Mailing list post
From: Gene W Smith (2002-07-31)
Subject: Four 10-note, 7-limit JI scales
If we take (10/9)^2 (15/14)^2 (16/15)^2 (21/20)^3 = 2 as scale steps, and
simplify the scale-finding problem by assuming 4/3 and 3/2 both belong to
the scale, we obtain four scales, the third and fourth of which are the
inverted forms of the first and second. A version of the major/minor
transformation, exchanging 10/9 with 16/15, which is equivalent to saying
2-->2, 3-->3, 5-->24/5, 7-->168/25, exchanges the first and second, as
well as the third and fourth. The first "decaa", and fourth, "decad", are
major versions, having two major tetrads and a minor tetrad, while
"decab" and "decac" have two minor and one major tetrad. In any system
where 50/49~1 the exchange transform sends tetrads to tetrads and can be
considered major/minor. In 22-et in particular, each scale becomes the
symmetrical decatonic. All of the scales have 23 intervals, 17 triads and
3 tetrads.
! decad.scl
! [15/14, 10/9, 21/20, 16/15, 15/14, 21/20, 10/9, 15/14, 16/15, 21/20]
inversion of decab
10
!
15/14
25/21
5/4
4/3
10/7
3/2
5/3
25/14
40/21
2/1
! decab.scl
! [21/20, 16/15, 15/14, 10/9, 21/20, 15/14, 16/15, 21/20, 10/9, 15/14]
(10/9) <==> (16/15) transform of decaa
10
!
21/20
28/25
6/5
4/3
7/5
3/2
8/5
42/25
28/15
2/1
! decac.scl
! [15/14, 16/15, 21/20, 10/9, 15/14, 21/20, 16/15, 15/14, 10/9, 21/20]
inversion of decaa
10
!
15/14
8/7
6/5
4/3
10/7
3/2
8/5
12/7
40/21
2/1
! decad.scl
! [15/14, 10/9, 21/20, 16/15, 15/14, 21/20, 10/9, 15/14, 16/15, 21/20]
inversion of decab
10
!
15/14
25/21
5/4
4/3
10/7
3/2
5/3
25/14
40/21
2/1
Full thread (2 messages)
From: Gene W Smith (2002-07-31)
Subject: Four 10-note, 7-limit JI scales
If we take (10/9)^2 (15/14)^2 (16/15)^2 (21/20)^3 = 2 as scale steps, and
simplify the scale-finding problem by assuming 4/3 and 3/2 both belong to
the scale, we obtain four scales, the third and fourth of which are the
inverted forms of the first and second. A version of the major/minor
transformation, exchanging 10/9 with 16/15, which is equivalent to saying
2-->2, 3-->3, 5-->24/5, 7-->168/25, exchanges the first and second, as
well as the third and fourth. The first "decaa", and fourth, "decad", are
major versions, having two major tetrads and a minor tetrad, while
"decab" and "decac" have two minor and one major tetrad. In any system
where 50/49~1 the exchange transform sends tetrads to tetrads and can be
considered major/minor. In 22-et in particular, each scale becomes the
symmetrical decatonic. All of the scales have 23 intervals, 17 triads and
3 tetrads.
! decad.scl
! [15/14, 10/9, 21/20, 16/15, 15/14, 21/20, 10/9, 15/14, 16/15, 21/20]
inversion of decab
10
!
15/14
25/21
5/4
4/3
10/7
3/2
5/3
25/14
40/21
2/1
! decab.scl
! [21/20, 16/15, 15/14, 10/9, 21/20, 15/14, 16/15, 21/20, 10/9, 15/14]
(10/9) <==> (16/15) transform of decaa
10
!
21/20
28/25
6/5
4/3
7/5
3/2
8/5
42/25
28/15
2/1
! decac.scl
! [15/14, 16/15, 21/20, 10/9, 15/14, 21/20, 16/15, 15/14, 10/9, 21/20]
inversion of decaa
10
!
15/14
8/7
6/5
4/3
10/7
3/2
8/5
12/7
40/21
2/1
! decad.scl
! [15/14, 10/9, 21/20, 16/15, 15/14, 21/20, 10/9, 15/14, 16/15, 21/20]
inversion of decab
10
!
15/14
25/21
5/4
4/3
10/7
3/2
5/3
25/14
40/21
2/1
From: Gene W Smith (2002-07-31)
Subject: Re: [tuning-math] Four 10-note, 7-limit JI scales
These scales also work well with the {225/224, 441/440} temperament,
whose mean square optimal values are essentially those of the 72-et. I
give a 72-et version of the first scale below (33 intervals 44 triads);
the third and fourth are modes of the first and second, so the second is
just a mode of the inversion of the first scale. Qm(3) is not knocked off
its perch, but these are a nice suppliment.
! mecaa.scl
! [5, 11, 7, 7, 5, 7, 11, 5, 7, 7]
{225/224, 441/440} tempering of decad, 72-et version
10
!
83.33333333
266.6666667
383.3333333
500.0000000
583.3333333
700.0000000
883.3333333
966.6666667
1083.333333
2/1