritone", which is 64:45 here instead of 10:7 : triangular lattice: A E B 5:3------5:4-----15:8 / \ / \ / \ / \ / \ / \ / \ / \ / \ Eb Bb F C G D 32:27----16:9-----4:3-----1:1------3:2-----9:8 / \ / \ / / \ / \ / / \ / \ / Gb Db Ab 64:45----16:15----8:5 -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3776 - Contents - Hide Contents Date: Sun, 3 Feb 2002 12:43:49 Subject: Gene's notation formula: alternate duodene? From: monz> 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)Hmmm ... i played around with Gene's notation formula for the simpler 5-limit case of the Duodene, Definitions of tuning terms: duodene, (c) 1998... * [with cont.] (Wayb.) and got this: 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] JI periodicity-block degree ratio ( 12 2/1 ) 11 15/8 10 9/5 9 5/3 8 8/5 7 3/2 6 36/25 5 4/3 4 5/4 3 6/5 2 9/8 1 16/15 0 1/1 triangular lattice: A E B 5:3-----5:4-----15:8 / \ / \ / \ / \ / \ / \ / \ / \ / \ F C G D 4:3-----1:1------3:2-----9:8 / \ / \ / \ / / \ / \ / \ / / \ / \ / \ / Db Ab Eb Bb 16:15----8:5-----6:5-----9:5 \ / \ / \ / Gb 36:25 I was surprised to see this result instead of the actual Duodene, which would have F# 45/32 instead of Gb 36/25 (the difference is one of the unison-vectors, the diesis 128:125 = [7 0 -3] ), and would thus have a perfect parallelogram lattice. Gene, can you explain why your formula gave this result instead of what i expected? Is there a "correction factor" involved? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3777 - Contents - Hide Contents Date: Sun, 3 Feb 2002 21:12:47 Subject: Re: Gene's PB formula, generalized (was: a notation for Schoenberg's...) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 9:03 PM > Subject: [tuning-math] Gene's PB formula, generalized (was: a notation for Schoenberg's...) > > >>> 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 >> >> >> 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) .oops ... that last bit should read * ... (ui/vi)^round(hi*n/d) . -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3778 - Contents - Hide Contents Date: Sun, 3 Feb 2002 13:05:58 Subject: Re: Gene's notation formula: alternate duodene? From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 12:43 PM > Subject: [tuning-math] Gene's notation formula: alternate duodene? > > > >> 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)Also, i just realized that this statement of the formula already has some specific values instead of variables, so let's generalize it: 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) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3779 - Contents - Hide Contents Date: Sun, 3 Feb 2002 21:16:16 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 9:11 PM > Subject: [tuning-math] Re: Gene's notation formula: alternate duodene? > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> 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 >> 25:24 is not a unison vector of the 12-tone scale at all. it is a > step vector. so if it's a step vector you want to use in gene's > formula that you generalized, 441:440 would sure seem like a poor > choice for the schoenberg case!ok ... i'm s t i l l confused about equivalence interval, period, and unison-vector, but i thought i had the difference between commatic and chromatic unison-vectors. now, what's the difference between a chromatic unison-vector and a step-vector? in the 12-tone scale, of which the duodene is an example, there is a difference between C and C#, and in this case that difference is most likely to be 25:24. so why is that a step (which i would equate with a "diatonic semitone") and not a chromatic unison-vector (which i would equate with a "chromatic semitone")? more confused ... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3780 - Contents - Hide Contents Date: Sun, 3 Feb 2002 13:14:03 Subject: Re: Gene's notation formula: alternate duodene? From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 1:05 PM > Subject: Re: [tuning-math] Gene's notation formula: alternate duodene? > > > Also, i just realized that this statement of the formula > already has some specific values instead of variables, > so let's generalize it: > > > 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)Oops! i also just realized that "n" is used as a variable twice now, so make that for a set of i rational unison-vectors {u1/v1, ... ui/vi}, -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3781 - Contents - Hide Contents Date: Sun, 3 Feb 2002 21:28:20 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 9:22 PM > Subject: [tuning-math] Re: Gene's notation formula: alternate duodene? > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> in the 12-tone scale, of which the duodene is an example, >> there is a difference between C and C#, and in this case >> that difference is most likely to be 25:24. >> but 25:24 is a chromatic unison vector with respect to the 7-tone PB > that is the diatonic scale, but not in the case of a 12-tone PB. >>> so why is that a step (which i would equate with a >> "diatonic semitone") and not a chromatic unison-vector >> (which i would equate with a "chromatic semitone")? >> clearly in this case, a 12-tone closed system, there is no difference > between a diatonic semitone and a chromatic semitone. in a 7-tone PB, > step vectors would include 16:15, 10:9, and 9:8.ok ... i think i got it, but it sure is confusing stuff ... Paul, i think you might be confusing some of the posts i sent in today. some are about the Schoenberg 12-tone PB, which is definitely meant to be 12-edo with all unison-vectors tempered out, but others are about the duodene, which Ellis meant to be JI with nothing tempered out. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3782 - Contents - Hide Contents Date: Sun, 3 Feb 2002 22:07:29 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 9:32 PM > Subject: [tuning-math] Re: Gene's notation formula: alternate duodene? > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> Paul, i think you might be confusing some of the posts >> i sent in today. some are about the Schoenberg 12-tone PB, >> which is definitely meant to be 12-edo with all unison-vectors >> tempered out, but others are about the duodene, which Ellis >> meant to be JI with nothing tempered out. >> 25:24 is still a step vector in the duodede. look! you can find it > between distinct pitches in the duodene!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. i do understand, however, how it is a step-vector in 12-edo. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3783 - Contents - Hide Contents Date: Sun, 3 Feb 2002 22:11:03 Subject: Re: Gene's PB formula, generalized From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 10:04 PM > Subject: [tuning-math] Re: Gene's PB formula, generalized > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> 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. >> I would call those hv(2), hw(2), hx(2), hy(2) and hz(2), > where the h's are vals, which you can equate to column vectors.thanks for stepping in with that, Gene. the feeling was nagging away at me that some reference to the prime-factors ought to be included in there somewhere, and now i see where. so how's this? ... 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(2)*n/d) * (u2/v2)^round(hy(2)*n/d) * ... (ui/vi)^round(hi(2)*n/d) . -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3784 - Contents - Hide Contents Date: Mon, 04 Feb 2002 01:23:16 Subject: Re: Gene's notation formula: alternate duodene? From: genewardsmith --- 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. 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 usingthis 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 addthree 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.

Message: 3785 - Contents - Hide Contents Date: Mon, 04 Feb 2002 06:12:57 Subject: Re: Gene's notation formula: alternate duodene? From: paulerlich --- 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. 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.

Message: 3786 - Contents - Hide Contents Date: Mon, 04 Feb 2002 06:13:23 Subject: Re: Gene's PB formula, generalized (was: a notation for Schoenberg's...) From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> So why is there an "extra vector" used in constructing the PB? Is > this just an arbitrary vector independent of the others, or is it a > step vector, or what?It's the step vector, the rest are the commas.

Message: 3787 - Contents - Hide Contents Date: Mon, 04 Feb 2002 06:14:11 Subject: Re: a notation for Schoenberg's rational implications From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> it's a chromatic unison vector with respect to a 7-tone PB in an MOS >> tuning, not with respect to a 12-tone PB in an equal tuning. In the >> latter it functions as a step vector -- but is a very unlikely choice >> for one. >> It worked, though.well clearly the formula wasn't depending on it to any great extent. no ratios of 11 or multiples of 11 even made it into the block.

Message: 3788 - Contents - Hide Contents Date: Mon, 04 Feb 2002 06:15:11 Subject: Re: Gene's PB formula, generalized From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> so how's this? ... > > > 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(2)*n/d) * (u2/v2)^round(hy(2)*n/d) > * ... (ui/vi)^round(hi(2)*n/d) .they can't all be unison vectors!

Message: 3789 - Contents - Hide Contents Date: Mon, 04 Feb 2002 04:34:10 Subject: Re: a notation for Schoenberg's rational implications From: paulerlich --- 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 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?that's generator, not period, i believe.> And as Paul predicted, this time Gene's formula worked > like a charm,can you remind me what you're referring to?> 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?what criteria? first of all, i have no idea what you did above, as you have too many unison vectors to define a PB.

Message: 3790 - Contents - Hide Contents Date: Mon, 04 Feb 2002 06:15:56 Subject: Re: Gene's PB formula, generalized (was: a notation for Schoenberg's...) From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> So why is there an "extra vector" used in constructing the PB? Is >> this just an arbitrary vector independent of the others, or is it a >> step vector, or what? >> It's the step vector, the rest are the commas.aha! monz please take note.

Message: 3791 - Contents - Hide Contents Date: Mon, 04 Feb 2002 04:36:04 Subject: Re: a notation for Schoenberg's rational implications From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > Take 2 ... > >>> From: monz <joemonz@y...> >> To: <tuning-math@y...> >> Sent: Sunday, February 03, 2002 11:43 AM >> Subject: [tuning-math] a notation for Schoenberg's rational implications >> >> >> I am here referring specifically to Schoenberg's 1911 theory >> as introduced in his _Harmonielehre_, >> <... etc> >> >> >> 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): >> >>> >> >> 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 > > > > Here's another JI PB for comparison, derived from 45:44, > which i had originally interpreted as a commatic > unison-vector -- thanks to the misprint i discovered in > the English translation (B instead of Bb for 11th/F), > and which led me down the wrong path for years -- but > which can now be used as a chromatic unison-vector. > > > kernel > 2 3 5 7 11 ratio ~cents > > [-2 2 1 0 -1] = 45:44 38.9057732 > [-5 2 2 -1 0] = 225:224 7.7115230 > [-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 53.2729432 > > > adjoint > > [12 -7 -2 7 12] > [19 -11 -3 11 19] > [28 -16 -5 16 28] > [34 -20 -6 19 34] > [41 -24 -7 24 42] > > > determinant = | 1 | > > > We still have the same two mappings to 12, as well as > the one to -2 (which i don't understand). > > But this time, the mappings to 5 and 19 disappear and > are replaced by two "diatonic" mappings to 7, which > differ only in their mapping of prime-factor 7: the > first one sends it to the 20th degree, which is a "7th", > and the second sends it to the 19th degree, which is > a "6th". > > > Am i right about this?i think so -- gene should check.> And the JI periodicity-block scale derived from thisagain, i'm baffled. how do you get a PB when you have one too many UVs?

Message: 3792 - Contents - Hide Contents Date: Mon, 04 Feb 2002 06:18:28 Subject: Re: Gene's PB formula, generalized From: genewardsmith --- 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. 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. "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.

Message: 3793 - Contents - Hide Contents Date: Mon, 04 Feb 2002 04:37:51 Subject: Re: Gene's notation formula: alternate duodene? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> 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) > > >> Also, i just realized that this statement of the formula > already has some specific values instead of variables, > so let's generalize it: > > > 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?

Message: 3794 - Contents - Hide Contents Date: Mon, 04 Feb 2002 07:36:00 Subject: Re: Gene's notation formula: alternate duodene? From: paulerlich --- 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.!

Message: 3795 - Contents - Hide Contents Date: Mon, 04 Feb 2002 04:57:07 Subject: Re: a notation for Schoenberg's rational implications From: paulerlich --- 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? anyway, how strange to call that a chromatic unison vector. are you sure you got that from me?

Message: 3796 - Contents - Hide Contents Date: Mon, 04 Feb 2002 07:52:16 Subject: Re: Gene's PB formula, generalized From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> 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-vectornot necessarily commatic. if we view this as an application of a smithian notation, these may be chromatic unison vectors.

Message: 3797 - Contents - Hide Contents Date: Mon, 04 Feb 2002 07:53:02 Subject: Re: Gene's notation formula: alternate duodene? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> 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. ?yes. and not only all the other steps, but all possible JI ratios.

Message: 3798 - Contents - Hide Contents Date: Mon, 04 Feb 2002 05:11:21 Subject: Re: Gene's notation formula: alternate duodene? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> 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 commatic25:24 is not a unison vector of the 12-tone scale at all. it is a step vector. so if it's a step vector you want to use in gene's formula that you generalized, 441:440 would sure seem like a poor choice for the schoenberg case!

Message: 3799 - Contents - Hide Contents Date: Mon, 04 Feb 2002 09:50:55 Subject: blast from the past From: paulerlich Yahoo groups: /tuning/message/7771 * [with cont.] if i had known about torsion then, you probably wouldn't be seeing 62- tone on the list . . . so, are all the 34-tone examples equivalent (i.e., same mapping)?

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