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.

--- 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.
>
>
>
> ------------------------ Yahoo! Groups
Sponsor ---------------------~-->
> Buy Stock for $4
> and no minimums.
> FREE Money 2002.
> http://us.click.yahoo.com/orkH0C/n97DAA/Ey.GAA/wHYolB/TM
> --------------------------------------------------------------------
-~->
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to
http://docs.yahoo.com/info/terms/
>
>