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Message: 2050 - Contents - Hide Contents

Date: Wed, 21 Nov 2001 06:53:15

Subject: Re: Yet another revised list

From: genewardsmith@xxxx.xxx

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



> [2401/2400, 81/80]   29.82023498



To get an in idea of what these extreme systems might be like, I took 
a look at this one. It is the 31&45 system, and has 3 5.79 cents 
flat, 5 1.66 cents flat, and 7 2.46 cents flat. The map to generators
I got from LLL reduction of 31 and 45 is

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

The first column generator is 19.99605624/31 and the second is
8.992112464/31, and it seems these might best be thought of as a 
generator system for the 31-et.
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Message: 2051 - Contents - Hide Contents

Date: Wed, 21 Nov 2001 10:05:44

Subject: Start of survey

From: genewardsmith@xxxx.xxx

I did the first three pairs on my list, and got the following. (All 
turned out to be Minkowski reduced according to Tenney height.)

<1728/1715, 2048/2025>

ets: 14, 22, 58, 80

LLL reduced map:

[ 0  2]
[-3  4]
[ 6  3]
[-5  7]

Generators: a = 0.1376381046 = 11.01104837 / 80; b = 1/2

Appromimately 58+22 in the 80-et.

Errors: 

3: 2.55
5: 4.68
7: 5.35

Extension of map to the 11-limit:

[ 0  2]
[-3  4]
[ 6  3]
[-5  7]
[ 7  5]

<225/224, 49/48>

ets: 9, 10, 19, 29

LLL-reduced map:

[-1  1]
[-2 -2]
[-2  5]
[-3  1]

Adjusted map:

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

Generator a = 0.1045573299 = 1.986589268 / 19

This system is closely related to 10+9 in the 19-et, and also related 
to 19+10.

Errors:

3: -3.83
5: -9.91
7: -19.76

<245/243, 50/49>

Map:

[-2 -2]
[-1  5]
[-1  9]
[-2  8]

Adjusted map:

[0 2]
[3 1]
[5 1]
[5 2]

Generator: 0.3629853525 = 7.985677755 / 22

Errors:

3:  4.79
5: -8.40
7:  9.09

This one may as well be taken as the generator 8/22 in the 22-et; 
this is a supermajor third (9/7), and we have two parallel chains 
separated by sqrt(2). This is a unique facet of the 22-et.
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Message: 2052 - Contents - Hide Contents

Date: Wed, 21 Nov 2001 10:21 +0

Subject: Re: Two versions of 12&34 in the 11-limit

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9ter9b+f1fe@xxxxxxx.xxx>
Aha, we have an algorithm!

This is the bit that concerns me:



> (7) We now have all we need so far as the 46-et goes; however we may 
> also detemper using the above map and linear programming or least 
> squares to find an optimal tuning. Using least squares in the 11-
> limit gives a generator a1 = .08700594368 = 4.002273409 / 46; this is 
> not much different from the 46-et and gives similar tuning errors. 
> However, the map to primes was only unique mod 23, and we might have 
> used instead -12 = 11 mod 23 for the number of steps we mapped 11 to.
> We obtain instead the map:



If you can choose two different mappings, that means the mapping isn't 
defined by the notation 12&34.  Then we get to:



>  Since the other maps to primes have a negative tendency, this seems 
> like it is probably the best plan.



If it's only "probably" the best, then you obviously don't have a 
deterministic algorithm.


                       Graham
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Message: 2053 - Contents - Hide Contents

Date: Wed, 21 Nov 2001 10:21 +0

Subject: Re: LLL definitions

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9tefqp+vq4h@xxxxxxx.xxx>
Gene wrote:



> The lattice reduction approach doesn't always give us 
> generator/period; it is optimizing steps to get to good intevals, and 
> that isn't always generator/period.



If Paul's conjecture is correct (and last time I mentioned it, you said 
you'd proved it) you should always be able to get the generator and period 
for a complete, linearly independent set of chromatic and commatic unison 
vectors.  I also have a program, the source code of which you can inspect, 
that does the job.  I did run into problems using an arbitrary chromatic 
unison vector last weekend, and will look at that next weekend.  It should 
be possible to get a mapping by generator and period, but the number of 
steps to either won't be defined.

See <Unison vector to MOS script * [with cont.]  (Wayb.)>.

None of this requires the lattice to be reduced.  That's for going in the 
other direction (mapping to unison vectors) and getting the simplest 
results.  As I'm having trouble getting this working, perhaps you could 
reduce this octave-specific basis for me.  Then I can see if it's what I 
want.

[46, -29, 0, 0, 0, 0]
[-14, 0, -29, 29, 0, 0]
[33, 0, 29, 0, -29, 0]
[7, 0, 0, 0, 29, -29]


                     Graham
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Message: 2054 - Contents - Hide Contents

Date: Wed, 21 Nov 2001 11:06:36

Subject: Re: LLL definitions

From: genewardsmith@xxxx.xxx

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



> If Paul's conjecture is correct (and last time I mentioned it, you 
said 
> you'd proved it) you should always be able to get the generator and 
period 
> for a complete, linearly independent set of chromatic and commatic 
unison 
> vectors. 



I certainly didn't mean to suggest I'd proven your statement; I had 
to put conditions on to get a proof.

As I'm having trouble getting this working, perhaps you could 


> reduce this octave-specific basis for me.  Then I can see if it's 
what I 
> want.
> 
> [46, -29, 0, 0, 0, 0]
> [-14, 0, -29, 29, 0, 0]
> [33, 0, 29, 0, -29, 0]
> [7, 0, 0, 0, 29, -29]



Since I don't know what inner product you want, I made no adjustment, 
and got the following:

[-14, 0, -29, 29, 0, 0]
[19, 0, 0, 29, -29, 0]
[7, 0, 0, 0, 29, -29]
[13, -29, -29, 0, 29, 0]
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Message: 2055 - Contents - Hide Contents

Date: Wed, 21 Nov 2001 11:09:20

Subject: Re: Two versions of 12&34 in the 11-limit

From: genewardsmith@xxxx.xxx

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



> If you can choose two different mappings, that means the mapping 
isn't 
> defined by the notation 12&34. 



Correct--you would need to decide what was an optimal version of the 
mod 23 reduced mapping.

 Then we get to:


> 


>> Since the other maps to primes have a negative tendency, this 
seems 
>> like it is probably the best plan.
> 


> If it's only "probably" the best, then you obviously don't have a 
> deterministic algorithm.



If you want a deterministic algorithm, you need to decide what to 
optimize. In fact, however, both systems work.
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Message: 2056 - Contents - Hide Contents

Date: Wed, 21 Nov 2001 11:34 +0

Subject: Re: LLL definitions

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9tg1rs+ddom@xxxxxxx.xxx>
Me:


> As I'm having trouble getting this working, perhaps you could 


>> reduce this octave-specific basis for me.  Then I can see if it's 
> what I 
>> want.
>> 
>> [46, -29, 0, 0, 0, 0]
>> [-14, 0, -29, 29, 0, 0]
>> [33, 0, 29, 0, -29, 0]
>> [7, 0, 0, 0, 29, -29]


Gene:


> Since I don't know what inner product you want, I made no adjustment, 
> and got the following:
> 
> [-14, 0, -29, 29, 0, 0]
> [19, 0, 0, 29, -29, 0]
> [7, 0, 0, 0, 29, -29]
> [13, -29, -29, 0, 29, 0]



Oh dear.  I wanted something like this:

(2, -1, 2, -1, 0)
(0, -3, 1, 1, 1)
(-3, 1, -1, -1, 1)
(-3, 0, 0, 1, -1)

So it looks like this off the shelf LLL algorithm isn't what I want at 
all.  Do you know of any way of doing that kind of reduction?  Or, more 
specifically, of getting a simple set of unison vectors for the consistent 
29+58 temperament?

mapping by period and generator:
[(29, 0), (46, 0), (67, 1), (81, 1), (100, 1), (107, 1)]

mapping by steps:
[(29, 0), (46, 0), (66, 1), (80, 1), (99, 1), (106, 1)]


             Graham
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Message: 2057 - Contents - Hide Contents

Date: Wed, 21 Nov 2001 11:34 +0

Subject: Re: Two versions of 12&34 in the 11-limit

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9tg210+ge72@xxxxxxx.xxx>
Gene wrote:



> If you want a deterministic algorithm, you need to decide what to 
> optimize. In fact, however, both systems work.



That's okay, so long as we agree that you can't uniquely define all linear 
temperaments with a notation like 12&34.

Have you tried doing an exhaustive search like mine, finding the best 
generator mappings for a list of equal temperaments?  This method should 
catch some that mine doesn't.


                         Graham
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Message: 2058 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 19:45:00

Subject: Re: Start of survey

From: Paul Erlich

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


> I did the first three pairs on my list, and got the following. (All 
> turned out to be Minkowski reduced according to Tenney height.)
> 
> <1728/1715, 2048/2025>
> 
> ets: 14, 22, 58, 80
> 
> LLL reduced map:
> 
> [ 0  2]
> [-3  4]
> [ 6  3]
> [-5  7]
> 
> Generators: a = 0.1376381046 = 11.01104837 / 80; b = 1/2
> 
> Appromimately 58+22 in the 80-et.
> 
> Errors: 
> 
> 3: 2.55
> 5: 4.68
> 7: 5.35



Complexity 22, max. error 5.35



> 
> <225/224, 49/48>
> 
> ets: 9, 10, 19, 29
> 
> LLL-reduced map:
> 
> [-1  1]
> [-2 -2]
> [-2  5]
> [-3  1]
> 
> Adjusted map:
> 
> [ 0  1]
> [-4  2]
> [ 3  2]
> [-2  3]
> 
> Generator a = 0.1045573299 = 1.986589268 / 19
> 
> This system is closely related to 10+9 in the 19-et, and also 
related 
> to 19+10.
> 
> Errors:
> 
> 3: -3.83
> 5: -9.91
> 7: -19.76



Complexity 7, max. error 19.76.



> 
> <245/243, 50/49>
> 
> Map:
> 
> [-2 -2]
> [-1  5]
> [-1  9]
> [-2  8]
> 
> Adjusted map:
> 
> [0 2]
> [3 1]
> [5 1]
> [5 2]
> 
> Generator: 0.3629853525 = 7.985677755 / 22 [b = 1/2]
> 
> Errors:
> 
> 3:  4.79
> 5: -8.40
> 7:  9.09
> 
> This one may as well be taken as the generator 8/22 in the 22-et; 
> this is a supermajor third (9/7), and we have two parallel chains 
> separated by sqrt(2). This is a unique facet of the 22-et.



Complexity 10, max. err. 17.49¢
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Message: 2059 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 19:49:10

Subject: Re: Survey II

From: Paul Erlich

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


> <2401/2400, 3136/3125>
> 
> ets: 31, 68, 99, 130
> 
> Map:
> 
> [ 1  2]
> [-1 14]
> [ 2  6]
> [ 2  9]
> 
> Adjusted map:
> 
> [ 0  1]
> [16 -1]
> [ 2  2]
> [ 5  2]
> 
> Generators a = .1615916143 = 15.99756982 / 99; b = 1
> 
> Related systems: 68+31 and 99+31
> 
> Errors:
> 
> 3: .604
> 5: 1.506
> 7: .724



Complexity 16, max. error 1.506

This was #1 on Graham's 7-limit list, right?



> <4000/3969, 245/243>
> 
> ets: 27, 41, 68, 109
> 
> Map:
> 
> [1   4]
> [1  -4]
> [1 -14]
> [2  -3]
> 
> Adjusted map:
> 
> [ 0  1]
> [ 8  1]
> [18  1]
> [11  2]
> 
> Generators: a = .0734545064 (~21/20) = 4.994906384 / 68
> 
> This is closely allied to the 27+41 system of the 68 et, but the 
> tuning is somewhat improved. It is also close to the 68+41 system 
of 
> the 109-et.
> 
> Errors:
> 
> 3: 3.21
> 5: 0.30
> 7: 0.77



Complexity 18, max. err. 3.21¢
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Message: 2060 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 19:53:34

Subject: Re: Survey III

From: Paul Erlich

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


> 
> <126/125, 49/48>
> 
> ets: 15, 19, 34
> 
> Map:
> 
> [1 -1]
> [0  6]
> [1  4]
> [2  1]
> 
> 
> Adjusted map:
> 
> [0 1]
> [6 0]
> [5 1]
> [3 2]
> 
> Generators: a = .263886711 (~6/5) = 5.013847509 / 19; b = 1
> 
> Related systems: 19+15, 15+4
> 
> Errors:
> 
> 3: -1.97
> 5: -2.99
> 7: -18.83



Complexity 6, max. err. 18.83¢


> 
> <3645/3584, 50/49>
> 
> ets: 12, 48
> 
> Map:
> 
> [ 0 12]
> [ 0 19]
> [-1 -2]
> [-1  4]
> 
> Adjusted map:
> 
> [ 0 12]
> [ 0 19]
> [-1 28]
> [-1 34]
> 
> Generators a = .01950640863 = 23.40769036 cents; b = 1/12 = 100 
cents
> 
> Errors:
> 
> 3: -1.96
> 5: -9.72
> 7:  7.77



Complexity 12, max. err. 17.49¢



> <6144/6125, 81/80>
> 
> ets: 7, 24, 31, 55
> 
> Map:
> 
> [1  1]
> [1  3]
> [0  8]
> [6 -5]
> 
> Adjusted map:
> 
> [ 0  1]
> [ 2  1]
> [ 8  0]
> [-11 6]
> 
> Generators a = .290240768 (~11/9) = 8.999116381 / 31; b = 1
> 
> This is for all intents and purposes the 24+7 system of the 31-et
> 
> Errors, and a comparison to the 31 et:
> 
> 3: -5.25 -5.18
> 5:   .51   .78
> 7:  -.71 -1.08



Complexity 19, max. error 5.76¢
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Message: 2061 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 21:44:54

Subject: Survey IV

From: genewardsmith@xxxx.xxx

<245/243, 64/63>

ets: 5, 17, 22, 27, 49

Map:

[-1  1]
[-1  2]
[ 3  6]
[-4  2]

Adjusted map:

[ 0  1]
[-1  2]
[-9  6]
[ 2  2]

Generators: a = .4080126504 (~4/3) = 19.99261987 / 49

Related systems: 27+22 and 22+5

Errors:

3: 8.43
5: 7.15
7: 10.40

Comparison to 27 and 49:

3: 9.16  8.25
5: 13.67 5.52
7: 8.95  10.77

This is essentially the 27+22 system of the 49 et

<126/125, 245/243>

Map:

[4 -1]
[3  1]
[5  1]
[5  2]

Generators a~9/7 = 17.005/46 and b~7/5 = 22.202/46

Adjusted map:

[0   1]
[7  -1]
[9  -1]
[13 -2]

Generators a = .3696836546 = 17.00544811 / 46; b = 1

Errors and 46-et errors:

3: 3.39  2.39
5: 6.27  4.99
7:-1.76 -3.61

Pretty close to the 27+19 system of 46-et

<225/224, 1728/1715>

ets: 9, 22, 31, 53, 84

Map (no adjustment needed)

[ 0  1]
[ 7  0]
[-3  3]
[ 8  1]

Generators: a = 18.99284544 / 84; b = 1 

This is the Orwell, of course.

Errors compared to 53 and 84

3: -2.670  -0.068  -1.955
5: -0.293  -1.408  -0.599
7:  1.785   4.759   2.603

For the 7-limit, the Orwell may as well be taken as the classic 
George 19/84; for the 11-limit the 53, which of course is also good, 
looks a little better.

<126/125, 81/80>

Map:

[ 1  1]
[ 1  2]
[ 0  4]
[-3  7]

Adjusted map:

[0    1]
[-1   2]
[-4   4]
[-10  7]

Generators: a = 0.4194600621 = 13.00326081 / 31 

This is, of course, meantone--essentially, 1/4-comma meantone

Errors and 31-et comparison

3: -5.31  -5.18
5:  0.28   0.78
7: -2.35  -1.08
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Message: 2062 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 01:43:24

Subject: Survey II

From: genewardsmith@xxxx.xxx

<2401/2400, 3136/3125>

ets: 31, 68, 99, 130

Map:

[ 1  2]
[-1 14]
[ 2  6]
[ 2  9]

Adjusted map:

[ 0  1]
[16 -1]
[ 2  2]
[ 5  2]

Generators a = .1615916143 = 15.99756982 / 99; b = 1

Related systems: 68+31 and 99+31

Errors:

3: .604
5: 1.506
7: .724

Comparison to 99:

3: 1.075
5: 1.565
7: .871

<5120/5103, 1728/1715>

ets: 53, 58, 111

Map:

[3 -1]
[4  0]
[3  6]
[9 -4]

Generators a = .3963765938 ~ 21/16; b = .189129781 ~ 8/7

Adjusted map:

[ 0  1]
[ 4  0]
[21 -6]
[-3  4]

Generators a and 1. Clearly there is something to be said for the 
original set of generators, which are approximately 44/111 and 21/111.
We have more precisely a = 43.99780191 / 111.

Errors:

3: .75
5: 2.88
7: 4.15

Comparion with 111-et:

3: .65
5: 2.38
7: 4.22

This is in effect the 58+53 system of the 111-et; it is also related 
to 53+5.

<64/63, 50/49>

ets: 10, 12, 22

Map:

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

Generators a = .4093213919 = 9.005070622 / 22 (~4/3) and b = 1/2.

This is the Paultone system, which retains two familiar comma 
relationships from the 12 et, while not being meantone.

Errors:

3:  6.86
5: -3.94
7: 13.55

Comparison with 22-et:

3:  7.14
5: -4.50
7: 12.99

The system is in effect 12+10 in the 22-et.

<4000/3969, 245/243>

ets: 27, 41, 68, 109

Map:

[1   4]
[1  -4]
[1 -14]
[2  -3]

Adjusted map:

[ 0  1]
[ 8  1]
[18  1]
[11  2]

Generators: a = .0734545064 (~21/20) = 4.994906384 / 68

This is closely allied to the 27+41 system of the 68 et, but the 
tuning is somewhat improved. It is also close to the 68+41 system of 
the 109-et.

Errors:

3: 3.21
5: 0.30
7: 0.77

Comparison to the 68-et:

3: 3.93
5: 1.92
7: 1.76
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Message: 2063 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 03:51:34

Subject: Re: Survey II

From: genewardsmith@xxxx.xxx

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


> <4000/3969, 245/243>
> 
> ets: 27, 41, 68, 109



I forgot to check these for Minkowski reduction. The rest are 
Minkowski reduced, but this one reduces to <2401/2400, 245/243>
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Message: 2064 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 06:33:31

Subject: Survey III

From: genewardsmith@xxxx.xxx

Again, everything checks as Minkowski reduced.

<3136/3125, 245/243>

ets: 19, 49, 68, 87

Map:

[ 2 -1]
[ 0  6]
[ 2  4]
[-1 13]

Adjusted map:

[  0  1]
[-12  6]
[-10  6]
[-25 12]

Generators a = .3677022284 (~9/7) = 25.00375153 / 68; b = 1

Related to 49+19 and 68+19

Errors:

3: 3.13
5: 1.26
7: 0.11

This improves on the 68-et:

3: 3.93
5: 1.92
7: 1.76

<126/125, 49/48>

ets: 15, 19, 34

Map:

[1 -1]
[0  6]
[1  4]
[2  1]


Adjusted map:

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

Generators: a = .263886711 (~6/5) = 5.013847509 / 19; b = 1

Related systems: 19+15, 15+4

Errors:

3: -1.97
5: -2.99
7: -18.83

Comparison to 19 and 34

3: -7.22    3.93
5: -7.37    1.92
7: -21.46 -15.88

<3645/3584, 50/49>

ets: 12, 48

Map:

[ 0 12]
[ 0 19]
[-1 -2]
[-1  4]

Adjusted map:

[ 0 12]
[ 0 19]
[-1 28]
[-1 34]

Generators a = .01950640863 = 23.40769036 cents; b = 1/12 = 100 cents

Errors:

3: -1.96
5: -9.72
7:  7.77

Two 12-et keyboards, a 23.4 cent comma apart, would allow this one to 
be tried. Is this a known idea? If not, it should be.

<6144/6125, 81/80>

ets: 7, 24, 31, 55

Map:

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

Adjusted map:

[ 0  1]
[ 2  1]
[ 8  0]
[-11 6]

Generators a = .290240768 (~11/9) = 8.999116381 / 31; b = 1

This is for all intents and purposes the 24+7 system of the 31-et

Errors, and a comparison to the 31 et:

3: -5.25 -5.18
5:   .51   .78
7:  -.71 -1.08
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Message: 2065 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 07:25:40

Subject: Re: Survey II

From: Paul Erlich

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


> <2401/2400, 3136/3125>
> 
> ets: 31, 68, 99, 130
> 
> Map:
> 
> [ 1  2]
> [-1 14]
> [ 2  6]
> [ 2  9]
> 
> Adjusted map:
> 
> [ 0  1]
> [16 -1]
> [ 2  2]
> [ 5  2]



Can you explain what "map" and "adjusted map" mean? What about the 
mapping from generators to primes (which seems most important of all)?



> <5120/5103, 1728/1715>
> 
> ets: 53, 58, 111
> 
> Map:
> 
> [3 -1]
> [4  0]
> [3  6]
> [9 -4]
> 
> Generators a = .3963765938 ~ 21/16; b = .189129781 ~ 8/7



Hmm . . . I thought one of the generators had to be 1/N octaves, N 
integer. So what's going on here?


> 
> <4000/3969, 245/243>
> 
> ets: 27, 41, 68, 109
> 
> Map:
> 
> [1   4]
> [1  -4]
> [1 -14]
> [2  -3]
> 
> Adjusted map:
> 
> [ 0  1]
> [ 8  1]
> [18  1]
> [11  2]
> 
> Generators: a = .0734545064 (~21/20) = 4.994906384 / 68
> 
> This is closely allied to the 27+41 system of the 68 et, but the 
> tuning is somewhat improved. It is also close to the 68+41 system 
of 
> the 109-et.
> 
> Errors:
> 
> 3: 3.21
> 5: 0.30
> 7: 0.77
> 
> Comparison to the 68-et:
> 
> 3: 3.93
> 5: 1.92
> 7: 1.76



It's not clear to me that the tuning's improved. 5/3, 7/3, and 7/5 
are all better in 68-tET.
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Message: 2066 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 07:27:40

Subject: Re: Survey II

From: Paul Erlich

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


> --- In tuning-math@y..., genewardsmith@j... wrote:
>> <4000/3969, 245/243>
>> 


>> ets: 27, 41, 68, 109
> 


> I forgot to check these for Minkowski reduction. The rest are 
> Minkowski reduced, but this one reduces to <2401/2400, 245/243>



Can you give the whole list, that you did LLL-reduced, Minkowski 
reduced instead? How about the ETs?
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Message: 2067 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 07:37:45

Subject: Re: Survey III

From: Paul Erlich

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



> <3645/3584, 50/49>
> 
> ets: 12, 48
> 
> Map:
> 
> [ 0 12]
> [ 0 19]
> [-1 -2]
> [-1  4]
> 
> Adjusted map:
> 
> [ 0 12]
> [ 0 19]
> [-1 28]
> [-1 34]
> 
> Generators a = .01950640863 = 23.40769036 cents; b = 1/12 = 100 
cents
> 
> Errors:
> 
> 3: -1.96
> 5: -9.72
> 7:  7.77
> 
> Two 12-et keyboards, a 23.4 cent comma apart, would allow this one 
to 
> be tried. Is this a known idea? If not, it should be.



I did work with this many years ago -- nice to see it again. The 10-
cent-flat major third, umm, takes some getting used to. I do talk 
about the 5-limit system of two 12-tET keyboards 15 cents apart a 
lot. What is the Minkowski-reduced basis for that latter system? It 
would seem that only one unison vector would be involved, but I can't 
see how.
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Message: 2068 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 07:59:42

Subject: Re: Survey II

From: genewardsmith@xxxx.xxx

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



> Can you explain what "map" and "adjusted map" mean? What about the 
> mapping from generators to primes (which seems most important of 
all)?



"Map" is what I get by LLL reducing the dual group to the two basis 
intervals. "Adjusted map" takes linear combinations of the columns of 
this matrix to get a generator/interval of repetition system. The 
columns are vals in prime notation, and hence are maps to primes.




>> <5120/5103, 1728/1715>
>> 
>> ets: 53, 58, 111
>> 
>> Map:
>> 
>> [3 -1]
>> [4  0]
>> [3  6]
>> [9 -4]
>> 
>> Generators a = .3963765938 ~ 21/16; b = .189129781 ~ 8/7



> Hmm . . . I thought one of the generators had to be 1/N octaves, N 
> integer. So what's going on here?



It's the difference between "map" and "adjusted map". This one seemed 
interesting to look at, as well as the adjusted one.



>> Errors:
>> 
>> 3: 3.21
>> 5: 0.30
>> 7: 0.77
>> Comparison to the 68-et:
>> 
>> 3: 3.93
>> 5: 1.92
>> 7: 1.76



> It's not clear to me that the tuning's improved. 5/3, 7/3, and 7/5 
> are all better in 68-tET.



Well, of course it's improved in a least-squares sense at any rate, 
but the close-to-exact values of the 5 and 7 would be good in wide 
interval contexts with sustained harmony, whereas the difference 
between 5/3 7/3 and 7/5 does not seem so interesting. In fact, 
however, this is another system which seems as if one might just as 
well leave it in an et.
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Message: 2069 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 08:14:50

Subject: Re: Survey II

From: Paul Erlich

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


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


>> Can you explain what "map" and "adjusted map" mean? What about 
the 
>> mapping from generators to primes (which seems most important of 
> all)?
> 


> "Map" is what I get by LLL reducing the dual group to the two basis 
> intervals.

No comprendo.

> "Adjusted map" takes linear combinations of the columns of 
> this matrix to get a generator/interval of repetition system. The 
> columns are vals in prime notation, and hence are maps to primes.



Oh, so the first column is the coefficient on a, and the second is 
the coefficient on b?



>> Hmm . . . I thought one of the generators had to be 1/N octaves, 
N 
>> integer. So what's going on here?
> 


> It's the difference between "map" and "adjusted map".



I'd like to understand this better.


> 
> Well, of course it's improved in a least-squares sense at any rate, 
> but the close-to-exact values of the 5 and 7 would be good in wide 
> interval contexts with sustained harmony, whereas the difference 
> between 5/3 7/3 and 7/5 does not seem so interesting.



Hmm . . . wide interval contexts? You mean bass-soprano dyadic 
counterpoint??
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Message: 2070 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 08:16:52

Subject: Re: Survey III

From: genewardsmith@xxxx.xxx

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



> I did work with this many years ago -- nice to see it again. The 10-
> cent-flat major third, umm, takes some getting used to. I do talk 
> about the 5-limit system of two 12-tET keyboards 15 cents apart a 
> lot. What is the Minkowski-reduced basis for that latter system? It 
> would seem that only one unison vector would be involved, but I 
can't 
> see how.



I think I can do this one without the computer's help; the 15-cents 
tells me that this is the 72 and 84 systems in the 5-limit, and they
both have the 12-et fifth, so they both have 3^12/2^19 in the kernel.
The map matrix has to be:

[ 0 12]
[ 0 19]
[-1 28]
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Message: 2071 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 08:29:09

Subject: Re: Survey II

From: genewardsmith@xxxx.xxx

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



>> "Map" is what I get by LLL reducing the dual group to the two 
basis 
>> intervals.

> No comprendo.



Usually this means I LLL reduce two of the ets I get in my list of 
ets, and the ets in question are ones which have both basis elements 
in the kernel. The intersection of the two kernels is the set of 
intervals generated by the two basis elements, which is a group of 
rank two. Dual to it is the group of rank two consisting of integral 
combinations of two ets. Lattice basis reduction reduces this from a 
two et basis to a basis of two vals which tell me the number of 
generator steps for a prime--they are generator vals, you might say.
If we use a Tenney metric on intervals, we induce a dual metric on 
the vals, which is a sup norm (maximum of absolute values) with an 
adjustment of dividing the 3 row by log2(3), etc.

Is this something we need to try to explain? I think if I could 
explain it to you (which I suspect this isn't quite doing yet) then 
we should be able to get the point across, but how theoretical should 
things get?



>> "Adjusted map" takes linear combinations of the columns of 
>> this matrix to get a generator/interval of repetition system. The 
>> columns are vals in prime notation, and hence are maps to primes.
> 


> Oh, so the first column is the coefficient on a, and the second is 
> the coefficient on b?



Yes, to get to each prime in turn. The first row tells you how many 
a's and how many b's to get a 2, and so forth.



>> Well, of course it's improved in a least-squares sense at any 
rate, 
>> but the close-to-exact values of the 5 and 7 would be good in 
wide 
>> interval contexts with sustained harmony, whereas the difference 
>> between 5/3 7/3 and 7/5 does not seem so interesting.
> 


> Hmm . . . wide interval contexts? You mean bass-soprano dyadic 
> counterpoint??



That's where this might be noticed. It's pretty small differences we 
are discussing.
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Message: 2072 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 08:31:03

Subject: Re: Survey III

From: Paul Erlich

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


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

>> It 


>> would seem that only one unison vector would be involved, but I 
> can't 
>> see how.
> 


> I think I can do this one without the computer's help; the 15-cents 
> tells me that this is the 72 and 84 systems in the 5-limit, and they
> both have the 12-et fifth, so they both have 3^12/2^19 in the 
kernel.



The Pythagorean comma . . . somehow when I thought about it before, 
it wasn't so obvious . . . I guess I tend to think in terms of an 
octave period, even though I'm so intimately familiar with a half-
octave-period system . . . but now it's completely obvious . . .



> The map matrix has to be:
> 
> [ 0 12]
> [ 0 19]
> [-1 28]

Naturally.
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Message: 2073 - Contents - Hide Contents

Date: Thu, 22 Nov 2001 12:08 +0

Subject: Re: LLL definitions

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9thca4+t6o3@xxxxxxx.xxx>
Gene wrote:



> The above does it; however you did most of the work, so I presume you 
> must have some idea how to proceed.



It took me a lot of work to get the "correct" lattice.  I had to guess a 
transformation matrix that would allow me to divide through by a common 
factor of 29, and check that it worked in Mathcad.  And I wasn't following 
a deterministic process.  I had to think quite hard about how the unison 
vectors could be combined to get the 29s, and still be linearly 
independent.  I expect I could get a computer to do the same job, but that 
would mean alot more work, and I don't want to go reinventing any wheels.  
I'm not that hung up on unison vectors anyway.  The program already does 
what I want it to.



> One can brute force it by first 
> getting a 13-limit notation with a basis of about the right size, 
> dual to a set of ets containing 29 and 58, and then searching for 
> elements of the kernel of 29&58--one should not need exponents beyond 
> +-2, so a search would be feasible. If we find something of rank 4 
> which can be extended to make a basis for the kernel of both 29 and 
> 58, we are ready to LLL reduce it. Since you did the above 
> calculation, however, perhaps you have another idea.



Yes, brute force would give the right results, but I'm not happy with it 
as a solution.  For the theory to be complete, we really need to prove 
that you can always find a set of simplified unison vectors, and that a 
particular algorithm will always produce them.  I think that'd be harder 
for brute force.

This may even be more important than going from unison vectors to a 
temperament.  History shows that good temperaments don't get discovered by 
somebody sitting down and saying "Hey, I wonder what would happen if I 
simplified all these intervals".  It's more likely that they'll hit on a 
good temperament, and then think about what simplifications it makes.


                              Graham
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Message: 2074 - Contents - Hide Contents

Date: Fri, 23 Nov 2001 19:18:56

Subject: Re: Survey V

From: Paul Erlich

--- 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?
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