eikohole6

Sixth eikohole ball

Open in Scale Workshop Scala analysis Download scl

Properties

Notes	54
Period	1200.0 ¢
Just	11-limit
Source	Mailing lists
Reference	https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_11776.html#11776
Thread	5 scales

Tone	Tone (¢)	Step	Step (¢)
56/55	31	56/55	31
126/121	70	45/44	39
21/20	84	121/120	14
16/15	112	64/63	27
12/11	151	45/44	39
11/10	165	121/120	14
28/25	196	56/55	31
9/8	204	225/224	8
112/99	214	896/891	10
8/7	231	99/98	18
63/55	235	441/440	4
7/6	267	55/54	32
196/165	298	56/55	31
6/5	316	99/98	18
40/33	333	100/99	17
27/22	355	81/80	22
56/45	379	1232/1215	24
63/50	400	81/80	22
14/11	418	100/99	17
9/7	435	99/98	18
72/55	466	56/55	31
4/3	498	55/54	32
147/110	502	441/440	4
27/20	520	99/98	18
224/165	529	896/891	10
15/11	537	225/224	8
168/121	568	56/55	31
7/5	583	121/120	14
63/44	621	45/44	39
16/11	649	64/63	27
22/15	663	121/120	14
3/2	702	45/44	39
84/55	733	56/55	31
14/9	765	55/54	32
63/40	786	81/80	22
8/5	814	64/63	27
18/11	853	45/44	39
33/20	867	121/120	14
42/25	898	56/55	31
56/33	916	100/99	17
12/7	933	99/98	18
189/110	937	441/440	4
96/55	964	64/63	27
7/4	969	385/384	5
16/9	996	64/63	27
98/55	1000	441/440	4
9/5	1018	99/98	18
20/11	1035	100/99	17
224/121	1066	56/55	31
28/15	1081	121/120	14
21/11	1119	45/44	39
64/33	1147	64/63	27
108/55	1168	81/80	22
2	1200	55/54	32

Parent scales

File	Notes	Max diff (¢)
hemienn82	72	4.3
edo-72	72	4.9
xen18-erlich-miracle-72	72	5.0
edo-55	55	10.5
edo-56	56	10.5
edo-58	58	10.2
edo-61	61	9.5
edo-57	57	10.5
octoid72	72	7.1
edo-62	62	9.4

Child scales

File	Notes	Max diff (¢)
eidohole5	42	0.0
eikohole3	20	0.0
eikohole2	18	0.0
eikobag	12	0.0
qujus6	12	0.0
decab	10	0.0
xen12-wilson-15-diamond-eikosany-intersection	10	0.0
al-farabi_chrom2	7	0.0
mavchrome6	7	0.0
synmav2	7	0.0

Mailing list post

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

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".

________________________________________________________________________

-- 
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.308 / Virus Database: 266.7.2 - Release Date: 11/3/05

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
________________________________________________________________________

-- 
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.308 / Virus Database: 266.7.2 - Release Date: 11/3/05

Raw file

! 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
!
! https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_11776.html#11776
!
! [info]
! source = Mailing lists
! file = tuning-math/messages/yahoo_tuning-math_messages_api_raw_9945-12429.json
! topic_id = 11776
! msg_id = 11776