Tuning-Math Archive Section 1: 675

Message: 675 - Contents - Hide Contents Date: Sat, 18 Aug 2001 04:00:42 Subject: Cents and cents ability From: genewardsmith@j... --- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> I mean, really... if +/- 5 cents is pretty much an accepted margin > of error in practice, then do we really need anything more accurate > than 240-EDO?, let alone 1200-EDO (which is exactly what my "Semitones" > specify). > > And if we *do* need more accuracy, why not use another widely accepted > standard with finer resolution, such as "cawapus", or even "midipus", > rather than having to deal with more decimal points after the cents?

Alas, I don't know what these are. For my own purposes, I often use a 612 system, which gives a sort of schisma as the basic unit. The advantage over cents is that some pretty good approximations are available where only integer values need to be remembered, for we have 2 = 612 s 3 = 969.997 s ~ 970 s 5 = 1421.02 s ~ 1421 s 7 = 1718.10 s ~ 1718 s 11 = 2264.67 s ~ 2265 s So I end up remembering stuff like 9/8 = 104 s, 10/9 = 93 s (and of course the equal tempered tone is exactly 102 s) and so forth. The major fifth is 358 s, and the equally tempered fifth is exactly 357 s, flat by one schisma (= s.) The major third is 197 s, whereas the equal tempered third is 204 s, sharp by 7 s.

Message: 676 - Contents - Hide Contents Date: Sat, 18 Aug 2001 05:09:14 Subject: Re: Hypothesis From: genewardsmith@j... --- In tuning-math@y..., graham@m... wrote:

> But if we could prove that all linear temperaments give something like an > MOS, that would prove the hypothesis.

I might try, but first I need some definitions. :)

Message: 678 - Contents - Hide Contents Date: Sat, 18 Aug 2001 17:45:40 Subject: Re: Microtemperament and scale structure From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:

> > Any such homomorphism is defined by its kernel, which are the > elements sent to the identity. In the case of h12, the kernel is > spanned by 81/80 (the diatonic comma) and 128/125 (the great diesis), > where we have h12(81/80) = h12(128/125) = 0.

What you call the "kernel" is what we call the set of unison vectors. I have a feeling your conception will prove to be more useful, since we've been using language like, "well, you could say that the two unison vectors of the 5-limit 12-tone periodicity block are the syntonic comma (81/80) and the small diesis (128/125), but you could also say that they're the syntonic comma and the diaschisma (2025/2048)", for example. We're well aware that any valid set comes from any other valid set simply by "adding" and "subtracting" the unison vectors from one another. But we've run into some pathological cases -- for example, the small diesis (128/125) and the schisma (32805/ 32768), while they can be derived from the same two unison vectors, define a periodicity block with 24, instead of 12, notes . . . and not a well-behaved periodicity block at that. Any insights?

> A tuning system which > does not contain the diatonic comma in its kernel (and this includes > just intonation!) will have a structure quite different that which > musicians normally expect. On the other hand one that does, such as > what we get from the 19 or 31 tone system, will seem more "normal".

We're all very cognizant of that, and you should be aware that many of the musicians on this list are knee-deep in exploring systems which rely specifically on having other commas in their kernal.

> > Consider the system h72(2) = 72, h72(3) = 114, h72(5) = 167, h72(7) = > 202, h72(11) = 249 (the last two values are irrelevant here, but they > do no harm and they cover the range of primes which makes the 72 > system interesting.)

You may know that we've been discussing this system in great detail, especially a few months ago on the tuning list, where we found some amazing 21-, 31-, and 41-tone subsets of 72-tET.

Message: 679 - Contents - Hide Contents Date: Sat, 18 Aug 2001 19:13:53 Subject: Re: Hypothesis From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:

> --- In tuning-math@y..., graham@m... wrote: >

>> But if we could prove that all linear temperaments give something > like an

>> MOS, that would prove the hypothesis. >

> I might try, but first I need some definitions. :)

A sketch of the proof of the Hypothesis is provided in post #591 on this list. Hopefully, all the terminology in that post should be self-explanatory, or explained by context. If not, Fokker periodicity blocks are explained in www.ixpres.com/interval/td/erlich/intropblock1.htm and the pages that follow. MOS is almost synonymous with WF (well- formed) and that concept is explained in many papers, such as 404 Not Found * [with cont.] Search for http://depts.washington.edu/~pnm/CLAMPITT.pdf in Wayback Machine except that in an MOS, the interval of repetition (which Clampitt calls interval of periodicity) can be a half, third, quarter, etc. of the interval of equivalence, and not necessarily equal to it. See what you can make of it all . . . I've ignored pathological cases, so hopefully you can come up with a mathematical framework that covers it all!

Message: 682 - Contents - Hide Contents Date: Sat, 18 Aug 2001 22:18:31 Subject: Re: Microtemperament and scale structure From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> What you call the "kernel" is what we call the set > of unison vectors. I have a feeling your conception > will prove to be more useful, since we've been > using language like, "well, you could say that the > two unison vectors of the 5-limit 12-tone > periodicity block are the syntonic comma (81/80) > and the small diesis (128/125), but you could also > say that they're the syntonic comma and the > diaschisma (2025/2048)", for example.

My way of saying things has the advantage of being standard mathematical terminology, which allows one to bring relevant concepts into play. I had noticed that unison vectors seemed to have something to do with the kernel, but I couldn't tell if it meant generators of the kernel or any element of the kernel, and I see by your comments that no one has really decided! The kernel of some homomorphism h is everything sent to the identity-- if this is the set of unison vectors then for instance 1 is *always* a unison vector, since h(1) = 0. On the other hand, unison vectors could be elements of a minimal set of generators for the kernel. In this case 81/80, 128/125 and 2048/2025 would all belong in the same kernel generated by any two of them. Depending on which set of generators you picked, two of them would be unison vectors and the other one would not be. 1 and 32805/32768 would also both be in the kernel, but neither would be unison vectors. Probably the simplest solution at this point would be to drop the terminology, but if you don't you need to decide what exactly it means. By the way, I called 128/125 a great diesis, and you call it a small diesis. Has this been decided? Which is it, and where does one go to find out? We're well

> aware that any valid set comes from any other > valid set simply by "adding" and "subtracting" the > unison vectors from one another. But we've run > into some pathological cases -- for example, the > small diesis (128/125) and the schisma (32805/ > 32768), while they can be derived from the same > two unison vectors, define a periodicity block with > 24, instead of 12, notes . . . and not a well-behaved > periodicity block at that. Any insights?

It's true that anything in the kernel is obtained by adding and subtracting, since the kernel of an abelian group homomorphism is an abelian group. It's not true that any linearly independent set of kernel elements which span the corresponding vector space over the rationals as a basis is also a minimal set of generators for the kernel, and that is what you have discovered. Let's take 81/80 and 128/125 to start with. I may write these additively, so that 81/80 = 2^-4 * 3^4 * 5^-1 is written [-4, 4, -1] and 128/125 becomes [7, 0, -3]. We can put these together into a 2x3 matrix, giving us [-4 4 -1] [ 7 0 -3] If we take the absolute value of the determinants of the minors of this matrix, we recover the homomorphism: abs(det([[4, -1],[0,-3]]) = 12, abs(det([[-4, -1],[7,-3]]) = 19, and abs(det([-4,4],[7,0]))= 28, recovering the homomorphism column vector [12] [19] [28] from the generators of the kernel. This computation shows these two "unison vectors" do in fact generate the kernel. If we perform a similar computation for 128/125 and 32805/32768 we first get the matrix [7 0 -3] [-15 8 1] The column vector we get from the absolute values of the determinants of the minors of this is: [24] [38] [56] In other words, these two define the kernel of a homomorphism to the 24 et division of the octave, which in the 5-limit is "pathological" in the sense that you have two separate 12 divisions a quarter-tone apart, and we cannot get from one to the other using relationships taken from 5-limit harmony, as all of the numbers in the above homomorphism are even.

Message: 683 - Contents - Hide Contents Date: Sat, 18 Aug 2001 23:19:58 Subject: Re: Microtemperament and scale structure From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >

>> What you call the "kernel" is what we call the set >> of unison vectors. I have a feeling your conception >> will prove to be more useful, since we've been >> using language like, "well, you could say that the >> two unison vectors of the 5-limit 12-tone >> periodicity block are the syntonic comma (81/80) >> and the small diesis (128/125), but you could also >> say that they're the syntonic comma and the >> diaschisma (2025/2048)", for example. >

> My way of saying things has the advantage of being standard > mathematical terminology, which allows one to bring relevant concepts > into play.

Absolutely. That's what I'm hoping for.

> I had noticed that unison vectors seemed to have something > to do with the kernel, but I couldn't tell if it meant generators of > the kernel or any element of the kernel, and I see by your comments > that no one has really decided!

Any n linearly dependent elements of the kernel will generate the kernel, in the context where the unison vectors are completely ignored or tempered out. Any of these elements can be called a unison vector of the system. Any such set of n can be called "the set of unison vectors defining" the system [oops, I realized later that this may not be correct]. In other contexts, the particular choice is relevant, such as in the construction of Fokker periodicity blocks in JI, and there, I suppose, it makes a difference which elements you call "the generators of the kernel". Am I understanding you correctly? [not really -- see below]

> > Probably the simplest solution at this point would be to drop the > terminology, but if you don't you need to decide what exactly it > means. By the way, I called 128/125 a great diesis, and you call it a > small diesis. Has this been decided? Which is it, and where does one > go to find out?

There are many lists of interval names out there. I've seen "small diesis" more often than "great diesis" for this interval -- "large diesis" refers to the difference between four minor thirds and an octave.

>> We're well >> aware that any valid set comes from any other >> valid set simply by "adding" and "subtracting" the >> unison vectors from one another. But we've run >> into some pathological cases -- for example, the >> small diesis (128/125) and the schisma (32805/ >> 32768), while they can be derived from the same >> two unison vectors, define a periodicity block with >> 24, instead of 12, notes . . . and not a well-behaved >> periodicity block at that. Any insights? >

> It's true that anything in the kernel is obtained by adding and > subtracting, since the kernel of an abelian group homomorphism is an > abelian group. It's not true that any linearly independent set of > kernel elements which span the corresponding vector space over the > rationals as a basis is also a minimal set of generators for the > kernel, and that is what you have discovered.

Aha -- so I was missing something above. In the case where the unison vectors are tempered out or completely ignored, there are several valid sets of _generators for the kernel_, but not all sets of n linearly independent elements of the kernel generate it. Can we unambiguously classify all elements of the kernel into those that can generate it and those that can't?

> Let's take 81/80 and 128/125 to start with. I may write these > additively, so that 81/80 = 2^-4 * 3^4 * 5^-1 is written [-4, 4, -1] > and 128/125 becomes [7, 0, -3]. We can put these together into a 2x3 > matrix, giving us > > [-4 4 -1] > [ 7 0 -3] > > If we take the absolute value of the determinants of the minors of > this matrix, we recover the homomorphism: > > abs(det([[4, -1],[0,-3]]) = 12, abs(det([[-4, -1],[7,-3]]) = 19, and > abs(det([-4,4],[7,0]))= 28, recovering the homomorphism column vector > > [12] > [19] > [28] > > from the generators of the kernel.

I think this is a computation that Graham Breed has done or was trying to do. Graham?

> This computation shows these > two "unison vectors" do in fact generate the kernel. If we perform a > similar computation for 128/125 and 32805/32768 we first get the > matrix > > [7 0 -3] > [-15 8 1] > > The column vector we get from the absolute values of the determinants > of the minors of this is: > > [24] > [38] > [56] > > In other words, these two define the kernel of a homomorphism to the > 24 et division of the octave,

I think you're jumping the gun with that intepretation. If you look at the pitches in the Fokker periodicity block defined by this matrix (without the 2's column), you'll see 12 roughly equally-spaced pairs of pitches separated by a syntonic comma (81/80). What is a syntonic comma in this system? Well, _two_ syntonic commas come out to be equal to the product of 128/125 and 32805/32768. But both of these are unison vectors, and must vanish. Therefore their product must vanish. So two syntonic commas vanishes, thus one syntonic comma must either vanish or be consided a half-octave. If the syntonic comma vanishes, then you have 12-tone equal temperament, not 24-tone equal temperament. If the syntonic comma is a half-octave, well that's kinda weird, but it still seems like you get 12-tone equal temperament . . . in case you're wondering about that half-octave, let's look an an analogous case where a half-octave is naturally involved. By the way, I don't think of 24-tET as pathological at all, I just don't think that one can say, in any useful sense, that it's _generated_ or _defined_ by 128/125 and 32805/32768. Here we move on to the 7-limit, and the unison vectors are [Note: I'm switching from "/" to ":" because of our convention to use ":" for intervals and "/" for pitches] 49:48 = [-4 -1 0 2] 64:63 = [ 6 -2 0 -1] 225:224 = [-5 2 2 -1] The absolute values of the determinants of the minors should be [10] [16] [23] [28] right? Here, the product of 64:63 and 225:224 is equal to _two_ 10:7s.And here, it's clearly a half-octave we're dealing with, not something that vanishes. So we really have a 10-tone scale here, and if all three unison vectors are tempered out uniformly, you get 10-tET. (In a case of particularly great interest to me, the 49:48 is _not_ tempered out, and you get a 10-tone system embedded in something very close to 22-tone equal temperament).

Message: 684 - Contents - Hide Contents Date: Sat, 18 Aug 2001 17:26:46 Subject: Re: Hi! Seeking advice From: monz

> From: <BobWendell@t...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, August 17, 2001 2:53 PM > Subject: [tuning-math] Re: Hi! Seeking advice > > > Or perhaps it has to do with my having only a BA in Mus Ed from a > small state university in Tennessee. I've notice that many academics > don't like to have their tea parties crashed by uncredentialed souls > like me, no matter what the quality of our musical product, > especially when our Website proclaims a unique approach to choral > training and we boast a small chamber choir of virtual choral > neophytes in most cases that sounds better than many of their choirs > richly stocked with music majors. It takes us longer to get our music > to market, so to speak, but it's usually great once we get it there > as long as the recent turnover rate has been reasonable. Hi Bob,

Don't be concerned about being "uncredentialed". I get plenty of respect from lots of people on these different tuning lists, and it's based entirely on my webpages and on what I've posted to those lists; and I have no degrees... nothing beyond a high school diploma. love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 685 - Contents - Hide Contents Date: Sun, 19 Aug 2001 05:26:56 Subject: Re: Microtemperament and scale structure From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> There are many lists of interval names out there. I've seen "small

diesis" more often than "great

> diesis" for this interval -- "large diesis" refers to the

difference between four minor thirds and an

> octave.

I use the table in Ellis' appendix to Helmholtz mostly. If there is nothing official, this seems like a good choice of _locus classicus_.

> Can we unambiguously classify all elements of the kernel into those

that can generate it and

> those that can't?

The identity 1 is never a generator, and any element which is a multiple of another element is never a generator. Beyond that we cannot go--being a generator is really a property of sets of elements, not of individual elements.

>> In other words, these two define the kernel of a homomorphism to the >> 24 et division of the octave,

> I think you're jumping the gun with that intepretation.

Well, I just showed it was true, though I haven't explained why the method works. However, I was unclear about one thing--I wasn't talking about periodicity blocks at all, simply about homomorphisms and ets. Any homomorphism uniquely determines a kernel, and vice- versa. If we take everything generated by the above two elements, we get a subgroup of our abelian group, and regarded as a kernel it will uniquely be associated to a homomorphism--which in this case has an image which happens to be the 24-et in the 5-limit. It is clear however that there is a very close relationship between sets of generators for a kernel and periodic blocks. In fact, we can uniquely associate a periodic block with a set of generators. Let {g1, g2, ... gk} be a set of generators. If we take equivalence classes by octaves, we get another set {h1, h2, ... hk} of generators for the group of equivalence classes (this is just another sort of homomorphism, actually.) If we take the absolute value of the determinant of {h1, h2, ... hk}, which is the same as the first minor with the 2-column removed we considered before, we get an integer N which is the content ("hypervolume") of the parallelepiped defined by {h1, h2, ... hk}--which fact if you think about its implications should make clear why the method for finding the homomorphism works. Now take the set S of all group elements a1*h1+a2*h2+ ... + ak*hk, where the ai are rational numbers 0 <= ai < 1. These are lattice elements in the parallelepiped, and because its content is N there are N of them. These N elements S define a periodic block uniquely associated to the set {h1, h2, ... hk} and hence to {g1, g2, ... gk}. We can make it a little more palatable by moving the identity to somewhere in the middle instead of at a corner, of course. :) Hence in the example you gave, you do get a periodic block with 24 elements, and I don't see anything pathological about it beyond the fact that some elements will be a comma apart. By the way, a periodic block is really a special case of a JT scale, and I don't know why it should be the most interesting case.

> By the way, I don't think of 24-tET as pathological at all, I just

don't think that one can say, in any

> useful sense, that it's _generated_ or _defined_ by 128/125 and 32805/32768.

Well, it is however--these two elements generate a kernel, and the kernel is *uniquely* associated to a homomorphism.

> Here we move on to the 7-limit, and the unison vectors are > > [Note: I'm switching from "/" to ":" because of our convention to

use ":" for intervals and "/" for

> pitches] > > 49:48 = [-4 -1 0 2] > 64:63 = [ 6 -2 0 -1] > 225:224 = [-5 2 2 -1] > > The absolute values of the determinants of the minors should be > > [10] > [16] > [23] > [28] > > right?

I don't know if they should be, but they are. :) Right! Here, the product of 64:63 and 225:224 is equal to _two_ 10:7s. ... minus an octave. (In a case of

> particularly great interest to me, the 49:48 is _not_ tempered out,

and you get a 10-tone

> system embedded in something very close to 22-tone equal temperament).

This is a tempered 10-tone scale, in other words.

Message: 686 - Contents - Hide Contents Date: Sun, 19 Aug 2001 08:42:28 Subject: Mea culpa From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I think you're jumping the gun with that intepretation.

You were right--my method doesn't work for finding the homomorphism in these cases. I'll try to sort this out tomorrow if I have time, but the image under the homomorphism has to have a 2-torsion part. I think the deal is h([a, b, c]) gets sent to [12*a+19*b+28*c, a+b+c (mod 2)].

Message: 687 - Contents - Hide Contents Date: Sun, 19 Aug 2001 11:05 +0 Subject: Re: Microtemperament and scale structure From: graham@m... Paul wrote:

>> Let's take 81/80 and 128/125 to start with. I may write these >> additively, so that 81/80 = 2^-4 * 3^4 * 5^-1 is written [-4, 4, -1] >> and 128/125 becomes [7, 0, -3]. We can put these together into a 2x3 >> matrix, giving us >> >> [-4 4 -1] >> [ 7 0 -3] >> >> If we take the absolute value of the determinants of the minors of >> this matrix, we recover the homomorphism: >> >> abs(det([[4, -1],[0,-3]]) = 12, abs(det([[-4, -1],[7,-3]]) = 19, and >> abs(det([-4,4],[7,0]))= 28, recovering the homomorphism column vector >> >> [12] >> [19] >> [28] >> >> from the generators of the kernel. >

> I think this is a computation that Graham Breed has done or was trying > to do. Graham? Hello!

Octave-specific vectors for an octave-invariant system are a fudge. So I add the octave to the kernel: [ 1 0 0] [-4 4 -1] [ 7 0 -3] Invert that and multiply by the determinant and you get: [-12 0 0] [-19 -3 1] [-28 0 4] The left hand column is the "homomorphism column vector" above (sign isn't important as long as it's consistent). It's identical to Gene's formula by the definition of matrix inversion. The 12 is Fokker's determinant. The other columns happen to be the generator mappings for the equivalent column being a chromatic unison vector. I don't think there's a proof for this always working yet, but it does. Note that this seems to be a different meaning of "generator" from above. Lots of background is at <Intonation information * [with cont.] (Wayb.)>. See the "temperaments from unison vectors" program. You'll need Numeric Python which I couldn't find on the FTP site last time I looked. Apparently, the ActiveState download includes it, so try that if you're on Windows. An aside: the operation "invert and multiply by determinant" could be made primary. It's actually simpler to calculate than a regular inverse, because the last thing you do is to divide by the determinant. This would give an algebra containing only integer matrices. I'd be interested to know if this exists in the mathematical literature anywhere.

>> This computation shows these >> two "unison vectors" do in fact generate the kernel. If we perform a >> similar computation for 128/125 and 32805/32768 we first get the >> matrix >> >> [7 0 -3] >> [-15 8 1] >> >> The column vector we get from the absolute values of the determinants >> of the minors of this is: >> >> [24] >> [38] >> [56] >> >> In other words, these two define the kernel of a homomorphism to the >> 24 et division of the octave, >

> I think you're jumping the gun with that intepretation.

This has been acknowledged now. The easy way you know it isn't 24-equal is that 24, 38 and 56 are all even numbers. So you're only defining every other note from 24-equal, which is identical to 12-equal.

> Here we move on to the 7-limit, and the unison vectors are > > [Note: I'm switching from "/" to ":" because of our convention to use > ":" for intervals and "/" for pitches] > > 49:48 = [-4 -1 0 2] > 64:63 = [ 6 -2 0 -1] > 225:224 = [-5 2 2 -1] > > The absolute values of the determinants of the minors should be > > [10] > [16] > [23] > [28] > > right? Here, the product of 64:63 and 225:224 is equal to _two_ > 10:7s.And here, it's clearly a half-octave we're dealing with, not > something that vanishes. So we really have a 10-tone scale here, and if > all three unison vectors are tempered out uniformly, you get 10-tET. > (In a case of particularly great interest to me, the 49:48 is _not_ > tempered out, and you get a 10-tone system embedded in something very > close to 22-tone equal temperament).

In this case one entry is and odd number, so it really is 10 notes and not 5. Graham

Message: 688 - Contents - Hide Contents Date: Sun, 19 Aug 2001 17:44:49 Subject: Re: Microtemperament and scale structure From: genewardsmith@j... --- In tuning-math@y..., graham@m... wrote:

> Octave-specific vectors for an octave-invariant system are a

fudge. So I

> add the octave to the kernel: > > [ 1 0 0] > [-4 4 -1] > [ 7 0 -3]

Adding the octave to the kernel changes the image from a rank one free group (something isomorphic to Z) to a cyclic group of order 12 (isomorphic to Z/12Z.) The homomorphism is simply h([a, b, c]) = 12*a+19*b+28*c (mod 12) = 7*b+4*c (mod 12). In other words, a fifth is 7 semitones, a major third is 4 semitones, and octaves we are ignoring.

> The other columns happen to be the generator mappings for the equivalent > column being a chromatic unison vector. I don't think there's a proof for > this always working yet, but it does. Note that this seems to be a > different meaning of "generator" from above.

As I've remarked already, one can't prove something without a statement of the conjectured result, and one can't understand the statement without definitions. So--what's the claim?

> An aside: the operation "invert and multiply by determinant" could be > made primary. It's actually simpler to calculate than a regular inverse, > because the last thing you do is to divide by the determinant. This would > give an algebra containing only integer matrices. I'd be interested to > know if this exists in the mathematical literature anywhere.

It's called the adjoint matrix. If M is a square matrix, you would notate it as adj(M). If R is any ring, we can define a ring M_n(R) of nxn matricies with entries in R, for any positive integer n. M_n(R) is actually an R- algebra, since scalar mulitplication by R is defined. If R is a commutative ring, we can define the determinant and hence the adjoint matrix in M_n(R). In particular if R is the integers Z, then we have an adjoint to any matrix with elements in Z, as you noted. We always have M*adj(M) = det(M)I = adj(M)*M, where "det" is the determinant mapping the matrix M to an element of R, "*" denotes matrix multiplication, and I is the identity matrix.

Message: 689 - Contents - Hide Contents Date: Sun, 19 Aug 2001 20:13:50 Subject: Re: Hypothesis From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> MOS is almost synonymous with WF (well- > formed) and that concept is explained in many > papers, such as > > 404 Not Found * [with cont.] Search for http://depts.washington.edu/~pnm/CLAMPITT.pdf in Wayback Machine > > except that in an MOS, the interval of repetition > (which Clampitt calls interval of periodicity) can be > a half, third, quarter, etc. of the interval of > equivalence, and not necessarily equal to it.

I looked at CLAMPITT.pdf, and it seems to me the argument that there is something interesting about WF scales is extremely unconvincing. Can anyone actually *hear* this? I notice that when you talk about periodiciy blocks, you ignore this stuff yourself, as well you might so far as I can see. What gives? Am I missing something?

Message: 690 - Contents - Hide Contents Date: Sun, 19 Aug 2001 21:36 +0 Subject: Re: Microtemperament and scale structure From: graham@m... genewardsmith@j... () wrote:

> --- In tuning-math@y..., graham@m... wrote: >

>> Octave-specific vectors for an octave-invariant system are a

> fudge. So I

>> add the octave to the kernel: >> >> [ 1 0 0] >> [-4 4 -1] >> [ 7 0 -3] >

> Adding the octave to the kernel changes the image from a rank one > free group (something isomorphic to Z) to a cyclic group of order 12 > (isomorphic to Z/12Z.) The homomorphism is simply > h([a, b, c]) = 12*a+19*b+28*c (mod 12) = 7*b+4*c (mod 12). In other > words, a fifth is 7 semitones, a major third is 4 semitones, and > octaves we are ignoring.

Well, this goes comfortably beyond what I know about group theory. Is Z the set of integers? Making the rank one like the (1) in SU(1)? A "cyclic group of order 12" would make sense if it means the same 12 notes repeat each octave. The rest I think I understand. So you weren't assuming octave invariance to start with?

>> The other columns happen to be the generator mappings for the > equivalent

>> column being a chromatic unison vector. I don't think there's a > proof for

>> this always working yet, but it does. Note that this seems to be a >> different meaning of "generator" from above. >

> As I've remarked already, one can't prove something without a > statement of the conjectured result, and one can't understand the > statement without definitions. So--what's the claim?

The concept of generator is defined in the Carey/Clampitt paper that Paul's already pointed you towards. Or maybe it's only referred to, but you'll get the idea. I'm claiming I can uniquely define a generated scale from a set of unison vectors. The full process is defined by a Python script. It's something like: Put the octave at the top of the matrix and the chromatic unison vector next. Invert the matrix and multiply by the lowest common denominator. The left hand column is the number of scale steps to each prime interval, maybe not in its lowest terms as we know. The second column is the generator mapping. The highest common factor gives you the number of periods (or intervals of periodicity) to an octave. Divide through by that and you have the numbers of generators that period-reduced give the prime intervals. Multiply through by -1 and it still works, but no other mapping will.

>> An aside: the operation "invert and multiply by determinant" could > be

>> made primary. It's actually simpler to calculate than a regular > inverse,

>> because the last thing you do is to divide by the determinant. > This would

>> give an algebra containing only integer matrices. I'd be > interested to

>> know if this exists in the mathematical literature anywhere. >

> It's called the adjoint matrix. If M is a square matrix, you would > notate it as adj(M).

Thanks. How could it be defined for a non-square matrix?

> If R is any ring, we can define a ring M_n(R) of nxn matricies with > entries in R, for any positive integer n. M_n(R) is actually an R- > algebra, since scalar mulitplication by R is defined. If R is a > commutative ring, we can define the determinant and hence the adjoint > matrix in M_n(R). In particular if R is the integers Z, then we have > an adjoint to any matrix with elements in Z, as you noted. We always > have

What's a ring?

> M*adj(M) = det(M)I = adj(M)*M, > > where "det" is the determinant mapping the matrix M to an element of > R, "*" denotes matrix multiplication, and I is the identity matrix.

Yes, that follows. Graham

Message: 691 - Contents - Hide Contents Date: Sun, 19 Aug 2001 14:29:47 Subject: Re: Microtemperament and scale structure From: monz

> From: <genewardsmith@j...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, August 18, 2001 10:26 PM > Subject: [tuning-math] Re: Microtemperament and scale structure > > > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >

>> There are many lists of interval names out there. I've seen >> "small diesis" more often than "great diesis" for this interval >> -- "large diesis" refers to the difference between four minor >> thirds and an octave. >

> I use the table in Ellis' appendix to Helmholtz mostly. If there is > nothing official, this seems like a good choice of _locus classicus_.

Hi Gene, and welcome to the tuning lists. I've come up against this confusion of terminology before, and tried to grapple with a solution for it in this post: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) Hope you find that helpful. This actually is in the archives of <Yahoo groups: /tuning * [with cont.] >, but there was only very limited response to it. I still think my ideas present a useful classification. love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 692 - Contents - Hide Contents Date: Sun, 19 Aug 2001 21:34:56 Subject: Some basic abstract algebra defintions From: genewardsmith@j... --- In tuning-math@y..., graham@m... wrote:

> Well, this goes comfortably beyond what I know about group theory. Is Z > the set of integers?

Z is the ring of integers (from German, "Zahlen" = numbers), but since you asked what a ring is let's make some definitions: (1) An abelian group (G, +, 0) is a set of elements G closed under an addition operation "+", with an identity element 0 and an inverse -a for any a in G, such that -a+a=0. The "+" operation is associative, a+(b+c)=(a+b)+c, and since the group is abelian, commutative also, so that a+b=b+a. We can also write the group multiplicatively as (G, *, 1) when that is convenient. (2) A ring (R, +, *, 0, 1) is an abelian group under "+", and closed under the associative (mulitplication) operation "*", with a multiplicative identity 1. It also satisfies the distributive laws: a*(b+c) = a*b+a*c (b+c)*a = b*a+c*a If the multiplication is commutative, so that a*b=b*a, we have a commutative ring. (3) A free abelian group of rank n, Z^n, is n copies of Z considered as an additive group: ZxZxZ ... xZ n times. Concretely, it consists of vectors of length n with integer values, under addition. (4) The cyclic group of order n, C(n) is the set of elements of Z modulo n, considered as an additive group; as a ring it is denoted Z_n or Z/nZ. Z considered as an abelian group is sometimes called the cyclic group of infinite order as well as the free abelian group of rank one. (5) The units of Z/nZ, represented by the elements relatively prime to n, form the unit group U(n) (not to be confused with the unitary group!) under multiplication. (I add this since you mentioned SU(1) below--that's pretty boring, by the way!--and that made me think of the two meanings of U(n).)

>> It's called the adjoint matrix. If M is a square matrix, you would >> notate it as adj(M).

> Thanks. How could it be defined for a non-square matrix?

You could define something using wedge products but I wouldn't recommend bothing with that now.

Message: 693 - Contents - Hide Contents Date: Mon, 20 Aug 2001 02:37:12 Subject: Re: Microtemperament and scale structure From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:

>>> In other words, these two define the kernel of a homomorphism to > the

>>> 24 et division of the octave, >

>> I think you're jumping the gun with that intepretation. >

> Well, I just showed it was true, though I haven't explained why the > method works. However, I was unclear about one thing--I wasn't > talking about periodicity blocks at all, simply about homomorphisms > and ets.

Well there may be an important difference then.

> > Hence in the example you gave, you do get a periodic block with 24 > elements, and I don't see anything pathological about it beyond the > fact that some elements will be a comma apart.

Right -- but here's what's pathological. Normally, if you temper out the defining unison vectors of the PB, you get an ET, where the number of notes is the determinant of the matrix of unison vectors. But, in this case, if you temper out the schisma and the diesis, you're tempering out their sum, which means you're tempering out _two_ syntonic commas . . . which means that you're either tempering out the syntonic comma, or setting it to half an octave. If you're tempering out the syntonic comma, then the number of notes in the ET is not the determinant (24), but only half that (12). If you're setting it to half an octave . . . whatever that means . . . it seems you still get only 12-tET.

> > By the way, a periodic block is really a special case of a JT scale, > and I don't know why it should be the most interesting case.

Tell me what JT means. And, as you can see in the _Gentle Introduction_, one could choose hexagons instead of parallelograms. In 3D, one could choose hexagonal prisms or rhombic dodecahedra instead of paralellepipeds. In fact, one can choose any weird shape that has one and only one element from the group (to the extent that group theory actually works here . . . ). But if my hypothesis is correct, then parallelograms will be the only way to construct hyper-MOS scales, I think -- since otherwise the period boundaries will not always be straight lines, and thus MOSs won't always result from tempering out all but one of the unison vectors.

>> By the way, I don't think of 24-tET as pathological at all, I just

> don't think that one can say, in any

>> useful sense, that it's _generated_ or _defined_ by 128/125 and > 32805/32768. >

> Well, it is however--these two elements generate a kernel, and the > kernel is *uniquely* associated to a homomorphism.

Try understanding my explanation above as to why, in a _musical_ context, I have a problem with seeing 24-tET as the result of this. Your math may be fine but the association with an ET with the full number of notes, I'm arguing, may not always be appropriate. How, in actual practice, could one take the infinite 5-limit just lattice, alter all the intervals slightly so that both 128/125 and 32805/32768 vanish, and end up with 24-tone equal temperament?

> Here, the product of 64:63 and 225:224 is equal to _two_ 10:7s. > > ... minus an octave.

Right . . . I guess I'm used to being cavalier about octaves in this context (though not in the context of lattice metrics) . . . I mean, each rung in the lattice is defined by an octave-invariant interval, for if it weren't, you wouldn't get anything like unisons between notes the edges of the periodicity block . . . right?

> (In a case of

>> particularly great interest to me, the 49:48 is _not_ tempered out,

> and you get a 10-tone

>> system embedded in something very close to 22-tone equal > temperament). >

> This is a tempered 10-tone scale, in other words.

Yes -- I call it the decatonic scale. It comes in two main forms -- the symmetrical decatonic (rotations of LssssLssss) and pentachordal decatonic (rotations of LsssssLsss). You'll see that it's especially rich in 4:5:6:7 and 1/7:1/6:1/5:1/4 tetrads.

Message: 694 - Contents - Hide Contents Date: Mon, 20 Aug 2001 02:38:38 Subject: Re: Mea culpa From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >

>> I think you're jumping the gun with that intepretation. >

> You were right--my method doesn't work for finding the homomorphism > in these cases.

Whew! I thought I was going crazy :)

> > I'll try to sort this out tomorrow if I have time, but the image > under the homomorphism has to have a 2-torsion part. I think the deal > is h([a, b, c]) gets sent to [12*a+19*b+28*c, a+b+c (mod 2)].

Well, I look forward to the explanation. Let me know if there's a reference I should study to make this all more comprehensible.

Message: 695 - Contents - Hide Contents Date: Mon, 20 Aug 2001 02:42:15 Subject: Re: Microtemperament and scale structure From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:

> > [-12 0 0] > [-19 -3 1] > [-28 0 4] > > The left hand column is the "homomorphism column vector" above (sign isn't > important as long as it's consistent). It's identical to Gene's formula > by the definition of matrix inversion. The 12 is Fokker's determinant. > > The other columns happen to be the generator mappings for the equivalent > column being a chromatic unison vector. > I don't think there's a proof for > this always working yet, but it does.

Can you show with examples?

Message: 696 - Contents - Hide Contents Date: Mon, 20 Aug 2001 02:46:56 Subject: Re: Hypothesis From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >

>> MOS is almost synonymous with WF (well- >> formed) and that concept is explained in many >> papers, such as >> >> 404 Not Found * [with cont.] Search for http://depts.washington.edu/~pnm/CLAMPITT.pdf in Wayback Machine >> >> except that in an MOS, the interval of repetition >> (which Clampitt calls interval of periodicity) can be >> a half, third, quarter, etc. of the interval of >> equivalence, and not necessarily equal to it. >

> I looked at CLAMPITT.pdf, and it seems to me the argument that there > is something interesting about WF scales is extremely unconvincing. > Can anyone actually *hear* this? I notice that when you talk about > periodiciy blocks, you ignore this stuff yourself, as well you might > so far as I can see. > > What gives? Am I missing something?

There are a tremendous number of arguments as to why there is something interesting about WF or MOS scales in the literature. Personally, I buy very few of them, if any. But there are some very powerful WF/MOS scales around, especially, of course, the usual diatonic scale, and the usual pentatonic scale. The whole point of my Hypothesis is to show that these scales, and perhaps ultimately the entire interest of WF/MOS scales, in fact has a deeper basis in just intonation and periodicity blocks.