Full thread (5 messages)
From: Gene Ward Smith (2005-03-11)
Subject: Eikosany ball series deep hole scales
These are the scales you get around the deep hole, using the Euclidean
metric with 5,7,9 and 11 equal. It's not clear to me that the eikosany
stands out; ball 2 with 18 notes and ball 4 with 24 notes are
permutation epimorphic. (Ball 1 is epimorphic in two different ways,
which is a fun fact we can find because of Manuel's improvements to
Scala, and which is also true of the 1-3-5-7 hexany.)
Here are balls one through six in Scala format.
! eikohole1.scl
First eikohole ball <6 9 13 17 20|-epimorphic
6
!
35/33
7/6
14/11
5/3
20/11
2
! eikohole2.scl
Second eikohole ball
18
!
56/55
21/20
12/11
63/55
6/5
14/11
4/3
7/5
16/11
3/2
84/55
8/5
18/11
56/33
9/5
28/15
21/11
2
! eikohole3.scl
Third eikohole ball = eikosany
20
!
56/55
21/20
12/11
63/55
7/6
6/5
14/11
72/55
4/3
7/5
16/11
3/2
84/55
8/5
18/11
56/33
9/5
28/15
21/11
2
! eikohole4
Fourth eikohole ball
24
!
21/20
77/72
11/10
7/6
6/5
11/9
77/60
4/3
11/8
7/5
77/54
22/15
3/2
14/9
8/5
77/48
33/20
77/45
7/4
9/5
11/6
28/15
77/40
2
! eidohole5.scl
Fifth eikohole ball
42
!
56/55
21/20
16/15
12/11
11/10
9/8
112/99
63/55
7/6
6/5
27/22
56/45
14/11
72/55
4/3
27/20
224/165
168/121
7/5
63/44
16/11
3/2
84/55
14/9
63/40
8/5
18/11
42/25
56/33
12/7
189/110
96/55
7/4
16/9
98/55
9/5
20/11
28/15
21/11
64/33
108/55
2
! eikohole6.scl
Sixth eikohole ball
54
!
56/55
126/121
21/20
16/15
12/11
11/10
28/25
9/8
112/99
8/7
63/55
7/6
196/165
6/5
40/33
27/22
56/45
63/50
14/11
9/7
72/55
4/3
147/110
27/20
224/165
15/11
168/121
7/5
63/44
16/11
22/15
3/2
84/55
14/9
63/40
8/5
18/11
33/20
42/25
56/33
12/7
189/110
96/55
7/4
16/9
98/55
9/5
20/11
224/121
28/15
21/11
64/33
108/55
2
From: Gene Ward Smith (2005-03-11)
Subject: Re: Eikosany ball series deep hole scales
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> These are the scales you get around the deep hole, using the Euclidean
> metric with 5,7,9 and 11 equal. It's not clear to me that the eikosany
> stands out; ball 2 with 18 notes and ball 4 with 24 notes are
> permutation epimorphic.
However, at ball 4 a 385/384 interval appears in the scale; before
that the intervals seem more reasonable.
From: Yahya Abdal-Aziz (2005-03-14)
Subject: RE: Eikosany ball series deep hole scales
-----Original Message-----
________________________________________________________________________
Date: Fri, 11 Mar 2005 08:54:18 -0000
From: "Gene Ward Smith" <gwsmith@svpal.org>
Subject: Eikosany ball series deep hole scales
[Gene]
These are the scales you get around the deep hole, using the Euclidean
metric with 5,7,9 and 11 equal. ...
[Yahya]
This is _very_ interesting indeed. It will be fun to see how each of
these scales works as a compositional resource. In line with my earlier
comments, how sensitive are the resulting scales to the choice of
metric? For example, what shells and scales arise from using the
following metrics? -
1. The "city-block" metric d = Sigma_i (|a_i| ?
2. The hyper-Euclidean metric d = (Sigma_i a_i^n)^(1/n),
where n is an integer >2 ?
3. The LCM metric d = LCM_i (a_i) ?
4. Your favourite metric here _________?
Notational note: a_i means "a subscript i". In all other cases, the
suffix _i means "over all i".
________________________________________________________________________
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From: Gene Ward Smith (2005-03-14)
Subject: Re: Eikosany ball series deep hole scales
--- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> This is _very_ interesting indeed. It will be fun to see how each of
> these scales works as a compositional resource. In line with my earlier
> comments, how sensitive are the resulting scales to the choice of
> metric? For example, what shells and scales arise from using the
> following metrics? -
> 1. The "city-block" metric d = Sigma_i (|a_i| ?
> 2. The hyper-Euclidean metric d = (Sigma_i a_i^n)^(1/n),
> where n is an integer >2 ?
> 3. The LCM metric d = LCM_i (a_i) ?
> 4. Your favourite metric here _________?
How sensitive they are depends in part on how big the ball is. A
useful non-Euclidean for the 7-limit is the Hahn norm, described here:
http://66.98.148.43/~xenharmo/hahn.htm
Scales which are of Hahn ball type are discussed here:
http://66.98.148.43/~xenharmo/crystal.htm
From: Yahya Abdal-Aziz (2005-03-15)
Subject: Re: Eikosany ball series deep hole scales
Gene,
Thanks for all the pointers!
I've just downloaded most of your Theory section as well
to read later.
Regards,
Yahya
-----Original Message-----
________________________________________________________________________
Date: Mon, 14 Mar 2005 07:15:56 -0000
From: "Gene Ward Smith" <gwsmith@svpal.org>
Subject: Re: Ball scales (was RE: Digest Number 3438-the diamond)
--- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> However, I feel that it may be more "natural" for
> 3, 5, 7, 9, and 11 each to be further from 1 than its
> predecessor odd number.
In some connections that might be best; for instance, we could make
3 of length log 3, 5 of length log 5 and so forth. One context in
which that is useful is discussed here:
http://66.98.148.43/~xenharmo/top.htm
However, at other times we want to maximize symmetry, and in this
case, we want to see if the eikosany, a symmetrical 11-limit scale,
can be seen as a scale of ball type, and the above metric would not
help us there.
________________________________________________________________________
Date: Mon, 14 Mar 2005 07:20:32 -0000
From: "Gene Ward Smith" <gwsmith@svpal.org>
Subject: Re: Eikosany ball series deep hole scales
--- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> This is _very_ interesting indeed. It will be fun to see how each of
> these scales works as a compositional resource. In line with my earlier
> comments, how sensitive are the resulting scales to the choice of
> metric? For example, what shells and scales arise from using the
> following metrics? -
> 1. The "city-block" metric d = Sigma_i (|a_i| ?
> 2. The hyper-Euclidean metric d = (Sigma_i a_i^n)^(1/n),
> where n is an integer >2 ?
> 3. The LCM metric d = LCM_i (a_i) ?
> 4. Your favourite metric here _________?
How sensitive they are depends in part on how big the ball is. A
useful non-Euclidean for the 7-limit is the Hahn norm, described here:
http://66.98.148.43/~xenharmo/hahn.htm
Scales which are of Hahn ball type are discussed here:
http://66.98.148.43/~xenharmo/crystal.htm
________________________________________________________________________
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