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Message: 6152 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_.
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Message: 6154 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 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. > > > > > > > > > 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 * > _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 6155 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 Setup *
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Message: 6156 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.
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Message: 6157 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 Setup *
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Message: 6159 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, (c) 2001 by Joe Monzo * 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
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Message: 6160 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 Setup *
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Message: 6161 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
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Message: 6162 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, (c) 2001 by Joe Monzo * > > 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 Setup *
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Message: 6163 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
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Message: 6164 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 Setup *
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Message: 6165 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, (c) 2001 by Joe Monzo * Definitions of tuning terms: positive system, (c) 2001 by Joe Monzo * 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 Setup *
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Message: 6166 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 Setup *
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Message: 6170 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
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Message: 6172 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: > > > Access Denied * > > Access Denied * > > 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: > > Access Denied * > > I think this does not work for our purposes. > > I checked out the entry for abelian group > > Access Denied * > > 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. > > > > > > 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 * > _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 6173 Date: Sat, 12 Jan 2002 04:18:04 Subject: Re: More proposed definitions From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > From: unidala <JGill99@i...> > > To: <tuning-math@y...> > > Sent: Friday, January 11, 2002 8:05 PM > > Subject: [tuning-math] Re: More proposed definitions > > > > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > > > > > I'd like to hear from some others besides Gene and Paul > > > ... should these math terms go directly into the > > > Tuning Dictionary, or should they live "off-campus"? > > > > > > > > > -monz > > > > > > J Gill: While it might not be the easiest task to implement, > > how about presenting it as non-esoterically as possible in > > the main definition, with links (much as Monz allready does) > > within that text which (heirarchically) enter the realms of > > complexity (deeper and deeper) if the reader is so inclined? > > > > That way the information is pre-compiled in levels of detail. > > > Hey J, thanks for your input ... and I think, thanks to your > suggestions, that I've already hit on the right way to do this: > simply include Gene's defintions "as is" as individual entries > in the Dictionary, but then have a "see also" link at the bottom > of each, which leads to an elementary tutorial webpage which > explains this brand of tuning math. > > Feedback? I don't think you'd want to do this for "pitch", "interval", etc., though . . .
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Message: 6174 Date: Sat, 12 Jan 2002 13:48:51 Subject: The International Linear Algebra Society From: monz I thought some folks here might appreciate this: The International Linear Algebra Society ILAS - The International Linear Algebra Society * -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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