I've already looked at <2,3,7> and <2,5,7>, and to this should probably

be added at minimum <2,5/3,7>, <2,3,7/5>,

<2,5,7/3>, <2,5/3,7/3>, <2,7/3,7/5> and the index two subgroups of the

7-limit given by <2,5,7,9>, <2,5/3,7,9>, <2,5,7/3,9>, <2,5/3,7/3,9>.

A search for ets supporting <2,5/3,7> turns up, for a log-flat badness

measure less than 0.5 and n to 5000, the following:

1 .2630344055

4 .45883938

11 .3949962817

15 .4272823563

57 .1451878461

114 .4106521743

213 .4874803032

270 .3169588282

327 .2215293311

384 .4759848469

3779 .4314215252

The 57-et stands out here. If we go to the index 2 subgroup <2,5/3,7,9>,

we get for badness less than 0.75 up to n = 2000 the following:

1 .3625700797

5 .5389365888

11 .2909521576

16 .7085795380

22 .7331534985

30 .6856035815

41 .7433989479

83 .7334574068

88 .6539733996

171 .3202175071

182 .7238985678

753 .5905099365

924 .4265227361

1095 .6996134010

Here 11 is clearly of special interest. If we call the 7-limit intervals

"blue" which can be expressed in terms of

2^a (5/3)^b 7^c 9^d

and the rest "red", then half of the 7-limit is colored blue (hence,

"index 2"). The MT reduced comma set for the

22-et in the 7-limit is <50/49, 64/63, 245/243>, as it happens, *all* of

these are blue, and so the 11-et has the same

MT reduced set of commas, and the kernel of the map from the 7-limit to

the 22-et is entirely blue.

We have

50/49 = 2 (5/3)^2 7^(-2) 9

64/63 = 2^6 7^(-1) 9^(-1)

245/243 = (5/3) 7^2 9^(-2)

While 50/49 is blue, 7/5 and 10/7 are red, so the meaning of 50/49~1 in

this subgroup has nothing to do with 7/5~10/7. However, we still have the

relationships 9/8~8/7 for 64/63 and (9/7)^2~5/3 for 245/243 as blue. The

connection between the 22-et and the 11-et therefore seems particularly

strong.

The corresponding periodicity blocks are

[1, 21/20, 8/7, 6/5, 9/7, 27/20, 40/27, 14/9, 5/3, 7/4, 40/21]

for the 11-et, and

[1, 28/27, 21/20, 10/9, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 27/20, 10/7, 40/27,

3/2, 14/9, 8/5, 5/3, 12/7, 7/4, 9/5, 40/21, 27/14]

for the 22-et; the 11-et scale is simply every other note of the 22-et

scale.

--- In tuning-math@y..., Gene W Smith <genewardsmith@j...> wrote:

> While 50/49 is blue, 7/5 and 10/7 are red, so the meaning of 50/49~1 in

> this subgroup has nothing to do with 7/5~10/7.

I forgot to add that (21/20)^2 ~ 9/8 is an entirely "blue" relationship.

Gene,

Neat. When I was looking at the 6-out-of-11 scale one of the things I

messed around with was rational interpretations that tried to maximize

the amount of simple consonant generators (7/4s in this case) in a way

that was somewhat analogous to the 3/2s in the syntonic diatonic.

One of these was a 2,7,11 prime interpretation where a 352/343 assumed

the role of a 81/80:

1/1 8/7 14/11 16/11 128/77 7/4 2/1

1/1 49/44 14/11 16/11 49/32 7/4 2/1

1/1 8/7 64/49 11/8 11/7 88/49 2/1

1/1 8/7 77/64 11/8 11/7 7/4 2/1

1/1 539/512 77/64 11/8 49/32 7/4 2/1

1/1 8/7 64/49 16/11 128/77 1024/539 2/1

take care,

--Dan Stearns

----- Original Message -----

From: "Gene W Smith" <genewardsmith@juno.com>

To: <tuning-math@yahoogroups.com>; <tuning@music.columbia.edu>

Sent: Friday, May 03, 2002 8:52 PM

Subject: [tuning-math] The <2,5/3,7> and <2,5/3,7,9> groups

> I've already looked at <2,3,7> and <2,5,7>, and to this should

probably

> be added at minimum <2,5/3,7>, <2,3,7/5>,

> <2,5,7/3>, <2,5/3,7/3>, <2,7/3,7/5> and the index two subgroups of

the

> 7-limit given by <2,5,7,9>, <2,5/3,7,9>, <2,5,7/3,9>, <2,5/3,7/3,9>.

>

> A search for ets supporting <2,5/3,7> turns up, for a log-flat

badness

> measure less than 0.5 and n to 5000, the following:

>

> 1 .2630344055

> 4 .45883938

> 11 .3949962817

> 15 .4272823563

> 57 .1451878461

> 114 .4106521743

> 213 .4874803032

> 270 .3169588282

> 327 .2215293311

> 384 .4759848469

> 3779 .4314215252

>

> The 57-et stands out here. If we go to the index 2 subgroup

<2,5/3,7,9>,

> we get for badness less than 0.75 up to n = 2000 the following:

>

> 1 .3625700797

> 5 .5389365888

> 11 .2909521576

> 16 .7085795380

> 22 .7331534985

> 30 .6856035815

> 41 .7433989479

> 83 .7334574068

> 88 .6539733996

> 171 .3202175071

> 182 .7238985678

> 753 .5905099365

> 924 .4265227361

> 1095 .6996134010

>

> Here 11 is clearly of special interest. If we call the 7-limit

intervals

> "blue" which can be expressed in terms of

>

> 2^a (5/3)^b 7^c 9^d

>

> and the rest "red", then half of the 7-limit is colored blue (hence,

> "index 2"). The MT reduced comma set for the

> 22-et in the 7-limit is <50/49, 64/63, 245/243>, as it happens,

*all* of

> these are blue, and so the 11-et has the same

> MT reduced set of commas, and the kernel of the map from the 7-limit

to

> the 22-et is entirely blue.

>

> We have

>

> 50/49 = 2 (5/3)^2 7^(-2) 9

>

> 64/63 = 2^6 7^(-1) 9^(-1)

>

> 245/243 = (5/3) 7^2 9^(-2)

>

> While 50/49 is blue, 7/5 and 10/7 are red, so the meaning of 50/49~1

in

> this subgroup has nothing to do with 7/5~10/7. However, we still

have the

> relationships 9/8~8/7 for 64/63 and (9/7)^2~5/3 for 245/243 as blue.

The

> connection between the 22-et and the 11-et therefore seems

particularly

> strong.

>

> The corresponding periodicity blocks are

>

> [1, 21/20, 8/7, 6/5, 9/7, 27/20, 40/27, 14/9, 5/3, 7/4, 40/21]

>

> for the 11-et, and

>

> [1, 28/27, 21/20, 10/9, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 27/20, 10/7,

40/27,

> 3/2, 14/9, 8/5, 5/3, 12/7, 7/4, 9/5, 40/21, 27/14]

>

> for the 22-et; the 11-et scale is simply every other note of the

22-et

> scale.

>

>

>

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