Tuning-Math Digests messages 1750 - 1774

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Message: 1750

Date: Fri, 5 Oct 2001 12:25 +01

Subject: Re: 3rd-best 11-limit temperament

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9pjjc9+5lu5@xxxxxxx.xxx>
Gene wrote:

> This is easy enough that I've been meaning to suggest that Manuel 
> consider putting into Scala a routine to calculate Gen(m, n, p) and 
> Mos(n,m,p) for two ets m and n and a prime limit p; in case m and n 
> are not relatively prime this needs to be adjusted by working inside 
> of the interval of repetition. Of course one can also think of this 
> in terms of the ets generated by linear combinations of hm and hn, as 
> for instance h53 = h22 + h31 and h72 = h31 + h41.

That's roughly what my Python module does, and Manuel's welcome to take 
that code as inspiration.  <############################################################################### *>  The 
temperament's returned by

temper.Temperament(m, n, temper.primes[:q])

where q is the number of prime intervals you're using, not including the 
2:1 which is the interval of equivalence.  So the temperament in question 
is

temper.Temperament(31, 22, temper.primes[:4])


                     Graham


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Message: 1751

Date: Fri, 05 Oct 2001 19:22:32

Subject: Re: Tetrachordality and Scala

From: Carl Lumma

> I'm afraid I've lost you on this. I defined omnitetrachordality on 
> the tuning list, and Gene said he was going to come up with some 
> theorems about it . . .

Do we want to enforce scale order on the transposed pitches, or
just let them fall where they are closest to the original pitches?
Earlier in this thread, I claimed we don't care about scale order,
and I still claim it.  But I was enforcing order in the example
in my last mail.

I don't like your definition of omnitetrachordality because:

() I can't tell if it's equivalent to What We Want^TM.  That is,
I can't tell what sort of semi-periodicity it enforces.  This
may be due to my utter lack of tools to eval. periodicity.  A
request for which spun this thread.

() I doesn't punish approximate fifths -- only admits or rejects.

-Carl


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Message: 1752

Date: Fri, 5 Oct 2001 12:25 +01

Subject: Re: 3rd-best 11-limit temperament

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9pibm5+jcmt@xxxxxxx.xxx>
Paul wrote:

> While the top two temperaments in Graham's 11-limit list are 
> essentially 31-out-of-72 and 46-out-of-72, the third one has the 
> lowest complexity measure of all in this list. Can anyone discuss 
> this, in terms of unison vectors, etc.?
> 
> 
> 12/53, 271.1 cent generator
> 
> basis:
> (1.0, 0.22594789337911292)
> 
> mapping by period and generator:
> [(1, 0), (0, 7), (3, -3), (1, 8), (3, 2)]
> 
> mapping by steps:
> [(31, 22), (49, 35), (72, 51), (87, 62), (107, 76)]
> 
> unison vectors:
> [[7, 8, 0, -7, 0], [-21, 3, 7, 0, 0], [-21, -2, 0, 0, 7]]
> 
> highest interval width: 17
> complexity measure: 17  (22 for smallest MOS)
> highest error: 0.007764  (9.317 cents)

This originally came out of Dave Keenan's spreadsheet.  Note that it's 
compatible because the period is an octave.  It was noted then that it was 
the simplest approximation -- better than meantone at 18 and schismic at 
19.

The unison vectors are not in their simplest terms.  Dan Stearns claimed 
before to have an algorithm for finding unison vectors, so I'd still like 
to see it.

    [[7, 8, 0, -7, 0]
+ 2*[-21, 3, 7, 0, 0]
  -------------------
    [-35 14 14 -7 0]
= 7*[ -5  2  2 -1 0]

    [-21, 3, 7, 0, 0]
- 2*[-21, -2, 0, 0, 7]]
  ---------------------
     [21  7  7  0 -14]
=  7*[ 3  1  1  0 -2]

    [[7, 8, 0, -7, 0]
- 3*[-21, -2, 0, 0, 7]]
  ---------------------
    [ 70 14 0  -7 -21]
= 7*[ 10  2 0  -1  -3]


so we have new unison vectors

[-5  2  2 -1  0]
[ 3  1  1  0 -2]
[10  2  0 -1 -3]

I don't know what you were planning to do with them.  To check the 
determinants

|3 -1  0  0|
|2  2 -1  0|  =  -31
|1  1  0 -2|
|2  0 -1 -3|

|0  2 -2  0|
|2  2 -1  0|  =  22
|1  1  0 -2|
|2  0 -1 -3|

and the adjoint of

| 1  0  2 -2  0|
|-4  4 -1  0  0|
|-5  2  2 -1  0|
| 3  1  1  0 -2|
|10  2  0 -1 -3|

is

| -31 -22  38  -36  24|
| -49 -35  60  -57  38|
| -72 -51  88  -84  56|
| -87 -62 107 -102  68|
|-107 -76 131 -124  83|


> Why is this better than an ME 22-out-of-46, which has a maximum error 
> of 8.6 cents in the 11-limit, probably reducable further in a non-ET 
> setting?

Presumably, you mean

>>> temper.Temperament(46, 22, temper.primes[:4])

3/34, 52.2 cent generator

basis:
(0.5, 0.043499613319368802)

mapping by period and generator:
[(2, 0), (3, 2), (5, -4), (5, 7), (7, -1)]

mapping by steps:
[(46, 22), (73, 35), (107, 51), (129, 62), (159, 76)]

unison vectors:
[[-3, -1, -1, 0, 2], [4, 0, -2, -1, 1], [0, -5, 1, 2, 0]]

highest interval width: 11
complexity measure: 22  (24 for smallest MOS)
highest error: 0.007005  (8.406 cents)

The increase in complexity is greater than the improved accuracy.  It's 
as complex as Miracle, but more than double the error!  It doesn't make 
limit11.key because it isn't unique.  Ah!  It is in limit11.mos.  Note 
that the diaschismic temperament including 46 and 58 does make the 
13-limit list (right at the bottom of mine) as does the 94+41 schismic.


                Graham


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Message: 1753

Date: Fri, 5 Oct 2001 21:05 +01

Subject: Re: 3rd-best 11-limit temperament

From: graham@xxxxxxxxxx.xx.xx

Paul Erlich wrote:

> > highest interval width: 11
> > complexity measure: 22  (24 for smallest MOS)
> 
> 24? What about the 22-tone MOS?

The number of otonal (or utonal) complete chords is always the number of 
notes in the scale minus the complexity measure.  So 24 notes gives you 2 
otonalities.  22 notes would give you no complete chords.  That follows 
from 22-22=0.


                  Graham


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Message: 1754

Date: Fri, 5 Oct 2001 16:04:23

Subject: Re: 3rd-best 11-limit temperament

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Graham wrote:
>That's roughly what my Python module does, and Manuel's welcome to take
>that code as inspiration.  <############################################################################### *>

It would be a good addition, however it's quite a big piece of code.
So probably I'd be quicker to rethink the algorithm myself. I already
have the code for minimax temperament in Ada, although I'm not sure
yet if it can be applied. I need to let this stuff sink in too.
I'd also want to make the set of consonant p-limit intervals user
definable.

Manuel


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Message: 1755

Date: Fri, 05 Oct 2001 20:17:48

Subject: Re: Tetrachordality and Scala

From: Paul Erlich

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> > I'm afraid I've lost you on this. I defined omnitetrachordality 
on 
> > the tuning list, and Gene said he was going to come up with some 
> > theorems about it . . .
> 
> Do we want to enforce scale order on the transposed pitches, or
> just let them fall where they are closest to the original pitches?

Don't know what you mean by "enforce scale order".

> Earlier in this thread, I claimed we don't care about scale order,
> and I still claim it.  But I was enforcing order in the example
> in my last mail.
> 
> I don't like your definition of omnitetrachordality because:
> 
> () I can't tell if it's equivalent to What We Want^TM.  That is,
> I can't tell what sort of semi-periodicity it enforces.  This
> may be due to my utter lack of tools to eval. periodicity.  A
> request for which spun this thread.

Hmmm . . .

> 
> () I doesn't punish approximate fifths -- only admits or rejects.
> 
Same here.


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Message: 1756

Date: Fri, 05 Oct 2001 20:20:19

Subject: Re: 3rd-best 11-limit temperament

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> Paul Erlich wrote:
> 
> > > highest interval width: 11
> > > complexity measure: 22  (24 for smallest MOS)
> > 
> > 24? What about the 22-tone MOS?
> 
> The number of otonal (or utonal) complete chords is always the 
number of 
> notes in the scale minus the complexity measure.  So 24 notes gives 
you 2 
> otonalities.  22 notes would give you no complete chords.  That 
follows 
> from 22-22=0.

So 22-out-of-46 MOS gives you no hexads? That's odd, since the 22-out-
of-46 omnitetrachordal scale (which is very similar) does. This may 
be the first time we're seeing an omnitetrachordal scale look 
harmonically better than its MOS counterpart. Veddy veddy 
interresteeng.


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Message: 1757

Date: Fri, 05 Oct 2001 20:25:52

Subject: Re: Tetrachordality and Scala

From: Carl Lumma

>> Do we want to enforce scale order on the transposed pitches, or
>> just let them fall where they are closest to the original pitches?
> 
> Don't know what you mean by "enforce scale order".

() Take the scale
() Transpose it by 3:2.
() Now compare the pitches of the two scales.

That's where we left off.  We were counting the number of changing
pitches.  But what if they all change slightly.  I suggest comparing
them statistically in log-freq. space.  Maybe just mean-deviation is
okay here.

The question is, when comparing the scales, do we insist on lining
up the pitches in order, for some rotation of the transposed scale.
Or do allow a re-ordering of pitches to get the closest matches.
For CS scales with a 3:2 in them, in order will always be the best
order, I think.

>> () I doesn't punish approximate fifths -- only admits or rejects.
>> 
> Same here.

That was supposed to be "it doesn't", referring to your measure.
As you can see above, I want to punish approximate fifths, or at
least formalize what we will allow as an approximate fifth.

-Carl


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Message: 1758

Date: Fri, 5 Oct 2001 21:33 +01

Subject: Re: 3rd-best 11-limit temperament

From: graham@xxxxxxxxxx.xx.xx

Paul wrote:

> What's limit11.mos?

It's one of a series of files on my website that use the smallest MOS as 
the complexity measure in the figure of demerit.  Check back through this 
forum and you'll find the discussion.  It happens that the 46+22 
temperament, with a smallest MOS of 24, performs much better here than 
with its standard complexity measure of 22.



                        Graham


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Message: 1759

Date: Fri, 05 Oct 2001 20:35:23

Subject: Re: Tetrachordality and Scala

From: Paul Erlich

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> >> Do we want to enforce scale order on the transposed pitches, or
> >> just let them fall where they are closest to the original 
pitches?
> > 
> > Don't know what you mean by "enforce scale order".
> 
> () Take the scale
> () Transpose it by 3:2.
> () Now compare the pitches of the two scales.
> 
> That's where we left off.  We were counting the number of changing
> pitches.  But what if they all change slightly.  I suggest comparing
> them statistically in log-freq. space.  Maybe just mean-deviation is
> okay here.
> 
> The question is, when comparing the scales, do we insist on lining
> up the pitches in order, for some rotation of the transposed scale.
> Or do allow a re-ordering of pitches to get the closest matches.
> For CS scales with a 3:2 in them, in order will always be the best
> order, I think.
> 
> >> () I doesn't punish approximate fifths -- only admits or rejects.
> >> 
> > Same here.
> 
> That was supposed to be "it doesn't", referring to your measure.
> As you can see above, I want to punish approximate fifths, or at
> least formalize what we will allow as an approximate fifth.
> 
> -Carl

How about a rule of thumb. 1.2% is what I'm currently using for both 
harmonic entropy and for these kinds of judgments. That's 20.775¢, or 
about 1 step in 58-tET.


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Message: 1760

Date: Fri, 05 Oct 2001 20:40:53

Subject: Question

From: Paul Erlich

What is the mapping from generators to primes in the 46-out-of-72 
temperament?


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Message: 1761

Date: Fri, 05 Oct 2001 22:38:38

Subject: Re: Tetrachordality and Scala

From: Carl Lumma

>How about a rule of thumb. 1.2% is what I'm currently using for
>both harmonic entropy and for these kinds of judgments. That's
>20.775¢, or about 1 step in 58-tET.

I'm perfectly happy to use a binary function like this, because
badness doesn't continue to increase smoothly as you deviate
from the 3:2 anyway -- you hit a wall before going towards 7:5,
or whatever.

How does it fit into the rest of the procedure, though?  For
cleanness I still like multiplying by a perfect 3:2 and seeing
how much stuff changes.

-Carl


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Message: 1762

Date: Sat, 06 Oct 2001 06:22:28

Subject: Re: More from 4/21/00: does this make sense?

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > Paul H., does this make any sense?

Gene, I have no idea what your response has to do with whether this:

"Conjecture: the bizarre, double-vision periodicity block I found 
could even
happen in 3 dimensions, if there are 4 unison vectors defining
truncated-octahedron equivalence regions, but due to the 
parallelopiped
basis of the periodicity block construction from three unison 
vectors, these
truncated-octahedron regions could only come up two at a time."

makes any sense or not, but I'll try to follow what you wrote anyway.
> 
> It certainly makes sense to look at the lattice-pair one gets by 
> taking the lattice of utonal tetrads together with the lattice of 
> otonal tetrads.

What does that mean?

> If we look at triples [a,b,c] with a quadradic form
> Q(a,b,c) = a^2+b^2+c62+ab+ac+bc, we have the symmetric lattice of 7-
> limit note-classes. 

Can you give the "for musician dummies" version of this statement?
> 
> The tetrads are defined as four lattice points, each of which is at 
> unit distance from the other three. The tetrad centroids are simply 
> the means of the lattice points, if we omit the division by 4, we 
get 
> for the 1-3-5-7 otonal tetrad
> [0 0 0]+[1 0 0]+[0 1 0]+[0 0 1] = [1 1 1], where 1+1+1=(-1) (mod 
4). 
> For the utonal tetrad which is its inversion, we get 
> [0 0 0]+[-1 0 0]+[0 -1 0]+[0 0 -1] = [-1 -1 -1], and -1-1-1=1 (mod 
4).
> 
> If we translate either of these tetrads by an arbitary [a b c] we 
end 
> up with the same result mod 4 (since we add four each of a, b, and 
> c.) Hence the otonal tetrads can be considered as a lattice with 
base 
> point [1 1 1], defined as [4a+1, 4b+1, 4c+1], or equivalently as 
> those triples [u v w] such that u+v+w=-1 (mod 4); the same goes for 
> the utonal lattice, with base point [-1 -1 -1] and u+v+w=1 (mod 4).

OK, I think I understand this, but what do tetrads have to do with 
what I was talking about?

> This generalizes easily to any p-limit, with the caveat that 
treating 
> all odd primes the same makes progressively less sense.

Well you know I agree with you on that!


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Message: 1763

Date: Sat, 6 Oct 2001 17:07 +01

Subject: Re: nugget from 4/21/00: first glimpse of "torsion"???

From: graham@xxxxxxxxxx.xx.xx

> --- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> I've been exploring some 13-limit periodicity blocks due to 
> Polychroni's
> questions, and I've found some which seem to contradict my conceptions
> periodicity blocks so far. For example, using the unison vectors 
> 243:242
> (7.1¢), 352:351 (4.9¢), 385:384 (4.5¢), 676:675 (2.6¢), 2401:2400 
> (0.7¢),
> and 3025:3024 (0.6¢), so that the Fokker matrix is
> 
>     -5     0     0     2     0
>      3     0     0    -1     1
>      3     2     0     0    -2
>      1     2    -4     0     0
>      3    -2     1    -2     0

This whole matrix needs to be multiplied through by -1 so that the 
intervals are all small, instead of slightly smaller than an octave.

> Instead of being an approximation of 20-tET, it's an double 
> approximation of
> 10-tET with 539:540 and 880:881 pairs.

...

> --- End forwarded message ---
> 
> discuss . . .

We know all about these now.  It's like that 24 note periodicity block 
defining 12-equal.  I've added it to the test cases at 
<#!/usr/local/bin/python *> and 
<http.microtonal.co.uk/vectors.out> if you're interested.

Is this "torsion" then?


                 Graham


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Message: 1764

Date: Sat, 06 Oct 2001 06:28:32

Subject: Re: nugget from 4/21/00: first glimpse of "torsion"???

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > discuss . . .
> 
> You can do the same calculation with these six commas as I did in 
> the "72 owns the 11-limit" article, with a similar result. Taking 
the 
> six combinations of these five at a time,

Wait a minute Gene. Although I did, for some strange reason, list six 
unison vectors, I only included five in the Fokker matrix. So what if 
I only had those five? Would I have to try all possible products and 
quotients of pairs out of that set of five?

> and adding an indeterminant 
> row [a b c d e f], we get a determinant of zero in two cases 
(linear 
> dependency), a determinant of h10 and of -h10, meaning we can 
> construct a 10-block, and two determinants of -2 h10, meaning we 
have 
> torsion. I'm afraid this sort of thing will happen a lot.

If you plug in indeterminants a, b, c, d, e, and f, how do you get 
something in terms of h to come out? I'd like to be able to do this 
calculation . . . but again, starting with only enough unison vectors 
to construct the Fokker matrix.


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Message: 1765

Date: Sat, 6 Oct 2001 17:07 +01

Subject: Re: 3rd-best 11-limit temperament

From: graham@xxxxxxxxxx.xx.xx

Me:

> > > 3/34, 52.2 cent generator
> > > 
> > > basis:
> > > (0.5, 0.043499613319368802)

Gene:

> > It seems to me he probably means the 22-24 system, with 22+24=46, 
> and 
> > not the 22-46 system, with 22+46=68. Paul?

Paul:

> I don't know. Graham got the right generator for the system I meant. 
> Does that mean you're wrong, Gene? I don't know. I do find it 
> interesting that though the 22-tone MOS has no 11-limit hexads, the 
> corresponding 22-tone omnitetrachordal scale has some.

This is the system generated from the consistent mappings of 46- and 
22-equal.  There's also a system consistent with 46 -and 58-equal which I 
called "diaschismic".  It does have a 22 note MOS, but it's too complex to 
be the 22 from 46 that Paul asked for.


               Graham


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Message: 1766

Date: Sat, 06 Oct 2001 06:59:16

Subject: Re: 72 owns the 11-limit

From: Paul Erlich

--- 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?


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Message: 1767

Date: Sat, 6 Oct 2001 17:07 +01

Subject: Re: Question

From: graham@xxxxxxxxxx.xx.xx

Paul wrote:

> How many distinct MOS scales are represented here? What are the 
> generators, and mapping from generators to primes, in each?

I've written a test script.  See 
<#!/usr/local/bin/python *> and the results at 
<[(-1, 2, 0), (-2, 0, -1), (-1, -2, 4)] *>.  In summary



1/4, 356.0 cent generator
[(1, 0), (1, 2), (5, -9), (4, -4)]

1/20, 116.6 cent generator
[(1, 0), (1, 6), (3, -7), (3, -2)]

1/10, 116.6 cent generator
[(1, 0), (1, 6), (3, -7), (3, -2)]


1/10, 55.2 cent generator
[(1, 0), (2, -6), (2, 7), (3, 2)]


?
[(?, 0), (?, -3), (?, 1), (?, 1)]



1/20, 55.2 cent generator
[(1, 0), (2, -6), (2, 7), (3, 2)]

9/20, 578.1 cent generator
[(1, 0), (-1, 6), (6, -7), (4, -2)]

9/20, 578.1 cent generator
[(1, 0), (-1, 6), (6, -7), (4, -2)]



25:24, 1029:1024 and 225:224 fail, apparently because it wants a 
half-octave generator, but doesn't give the usual clue.  I think it should 
come out like this:


1/9, 66.7 cent generator

basis:
(0.5, 0.0555981053341)

mapping by period and generator:
[[2, 0], [4, -3], [5, 1], [6, 1]]

mapping by steps:
[(10, 8), (17, 13), (26, 21), (31, 25)]

unison vectors:
[[1, 0, 2, -2], [10, -1, -2, -1]]

highest interval width: 4
complexity measure: 8  (10 for smallest MOS)
highest error: 0.248243  (297.892 cents)



                        Graham


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Message: 1768

Date: Sat, 06 Oct 2001 07:02:06

Subject: Re: Question

From: genewardsmith@xxxx.xxx

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

> Thanks a lot Bob.

You're welcome. Who's Bob?

> So the 20-tone-per-octave MOS scale will have three 1:3:7:11 and 
> three 1/(1:3:7:11) chords (right?). Not enough to warrant too much 
> interest in this 20-tone-per-octave MOS at this point . . .

I think you are asking for trouble in the form of torsion with this 
20 business. Why 20? I could check all 7 choose 3 subsets of the 
commas you give below, but is there a reason to think this will work?


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Message: 1769

Date: Sat, 06 Oct 2001 18:33:02

Subject: Re: 72 owns the 11-limit

From: genewardsmith@xxxx.xxx

--- 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.


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Message: 1770

Date: Sat, 06 Oct 2001 07:20:32

Subject: Re: 72 owns the 11-limit

From: Paul Erlich

> --- 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? :)


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Message: 1771

Date: Sat, 06 Oct 2001 18:40:08

Subject: Re: 72 owns the 11-limit

From: genewardsmith@xxxx.xxx

--- 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.


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Message: 1772

Date: Sat, 06 Oct 2001 07:23:42

Subject: Re: Question

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > Thanks a lot Bob.
> 
> You're welcome. Who's Bob?

Oops, sorry Gene. It's late!
> 
> > So the 20-tone-per-octave MOS scale will have three 1:3:7:11 and 
> > three 1/(1:3:7:11) chords (right?). Not enough to warrant too 
much 
> > interest in this 20-tone-per-octave MOS at this point . . .
> 
> I think you are asking for trouble in the form of torsion with this 
> 20 business. Why 20? I could check all 7 choose 3 subsets of the 
> commas you give below,

8 choose 3?

> but is there a reason to think this will work?

If there is one without torsion, then we have something to look at 
that is potentially as interesting as Blackjack.

Do you have a reason do think there won't be one?


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Message: 1773

Date: Sat, 06 Oct 2001 21:09:12

Subject: Re: nugget from 4/21/00: first glimpse of "torsion"???

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> > --- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> > I've been exploring some 13-limit periodicity blocks due to 
> > Polychroni's
> > questions, and I've found some which seem to contradict my conceptions
> > periodicity blocks so far. For example, using the unison vectors 
> > 243:242
> > (7.1¢), 352:351 (4.9¢), 385:384 (4.5¢), 676:675 (2.6¢), 2401:2400 
> > (0.7¢),
> > and 3025:3024 (0.6¢), so that the Fokker matrix is
> > 
> >     -5     0     0     2     0
> >      3     0     0    -1     1
> >      3     2     0     0    -2
> >      1     2    -4     0     0
> >      3    -2     1    -2     0
> 
> This whole matrix needs to be multiplied through by -1 so that the 
> intervals are all small, instead of slightly smaller than an octave.

What are you talking about? There's no 2 column here, and there's no reason=

 to define the 
intervals downward instead of upward. Anyway, multiplying by -1 won't affec=

t the determinant, 
or anything else significant . . . will it?
> 
> Is this "torsion" then?

Yes . . . Gene explained this a while back . . . I need to go out and get a=

n abstract algebra 
textbook . . . I used to be a champion math student :(


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Message: 1774

Date: Sat, 06 Oct 2001 07:27:00

Subject: Re: Question

From: Paul Erlich

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

> > I could check all 7 choose 3 subsets of the 
> > commas you give below,
> 
> 8 choose 3?
> 
Wait a minute . . . there are only 8 cases to check . . . I gave 8 
triplets with a Fokker determinant of 20.


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