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Message: 1250 - 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: 1251 - 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: 1252 - Contents - Hide Contents

Date: Sat, 18 Aug 2001 19:15:34

Subject: Re: Microtemperament and scale structure

From: Paul Erlich

I wrote:


> 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.


Sorry . . . that should have been the diaschisma 
(2048/2025) and the schisma (32805/32768).

Message: 1253 - Contents - Hide Contents

Date: Sat, 18 Aug 2001 19:36:00

Subject: Re: Microtemperament and scale structure

From: Paul Erlich

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

> I wrote:
> 

>> 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.
> 

> Sorry . . . that should have been the diaschisma 
> (2048/2025) and the schisma (32805/32768).


Naah . . . I was right the first time . . . sorry!

Message: 1254 - Contents - Hide Contents

Date: Sat, 18 Aug 2001 22:18:31

Subject: Re: Microtemperament and scale structure

From: genewardsmith@xxxx.xxx

--- 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: 1255 - 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: 1256 - Contents - Hide Contents

Date: Sat, 18 Aug 2001 03:06:12

Subject: Re: Microtemperament and scale structure

From: genewardsmith@xxxx.xxx

Apparently the math stuff belongs here, so I am reposting it in case 
anyone wants to follow up.

--- In tuning@y..., kalleaho@m... wrote:


> I understand that with microtemperament one can achieve greater 
> number of consonant intervals in a scale that is originally tuned 
in 
> strict Just Intonation. 

> This can be a great method for generating new and interesting 
scales 
> but doesn't it destroy the structure of the original scale?


When you finally get to the point where you can't tell the 
difference, the question becomes moot. However, the answer is yes--
introducing approximations increases the flexibility of harmonic 
relationships and thereby changes the structure. The question is 
really best understood in terms of group theory.

The intervals of any form of just intonation are by definition 
positive rational numbers, and so an element of the abelian group of 
positive rationals under multiplication (since our hearing, and hence 
musical structure, is multiplicative.) We are never really interested 
in all rational numbers, but can content ourselves with a finitely 
generated subgroup--for one easy example, the group G of all the 
numbers of the form 2^a * 3^b * 5^c, where a, b and c are integers, 
are the numbers generated by the first three primes, 2, 3 and 5. 

An approximate tuning system can very often be seen as a (subset of 
a) group homomorphism to a group of smaller rank, and most 
significantly to a group of rank 1. For another easy example, the 
rank 3 free group G can be sent to a rank 1 free group by a 
homomorphism h12(2) = 12, h12(3) = 19, h12(5) = 28. A subset of 88 
contiguous elements of this group associated to notes tuned as usual 
by setting g12(1) = 440 hz, g12(2) = 2*440 hz, g12(3) = 2^19/12*440 
hz, g12(5) = 2^28/12*440 hz is the standard keyboard. 

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. 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". 

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.) This 72 system has a 12 system tuning embedded 
in it, so we could suppose that structurally they are very similar. 
However, intersecting the kernels shows they are remotely related; in 
particular h72(81/80) = 1 and h72(128/125) = 3, the second is not so 
important but the first shows that the 72 system is fundamentally 
different in structure from the 12 system. In contrast, h31(2) = 31, 
h31(3) = 49, h31(5) = 72, h31(7) = 87 and h31(11) = 107 *does* have 
the property that h31(81/80) = 0; and while h31(128/125) = 1 we still 
find h31 is much closer in structre to h12 than is h72.

Message: 1257 - Contents - Hide Contents

Date: Sat, 18 Aug 2001 17:26:46

Subject: Re: Hi! Seeking advice

From: monz

> From: <BobWendell@xxxxxxxxxxx.xxx>
> 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: 1258 - Contents - Hide Contents

Date: Sat, 18 Aug 2001 04:00:42

Subject: Cents and cents ability

From: genewardsmith@xxxx.xxx

--- 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: 1259 - Contents - Hide Contents

Date: Sat, 18 Aug 2001 05:09:14

Subject: Re: Hypothesis

From: genewardsmith@xxxx.xxx

--- 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: 1260 - Contents - Hide Contents

Date: Sun, 19 Aug 2001 17:44:49

Subject: Re: Microtemperament and scale structure

From: genewardsmith@xxxx.xxx

--- 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: 1261 - Contents - Hide Contents

Date: Sun, 19 Aug 2001 20:13:50

Subject: Re: Hypothesis

From: genewardsmith@xxxx.xxx

--- 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: 1262 - Contents - Hide Contents

Date: Sun, 19 Aug 2001 21:36 +0

Subject: Re: Microtemperament and scale structure

From: graham@xxxxxxxxxx.xx.xx

genewardsmith@xxxx.xxx () 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: 1263 - Contents - Hide Contents

Date: Sun, 19 Aug 2001 14:29:47

Subject: Re: Microtemperament and scale structure

From: monz

> From: <genewardsmith@xxxx.xxx>
> 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: 1264 - Contents - Hide Contents

Date: Sun, 19 Aug 2001 21:34:56

Subject: Some basic abstract algebra defintions

From: genewardsmith@xxxx.xxx

--- 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: 1265 - Contents - Hide Contents

Date: Sun, 19 Aug 2001 05:26:56

Subject: Re: Microtemperament and scale structure

From: genewardsmith@xxxx.xxx

--- 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: 1266 - Contents - Hide Contents

Date: Sun, 19 Aug 2001 08:42:28

Subject: Mea culpa

From: genewardsmith@xxxx.xxx

--- 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: 1267 - Contents - Hide Contents

Date: Sun, 19 Aug 2001 11:05 +0

Subject: Re: Microtemperament and scale structure

From: graham@xxxxxxxxxx.xx.xx

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: 1268 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 11:45 +0

Subject: Re: Microtemperament and scale structure

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9lq0ik+d44m@xxxxxxx.xxx>
In article <9lq0ik+d44m@xxxxxxx.xxx>, genewardsmith@xxxx.xxx () wrote:


> At last we are making progress! I don't see much role for 
> the "chromatic" element here, though. If the n-1 unison vectors are 
> linearly independent, we've already seen recently how to tell if they 
> generate a kernel of something mapping to Z: compute the gcd of the 
> determinant minors, and see if it is 1 or not. If they have no common 
> factor, then they define such a mapping, and the "chromatic vector" 
> will go to a certain number of steps in this mapping--hopefully 1, 
> but perhaps 2, 3, 4 ... etc.


Yes, the chromatic UV should be redundant.  That reminds me of two 
hypotheses I didn't get round to implementing in Python code:

1) If you use an octave-invariant matrix, with the chromatic UV at the 
top, the left hand column of the adjoint matrix is the mapping by 
generator, like the second column was before.  (May need to be divided 
through by the GCD.)  That makes the adjoint matrix a list of generator 
mappings (the octave being a special case where it's specific).

2) Any simple interval will do for the chromatic UV, so long as all the 
UVs are linearly independent.  I don't have strict criteria for 
"simple" here, any more than criteria for what work as unison vectors in 
the first place.  But trying [1 0 0 ...], [0 1 0 ...], etc until something 
works (non-zero determinant) should do the trick.  I don't think such 
things can be described as unison vectors, but I believe they do work in 
this context.

So are you saying you have an equivalent to (2)?  I don't think the GCD 
has to be 1.


> As for temperment, that has to do with tuning and you cannot draw any 
> conclusions about tuning unless you introduce it into your statement
> somewhere--nothing in, nothing out.


I suppose it depends on how you define "temperament".  Is "meantone" a 
temperament or a class of temperaments?  The chromatic UV is used to 
define the tuning.  If you want to push the definition and make a third a 
unison vector, you can define quarter comma meantone by setting it just.  
So the commatic UVs define the temperament class and the chromatic UV is 
used to define the specific tuning.  I make the octave explicit for the 
same reason.


>> The strong form says that if you construct the 
>> Fokker (hyperparallelepiped) periodicity block 
>> from the n unison vectors, and again 1 is 
>> chromatic and n-1 are commatic, then the notes in 
>> the PB form an MOS scale.
> 

> PB I presume means periodicity block, and MOS is some kind of jumped-
> up well-formed scale, I understand. Could you similarly define MOS 
> (and WF while you are at it?)


Whatever they mean, MOS and WF are the same thing: a generated scale with 
only two step sizes.  It may be a useful property for alternative systems 
of tonality, but we don't have enough examples to pronounce on that yet.  
It's certainly useful to think about when designing generalized keyboards 
or alternative notation systems.  (See 
<http://www.anaphoria.com/xen3b.PDF - Type Ok * [with cont.]  (Wayb.)> if you haven't already.)

                 Graham

Message: 1269 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 11:45 +0

Subject: Re: Microtemperament and scale structure

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9lpte7+27ap@xxxxxxx.xxx>
Paul wrote:


>> 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?


It's what <Unison vector to MOS script * [with cont.]  (Wayb.)> is all about.  
<Unison vectors * [with cont.]  (Wayb.)> is a list of examples.

                   Graham

Message: 1271 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 18:28:26

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:
> 

>> Gene -- first of all, start with a set of n unison 
>> vectors. The unison vectors that are tempered out 
>> or completely ignored are called "commatic unison 
>> vectors". The unison vectors that amount to a 
>> musically significant difference, but not (often) 
>> large enough to move you from one scale step to 
>> the next, are called "chromatic unison vectors".
> 

> Thanks! I'd guessed that was what it meant. I think you are adding 
to 
> the confusion by calling both of them "unison vectors", though--why 
> not unison and step vectors instead?

Three reasons:


1) The number of notes in the scale should be (normally) the 
determinant of the matrix of unison vectors. One has to include both 
the chromatic and the commatic unison vectors in this calculation.

2) In the "prototypical" case, the commatic unison vector is "the 
comma", 81:80; and the chromatic unison vector is "the chromatic 
unison" or "augmented unison", 25:24. These define a 7-tone 
periodicity block: the diatonic scale. You see how the terminology is 
just a generalization of this case.

3) "Step vectors" would refer to something else. In the prototypical 
example, the step vectors would be 16:15, 10:9, and 9:8.

> 

>> The weak form of the hypothesis simply says that 
>> if there is 1 chromatic unison vector, and n-1 
>> commatic unison vectors, then what you have is a 
>> linear temperament, with some generator and 
>> interval of repetition (which is usually equal to the 
>> interval of equivalence, but sometimes turns out to 
>> be half, a third, a quarter . . . of it).
> 

> At last we are making progress! I don't see much role for 
> the "chromatic" element here, though.


You're right . . . it plays no role here.


> If the n-1 unison vectors are 
> linearly independent, we've already seen recently how to tell if 
they 
> generate a kernel of something mapping to Z:


No -- you did that with n unison vectors -- I'm not counting the 2 
axis as a "dimension" here.

> 

>> The strong form says that if you construct the 
>> Fokker (hyperparallelepiped) periodicity block 
>> from the n unison vectors, and again 1 is 
>> chromatic and n-1 are commatic, then the notes in 
>> the PB form an MOS scale.
> 

> PB I presume means periodicity block, and MOS is some kind of 
jumped-
> up well-formed scale, I understand. Could you similarly define MOS 
> (and WF while you are at it?)


MOS means that there is an interval of repetition (normally equal to 
the interval of equivalence [usually octave], but sometimes it comes 
out as half, third, quarter . . . of the IE). The scale repeats 
itself exactly within each interval of repetition. Within each, there 
is a generating interval, which is iterated some number of times such 
that the scale has two step sizes.

Examples:

The diatonic scale (LsssLss) is MOS: the IoR is an octave, and the 
generator is L+s+s.

The melodic minor scale (LssssLs) is not MOS: there is no generator 
that produces all the notes and no others.

Message: 1272 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 18:39:04

Subject: Re: Hypothesis

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:

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

>> 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. 
> 

> Unless I am missing something (highly likely at this point!) the 
> pentatonic and diatonic scales are WF in mean tone intonation but 
not 
> in just intonation. Is that right? If it is right, doesn't that 
serve 
> to make the whole idea seem fishy?


Strict, fixed-pitch just intonation has almost never been used in 
actual music with these scales. This is because of the so-
called "comma problem". Don't let the JI advocates fool you: 
Pythagorean tuning and various meantone-like temperaments have been 
far more important than fixed-pitch 5-limit just intonation for the 
actual performance of these scales -- even in China!

One thing I forgot to mention about the hypothesis: if you don't use 
the parallelepiped, you might end up with a scale that is not MOS, 
but has the same number of notes as the MOS that comes from the 
parallelepiped. I conjecture that in some precise sense, the MOS has 
more consonant structures (intervals and/or chords) than the 
corresponding non-MOS. This is seen, for example, in the decatonic 
case, where the chromatic unison vector is one member of the set 
{25:24, 28:27, 49:48}, and the commatic unison vectors are two 
members of the set {50:49, 64:63, 225:224}. The MOS is LssssLssss, 
which has 8 consonant 7-limit tetrads (4 4:5:6:7s and 4 
1/7:1/6:1/5:1/4s); while a melodically superior non-MOS scale, 
LsssssLsss, has only 6.

Message: 1273 - 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: 1274 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 18:41:50

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:
> 

>> 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.
> 

> I'm afraid that is where the "torsion" I was talking about comes 
in. 
> Suppose you color all 5-limit notes either green or red, by making 
> [a,b,c] green if a+b+c is even, and red if it is odd. Then two reds 
> add up to a green, a green and a red to a red, and two greens a 
green.
> 
> Your two generators are green, but the comma is red. The generators 
> generate only greens, but you need two reds to get a green. Hence 
the 
> image under the homomorphism goes to a 12 et note, but there is a 
red 
> keyboard and a green keyboard!


Are you saying that both keyboards are tuned identically, or that 
there may be an offset?

> 

>> Tell me what JT means. 
> 

> To me, something defined in terms of rational numbers. What does it 
> mean to you?


I was just asking what it stood for. "Just Tuning"?

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