72 owns the 11-limit

I just finished an interesting calculation, where I took the nine
smallest superparticular ratios belonging to the 11-limit, namely
225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024,
4375/4374 and 9801/9800. I then checked all 126 4-element subsets,
finding 69 of rank 4. Astonisingly, all 69 had the 72 et as the
generator of its null space; in 59 of those cases the determinant of
the 5x5 matrix with 5 indeterminates for the first row was +-h72.

It seems to me I had better check some other prime limits, and it
would be helpful if someone could post some of the Xenharmonicon data
so I don't need to recalculate it. At the moment, at least until my
mother has her brain surgery, I don't feel I am able to leave the
house for long and do this myself.

--- In tuning-math@y..., genewardsmith@j... wrote:
> I just finished an interesting calculation, where I took the nine
> smallest superparticular ratios belonging to the 11-limit, namely
> 225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024,
> 4375/4374 and 9801/9800. I then checked all 126 4-element subsets,
> finding 69 of rank 4. Astonisingly, all 69 had the 72 et as the
> generator of its null space; in 59 of those cases the determinant
of
> the 5x5 matrix with 5 indeterminates for the first row was +-h72.

What was it in the other 10 cases?

I did some similar calculations for the 7-limit a couple of years
ago; I found that, for the smallest of Fokker's unison vectors, 22
and 171 were the "co-owners".
>
> At the moment, at least until my
> mother has her brain surgery,

My prayers for a successful operation and a full recovery.

--- In tuning-math@y..., genewardsmith@j... wrote:
> I just finished an interesting calculation, where I took the nine
> smallest superparticular ratios belonging to the 11-limit, namely
> 225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024,
> 4375/4374 and 9801/9800. I then checked all 126 4-element subsets,
> finding 69 of rank 4. Astonisingly, all 69 had the 72 et as the
> generator of its null space; in 59 of those cases the determinant
of
> the 5x5 matrix with 5 indeterminates for the first row was +-h72.

So 10 had torsion . . . I get 144 as the Fokker determinant for these
10.

Now . . . as to the justification of this calculation . . . I
understand why superparticulars for unison vectors might be appealing
in their own right, but when joining them together in triples, don't
some of the triplets correspond to a basis that is more "skewed" than
that of some triplet that includes at least one non-superparticular?
What I mean by "skewed" is that the unison vectors all lie at a very
small angle to some subspace (I'm thinking especially in my
triangular lattice), so that certain consonant intervals and chords
(small structures in the lattice) will require one to invoke many of
the unison vectors to construct them . . . thus some effective unison
vector, a product of some powers of some of the nominal unison
vectors, will be more immediately relevant . . . and these won't
necessarily be superparticular.

Is there any way to determine the "canonical" set of unison vectors
for a PB . . . perhaps this would be the set that minimizes the sizes
of the numbers in the ratios representing the unison vectors . . . ?

In the above I'm thinking of _all_ the unison vectors as
commatic . . . but what if one is chromatic?

> --- In tuning-math@y..., genewardsmith@j... wrote:
> > I just finished an interesting calculation, where I took the nine
> > smallest superparticular ratios belonging to the 11-limit, namely
> > 225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400,
3025/3024,
> > 4375/4374 and 9801/9800. I then checked all 126 4-element
subsets,
> > finding 69 of rank 4. Astonisingly, all 69 had the 72 et as the
> > generator of its null space; in 59 of those cases the determinant
> of
> > the 5x5 matrix with 5 indeterminates for the first row was +-h72.

Just for fun, I tried the 10 smallest, and I got the following Fokker
determinants:

freq. determinant
1 7
1 10
1 12
1 19
1 24
1 54
1 68
1 99
1 116
3 22
3 53
3 80
3 126
4 62
5 27
5 58
8 34
8 46
10 144
24 31
60 72
65 0

Which of these have torsion? (or how do I find out for myself)

If I take all of the 11-limit superparticular ratios smaller than
20.7¢ (thus those smaller than 17.6¢), I get

freq. determinant
1 1
1 9
1 11
1 29
1 49
1 51
1 60
1 64
1 65
1 79
1 91
1 96
2 90
2 116
3 20
3 23
3 24
3 37
3 48
3 50
3 68
3 80
3 99
3 126
5 28
5 82
8 10
8 18
11 4
11 144
13 58
14 45
14 54
17 15
17 53
19 7
19 19
20 38
21 12
21 26
24 41
24 62
34 14
40 22
40 46
41 27
45 8
46 34
63 72
187 0
188 31

If I take all of the 11-limit superparticular ratios smaller than
35¢, I get

freq. determinant
1 30
1 59
1 63
1 65
1 66
1 76
1 79
1 81
1 86
1 91
1 92
1 93
2 56
2 78
2 96
2 116
3 13
3 21
3 32
3 40
3 47
3 49
3 61
3 80
3 99
3 126
4 1
4 29
4 33
4 51
4 60
5 11
5 42
5 82
5 88
6 108
7 90
9 64
11 6
11 39
11 48
11 50
11 144
14 58
16 37
16 52
17 53
18 23
18 36
20 2
20 9
27 3
27 41
32 68
33 45
39 18
40 44
40 46
40 62
43 5
47 28
53 38
54 16
54 54
65 72
68 17
81 26
83 20
90 4
91 24
92 19
131 15
133 27
145 34
164 7
175 14
176 22
198 8
218 10
251 12
298 31
582 0

Which of these have torsion? (or how do I find out for myself)

So maybe 31 really owns the 11-limit? :)

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

> Just for fun, I tried the 10 smallest, and I got the following
Fokker
> determinants:
>
> freq. determinant
> 1 7
> 1 10
> 1 12
> 1 19
> 1 24
> 1 54
> 1 68
> 1 99
> 1 116
> 3 22
> 3 53
> 3 80
> 3 126
> 4 62
> 5 27
> 5 58
> 8 34
> 8 46
> 10 144
> 24 31
> 60 72
> 65 0
>
> Which of these have torsion? (or how do I find out for myself)

It's a pretty safe guess that all the ones with determinants 24, 54,
116, 126, 62 and 144 have torsion, but to really find out you should
add the 2 column and take the gcd of the minors, or equivalently, add
the 2 column and a row of indeterminants, if whatever you are using
allows you to work with those, and take the gcd of the coefficients.

> If I take all of the 11-limit superparticular ratios smaller than
> 20.7¢ (thus those smaller than 17.6¢), I get

> freq. determinant

> 1 9
> 1 11
> 3 23
> 3 37

Some fairly exotic scale possibilities in here!

> So maybe 31 really owns the 11-limit? :)

It makes a good run at it, but smaller is better when it comes to
commas.

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

> It's a pretty safe guess that all the ones with determinants 24,
54,
> 116, 126, 62 and 144 have torsion...

Come to think of it, this is the 11-limit and the 24 may *not* have
torsion; anyway, the only way to be sure is to check.

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

>
> > So maybe 31 really owns the 11-limit? :)
>
> It makes a good run at it, but smaller is better when it comes to
> commas.

Well if that's so, wouldn't 250000:250047 be better than all the superparticular commas? No, I
think larger commas with smaller numbers can often be better because they portend simpler
systems.

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

> but to really find out you should
> add the 2 column and take the gcd of the minors,

I'd really like to be able to do this. The gcd of the minors . . . can you explicate exactly what the
procedure is . . . I have Matlab.

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

> > freq. determinant
>
> > 1 9
> > 1 11
> > 3 23
> > 3 37
>
> Some fairly exotic scale possibilities in here!

Well, the 11- and 23-tone ones especially would correspond to "skewed" periodicity blocks, a
concept I was trying to explain early this morning (which means that some of the simple
consonant intervals will involve many unison vectors and thus be quite out-of-tune). Did that
make sense to you? Is there any way to measure this "skewness", say with dot products of the
vectors, but modified to conform with the tetrahedral-octahedral lattice instead of the Cartesian
one? What about the idea of a "canonical" basis? Perhaps I need to go off and get some math
textbooks and figure this all out for myself . . .

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

> I'd really like to be able to do this. The gcd of the minors . . .
can you explicate exactly what the
> procedure is . . . I have Matlab.

Matlab actually has a lot of Maple stuff in it now, since they bought
some Maple functionality from the Maple people. Unfortunately, I
don't know Matlab so I don't exactly know how it works.

The function igcd(n1, n2, ..., nk) in Maple returns the greatest
common divisor of k integers; this may be callable in Matlab, or
Matlab may have its own number theory functions. If you have a matrix
with k-1 rows and k columns, you can produce k different square
matricies by removing one of the columns; these (or the determinants
of these) are called minors. Since Matlab is strong on linear algebra
I assume it can do this, if not Maple can, and again that may be
callable. The proceedure would be to get the k different integers
det(min_j), 1<=j<=k, where min_j is the matrix you get by removing
the jth column, and take igcd(det(min_1), ..., det(min_k)). If this
is not 1, then you have torsion.

I confess I haven't been following this thread, but gcd has a
nice recursive definition based on the Euclidean algorithm.
Or something. Here's lisp that'll do it:

(define gcd
(lambda (ls)
(let ((y (car (cdr ls))))
(cond
[(zero? y) (car ls)]
[else (gcd (cons y (cons (modulo (car ls) y) '())))]))))

-Carl

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > I'd really like to be able to do this. The gcd of the
minors . . .
> can you explicate exactly what the
> > procedure is . . . I have Matlab.
>
> Matlab actually has a lot of Maple stuff in it now, since they
bought
> some Maple functionality from the Maple people. Unfortunately, I
> don't know Matlab so I don't exactly know how it works.
>
> The function igcd(n1, n2, ..., nk) in Maple returns the greatest
> common divisor of k integers; this may be callable in Matlab, or
> Matlab may have its own number theory functions. If you have a
matrix
> with k-1 rows and k columns, you can produce k different square
> matricies by removing one of the columns; these (or the
determinants
> of these) are called minors. Since Matlab is strong on linear
algebra
> I assume it can do this, if not Maple can, and again that may be
> callable. The proceedure would be to get the k different integers
> det(min_j), 1<=j<=k, where min_j is the matrix you get by removing
> the jth column, and take igcd(det(min_1), ..., det(min_k)). If this
> is not 1, then you have torsion.

OK! This time, eliminating all PBs with torsion:

The set of 11-limit superparticular unison vectors smaller than 20.7
cents (thus smaller than 17.6 cents) yields the following, taking 4
at a time:

freq. determinant
1 1
1 9
1 11
1 29
1 49
1 51
1 60
1 62
1 64
1 65
1 79
1 82
1 91
1 96
1 144
3 20
3 23
3 24
3 37
3 48
3 50
3 68
3 80
3 99
3 126
8 10
8 18
8 54
11 4
13 58
14 45
17 15
17 53
19 7
19 19
20 38
21 12
21 26
24 41
34 14
40 22
40 46
41 27
45 8
46 34
63 72
188 31

So there are three that legitimately have 126 notes, three that
legitimately have 24, and one with 144.

The set of 11-limit superparticular unison vectors smaller than 35
cents yields the following, taking 4 at a time:

1 30
1 40
1 59
1 63
1 65
1 66
1 76
1 79
1 82
1 86
1 91
1 92
1 96
1 144
2 78
3 13
3 21
3 32
3 47
3 49
3 61
3 62
3 80
3 88
3 90
3 99
3 126
4 1
4 29
4 33
4 42
4 51
4 52
4 60
5 11
5 28
6 68
9 64
10 48
11 6
11 39
11 50
13 36
14 58
15 44
16 37
17 53
18 16
18 23
20 2
20 9
21 54
27 3
27 41
33 45
39 18
40 46
43 5
48 38
56 20
65 72
67 24
68 17
81 26
90 4
92 19
131 15
133 27
137 34
164 7
169 14
176 22
188 8
218 10
251 12
298 31

A lot of legitimate 24's in this list!

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

> A lot of legitimate 24's in this list!

Depending on what you mean by "legitimate". I think it would also be
interesting to see what you get after culling everything which does
not pass my validity test, by taking the product of all four
superparticular ratios, raising the result to the power of the number
of notes in the block (the number found by the absolute value of the
determinant, so that in the above case it would be 24), and removing
everything where the result is greater than 2.

By the way, taking the Breedsma, kalisma, ragisma and schisma
as unison vectors gives a truly big 11-limit PB of 342 tones.
Using the xenisma in addition gives the same PB.

Manuel

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > A lot of legitimate 24's in this list!
>
> Depending on what you mean by "legitimate". I think it would also
be
> interesting to see what you get after culling everything which does
> not pass my validity test, by taking the product of all four
> superparticular ratios, raising the result to the power of the
number
> of notes in the block (the number found by the absolute value of
the
> determinant, so that in the above case it would be 24), and
removing
> everything where the result is greater than 2.

What does this test show?

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> By the way, taking the Breedsma, kalisma, ragisma and schisma
> as unison vectors gives a truly big 11-limit PB of 342 tones.

342-tET comes up in the list of the simplest ETs to achieve level-3
consistency in various odd limits:

3-limit -> 5-tET
5-limit -> 12-tET
7-limit -> 31-tET
9-limit -> 171-tET
11-limit -> 342-tET
13-limit -> 5585-tET
15-limit -> 5585-tET

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> By the way, taking the Breedsma, kalisma, ragisma and schisma
> as unison vectors gives a truly big 11-limit PB of 342 tones.
> Using the xenisma in addition gives the same PB.

One can extend this to an 11-limit notation by adding 385/384,
441/440 or 540/539; unfortunately, none of these seem to have names.
We have:

(32805/32768, 2401/2400, 4375/4374, 9801/9800, 385/384)^(-1) =

[-h72, -h118, -h212, -h171, h342]

(ditto, 441/440)^(-1) = [-h72, -h118, h130, h171, h342]

(ditto, 540/539)^(-1) = [-h72, h224, h130, h171, h342]

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

> What does this test show?

If something passes the test then the Paul Theorem will work; it
isn't a necessary condition, however.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > What does this test show?
>
> If something passes the test then the Paul Theorem will work; it
> isn't a necessary condition, however.

I suspect a necessary condition would have to distinguish between
commatic and chromatic unison vectors. Does the Paul Theorem work for
all the examples in my list?

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> > By the way, taking the Breedsma, kalisma, ragisma and schisma
> > as unison vectors gives a truly big 11-limit PB of 342 tones.
> > Using the xenisma in addition gives the same PB.
>
> One can extend this to an 11-limit notation by adding 385/384,
> 441/440 or 540/539; unfortunately, none of these seem to have
names.

I recommend naming 385/384 after Dave Keenan . . . it figures
particularly heavily in his many postings about microtemperament.
Keenan's kleisma?

896/891 has come up a lot in my own investigations -- the best Fokker
PB fit to Partch's scale seems to differ from it merely by a few
896/891 deviations, and the shrutar tuning Dave Keenan worked out for
me involves tempering some pitches by fractions of the diaschisma,
and other pitches by fractions of the 896/891. No idea what we'd call
it . . . undecimal comma is taken . . .

>I recommend naming 385/384 after Dave Keenan . . . it figures
>particularly heavily in his many postings about microtemperament.
>Keenan's kleisma?

That's fine with me.

>896/891: No idea what we'd call it . . . undecimal comma is taken . .

I vote undecimal semicomma. It's close in size to Rameau's and
Fokker's semicomma, about half a syntonic comma.

Manuel