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Message: 3150 - Contents - Hide Contents Date: Fri, 11 Jan 2002 04:59:24 Subject: Some 7-tone 72-et scales from 385/384 From: genewardsmith We have only three sizes of steps instead of four, but this does come from the 11-limit comma 385/384. Even so, 11 does not figure very prominately inmost of these. [0, 7, 14, 30, 37, 49, 56] [7, 7, 16, 7, 12, 7, 16] edges 10 15 17 17 connectivity 2 4 4 4 [0, 7, 14, 26, 33, 49, 56] [7, 7, 12, 7, 16, 7, 16] edges 9 13 16 17 connectivity 1 3 4 4 [0, 7, 14, 26, 42, 49, 56] [7, 7, 12, 16, 7, 7, 16] edges 8 14 16 16 connectivity 1 3 4 4 [0, 7, 14, 21, 37, 44, 56] [7, 7, 7, 16, 7, 12, 16] edges 7 14 15 16 connectivity 0 3 3 4 [0, 7, 14, 21, 33, 49, 56] [7, 7, 7, 12, 16, 7, 16] edges 6 12 14 16 connectivity 0 2 3 4 [0, 7, 14, 26, 33, 40, 56] [7, 7, 12, 7, 7, 16, 16] edges 6 10 14 16 connectivity 0 2 3 4 [0, 7, 14, 21, 33, 40, 56] [7, 7, 7, 12, 7, 16, 16] edges 5 10 13 16 connectivity 0 2 2 4 [0, 7, 14, 21, 28, 44, 56] [7, 7, 7, 7, 16, 12, 16] edges 4 11 12 14 connectivity 0 2 2 3 [0, 7, 14, 21, 28, 40, 56] [7, 7, 7, 7, 12, 16, 16] edges 3 9 11 14 connectivity 0 1 2 3
Message: 3151 - Contents - Hide Contents Date: Fri, 11 Jan 2002 14:04:16 Subject: Re: [tuning] Re: badly tuned remote overtones From: monz First, I'd like to start this post off with a link to my "rough draft" of a lattice of the periodicity-block Gene calculated for Schoenberg's theory: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) This shows the 12-tone periodicity-block (primarily 3- and 5-limit, with one 11-limit pitch), and its equivalent p-block cousins at +/- each of the four unison-vectors. Now to respond to Paul...> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning@xxxxxxxxxxx.xxx> > Sent: Friday, January 11, 2002 12:47 PM > Subject: [tuning] Re: badly tuned remote overtones > > > You seem to be brushing some of the unison vectors you had > previously reported, and from which Gene derived 7-, 5-, and 2-tone > periodicity blocks, under the rug.Ah ... so then this, from Gene:> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 26, 2001 3:25 PM > Subject: [tuning-math] Re: Gene's notation & Schoenberg lattices > > ... This matrix is unimodular, meaning it has determinant +-1. > If I invert it, I get > > [ 7 12 7 -2 5] > [11 19 11 -3 8] > [16 28 16 -5 12] > [20 34 19 -6 14] > [24 42 24 -7 17] >actually *does* specify "7-, 5-, and 2-tone periodicity blocks". Yes?> Face it, Monz -- without some careful "fudging", Schoenberg's > derviation of 12-tET as a scale for 13-limit harmony is not > the rigorous, unimpeachable bastion of good reasoning that > you'd like to present it as.Your point is taken, but please try to understand my objectives more clearly. I agree with you that "Schoenberg's derviation ... is not the rigorous, unimpeachable bastion of good reasoning" etc. I'm simply trying to get a foothold on what was in his mind when he came up with his radical new ideas for using 12-tET to represent higher-limit chord identities. I've seen it written (can't remember where right now) that without the close personal attachment to Schoenberg that his students had, it's nearly impossible to understand all the subtleties of his teaching. I'm just trying to dig into that scenario a bit, and in a sense to "get closer" to Schoenberg and his mind.> The contradictions in Schoenberg's arguments were known at least > as early as Partch's Genesis, and he isn't going to weasel his way > out of them now :) If 12-tET can do what you and Schoenberg are > trying to say it can, it can do _anything_, and there would be > no reason ever to adopt any other tuning system.Ahh ... well, I think you've put on finger on the crux of the matter. Schoenberg consciously rejected microtonality and also made a conscious decision to use the 12-tET tuning as tho it *could* do "_anything_". As I've documented again and again, he *did* have a favorable attitude towards adopting other tuning systems, but was of the opinion that only in the future would the time be right for that. With us now living *in* that future, it seems to me that perhaps he was right after all. Perhaps it's even possible that Schoenberg's actions in adopting the "new version" of 12-tET ("atonality") helped to precipitate the current trend towards microtonality and alternative tunings. ...? Always curious about these things, -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3152 - Contents - Hide Contents
Date: Fri, 11 Jan 2002 22:08:47
Subject: More proposed definitions
From: genewardsmith
algebraic number
Algebraic numbers are the roots of polynomial equations with <integer.htm> coefficients.
A polynomial equation with integer coefficients is
a_0 x^n + a_1 x^{n-1} + ... + a_0
where the a_i. . . are integers.
x is algebraic if and only if it is the solution to such an equation.
algebraic integer
An algebraic number which satisifies a polynomial equation with integer coefficients such that the leading coefficient a_0 is 1.
algebraic number field
If
r satisfies an irreducible (non-factoring) polynomial equation with
integer coefficients of degree d, then the algebraic number field Q(r)
is defined as the set of elements a_0 + a_1 r + ... + a_{d-1} r^{d-1},
where the coefficients a_i are rational numbers. An example of an
algebraic number field would be all numbers of the form
a + b r, where r = (1+sqrt(5))/2 is the golden ratio. The elements of
an algebraic number field form a field--they may be added, subtracted,
multiplied, and divided.
Message: 3153 - Contents - Hide Contents Date: Fri, 11 Jan 2002 07:16:28 Subject: Re: For Joe--proposed definitions From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote: The purely musical definition> of "scale" would have to begin with "... a set of musical > *pitches* ...", etc. Yours would be a #2 definition.I was aware it wouldn't fit all usages. I wanted a defintion which was precise enough to use mathematically, and this is a start. In fact, you might want to be more restrictive and not allow a lot of the things I am calling scales here--that could be discussed.
Message: 3154 - Contents - Hide Contents Date: Fri, 11 Jan 2002 22:15:13 Subject: Re: For Joe--proposed definitions From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Are they? I would think that they would be useful mainly for a >> mathematician who may not know anything about music but who still may >> wish to understand Gene's research. >> They are useful for anyone (eg, me) who might want to state and >prove theorems. You can't do that without definitions.Exactly. Note I said _may_.
Message: 3155 - Contents - Hide Contents Date: Fri, 11 Jan 2002 07:21:20 Subject: Re: For Joe--proposed definitions From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Thanks, Gene! This is great. One quibble... > >> Scale >>>> A discrete set of real numbers, containing 1, and such >> that the distance between sucessive elements of the scale >> is bounded both below and above by positive real numbers.Incidentally, this is incorrect as written since we need to define "distance" logarithmically. If we take a scale as defined above and send scale element s to 2^s, then we get a scale considered as pitch values. In other words, these scale elements should be considered as cents or something of that sort. I'll rewrite it if you like, but perhaps we should hear from other people first.
Message: 3156 - Contents - Hide Contents Date: Fri, 11 Jan 2002 14:18:26 Subject: Re: More proposed definitions From: monz Gene, I thank you much for all these definitions. (Won't be able to upload them until tonight at the earliest.) But ... isn't it getting to be more about "mathematics" and less about "tuning"? I hesitate to put all of these definitions directly into the Tuning Dictionary. Perhaps there should be a separate "mathematical supplement"? Some of you others, please comment on this. -monz ----- Original Message ----- From: genewardsmith <genewardsmith@xxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Friday, January 11, 2002 2:08 PM Subject: [tuning-math] More proposed definitions> algebraic number > > Algebraic numbers are the roots of polynomial equations with <integer.htm> coefficients. > A polynomial equation with integer coefficients is > a_0 x^n + a_1 x^{n-1} + ... + a_0 > where the a_i. . . are integers. > x is algebraic if and only if it is the solution to such an equation. > > algebraic integer > > An algebraic number which satisifies a polynomial equation with integercoefficients such that the leading coefficient a_0 is 1.> > algebraic number field > > If r satisfies an irreducible (non-factoring) polynomial equation withinteger coefficients of degree d, then the algebraic number field Q(r) is defined as the set of elements a_0 + a_1 r + ... + a_{d-1} r^{d-1}, where the coefficients a_i are rational numbers. An example of an algebraic number field would be all numbers of the form> a + b r, where r = (1+sqrt(5))/2 is the golden ratio. The elements of analgebraic number field form a field--they may be added, subtracted, multiplied, and divided.> > > > > > > > > To unsubscribe from this group, send an email to: > tuning-math-unsubscribe@xxxxxxxxxxx.xxx > > > > Your use of Yahoo! Groups is subject to Yahoo! Terms of Service * [with cont.] (Wayb.) > _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3157 - Contents - Hide Contents Date: Fri, 11 Jan 2002 00:31:11 Subject: Re: For Joe--proposed definitions From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 10, 2002 11:21 PM > Subject: [tuning-math] Re: For Joe--proposed definitions > > > [re: definition of "scale"] > I'll rewrite it if you like, but perhaps we should hear > from other people first.Can I have both? :) I'd like to have the input of others (especially Paul, Dave, Graham, Manuel) on this. But feel free to rewrite it as you see fit. I'm not yet up to speed with your work here over the past few months, so I really can't comment until I understand more. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3158 - Contents - Hide Contents Date: Fri, 11 Jan 2002 22:21:19 Subject: Re: More proposed definitions From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Gene, I thank you much for all these definitions. > (Won't be able to upload them until tonight at the > earliest.) > > But ... isn't it getting to be more about "mathematics" > and less about "tuning"? I hesitate to put all of > these definitions directly into the Tuning Dictionary. > Perhaps there should be a separate "mathematical > supplement"?That's exactly what I was thinking.
Message: 3159 - Contents - Hide Contents Date: Fri, 11 Jan 2002 01:13:43 Subject: [tuning] Re: badly tuned remote overtones From: monz Hi Paul and Gene,> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning@xxxxxxxxxxx.xxx> > Sent: Thursday, January 10, 2002 10:10 AM > Subject: [tuning] Re: badly tuned remote overtones > > > --- In tuning@y..., "monz" <joemonz@y...> wrote: >>> The periodicity-blocks that Gene made from my numerical analysis >> of Schoenberg's 1911 and 1927 theories are a good start. >> Well, given that most of the periodicity blocks imply not 12-tone, > but rather 7-, 5-, and 2-tone scales, it strikes me that Schoenberg's > attempted justification for 12-tET, at least as intepreted by you, > generally fails. No?I originally said:> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, December 25, 2001 3:44 PM > Subject: [tuning-math] lattices of Schoenberg's rational implications > > > Unison-vector matrix: > > 1911 _Harmonielehre_ 11-limit system > > ( 1 0 0 1 ) = 33:32 > (-2 0 -1 0 ) = 64:63 > ( 4 -1 0 0 ) = 81:80 > ( 2 1 0 -1 ) = 45:44 > > Determinant = 7 > > ... <snip> ... > > But why do I get a determinant of 7 for the 11-limit system? > Schoenberg includes Bb and Eb as 7th harmonics in his description, > which gives a set of 9 distinct pitches. But even when > I include the 15:14 unison-vector, I still get a determinant > of -7. And if I use 16:15 instead, then the determinant > is only 5.But Paul, you yourself said:> From: Paul Erlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, July 19, 2001 12:43 PM > Subject: [tuning-math] Re: lattices of Schoenberg's rational implications > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> Could anyone out there do some periodicity-block >> calculations on this theory and say something about that? >> It's pretty clear that Schoenberg's theory implies a 12-tone > periodicity block.That was quite a while ago ... have you changed your position on that? I thought that Gene showed clearly that a 12-tone periodicity-block could be constructed out of Schoenberg's unison-vectors.> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 26, 2001 12:27 AM > Subject: [tuning-math] Re: lattices of Schoenberg's rational implications > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> Can someone explain what's going on here, and what candidates >> may be found for unison-vectors by extending the 11-limit system, >> in order to define a 12-tone periodicity-block? Thanks. >> See if this helps; > > We can extend the set {33/32,64/63,81/80,45/44} to an > 11-limit notation in various ways, for instance > > <56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5] > > where g7 differs from h7 by g7(7)=19.Gene, how did you come up with 56/55 as a unison-vector? Why did I get 5 and 7 as matrix determinants for the scale described by Schoenberg, but you were able to come up with 12?> Using this, we find the corresponding block is > > (56/55)^n (33/32)^round(12n/7) (64/63)^n (81/80)^round(-2n/12) > (45/44)^round(5n/7), or 1-9/8-32/27-4/3-3/2-27/16-16/9; the > Pythagorean scale. We don't need anything new to find a > 12-note scale; we get > > 1--16/15--9/8--32/27--5/4--4/3--16/11--3/2--8/5--5/3--19/9--15/8 > > or variants, the variants coming from the fact that 12 > is even, by using 12 rather than 7 in the denominator.Can you explain this business about variants in a little more detail? I understand the general concept, having seen it in periodicity-blocks I've constructed on my spreadsheet, but I'd like your take on the particulars for this case. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3160 - Contents - Hide Contents Date: Fri, 11 Jan 2002 22:24:43 Subject: Re: More proposed definitions From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> But ... isn't it getting to be more about "mathematics" > and less about "tuning"?Possibly, but I thought you asked me to give you a definition of algebraic number field for your dictionary. Paul already gave a (correct, even thoughI fiddled with it) definition of algebraic number, and also of transcendental number. More to the point would be definitions of vector, vector space, lattice, bilinear form, group, abelian group, homomophism, kernel, equivalence relation, equivalence class, quotient group, graph, wedge product, and determinant, but this would definately start to look like mathematics.
Message: 3161 - Contents - Hide Contents Date: Fri, 11 Jan 2002 10:49:52 Subject: Re: Dictionary query From: manuel.op.de.coul@xxxxxxxxxxx.xxx Paul, Gene, Joe, You've missed or ignored my answer to Joe's question, which was the most concise I could give. The borderline is the point where the Pythagorean comma vanishes: 700 cents. This choice is not 12-tET centric in my view.> Thanks very much for that, Paul. So how does it look now? > Definitions of tuning terms: positive system, ... * [with cont.] (Wayb.)You could add that systems with p=1 (scale steps) are called singly positive, with p=2 doubly positive, p=-1 singly negative, etc. Manuel
Message: 3162 - Contents - Hide Contents Date: Fri, 11 Jan 2002 01:59:34 Subject: Re: badly tuned remote overtones From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning@xxxxxxxxxxx.xxx> > Sent: Thursday, January 10, 2002 10:10 AM > Subject: [tuning] Re: badly tuned remote overtones > > > --- In tuning@y..., "monz" <joemonz@y...> wrote: >>> The periodicity-blocks that Gene made from my numerical analysis >> of Schoenberg's 1911 and 1927 theories are a good start. >> Well, given that most of the periodicity blocks imply not 12-tone, > but rather 7-, 5-, and 2-tone scales, it strikes me that Schoenberg's > attempted justification for 12-tET, at least as intepreted by you, > generally fails. No?Ahh ... actually Paul ... no. Now I realize my mistake: I had failed to take into consideration the 5-limit enharmonicity required by Schoenberg. To construct a periodicity-block according to his descriptions, one would have to temper out one of the "enharmonic equivalents". We may choose 2048:2025 = [2] [3] * [11 -4 -2] [5] Plugging that into the unison-vector matrix I had already derived before: 2 3 5 7 11 unison vectors ~cents [ 11 -4 -2 0 0] = 2048:2025 19.55256881 [ -5 1 0 0 1] = 33:32 53.27294323 [ 6 -2 0 -1 0] = 64:63 27.2640918 [ -4 4 -1 0 0] = 81:80 21.5062896 inverse (without powers of 2) = [-1 0 0 2] [-4 0 0 -4] 1 [ 2 0 -12 -4] * -- [ 1 12 0 -2] 12 So it looks to me like Schoenberg's explanation in _Harmonielehre_ definitely implies a 12-tone periodicity-block. I'd venture to say that Schoenberg had a good intuitive grasp of all this, without actually knowing anything about periodicity-block theory. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3163 - Contents - Hide Contents Date: Fri, 11 Jan 2002 10:59:44 Subject: Re: For Joe--proposed definitions From: manuel.op.de.coul@xxxxxxxxxxx.xxx I've never felt the need for a mathematical definition of "scale". Never looked it up in a dictionary either. Perhaps "things like do re mi fa sol la ti do" will do. Manuel
Message: 3164 - Contents - Hide Contents Date: Fri, 11 Jan 2002 02:02:32 Subject: Re: Dictionary query From: monz> From: <manuel.op.de.coul@xxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, January 11, 2002 1:49 AM > Subject: Re: [tuning-math] Re: Dictionary query > > > Paul, Gene, Joe, > > You've missed or ignored my answer to Joe's question, > which was the most concise I could give. > The borderline is the point where the Pythagorean comma > vanishes: 700 cents. This choice is not 12-tET centric > in my view.Ahh ... so then it's not 12-*tET* centric, but it *is* 12-*tone* centric, because the Pythagorean comma is ("8ve"-invariant) 3^12.>> Thanks very much for that, Paul. So how does it look now? >> Definitions of tuning terms: positive system, ... * [with cont.] (Wayb.) >> You could add that systems with p=1 (scale steps) are > called singly positive, with p=2 doubly positive, > p=-1 singly negative, etc.Thanks, Manuel, good idea ... but what's "p"? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3165 - Contents - Hide Contents Date: Fri, 11 Jan 2002 11:08:44 Subject: Re: Dictionary query From: manuel.op.de.coul@xxxxxxxxxxx.xxx>Ahh ... so then it's not 12-*tET* centric, but it *is* >12-*tone* centric, because the Pythagorean comma is >("8ve"-invariant) 3^12. Yup. >Thanks, Manuel, good idea ... but what's "p"?Symbol for the Pythagorean comma. If v is the size of the fifth, and a the size of the octave, then p = 12 v - 7 a. For example in 31-tET, v=18 and a=31, so p=-1. Manuel
Message: 3166 - Contents - Hide Contents Date: Fri, 11 Jan 2002 02:12:26 Subject: Re: Dictionary query From: monz> From: <manuel.op.de.coul@xxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, January 11, 2002 2:08 AM > Subject: Re: [tuning-math] Re: Dictionary query > >>> Thanks, Manuel, good idea ... but what's "p"? >> Symbol for the Pythagorean comma. > If v is the size of the fifth, and a the size of the > octave, then p = 12 v - 7 a. > For example in 31-tET, v=18 and a=31, so p=-1.OK, got it! Thanks for providing the concrete 31-tET example ... now I understand. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3167 - Contents - Hide Contents Date: Fri, 11 Jan 2002 03:06:47 Subject: updated "positive" and "negative" definitions From: monz OK, have a look now: Definitions of tuning terms: negative system, ... * [with cont.] (Wayb.) Definitions of tuning terms: positive system, ... * [with cont.] (Wayb.) But I'm confused about one thing: on the "negative" page, I have 53- and 65-EDO listed as negative temperaments, and they do indeed have negatively-tempered "5ths", but they both have p = +1. ????? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3168 - Contents - Hide Contents Date: Fri, 11 Jan 2002 03:12:38 Subject: Re: badly tuned remote overtones From: monz Hi Gene,> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx>; <tuning@xxxxxxxxxxx.xxx> > Sent: Friday, January 11, 2002 1:59 AM > Subject: [tuning-math] Re: badly tuned remote overtones > > > Now I realize my mistake: I had failed to take into > consideration the 5-limit enharmonicity required by Schoenberg. > To construct a periodicity-block according to his descriptions, > one would have to temper out one of the "enharmonic equivalents". > > We may choose 2048:2025 = > > [2] > [3] * [11 -4 -2] > [5] > > > Plugging that into the unison-vector matrix I had already > derived before: > > 2 3 5 7 11 unison vectors ~cents > > [ 11 -4 -2 0 0] = 2048:2025 19.55256881 > [ -5 1 0 0 1] = 33:32 53.27294323 > [ 6 -2 0 -1 0] = 64:63 27.2640918 > [ -4 4 -1 0 0] = 81:80 21.5062896 > > > inverse (without powers of 2) = > > [-1 0 0 2] > [-4 0 0 -4] 1 > [ 2 0 -12 -4] * -- > [ 1 12 0 -2] 12 >How does this compare with the other 12-tone periodicity-block you calculated for Schoenberg? Can you please give a listing of the pitches inside *this* PB? Thanks. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3170 - Contents - Hide Contents Date: Sat, 12 Jan 2002 20:59:50 Subject: Re: More proposed definitions From: genewardsmith --- In tuning-math@y..., graham@m... wrote: If Gene wants to express musical concepts in> mathematical terms that's fine. But it shouldn't be in a general tuning > dictionary. If Gene wants space to write his own dictionary, he can have > it at microtonal.co.uk.Actually, I've thought of doing that, but I couldn't get it to work--I could upload files, but could not find them afterwards. However, some of the concepts, eg val, are of the sort that a word ought to be coined for it, and made a part of the tuning vocabulary. It doesn't need to be my word, but it should be some word.> Really, Gene seems to be aiming at either creating a new branch of > mathematics relating to tuning theory, or defining aspects of tuning > theory in such a way that they become isomorphic to an existing branch of > mathematics.The latter--it's applied math. Of couse, as with eg coding theory, that *can* introduce new mathematical objects for consideration.
Message: 3172 - Contents - Hide Contents Date: Sat, 12 Jan 2002 18:37:17 Subject: Re: dict/genemath.htm From: clumma> Gene wrote:>> I just spent some time trying to discover what Lumma Stability >> was, and failing. >> If you open the Scala file tips.par in a text editor and search > for "stability" you will find a definition.I couldn't have said it better myself. The def. of Rothenberg stability, though... isn't this the portion of intervals breaking strict propriety, rather than just the intervals appearing in more than one class? -Carl
Message: 3173 - Contents - Hide Contents Date: Sat, 12 Jan 2002 21:20:35 Subject: Re: tuning dictionaries vs math dictionaries From: genewardsmith --- In tuning-math@y..., jon wild <wild@f...> wrote:> http://mathworld.wolfram.com/AlgebraicNumber.html * [with cont.] > http://mathworld.wolfram.com/Determinant.html * [with cont.]This looks like a good plan, when it works, but it doesn't always work. The definition for algebraic number said everything Paul or I said, and more. The definition of determinant was the usual one--I would have defined it in terms of wedge products after defining the wedge product--but should certainly do. The defintion of wedge product, unfortunately, does not exist, or at least I couldn't find it. Here is a definition of Clifford algbera: http://mathworld.wolfram.com/CliffordAlgebra.html * [with cont.] I think this does not work for our purposes. I checked out the entry for abelian group Abelian Group -- from MathWorld * [with cont.] and I think this would be far more confusing than a definition written with musical applications in mind--it assumes a large amount of irrelevant group theory knowledge.
Message: 3174 - Contents - Hide Contents Date: Sat, 12 Jan 2002 13:25:35 Subject: Re: tuning dictionaries vs math dictionaries From: monz I'll make use of mathworld links where appropriate, but I'd still like the tuning-specific dope from Gene. My whole intention is to understand, and help others to understand, the work that's gone on at tuning-math for the last several months. Gene, Paul, Graham, and Dave are the only members posting who seem to follow it. After more thought, I'm more hesitant to split the Dictionary up. But if there is a way to make a simplified tuning-specific definition as well as a more comprehensive and more general one, I'll upload them both and link them together. That way people innocently surfing into the Dictionary won't get overwhelmed, and those who want more can still get it. -monz ----- Original Message ----- From: genewardsmith <genewardsmith@xxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Saturday, January 12, 2002 1:20 PM Subject: [tuning-math] Re: tuning dictionaries vs math dictionaries> --- In tuning-math@y..., jon wild <wild@f...> wrote: > >> http://mathworld.wolfram.com/AlgebraicNumber.html * [with cont.] >> http://mathworld.wolfram.com/Determinant.html * [with cont.] >> This looks like a good plan, when it works, but it doesn't always work.The definition for algebraic number said everything Paul or I said, and more. The definition of determinant was the usual one--I would have defined it in terms of wedge products after defining the wedge product--but should certainly do. The defintion of wedge product, unfortunately, does not exist, or at least I couldn't find it. Here is a definition of Clifford algbera:> > http://mathworld.wolfram.com/CliffordAlgebra.html * [with cont.] > > I think this does not work for our purposes. > > I checked out the entry for abelian group > > Abelian Group -- from MathWorld * [with cont.] > > and I think this would be far more confusing than a definition writtenwith musical applications in mind--it assumes a large amount of irrelevant group theory knowledge.> > > > > > To unsubscribe from this group, send an email to: > tuning-math-unsubscribe@xxxxxxxxxxx.xxx > > > > Your use of Yahoo! Groups is subject to Yahoo! Terms of Service * [with cont.] (Wayb.) > _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
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