Tuning-Math Digests messages 2075 - 2099

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Message: 2075

Date: Fri, 23 Nov 2001 21:46:43

Subject: Re: Survey V

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
> 
> > <1029/1024, 686/675> Minkowski reduced
> > 
> > Map (no adjustment)
> > 
> > [ 0  2]
> > [-3  5]
> > [ 6  1]
> > [ 1  5]
> > 
> > Generators a = .3040426304 (~100/81) = 13.985961 / 46; b = 1/2
> 
> What about the errors?

Did I forget those? I was supposed to include them, with a comparison 
to the 46 et:

3:  3.49  2.39
5:  2.79  4.99
7: -3.98 -3.61

There doesn't seem to be much gained by not using 46-et for this. I 
like this 686/675 in the kernel--it gives it character. It's also a 
good system, with lots of 7-limit stuff in it.


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Message: 2076

Date: Fri, 23 Nov 2001 09:44:06

Subject: Survey V

From: genewardsmith@xxxx.xxx

<2401/2400, 2048/2025> Minkowski reduced

ets: 10, 58, 68

Map (no adjustment needed)

[ 0  2]
[-4  4]
[ 8  3]
[ 3  5]

a = .1030320504 (~15/14) = 7.006179427 / 68; b = 1/2

Error compared to 68:

3: 3.39  3.93
5: 2.79  1.92
7: 2.09  1.76

This system is effectively 58+10

<1029/1024, 686/675> Minkowski reduced

Map (no adjustment)

[ 0  2]
[-3  5]
[ 6  1]
[ 1  5]

Generators a = .3040426304 (~100/81) = 13.985961 / 46; b = 1/2

<4375/4374, 6144/6125>

Minkowski reduction: <4375/4374, 5120/5103>

ets: 7, 39, 46, 53, 99

Map:

[ 1  -2]
[ 3  -1]
[ 6   1]
[-2  13]

Adjusted map:

[  0  1]
[ -5  3]
[-13  6]
[ 17 -2]

Generators: a = .2828456082 (~17/14, ~~11/9) = 28.00171521 / 99; b=1

This is essentially the 53+46 system of the 99-et; it's also related 
to 46+7

Errors compared to 99:

3: 0.9713  1.0753
5: 1.2948  1.5651
7: 1.2245  0.8711

<3136/3125, 49/48>

Minkowski reduction: <3125/3072, 49/48>

Map:

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

Adjusted map:

[ 0  1]
[10  0]
[ 2  2]
[ 5  2]

Generator: a = .1584971341 (~10/9, exactly 3^(1/10)) 
= 3.011445548 / 19

Errors compared to 19:

3:     0   -7.22
5:  -5.92  -7.37
7: -17.84 -21.46

This is not your father's 19-et! In fact, with its perfect fifths, 
it's not much like either the 19 or 25 et it can be played in. This 
one is a genuine original, a temperament best left as a temperament. 
Definately worth a look as a way of tempering 19 notes.


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Message: 2077

Date: Sat, 24 Nov 2001 19:59:21

Subject: Re: Survey

From: genewardsmith@xxxx.xxx

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

> <input vectors *>

input vectors
49:50
4374:4375

calculated vectors
17496:16807
9765625:9565938

Yipe! How are these being calculated?


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Message: 2078

Date: Sat, 24 Nov 2001 21:01 +0

Subject: Re: Survey

From: graham@xxxxxxxxxx.xx.xx

genewardsmith@xxxx.xxx () wrote:

> --- In tuning-math@y..., Graham Breed <graham@m...> wrote:
> 
> > <input vectors *>
> 
> input vectors
> 49:50
> 4374:4375
> 
> calculated vectors
> 17496:16807
> 9765625:9565938
> 
> Yipe! How are these being calculated?

I've included the code below.  It looks at the mapping, finds a unison 
involving each prime interval, and then tries to simplify it.  The results 
here in vector form are [3, 7, 0, -5] and [-1, -14, 10, 0].  However it 
combines them, even apparently if it used an LLL algorithm, it couldn't 
get rid of the common factors of 7, 10 and 5 in the last three columns.

I can't even work out how to simplify the original vectors of
[-1, 0, -2, 2] and [1, 7, -4, -1].  I'm worryingly close to deciding that 
it can't be done.


  def getUnisonVectors(self):
  
    # find some vectors that work
    H = [1]+self.primes
    hcf = self.mapping[0][0]
    fifthIndex = 1
    while self.mapping[fifthIndex][1]==0:
      fifthIndex = fifthIndex + 1
      #okay, so this won't index the fifth any more
    fifth = self.mapping[fifthIndex]
    denom = -hcf*fifth[1]
    vectors = []
    for index in range(1,len(self.mapping)):
      if index == fifthIndex:
        continue
      m, n = self.mapping[index]
      vector = [0]*len(H)
      vector[0] = fifth[1]*m-fifth[0]*n
      vector[fifthIndex] = hcf*n
      vector[index] = denom
      vectors.append(normalizeInterval(vector, H))

    # now simplify them
    nOthers = len(vectors)-1
    cmpfn = intervalCompare(H)
    while nOthers:
      # loop until broken if there are alternatives
      vectors.sort(cmpfn)
      worst = vectors[nOthers]
      for index in range(nOthers):
        alternative = normalizeInterval(
            map(operator.add, vectors[index],worst))
        if cmpfn(alternative, worst)<0:
          vectors[nOthers]=alternative
          break
        alternative = normalizeInterval(
            map(operator.sub, vectors[index],worst), H)
        if cmpfn(alternative, worst)<0:
          vectors[nOthers]=alternative
          break        
      else:
        # can't be improved
        break;
    return vectors


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Message: 2079

Date: Sat, 24 Nov 2001 06:08:12

Subject: Survey VI

From: genewardsmith@xxxx.xxx

<225/224, 4000/3969>

Minkowski reduction: <335/224, 3125/3087>

Ets: 12, 29, 41, 53, 94

Map:

[ 1  2]
[ 2  3]
[-1  6]
[-3  8]

Reduced map:

[ 0  1]
[-1  2]
[ 8 -1]
[14 -3]

Generators: a = .4148831986 (~4/3) = 38.99902067 / 94; b = 1

Septimal schismic system; effectively this is 53+41

Errors and 94-et:

3:  .185  .173
5: -3.44 -3.33
7:  1.21  1.39

<4375/4374, 225/224> Minkowski reduced

Ets: 19, 53, 72

Map:

[ 3  1]
[ 6  0]
[ 8  1]
[13 -3]

Adjusted map:

[ 0  1]
[ 6  0]
[ 5  1]
[22 -3]

Generators: a = .2639365655 (~6/5) = 19.00343272 / 72; b = 1

This is pretty much the 53+19 system of the 72 et, but a bit better 
in tune; in particular the long string of generators to 7 helps to 
get it in better tune.

Errors and 72-et:

3: -1.61  -1.96
5: -2.69  -2.98
7: -0.90  -2.16

Because the 7's are so inacessible, it's tempting to treat this as a 
purely 5-limit system, in which case it's closer to the 53 et, with a 
generator of 14.00435/53, and triads which are very good; this 
version is the kleismic temperament. Another possibility would be a 
planar temperament.


<1728/1715, 81/80>

Minkowski reduction: <1029/1024, 81/80>

Ets: 5, 26, 31, 57

Map (no adjustment)

[ 0  1]
[ 3  1]
[ 12 0]
[-1  3]

Generators: a = .1934362896 (~8/7) = 5.996524978 / 31; b = 1

Errors and 31-et

3: -5.58 -5.18
5: -0.83  0.78
7: -0.95 -1.08

Pretty much the 26+5 system of the 31-et.

<64/63, 686/685> Minkowski reduced

Ets: 10, 27

Map:

[1 -1]
[1 -3]
[5  4]
[4  0]

Adjusted map:

[ 0  1]
[ 2  1]
[-9  5]
[-4  4]

Generators: a = .2969233588 (~11/9) = 8.016930688 / 27; b = 1

Errors and 27-et

3: 10.66  9.15
5:  6.91 13.68
7:  5.94  8.95

This is one version of a neutral third temperament, though Graham 
does not even mention the 27-et in connection to neutral thirds, so 
it's hardly canonical.


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Message: 2080

Date: Sat, 24 Nov 2001 08:06:19

Subject: Re: Survey VI

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> <225/224, 4000/3969>
> 
> Minkowski reduction: <335/224, 3125/3087>

That 335 should really be 225, right?
> 
> <64/63, 686/685> Minkowski reduced

That 685 should be 675, right?


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Message: 2081

Date: Sat, 24 Nov 2001 08:10:03

Subject: Re: Survey VI

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
> > <225/224, 4000/3969>
> > 
> > Minkowski reduction: <335/224, 3125/3087>
> 
> That 335 should really be 225, right?
> > 
> > <64/63, 686/685> Minkowski reduced
> 
> That 685 should be 675, right?

Yes to both. Bad fingers, bad!


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Message: 2082

Date: Sat, 24 Nov 2001 12:24:45

Subject: Survey

From: Graham Breed

I've caught up with this.  Scripts are at

<############################################################################### *>
<#!/usr/local/bin/python *>
<#!/usr/local/bin/python *>
<Gene, it looks, from your post on the tuning list, that a basis need *>

And results at

<input vectors *>


There's still a problem with calculating the containing MOS.  I've managed to 
work around that by not calculating the containing MOS.  It looks right as 
far as I can tell.  Unison vectors aren't LLL reduced, and some might be very 
wrong.  If you want other combinations done, it's very easy to feed it a 
different input file.  Only 7-limit currently.


                        Graham


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Message: 2084

Date: Sun, 25 Nov 2001 21:00:16

Subject: Survey VII

From: genewardsmith@xxxx.xxx

<1029/1024, 245/243> Minkowski reduced

Ets: 5, 41, 46, 87

Map:

[ 1   1]
[ 1   4]
[-1  16]
[ 3   2]

Adjusted map:

[ 0   1]
[ 3   1]
[ 17 -1]
[-1   3]

Generators: a = .1953420071 (~8/7) = 16.99475462 / 87; b = 1

Error and 87-et:

3:  1.28    1.49
5: -1.34   -0.11
7: -3.24   -3.31

This is pretty nearly the 46+41 system of the 87-et, which fails to 
take much advantage of the excellent thirds, but does have a lot of
1--3/2--7/4 chords to play with.

<4375/4374, 2401/2400> Minkowski reduced

Ets: 27, 72, 99, 171, 270, 441, 612

Map:

[ 0   9]
[-2  -1]
[-3  -2]
[-2  10]

Adjusted map:

[ 0   9]
[-2  15]
[-3  22]
[-2  26]

Generators: a = .04083262537 (~36/35) = 24.98956673 / 612; b = 1/9
(~27/25) = 68/612

Alternative adjusted map:

[0   9]
[2  13]
[3  19]
[2  24]

a' = 43.01043329/612 (~21/20) as generator

Errors vs 441 and 612 ets:

3:  .0467  .0858  .0058
5:  .0222  .0808 -.0392
7: -.1575 -.1184 -.1985

In case anyone cares about errors that small, it can be seen that 
both 441 and 612 do a fine job with this system, which is another 
familiar face, the ennealimmal temperament. I'd remark that it's good 
enough even for Harry Partch but for the fact that it would get me in 
trouble, so I'll point out that it's good enough even for Harry 
Potter. Good enough for what I leave to Hogwarts to figure out, but 
tempering the 72 notes remains one possibility.

<875/864, 50/49> Minkowski reduced

Ets: 22, 26, 48

Map:

[2  2]
[1 -3]
[3  0]
[4  1]

Adjusted map:

[0  2]
[4  1]
[3  3]
[3  4]

Generators: a = .2713428065 (~6/5) = 13.0244547 / 48; b = 1/2

Errors and the 48-et:

3:  0.49   -1.96
5: -9.48  -11.31
7:  8.01    6.17

Here's another system with very flat major thirds, and Paul, I 
suppose, would prefer it in the 18+4 version of the 22-et, or 22+4 of 
the 26 et, over 26+22 in the 48-et.


<4000/3969, 2048/2025>

Minkowski reduction: <3136/3125, 2048/2025>

Map (no adjustment)

[ 0  4]
[-1  8]
[ 2  6]
[ 5  3]

Generators: a = .4118714296 (~4/3) = 28/00725721 / 68; b = 1/4

Errors compared to 68:

3: 3.80  3.93
5: 2.18  1.92
7: 2.40  1.76

Effectively the 56+12 system of the 68-et.


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Message: 2085

Date: Sun, 25 Nov 2001 02:20:24

Subject: Re: Survey

From: genewardsmith@xxxx.xxx

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

> I can't even work out how to simplify the original vectors of
> [-1, 0, -2, 2] and [1, 7, -4, -1].  I'm worryingly close to 
deciding that 
> it can't be done.

I get that <49/48, 4375/4374> is already Minkowski reduced, and when 
I LLL reduced it, I got <49/48, 5103/5000> which hardly seems like an 
improvement.

Why don't you write something to Minkowski reduce a pair of 7-limit 
intervals according to Tenney height? There may not be an intelligent 
algorithm, but the problem is so small you don't need one. You can 
calculate bounds on how far you need to search, and simply search 
that region. It's been working well for me.


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Message: 2086

Date: Sun, 25 Nov 2001 08:12:54

Subject: Re: Survey

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., graham@m... wrote:
> 
> > I can't even work out how to simplify the original vectors of
> > [-1, 0, -2, 2] and [1, 7, -4, -1].  I'm worryingly close to 
> deciding that 
> > it can't be done.
> 
> I get that <49/48, 4375/4374> is already Minkowski reduced, and 
when 
> I LLL reduced it, I got <49/48, 5103/5000> which hardly seems like 
an 
> improvement.

Haven't I convinced you that Minkowski is that way to go rather than 
LLL? In either case, could you please explain the mathematical 
criterion that defines "Minkowski reduced", as you did for LLL?


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Message: 2087

Date: Sun, 25 Nov 2001 12:24:17

Subject: Re: Survey

From: genewardsmith@xxxx.xxx

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

> Haven't I convinced you that Minkowski is that way to go rather 
than 
> LLL? 

Sure.

In either case, could you please explain the mathematical 
> criterion that defines "Minkowski reduced", as you did for LLL?

Let p/q be reduced to lowest terms; then T(p/q) = pq. A pair of 
intervals {p/q, r/s} with p/q>1, r/s>1, T(p/q) < T(r/s) and p/q and 
r/s independent is Minkowski reduced iff the only numbers in the set 
{(p/q)^i (r/s)^j} such that T(t/u) < T(r/s) are powers of p/q.


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Message: 2088

Date: Mon, 26 Nov 2001 17:12 +0

Subject: Re: Tormented by Torsion again

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <memo.663318@xxx.xxxxxxxxx.xx.xx>
I wrote:

> Incidentally, you did say 256:243 and 65536:59049 before, but my 
> comments were for 256:243 and 16807:15552, although I didn't point that 
> out.  65536:59049 is (256:243)^2.  So 256:243 and 16807:15552 aren't 
> Minkowski reduced, but there's no particular reason why they should be.

Aaaaaaaaargh!

256:243 and 16807:15552 are Minkowski reduced.

256:243 and 65536:59049 are not (they reduce to 256:243 and 1:1).


                    Graham


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Message: 2089

Date: Mon, 26 Nov 2001 17:54 +0

Subject: Re: Tormented by Torsion again

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9tttj7+v8k6@xxxxxxx.xxx>
Paul wrote:

> > Okay, we'll assume 256:243 and 16807:15552 are Minkowski reduced 
> then.  
> > 63:64 and 49:48 describe the same temperament.
> 
> Hmm . . . so this is one of the torsion-spawning cases? Can you show 
> how this works?

How I generated <input vectors *>:

I went through each pair of unison vectors in your list.  (Some list, Gene 
seems to have a longer one.)  I worked out the linear temperament 
consistent with each pair of unison vectors, and optimized it.  Then I 
used the method that Temperament objects already have to return the 
simplest unison vectors I could find.

That's what <#!/usr/local/bin/python *> does.

Unfortunately, that method sometimes throws up unisons that are way too 
complex, and it's related to torsion.  So I went through and listed them.

256:243 and 16807:15552 are [8, -5, 0, 0] and [-6, -5, 0, 5] in vector 
form.  63:64 and 49:48 become [-6, 2, 0, 1] and [-4, -1, 0, 2].

Both pairs give the same mapping by period and generator: [(5, 0), (8, 0), 
(10, 1), (14, 0)].  We can verify this

[ 8 -5 0 0][ 5 0]   [0 0]
[-6 -5 0 5][ 8 0] = [0 0]
[-6  2 0 1][10 1]   [0 0]
[-4 -1 0 2][14 0]   [0 0]

We can also transform the first pair into the second pair

([-6, -5, 0, 5]-3*[8, -5, 0, 0])/5 = [-6 2 0 1]

This would be i=-3/5, j=1/5

(2*[-6, -5, 0, 5]-[8, -5, 0, 0])/5 = [-4 -1 0 2]

Which is i=-1/5, j=2/5.

However, the Minkowski criterion as we currently understand it, and my 
reduction method, don't allow for fractional values of i and j.  The 
practical difficulty in implementing this is that the vectors have to get 
bigger before they can be simplified by dividing through by a common 
factor.  That makes the search harder.

A brute force approach would work, trying every possible vector and seeing 
if it approximates to a unison.  I'll look at this if I find the time.  
Another idea would be to list all members of the second-order odd limit 
that approximate to be the same.  Any more complex unison vectors probably 
won't be that interesting anyway.


                     Graham


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Message: 2090

Date: Mon, 26 Nov 2001 01:07:34

Subject: Survey VIII

From: genewardsmith@xxxx.xxx

<4375/4374, 3136/3125> Minkowski reduced

Ets: 19, 80, 99, 118

Map:

[3  -2]
[2   3]
[4   2]
[1  11]

The adjusted mappings are pretty horrendous, and this might be used 
instead, with generators a' = 46.9967/99 and b' = 20.9953/99.

Adjusted map:

[ 0   1]
[-13  5]
[-14  6]
[-35 12]

Generators: a = .2626420944 (~6/5) = 26.00156735 / 99; b = 1

Errors and 99 et:

3: 0.828  1.975
5: 1.299  1.565
7: 0.206  0.871

<2048/2025, 50/49> Minkowski reduced

Ets: 10, 12, 22

Map (no adjustment)

[ 0  2]
[-1  4]
[ 2  3]
[ 2  4]

Generators: a = .4093213919 (~4/3) = 9.995070622 / 22; b = 1/2

Errors and 22 et:

3:  6.86   7.14
5: -3.94  -4.50
7: 13.55  12.99

This is, of course, paultone, and it can be seen that we don't get 
much milage out of using anything but 12+10 in the 22-et for it.

<3136/3125, 64/63>

Minkowski reduction <3125/3087, 64/63>

Ets: 12,13,25,37,49

Map (no adjustment)

[ 0   1]
[-5   2]
[ 4   2]
[ 10  2]

Generators: a = .08140287107 = 3.01190623 / 37; b = 1

Errors and 37-et

3: 9.63  11.56
5: 4.42   2.88
7: 2.81   4.15

A 12-tone temperament for the adventurous

<4375/4374, 2048/2025> Minkowski reduced

Ets: 46, 80, 126

Map (no adjustment)

[ 0   2]
[ 1   3]
[-2   5]
[15   3]

Generators: a = .08730149627 (~17/16) = 10.99998853 / 126

Errors and 126-et

3: 2.81  2.81
5: 4.16  4.16
7: 2.60  2.60

The difference between this and the 126-et is far below the limits of 
perceptibility, so this may as well be called 80+46. Of course the 
version in the 46-et is perfectly fine, and 17/16 is so close to 
2^(11/126) that it can be used also.


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Message: 2091

Date: Mon, 26 Nov 2001 17:19:35

Subject: Re: Tormented by Torsion again

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9ttq3j+5tck@e...>
> Me:
> > > Well, that's up to you.  I don't see what we gain by enforcing 
the 
> > square 
> > > rule.
> 
> Paul:
> > It's not a square rule. But what we gain is simplicity.
> 
> How?

Because instead of the rule for Minkowski reduction, we use an even 
simpler rule.

> And why is this worth losing perfectly valid results?

You can also get additional valid results by increasing the number of 
unison vectors in our list. But shouldn't there be 
some "reasonableness" criteria applied? Otherwise, you'll get an 
infinite list of linear temperaments!

> 
> Me:
> > > What's a contrapositive?  I've given an example that's fine 
when 
> > the rule 
> > > isn't satisfied, and lots that are fine even though it isn't.
> 
> Paul:
> > Right -- but I'm claiming you won't find an example that's _not_ 
fine 
> > when the rule is satisfied.
> 
> Sorry, typo.
> 
> Me:
> > > {(p/q)^i (r/s)^j}
> > > 
> > > assuming i and j are integers.
> 
> Paul:
> > Yes, they are integers -- I thought you said 49:48 and 63:64 was 
> > equivalent to this, and I took "your" word for it. So what _do_ 
you 
> > get when you Minkowski-reduce this system? Gene?
> 
> Okay, we'll assume 256:243 and 16807:15552 are Minkowski reduced 
then.  

Let's call this statement G. See below.

> 63:64 and 49:48 describe the same temperament.

Hmm . . . so this is one of the torsion-spawning cases? Can you show 
how this works?
 
> Incidentally, you did say 256:243 and 65536:59049 before, but my 
comments 
> were for 256:243 and 16807:15552, although I didn't point that out. 

I know, my typo.

> 65536:59049 is (256:243)^2.

OK, forget my typo, please.

> So 256:243 and 16807:15552 aren't Minkowski 
> reduced,

Now you're saying not G. You've claimed both G and not G, but you 
haven't given evidence for either.

> but there's no particular reason why they should be.

> 
> The vectors are
> 
> [8, -5, 0, 0]
> [-6, -5, 0, 5]
> 
> How could you combine them to get anything simpler?  My program's 
already 
> checking the simplest cases.  [14, 0, 0, -5] being the obvious one.
> 
> >>> 16384*16807
> 275365888
> >>> 16807*15552
> 261382464
> 
> I'm certain these are Minkowski reduced, if I've got the definition 
right.

OK, so now you're saying G again, and you've given a reason why.


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Message: 2092

Date: Mon, 26 Nov 2001 21:52:00

Subject: Re: Tormented by Torsion again

From: genewardsmith@xxxx.xxx

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

> We can also transform the first pair into the second pair
> 
> ([-6, -5, 0, 5]-3*[8, -5, 0, 0])/5 = [-6 2 0 1]
> 
> This would be i=-3/5, j=1/5
> 
> (2*[-6, -5, 0, 5]-[8, -5, 0, 0])/5 = [-4 -1 0 2]
> 
> Which is i=-1/5, j=2/5.

Which gets rid of the 5-torsion.

> However, the Minkowski criterion as we currently understand it, and 
my 
> reduction method, don't allow for fractional values of i and j.  

That's because it gives a different lattice--however, if we have 
torsion then we *want* a different lattice.


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Message: 2093

Date: Mon, 26 Nov 2001 08:00:09

Subject: Re: Survey VIII

From: Paul Erlich

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

> <2048/2025, 50/49> Minkowski reduced
> 
> Ets: 10, 12, 22
> 
> Map (no adjustment)
> 
> [ 0  2]
> [-1  4]
> [ 2  3]
> [ 2  4]
> 
> Generators: a = .4093213919 (~4/3) = 9.995070622 / 22; b = 1/2
> 
> Errors and 22 et:
> 
> 3:  6.86   7.14
> 5: -3.94  -4.50
> 7: 13.55  12.99
> 
> This is, of course, paultone,

I thought the minkowski reduced basis for paultone was <64/63, 
50/49>. How can there be two minkowski reduced bases for paultone?


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Message: 2094

Date: Mon, 26 Nov 2001 17:20:10

Subject: Re: Tormented by Torsion again

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <memo.663318@c...>
> I wrote:
> 
> > Incidentally, you did say 256:243 and 65536:59049 before, but my 
> > comments were for 256:243 and 16807:15552, although I didn't 
point that 
> > out.  65536:59049 is (256:243)^2.  So 256:243 and 16807:15552 
aren't 
> > Minkowski reduced, but there's no particular reason why they 
should be.
> 
> Aaaaaaaaargh!
> 
> 256:243 and 16807:15552 are Minkowski reduced.
> 
> 256:243 and 65536:59049 are not (they reduce to 256:243 and 1:1).
> 
> 
>                     Graham

Oh, OK, gotcha.


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Message: 2095

Date: Mon, 26 Nov 2001 22:00:35

Subject: Re: Tormented by Torsion again

From: genewardsmith@xxxx.xxx

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

> > Okay, we'll assume 256:243 and 16807:15552 are Minkowski reduced 
> then.  

> Let's call this statement G. See below.

It's true.

> > 63:64 and 49:48 describe the same temperament.
> 
> Hmm . . . so this is one of the torsion-spawning cases? 

Indeed it is. I just wrote a wedge product routine, and here is what 
I get:

16807/15552^256/243 = [70, 0, -40, 0, 25, 0]
64^63^49/48 = [-14, 0, 8, 0, -5,0]

The first has coefficients with a gcd of 5, telling us there is 
5-torsion; the second has a gcd of 1, telling us there is no torsion.


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Message: 2096

Date: Mon, 26 Nov 2001 08:13:01

Subject: Tormented by Torsion again

From: genewardsmith@xxxx.xxx

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

> I thought the minkowski reduced basis for paultone was <64/63, 
> 50/49>. How can there be two minkowski reduced bases for paultone?

I was just about to post using the above subject line when I saw 
this. I'm bummed :) The reason is that 2048/2025 * 50/49 = (64/64)^2,
but no product of 2048/2025 and 50/49 makes 64/63. Neither LLL nor 
Minkowski got rid of this problem, so I will need to check all the 
results and see how to cure this disease.


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Message: 2097

Date: Mon, 26 Nov 2001 22:08:29

Subject: Re: Tormented by Torsion again

From: genewardsmith@xxxx.xxx

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

> > > Ah, but how about 64:63 and 3125:3087?
> > > 
> > > [0, -2, 5, -3]
> > > [6, -2, 0, -1]
> > > 
> > > Lots of common factors, but (64:63)^2 = 4096:3969.

> Paul:
> > So this is a good one. Why do you bring this example up? Isn't it 
> > just a normal example?
> 
> I don't know, what is "normal"?  You can't combine these vectors to 
get a 
> 1 in any column but the last.  That has something to do with 
torsion, but 
> I don't think it's the real problem.

I get 64/63^3125/3087 = [-12, 30, -18, -10, 4, 5], and so no torsion. 
I'm with Paul--this seems quite normal to me.


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Message: 2098

Date: Mon, 26 Nov 2001 08:38:54

Subject: Wedge products and the torsion mess

From: genewardsmith@xxxx.xxx

Perhaps wedge products are the best way of cleaning this up. If we 
write 2^a 3^b 5^c 7^d as a e2 + b e3 + c e5 + d e7 we can take wedge 
products by the following rule ei^ei = 0, and if i != j, then
e1^ej = - ej^ei. In the 5-limit case, the wedge product will be, in 
effect, the correspodning val. In the 7-limit case, we get something 
six dimensional, which if we added another interval would give us a 
val. However, it still can be used to test for torsion.

50/49 = e2+2e5-2e7, 2048/2025 = 11e2+4e3-2e5. Taking the wedge 
product gives us 50/49^2048/2025 = 
4e2^e3 - 24e2^e5 - 8 e3^e5 - 4 e5^e7. This has a common factor of 4. 
On the other hand 50/49^54/63 = -2 e2^e3 - 12 e2^e5 + 5 e2^e7
+ 4 e3^e5 + 2 e3^e7 - 2 e5^e7, with a gcd of 1 for the coefficients.
All is, therefore, not lost, I think. I'll ponder the question 
further.


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Message: 2099

Date: Mon, 26 Nov 2001 08:54:59

Subject: Re: Wedge products and the torsion mess

From: genewardsmith@xxxx.xxx

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

>I'll ponder the question 
> further.

One way to see what is going on is this: if the wedge product has a 
common factor, then whatever we pick as another basis interval in 
order to compute the corresponding val will also have a common factor 
when we take determinants, and hence show torsion according to our 
usual test of the gcd of the coefficients of the val. Therefore the 
torsion is already present in the two elements we started with. 
2048/2025 and 50/49 cannot be extended in a non-torsion way to three 
7-limit intervals, in other words, which would be suitable for a 
block. This is the same problem as before, in a more insidious form.


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