Let's try chromatic unison vectors between 100 cents and 35 cents,

commatic unison vectors smaller than 35 cents but with numerator and

denominator less than 3000, disallow torsion, and select unison

vectors only from this list:

http://www.kees.cc/tuning/s2357.html

The roundup:

freq. determinant

1 23

1 31

2 6

2 13

3 1

3 15

3 16

3 17

4 21

5 2

5 8

5 11

5 19

7 3

7 4

7 14

8 12

11 7

14 5

18 10

Most of the 10-tone scales correspond to the ones that carry my name

in Scala (10-out-of-22); true for the 14-tone ones as well (14-out-of-

26).

How can we try to eliminate "skewed" blocks from this list? Let's be

crude and ignore the difference between a taxicab metric on the Kees

van Prooijen lattice, and a Euclidean metric. So the "length" of a

unison vector is log(numerator), while the volume the three subtend

is simply the number of notes. A non-skewed block would have volume

equal to the product of the three lengths. Define the "straightness"

as the number of notes divided by this would-be volume. Here are the

rankings by straightness:

#1: 14 notes; commas 81:80, 50:49; chroma 25:24 (Erlich 14-of-~26)

#2: 31 notes; commas 1029:1024, 245:243; chroma 25:24 (v. improper!)

#3: 17 notes; commas 245:243, 64:63; chroma 25:24

#4: 19 notes; commas 126:125, 81:80; chroma 49:48

#5: 15 notes; commas 126:125, 64:63; chroma 28:27

#6: 21 notes; commas 1029:1024, 81:80; chroma 25:24 (not blackjack!)

#7: 14 notes; commas 81:80, 50:49; chroma 49:48 (Erlich 14-of-~26)

#8: 12 notes; commas 64:63, 50:49; chroma 36:35 (Erlich 12-of-22)

#9: 19 notes; commas 225:224, 81:80; chroma 49:48 (19-out-of-31???)

#10: 14 notes; commas 245:243, 50:49; chroma 25:24

#11: 12 notes; commas 81:80, 50:49; chroma 36:35

#12: 15 notes; commas 126:125, 64:63; chroma 49:48

#13: 10 notes; commas 64:63, 50:49; chroma 25:24 (Erlich 10-of-22)

#14: 19 notes; commas 225:224, 126:125, chroma 49:48

#15: 10 notes; commas 64:63, 50:49; chroma 28:27 (Erlich 10-of-22)

#16: 19 notes; commas 245:243, 126:125; chroma 49:48

#17: 23 notes; commas 2401:2400, 126:125; chroma 28:27

#18: 14 notes; commas 245:243, 81:80; chroma 25:24

#19: 16 notes; commas 245:243, 225:224; chroma 21:20

#20: 14 notes; commas 245:243, 50:49; chroma 49:48

#21: 12 notes; commas 126:125, 64:63; chroma 36:35

#22: 16 notes; commas 1029:1024, 50:49; chroma 36:35

#23: 19 notes; commas 245:243, 225:224; chroma 49:48

#24: 12 notes; commas 225:224, 50:49; chroma 36:35

#25: 10 notes; commas 64:63, 50:49; chroma 49:48

#26: 12 notes; commas 126:125, 81:80; chroma 36:35

#27: 21 notes; commas 1029:1024, 225:224; chroma 36:35 (Blackjack)

#28: 12 notes; commas 225:224, 64:63; chroma 36:35

#29: 10 notes; commas 225:224, 50:49; chroma 25:24 (Erlich 10-of-22)

#30: 17 notes; commas 2401:2400, 64:63; chroma 36:35

#31: 10 notes; commas 225:224, 50:49; chroma 28:27 (Erlich 10-of-22)

#32: 12 notes; commas 225:224, 81:80; chroma 36:35

#33: 21 notes; commas 2401:2400, 225:224; chroma 36:35 (Blackjack)

The list continues:

Straightness # of notes numerators of commas numeratr'chroma

0.13869 17 2401 81 36

0.13792 10 225 64 25

0.1358 11 245 126 21

0.13323 10 225 64 28

0.1331 16 1029 126 36

0.12844 11 245 126 25

0.12784 12 225 126 36

0.1269 8 126 50 28

0.12127 10 225 50 49

0.11899 7 81 64 25

0.11489 15 1029 126 49

0.11449 10 1029 50 25

0.11407 10 225 64 49

0.11156 8 245 50 28

0.10854 21 1029 2401 36

0.10818 7 126 81 21

0.10769 10 1029 64 25

0.10517 14 2401 81 49

0.10373 8 245 50 36

0.10232 7 126 81 25

0.098844 7 126 81 28

0.098556 10 2401 50 28

0.096602 7 225 81 21

0.096174 11 1029 225 21

0.094692 10 1029 50 49

0.092706 10 2401 64 28

0.091573 6 245 50 21

0.09137 7 225 81 25

0.09111 13 245 2401 28

0.090237 8 245 126 28

0.089861 5 81 64 21

0.088262 7 225 81 28

0.087777 7 225 126 21

0.085705 11 2401 225 21

0.084721 13 245 2401 36

0.08401 14 245 2401 49

0.083909 8 245 126 36

0.083022 7 225 126 25

0.082103 5 81 64 28

0.080199 7 225 126 28

0.079376 10 2401 64 49

0.072627 6 1029 50 21

0.071026 7 2401 64 21

0.070297 5 81 64 49

0.069443 4 126 50 21

0.068396 10 1029 225 49

0.067934 5 245 81 21

0.066921 11 1029 2401 21

0.065681 4 126 50 25

0.065584 5 245 64 28

0.062069 5 245 81 28

0.060985 5 245 64 36

0.060951 10 2401 225 49

0.057716 5 245 81 36

0.05693 5 1029 64 21

0.056154 5 245 64 49

0.054324 4 126 50 49

0.048991 3 126 64 21

0.048367 5 1029 64 36

0.047592 10 1029 2401 49

0.046337 3 126 64 25

0.045775 5 1029 81 36

0.043148 4 2401 50 21

0.042149 5 1029 81 49

0.040377 2 64 50 21

0.038212 2 81 50 21

0.036658 4 2401 50 36

0.034913 2 81 50 28

0.033669 5 1029 245 49

0.03128 3 245 225 25

0.031004 2 225 50 21

0.030216 3 245 225 28

0.029652 4 2401 126 36

0.029164 2 225 64 21

0.028808 3 2401 81 21

0.028097 3 245 225 36

0.027303 4 2401 126 49

0.026321 3 2401 81 28

0.0097913 1 1029 126 21

0.0092609 1 1029 126 25

0.0076707 1 245 2401 21

This shows that in general, the PBs with a lower number of notes

often tend to be more "skewed" and thus less interesting. But there

seem to be lots of interesting new scales that are quite "straight"!

Can someone please provide the generator, interval of repetition,

mapping from generators to primes (3,5,7), and maximum 7-limit error,

for at least the top 32 in the rankings, and preferably many more?

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

>Here are the

> rankings by straightness:

Here are some other measures for the top ten on your list; the first

is a solid angle measure, the area on the unit sphere corresponding

to the three vectors. The second is my validity condition; this is a

sufficient condition, not a necessary one, but one might well ask how

many of these correctly order the notes in the block--#2, the "very

improper" one, has a validity measure over 5. The last measure is the

most like your measure; it is the volume (which is to say, the

determinant) divided by the product of the lengths of the sides.

Since a unit volume is the volume of the parallepiped with sides 3,

5, and 7, and not a unit cube, the maximum this can attain is

actually sqrt(2), which is what a cube would give.

#1: 14 notes; commas 81:80, 50:49; chroma 25:24 (Erlich 14-of-~26)

1.337003903 2.796199310 1.120897076

#2: 31 notes; commas 1029:1024, 245:243; chroma 25:24 (v. improper!)

1.208253216 5.315252144 1.203940238

#3: 17 notes; commas 245:243, 64:63; chroma 25:24

0.625778711 3.007295043 0.8997354106

#4: 19 notes; commas 126:125, 81:80; chroma 49:48

1.036438116 2.179705030 1.149932312

#5: 15 notes; commas 126:125, 64:63; chroma 28:27

1.397360786 2.462710473 0.899238708

#6: 21 notes; commas 1029:1024, 81:80; chroma 25:24 (not blackjack!)

0.907922503 3.388625357 0.9326427427

#7: 14 notes; commas 81:80, 50:49; chroma 49:48 (Erlich 14-of-~26)

2.405549309 2.107333077 1.120897076

#8: 12 notes; commas 64:63, 50:49; chroma 36:35 (Erlich 12-of-22)

1.714143895 2.158607408 1.309307341

#9: 19 notes; commas 225:224, 81:80; chroma 49:48 (19-out-of-31???)

0.420158312 2.038933737 1.014144974

#10: 14 notes; commas 245:243, 50:49; chroma 25:24

0.71923786 2.635572871 0.9810960584

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

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

>

> >Here are the

> > rankings by straightness:

>

> Here are some other measures for the top ten on your list; the

first

> is a solid angle measure, the area on the unit sphere corresponding

> to the three vectors.

In a Cartesian lattice with 3, 5, and 7 axes? Also, doesn't this

depend in an arbitrary way on the signs of the unison vectors?

> The second is my validity condition; this is a

> sufficient condition, not a necessary one, but one might well ask

how

> many of these correctly order the notes in the block--#2, the "very

> improper" one, has a validity measure over 5.

Can you explain what this validity condition is about?

> The last measure is the

> most like your measure; it is the volume (which is to say, the

> determinant) divided by the product of the lengths of the sides.

Lengths measured with Euclidean distance in the Cartesian lattice

with 3, 5, and 7 axes?

> Since a unit volume is the volume of the parallepiped with sides 3,

> 5, and 7,

A rectangular prism? Can you flesh this out for me please?

>

>

> #1: 14 notes; commas 81:80, 50:49; chroma 25:24 (Erlich 14-of-~26)

>

> 1.337003903 2.796199310 1.120897076

>

> #2: 31 notes; commas 1029:1024, 245:243; chroma 25:24 (v. improper!)

>

> 1.208253216 5.315252144 1.203940238

>

> #3: 17 notes; commas 245:243, 64:63; chroma 25:24

>

> 0.625778711 3.007295043 0.8997354106

>

> #4: 19 notes; commas 126:125, 81:80; chroma 49:48

>

> 1.036438116 2.179705030 1.149932312

>

> #5: 15 notes; commas 126:125, 64:63; chroma 28:27

>

> 1.397360786 2.462710473 0.899238708

>

> #6: 21 notes; commas 1029:1024, 81:80; chroma 25:24 (not blackjack!)

>

> 0.907922503 3.388625357 0.9326427427

>

> #7: 14 notes; commas 81:80, 50:49; chroma 49:48 (Erlich 14-of-~26)

>

> 2.405549309 2.107333077 1.120897076

>

> #8: 12 notes; commas 64:63, 50:49; chroma 36:35 (Erlich 12-of-22)

>

> 1.714143895 2.158607408 1.309307341

>

> #9: 19 notes; commas 225:224, 81:80; chroma 49:48 (19-out-of-31???)

>

> 0.420158312 2.038933737 1.014144974

>

> #10: 14 notes; commas 245:243, 50:49; chroma 25:24

>

> 0.71923786 2.635572871 0.9810960584

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

> Let's try chromatic unison vectors between 100 cents and 35 cents,

> commatic unison vectors smaller than 35 cents but with numerator

and

> denominator less than 3000, disallow torsion, and select unison

> vectors only from this list:

>

> http://www.kees.cc/tuning/s2357.html

I thought natural to see what ETs come up when we take three commatic

unison vectors at a time, still defined as above, rather than the

MOSs that come from one chromatic and two commatic. The results

(ranked by straightness):

Straighness ET numerators of unison vectors

0.24926 46 1029 245 126

0.2458 22 245 64 50

0.24401 27 245 126 64

0.24116 36 1029 245 50

0.21804 26 1029 81 50

0.21029 31 1029 126 81

0.19838 41 1029 245 225

0.18874 22 245 225 50

0.18778 31 1029 225 81

0.1874 31 2401 126 81

0.17754 22 245 225 64

0.17679 41 245 2401 225

0.17246 27 2401 126 64

0.17062 31 1029 225 126

0.16784 12 81 64 50

0.16734 31 2401 225 81

0.16251 19 245 126 81

0.15251 12 126 64 50

0.15205 31 2401 225 126

0.15162 27 245 2401 64

0.14803 14 245 81 50

0.14511 19 245 225 81

0.14433 12 126 81 50

0.13804 41 1029 245 2401

0.13577 12 126 81 64

0.13185 19 245 225 126

0.13066 31 1029 2401 81

0.13038 27 245 2401 126

0.12888 12 225 81 50

0.12192 16 1029 126 50

0.12123 12 225 81 64

0.1195 17 2401 81 64

0.11872 31 1029 2401 126

0.11711 12 225 126 50

0.11016 12 225 126 64

0.10752 15 1029 126 64

0.10463 14 2401 81 50

0.088612 10 1029 64 50

0.083576 14 245 2401 50

0.078966 10 2401 64 50

0.076863 8 245 126 50

0.074401 14 245 2401 81

0.068043 10 1029 225 50

0.064004 10 1029 225 64

0.060636 10 2401 225 50

0.057037 10 2401 225 64

0.049731 5 245 81 64

0.047346 10 1029 2401 50

0.044536 10 1029 2401 64

0.039442 5 1029 81 64

0.031507 5 1029 245 64

0.029818 5 1029 245 81

0.027162 4 2401 126 50

I'm shocked that 36 shows up so close to the top! Discuss.

No one calculated the information I requested (generators, mappings

from primes to generators, minimax error). Gene, perhaps you can

provide me with an algorithm to find the generator, and mapping from

primes to generators, given the chromatic unison vector and set of

commatic unison vectors? I'm not going to try to understand how it

works now -- just want to get some results.

From my ET posting today, I can try to understand what ETs some of

these MOS scales (and others with the same generators) might be

embeddable in:

>

> #1: 14 notes; commas 81:80, 50:49; chroma 25:24 (Erlich 14-of-~26)

26-tET of course!

> #2: 31 notes; commas 1029:1024, 245:243; chroma 25:24 (v. improper!)

46-tET, 36-tET, 41-tET . . . if it's got more than 31 notes, try 31

anyway!

> #3: 17 notes; commas 245:243, 64:63; chroma 25:24

22-tET, 27-tET.

> #4: 19 notes; commas 126:125, 81:80; chroma 49:48

31-tET -- is this the famous 19-out-of-31?

> #5: 15 notes; commas 126:125, 64:63; chroma 28:27

27-tET -- this must be the 15-out-of-27 Gene was talking about when

he first joined this list.

> #6: 21 notes; commas 1029:1024, 81:80; chroma 25:24 (not blackjack!)

26-tET, 31-tET

> #7: 14 notes; commas 81:80, 50:49; chroma 49:48 (Erlich 14-of-~26)

Same as #1.

> #8: 12 notes; commas 64:63, 50:49; chroma 36:35 (Erlich 12-of-22)

22-tET, of course!

> #9: 19 notes; commas 225:224, 81:80; chroma 49:48 (19-out-of-31???)

31-tET

> #10: 14 notes; commas 245:243, 50:49; chroma 25:24

22-tET, 36-tET . . . what's this?

> #11: 12 notes; commas 81:80, 50:49; chroma 36:35

26-tET, 14-tET . . . complement of Erlich 14-out-of-26

> #12: 15 notes; commas 126:125, 64:63; chroma 49:48

Same as #5

> #13: 10 notes; commas 64:63, 50:49; chroma 25:24 (Erlich 10-of-22)

22-tET, of course!

> #14: 19 notes; commas 225:224, 126:125, chroma 49:48

31-tET

> #15: 10 notes; commas 64:63, 50:49; chroma 28:27 (Erlich 10-of-22)

Same as #13

> #16: 19 notes; commas 245:243, 126:125; chroma 49:48

46-tET, 27-tET . . . some kind of 8-tone scheme behind this . . .

> #17: 23 notes; commas 2401:2400, 126:125; chroma 28:27

31-tET, 27-tET . . . some kind of 4-tone scheme behind this . . .

kleismic???

> #18: 14 notes; commas 245:243, 81:80; chroma 25:24

NOTHING ON MY LIST! WHAT IS THIS?

> #19: 16 notes; commas 245:243, 225:224; chroma 21:20

41-tET, 22-tET, 19-tET . . . is this Graham's MAGIC thing?

> #20: 14 notes; commas 245:243, 50:49; chroma 49:48

Same as #10

> #21: 12 notes; commas 126:125, 64:63; chroma 36:35

Same as #5

> #22: 16 notes; commas 1029:1024, 50:49; chroma 36:35

36-tET, 26-tET . . . curious

> #23: 19 notes; commas 245:243, 225:224; chroma 49:48

Same as #19 but longer chain

> #24: 12 notes; commas 225:224, 50:49; chroma 36:35

22-tET -- Erlich 12-out-of-22

> #25: 10 notes; commas 64:63, 50:49; chroma 49:48 [Erlich 10-of-22]

Same as #13

> #26: 12 notes; commas 126:125, 81:80; chroma 36:35

Complement of #4?

> #27: 21 notes; commas 1029:1024, 225:224; chroma 36:35 (Blackjack)

41-tET, 31-tET

> #28: 12 notes; commas 225:224, 64:63; chroma 36:35

22-tET

> #29: 10 notes; commas 225:224, 50:49; chroma 25:24 (Erlich 10-of-22)

22-tET

> #30: 17 notes; commas 2401:2400, 64:63; chroma 36:35

27-tET . . . some kind of 10-tone scheme behind this . . .

> #31: 10 notes; commas 225:224, 50:49; chroma 28:27 (Erlich 10-of-22)

22-tET

> #32: 12 notes; commas 225:224, 81:80; chroma 36:35

Complement of #9?

> #33: 21 notes; commas 2401:2400, 225:224; chroma 36:35 (Blackjack)

41-tET, 31-tET

Now, what _are_ the (optimal) generators?

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

Here is the first triple on your list, 81/80, 50/49, 25/24 worked out

as an example; let's see if you have any questions.

(1) Step one is to find the val generating the dual group, and that

means finding the minors or something equivalent, such as taking the

determinant of

[ a b c d]

[-4 4 -1 0]

[ 1 0 2 -2]

[-3 -1 2 0]

which is g = 14a + 22b + 32c + 39d.

(2) Step two is finding a val v such that v(81/80) = v(50/49) = 0, and

v(25/24) = 1. If we solve the linear system of equations

-4a + 4b - c = 0

a + 2c - 2d = 0

-3a - b + 2c = 1

we get a = (7c-4)/16, b = (11c-4)/16, d = (39c-4)/32; since we want

integter solutions we want 39c = 4 (mod 32), which means c = -4 will

work. This gives us -2a-3b-4c-5d; we get the same linear span by

adding this to our previous val, getting h12 = 12a + 19b + 28c + 34d.

Note that while g+h12=h26, g is not h14, since g(5)=32, not 33.

(3) We now want to find A and B such that A^g(q) B^h12(q) give good

appoximations to 3,5,7,5/3,7/3 and 7/5. Since I don't want to mess

around finding out how Maple's linear programming routines work and

since least squares is so easy, I'll use that. Optimizing by least

squares and assuming octaves are pure, I get A = 38.098 cents and

B = 55.557 cents.

(4) 14/12 = 7/6 = 1+1/6, and the convergent to 7/6 is 1. We therefore

might choose A + B = 93.651 cents as our generator; this happens to

be reasonably close to 1200/13 = 92.308 cents; not much of a surprise

since g+h12=h26 and gcd(12,14)=2. If we use 1200/13 as our generator,

with a period of half an octave, we have:

(5) h26(3) = 41 = 2 (mod 13)

h26(5) = 60 = -5 (mod 13)

h26(7) = 73 = -5 (mod 13)

The complexity is therefore 2*7 = 14. The example is actually a

little too easy at this point, since the generator is 1/13 of an

octave I don't need to do any mod 13 division.

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

> In a Cartesian lattice with 3, 5, and 7 axes? Also, doesn't this

> depend in an arbitrary way on the signs of the unison vectors?

No, in the symmetrical lattice. However, I wouldn't pay much

attention to this; for one thing there are six eight triangles for

every three verticies, and which one to you pick?

> > The second is my validity condition; this is a

> > sufficient condition, not a necessary one, but one might well ask

> how

> > many of these correctly order the notes in the block--#2,

the "very

> > improper" one, has a validity measure over 5.

> Can you explain what this validity condition is about?

With a high number like that, it seems likely that the val does not

order the block linearly, and hence that "Paul" doesn't work.

> > The last measure is the

> > most like your measure; it is the volume (which is to say, the

> > determinant) divided by the product of the lengths of the sides.

> Lengths measured with Euclidean distance in the Cartesian lattice

> with 3, 5, and 7 axes?

No, lengths with the Euclidean distance where 1, 3 5 and 7 are the

verticies of a regualar tetrahedron.

> > Since a unit volume is the volume of the parallepiped with sides

3,

> > 5, and 7,

> A rectangular prism? Can you flesh this out for me please?

We want volume to correspond to number of lattice points in a region,

so we want to make the vectors 3, 5, and 7, which give us the

identity matrix, define a volume of 1; this means a cube with sides

of length 1 has a measure of sqrt(2), but there is no reason to let

that worry us. The 3, 5, 7 is not a rectangular prism, but a

parallepidped; we have 3.5 = 3.7 = 5.7 = 1/2 so they are at 60 degree

angles.

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

> I'm shocked that 36 shows up so close to the top! Discuss.

I also got a good number (1.2108) for my own straightness measure of

36 divided by the product of the lengths of 1029/1024, 245/243 and

50/49. However, my validity measure, which is the product of those

three raised to the 36th power, was not very good: 3.3126. That

measure may be more significant.

In-Reply-To: <9pt7ch+ucrd@eGroups.com>

Paul wrote:

> No one calculated the information I requested (generators, mappings

> from primes to generators, minimax error). Gene, perhaps you can

> provide me with an algorithm to find the generator, and mapping from

> primes to generators, given the chromatic unison vector and set of

> commatic unison vectors? I'm not going to try to understand how it

> works now -- just want to get some results.

If you look at <http://x31eq.com/vectors.paul.py>, all you

need to do is alter it to use different unison vectors (chromatic on top,

all larger than a unison). If downloading ActivePython's too much trouble

for you, I can run it at home sometime.

Graham

In-Reply-To: <9ptkum+fe30@eGroups.com>

Paul wrote:

> > #19: 16 notes; commas 245:243, 225:224; chroma 21:20

>

> 41-tET, 22-tET, 19-tET . . . is this Graham's MAGIC thing?

Yes, that's Magic.

Graham

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

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

>

> > In a Cartesian lattice with 3, 5, and 7 axes? Also, doesn't this

> > depend in an arbitrary way on the signs of the unison vectors?

>

> No, in the symmetrical lattice.

Oh, goody!

> However, I wouldn't pay much

> attention to this; for one thing there are six eight triangles for

> every three verticies,

Six eight triangles?

> and which one to you pick?

Well, once you're tempering out the commatic unison vectors, much of

the choice becomes irrelevant -- right?

>

> > > The second is my validity condition; this is a

> > > sufficient condition, not a necessary one, but one might well

ask

> > how

> > > many of these correctly order the notes in the block--#2,

> the "very

> > > improper" one, has a validity measure over 5.

>

> > Can you explain what this validity condition is about?

>

>

> With a high number like that, it seems likely that the val does not

> order the block linearly, and hence that "Paul" doesn't work.

Well, I guess I'll have to try programming the algorithm you just

gave me and see what comes out as the generator, etc.

>

> > > The last measure is the

> > > most like your measure; it is the volume (which is to say, the

> > > determinant) divided by the product of the lengths of the sides.

>

> > Lengths measured with Euclidean distance in the Cartesian lattice

> > with 3, 5, and 7 axes?

>

> No, lengths with the Euclidean distance where 1, 3 5 and 7 are the

> verticies of a regualar tetrahedron.

I'd prefer an "isosceles tetrahedron", but this is good too . . .

>

> > > Since a unit volume is the volume of the parallepiped with

sides

> 3,

> > > 5, and 7,

>

> > A rectangular prism? Can you flesh this out for me please?

>

> We want volume to correspond to number of lattice points in a

region,

> so we want to make the vectors 3, 5, and 7, which give us the

> identity matrix, define a volume of 1; this means a cube with sides

> of length 1 has a measure of sqrt(2), but there is no reason to let

> that worry us. The 3, 5, 7 is not a rectangular prism, but a

> parallepidped; we have 3.5 = 3.7 = 5.7 = 1/2 so they are at 60

degree

> angles.

OK -- when you said "sides 3, 5, and 7", I thought you meant lengths

3, 5, and 7. Sorry.

Have you given any thought to the idea of a "canonical basis" for the

case where all the unison vectors are chromatic, and the case where

one is commatic? A lot of the measures we've come up here depend on

the specific unison vectors we name as the basis, even though many

other sets yield exactly the same temperament.

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

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

>

> > I'm shocked that 36 shows up so close to the top! Discuss.

>

> I also got a good number (1.2108) for my own straightness measure

of

> 36 divided by the product of the lengths of 1029/1024, 245/243 and

> 50/49. However, my validity measure, which is the product of those

> three raised to the 36th power, was not very good: 3.3126. That

> measure may be more significant.

Your validity measure is equivalent to something I proposed a couple

of years ago on the tuning list. If you're constructing the

parallelepiped PB in JI (i.e. a hyper-MOS with no unison vectors

tempered out), then the largest interval functioning as a unison, and

directly affecting what notes do and do not appear in the block,

should be smaller than the smallest step in the block. Clearly if

that interval greater than 1/N octave, it can't be smaller than the

smallest step. The largest interval functioning as a unison, etc.,

spans a diagonal of the block -- specifically that diagonal which is

the result of adding the unison vectors when all are taken as

ascending intervals. So in this case, we have

1029/1024 = 8.4327¢

245/243 = 14.1905¢

50/49 = 34.9756¢

sum = 57.5989¢ > 33.3333¢

HOWEVER, when you're tempering out unison vectors, this validity

measure ceases to mean very much. For example, 12-tET can be defined

using the minor diesis and the major diesis, but clearly fails the

validity measure with respect to these unison vectors. So what? You

see how a "canonical basis" might be useful.

Now, 36-tET only showed up once in our list -- since we stuck to the

smallest unison vectors for their lengths, we probably found

the "canonical" basis for 36-tET, and it turned out invalid. OK. But

ultimately I'd like to drop the restriction on taking unison vectors

from Kees' list -- Herman Miller has made wonderful music using the

maximal diesis (250:243) as a commatic unison vector!

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

> In-Reply-To: <9pt7ch+ucrd@e...>

> Paul wrote:

>

> > No one calculated the information I requested (generators,

mappings

> > from primes to generators, minimax error). Gene, perhaps you can

> > provide me with an algorithm to find the generator, and mapping

from

> > primes to generators, given the chromatic unison vector and set

of

> > commatic unison vectors? I'm not going to try to understand how

it

> > works now -- just want to get some results.

>

> If you look at <http://x31eq.com/vectors.paul.py>, all

you

> need to do is alter it to use different unison vectors (chromatic

on top,

> all larger than a unison). If downloading ActivePython's too much

trouble

> for you, I can run it at home sometime.

>

If that's not too much trouble, I'd be in your debt!

Using the validity measure _without_ a "straightness" measure is not

recommended. All those 1-tone PBs look awfully good on the validity

measure!

In-Reply-To: <9pv7i6+6ddv@eGroups.com>

Okay, I've made it even easier to run my unison vectors to temperament

program. You can feed it an input file containing the ratios as ratios.

So I've run it over the examples Paul gave in

</tuning-math/message/1223>.

Code is at

<http://x31eq.com/temper.py>

<http://x31eq.com/vectors.py>

<http://x31eq.com/findTemperaments.py>

You need Numeric Python. From the latest ActivePython release, you do

"ppm install Numeric". But that is quite a download. If you can find

where Numeric Python lives you only need that and the standard Python

distribution from <http://www.python.org> (or a minimal distribution

without TKInter, if you can find one).

Here's the example input file:

<http://x31eq.com/paul.limit7.vectors>

It assumes sets of ratios are separated by a line with no ratios in it.

If it's always going to be one set per line, the program can be

simplified. I'm taking the chromatic UV as the *last* on the list, as

that's the way you seem to do it.

To run this example, "python findTemperament.py paul.7limit" in the folder

that has the scripts and input file. That produces the output file

paul.7limit.out which you can also find at my website. I've also run this

over the vectors Gene posted recently.

<http://x31eq.com/paul.7limit.out>

<http://x31eq.com/gene.7limit.vectors>

<http://x31eq.com/gene.7limit.out>

Sometime I'll get it to use the octave-specific vectors. For now, it

converts them to octave-equivalent, and then back again.

Paul did ask about this one before:

[(-1, 2, 0), (0, 2, -2), (-5, 1, 2)]

5/7, 434.0 cent generator

basis:

(0.5, 0.3616541669070521)

mapping by period and generator:

[(2, 0), (1, 3), (1, 5), (2, 5)]

mapping by steps:

[[8, 6], [13, 9], [19, 14], [23, 17]]

unison vectors:

[[1, 0, 2, -2], [0, -5, 1, 2]]

highest interval width: 5

complexity measure: 10 (14 for smallest MOS)

highest error: 0.014573 (17.488 cents)

Graham

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

> That produces the output file

> paul.7limit.out which you can also find at my website.

>

> <http://x31eq.com/paul.7limit.out>

This URL doesn't seem to work :(

Paul wrote:

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

>

> > That produces the output file

> > paul.7limit.out which you can also find at my website.

> >

> > <http://x31eq.com/paul.7limit.out>

>

> This URL doesn't seem to work :(

Oops! Should be paul.limit7.out, etc. I'll get it all correct at

<http://x31eq.com/vectors.html>.

Graham

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

> Paul wrote:

>

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

> >

> > > That produces the output file

> > > paul.7limit.out which you can also find at my website.

> > >

> > > <http://x31eq.com/paul.7limit.out>

> >

> > This URL doesn't seem to work :(

>

> Oops! Should be paul.limit7.out, etc. I'll get it all correct at

> <http://x31eq.com/vectors.html>.

>

>

> Graham

Thanks a bunch Graham. You're my hero!

Thanks to Graham, I'm now in a position to answer my questions:

>

> > #3: 17 notes; commas 245:243, 64:63; chroma 25:24

>

> 22-tET, 27-tET.

Generator 710.7 cents -- this is one of the three sizes of fifth that

Dave Keenan recognizes as generating a good 7-limit scale with a

single chain of fifths wrapped within the octave:

http://www.uq.net.au/~zzdkeena/Music/1ChainOfFifthsTunings.htm

>

> > #4: 19 notes; commas 126:125, 81:80; chroma 49:48

>

> 31-tET -- is this the famous 19-out-of-31?

Sure is!

>

> > #5: 15 notes; commas 126:125, 64:63; chroma 28:27

>

> 27-tET -- this must be the 15-out-of-27 Gene was talking about when

> he first joined this list.

It sure is!

>

> > #6: 21 notes; commas 1029:1024, 81:80; chroma 25:24 (not

blackjack!)

>

> 26-tET, 31-tET

This one is fascinating. It's nothing like Blackjack, yet it has the

same complexity measure and same number of notes per octave, and only

slightly larger errors. The optimal generator achieves proper MOSs at

5, 26, and 31 notes -- and then again only at 584 notes per octave!

Score another one for 31-tET.

>

> > #9: 19 notes; commas 225:224, 81:80; chroma 49:48 (19-out-of-

31???)

Yup.

>

> > #10: 14 notes; commas 245:243, 50:49; chroma 25:24

>

> 22-tET, 36-tET . . . what's this?

Interesting . . . a 7:9 generator in a half-octave . . .

Hey Graham . . . why does #11 open in your output with '5/6' while

the otherwise identical #1 opens with '1/7'?

>

> > #14: 19 notes; commas 225:224, 126:125, chroma 49:48

>

> 31-tET

Good ol' meantone again.

>

> > #16: 19 notes; commas 245:243, 126:125; chroma 49:48

>

> 46-tET, 27-tET . . . some kind of 8-tone scheme behind this . . .

Curious one this! 27-tET doesn't really do it justice . . . but I

suppose I could live with it . . .

>

> > #17: 23 notes; commas 2401:2400, 126:125; chroma 28:27

>

> 31-tET, 27-tET . . . some kind of 4-tone scheme behind this . . .

> kleismic???

This one actually looks really good . . . yes, a minor third

generator here . . . score another one for 31-tET!

> > #18: 14 notes; commas 245:243, 81:80; chroma 25:24

>

> NOTHING ON MY LIST! WHAT IS THIS?

This works in 19-tET . . .

>

> > #22: 16 notes; commas 1029:1024, 50:49; chroma 36:35

>

> 36-tET, 26-tET . . . curious

This is quite an interesting one . . . after 26, the next proper MOS

is at 110 . . . score one for 26-tET!

>

> > #26: 12 notes; commas 126:125, 81:80; chroma 36:35

>

> Complement of #4?

Yup!

> > #30: 17 notes; commas 2401:2400, 64:63; chroma 36:35

>

> 27-tET . . . some kind of 10-tone scheme behind this . . .

Yup . . . score another for 27-tET . . .

>

> > #32: 12 notes; commas 225:224, 81:80; chroma 36:35

>

> Complement of #9?

Yup.

Alright, I think I'm getting a classical guitar (which tolerates

greater mistuning in the fifths than other guitars) outfitted with

Mark Rankin's fingerboards, two of which will be in 26-tET and 27-

tET. I already have 22-tET and 31-tET electric guitars.

In-Reply-To: <9q2f10+h9nk@eGroups.com>

Paul wrote:

> > > #10: 14 notes; commas 245:243, 50:49; chroma 25:24

> >

> > 22-tET, 36-tET . . . what's this?

>

> Interesting . . . a 7:9 generator in a half-octave . . .

>

> Hey Graham . . . why does #11 open in your output with '5/6' while

> the otherwise identical #1 opens with '1/7'?

One gives a 12 note periodicity block, the other 14. I don't know why

it's 5/6 instead of 1/6. I used to always take the smallest generator,

but the definition of "smallest" can be different for different

periodicity blocks. So I use an arbitrary rule which works in this case

(the mappings are the same) but doesn't always when the period isn't an

octave.

Graham

In-Reply-To: <9q2f10+h9nk@eGroups.com>

Paul wrote:

> > > #22: 16 notes; commas 1029:1024, 50:49; chroma 36:35

> >

> > 36-tET, 26-tET . . . curious

>

> This is quite an interesting one . . . after 26, the next proper MOS

> is at 110 . . . score one for 26-tET!

You're very keen to take ETs here. Why don't you work out what ETs are

consistent with each unison vector, like at

<http://x31eq.com/paul.limit7.et> and see how often each comes

up?

And in this case, you take the different sets

(-10, 1, 0, 3), 1029:1024

5 10 15 16 26 31 36 41 46 56 57 62 72 77 82 87 ...

(1, 0, 2, -2), 50:49

4 6 10 12 16 18 22 26

and check the intersection

5 10 15 16 26 31 41 ...

4 6 10 12 16 18 22 26

Graham

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

> In-Reply-To: <9q2f10+h9nk@e...>

> Paul wrote:

>

> > > > #22: 16 notes; commas 1029:1024, 50:49; chroma 36:35

> > >

> > > 36-tET, 26-tET . . . curious

> >

> > This is quite an interesting one . . . after 26, the next proper

MOS

> > is at 110 . . . score one for 26-tET!

>

> You're very keen to take ETs here. Why don't you work out what ETs

are

> consistent with each unison vector, like at

> <http://x31eq.com/paul.limit7.et> and see how often each

comes

> up?

Well, if I can have a single guitar which will work for a lot of

these good MOS scales, that saves money, plus I can switch from one

to another withing a single composition.

>

> And in this case, you take the different sets

>

> (-10, 1, 0, 3), 1029:1024

> 5 10 15 16 26 31 36 41 46 56 57 62 72 77 82 87 ...

>

> (1, 0, 2, -2), 50:49

> 4 6 10 12 16 18 22 26

>

> and check the intersection

>

> 5 10 15 16 26 31 41 ...

> 4 6 10 12 16 18 22 26

>

Not sure what that tells me.