Topic: The 18 qujus scales

18 scales

Thread (1 messages)

From: Gene Ward Smith (2004-04-05)
Subject: The 18 qujus scales

By a qujus scale I mean a Fokker block with commas a QUartertone of 
36/35, a JUbilee comma of 50/49, and a Septimal comma of 64/63. Below 
I list all 18 of them, in order of decreasing Lumma stability. On the 
line describing it, the first number is the minimum maximum Tenney 
height over all the transpositions, where the transposition chosen is 
the unique one giving the minimax height. The next four numbers, 
seperated by commas, are the number of o/utonal tetrads and 
supermajor/subminor tetrads. There are always two each of the otonal 
and utonal tetrads, but some scales have one subminor and supermajor, 
and some have two. Following this is the ratio between the largest 
and smallest scale step, then the lumma stability and finally 
propriety. The first listed scale, qujus1, is clearly the most 
regular, and is equivalent under transposition to 
parizekj1.scl, "Petr Parizek, 12-tone septimal tuning, 2002" in 
Manuel's database.


! qujus1.scl
scale 1 420 2,2,2,2 1.897095 0.538729 strictly proper
12
!
21/20
10/9
7/6
5/4
4/3
7/5
3/2
14/9
5/3
7/4
28/15
2

! qujus2.scl
scale 2 840 2,2,2,2 2.330127 0.344658 strictly proper
12
!
21/20
8/7
6/5
9/7
4/3
10/7
3/2
8/5
12/7
9/5
40/21
2

! qujus3.scl
scale 3 840 2,2,2,2 2.330127 0.344658 strictly proper
12
!
21/20
10/9
7/6
5/4
4/3
7/5
3/2
14/9
5/3
7/4
40/21
2

! qujus4.scl
scale 4 840 2,2,2,2 2.330127 0.308814 strictly proper
12
!
21/20
8/7
6/5
9/7
4/3
7/5
3/2
8/5
12/7
9/5
40/21
2

! qujus5.scl
scale 5 840 2,2,2,2 2.330127 0.308814 strictly proper
12
!
21/20
10/9
7/6
5/4
4/3
10/7
3/2
14/9
5/3
7/4
40/21
2

! qujus6.scl
scale 6 420 2,2,2,2 2.330127 0.272970 strictly proper
12
!
21/20
8/7
6/5
9/7
4/3
7/5
3/2
8/5
12/7
9/5
28/15
2

! qujus7.scl
scale 7 840 2,2,1,1 4.109804 0.266088 strictly proper
12
!
21/20
8/7
7/6
5/4
4/3
7/5
3/2
8/5
5/3
7/4
40/21
2

! qujus8.scl
scale 8 840 2,2,1,1 4.109804 0.266088 strictly proper
12
!
21/20
8/7
6/5
5/4
4/3
10/7
3/2
8/5
12/7
7/4
40/21
2

! qujus9.scl
scale 9 420 2,2,1,1 4.109804 0.27795 strictly proper
12
!
21/20
8/7
7/6
5/4
4/3
7/5
3/2
8/5
5/3
7/4
28/15
2

! qujus10.scl
scale 10 840 2,2,1,1 4.109804 0.207795 strictly proper
12
!
21/20
8/7
7/6
5/4
4/3
10/7
3/2
8/5
5/3
7/4
40/21
2

! qujus11.scl
scale 11 840 2,2,1,1 4.109804 0.207795 strictly proper
12
!
21/20
8/7
6/5
5/4
4/3
7/5
3/2
8/5
12/7
7/4
40/21
2

! qujus12.scl
scale 12 840 2,2,1,1 4.109804 0.207795 strictly proper
12
!
15/14
8/7
6/5
5/4
4/3
10/7
3/2
8/5
12/7
7/4
40/21
2

! qujus13.scl
scale 13 420 2,2,1,1 4.109804 0.194943 strictly proper
12
!
21/20
8/7
6/5
5/4
4/3
7/5
3/2
8/5
12/7
7/4
28/15
2

! qujus14.scl
scale 14 840 2,2,1,1 4.109804 0.194943 strictly proper
12
!
15/14
8/7
7/6
5/4
4/3
10/7
3/2
8/5
5/3
7/4
40/21
2

! qujus15.scl
scale 15 1568 2,2,1,1 2.801371 0.193741 impropriety 0.044238
12
!
21/20
35/32
7/6
5/4
21/16
10/7
3/2
49/32
5/3
7/4
15/8
2

! qujus16.scl
scale 16 1568 2,2,1,1 4.109804 0.193741 impropriety 0.044238
12
!
25/24
35/32
7/6
5/4
21/16
10/7
35/24
49/32
5/3
7/4
15/8
2

! qujus17.scl
scale 17 1568 2,2,1,1 4.109804 0.135448 impropriety 0.044238
12
!
15/14
35/32
7/6
5/4
21/16
10/7
3/2
49/32
5/3
7/4
15/8
2

! qujus18.scl
scale 18 1960 2,2,1,1 4.109804 0.135448 improriety 0.044238
12
!
21/20
8/7
7/6
49/40
4/3
7/5
3/2
8/5
49/30
7/4
28/15
2