OTC 2L ns

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Omnitetrachordal scales of the form 2L+ns -- a special case?

An OTC scale of this form (example: LssssLsssssss) has been confirmed for every scale size from 3 to 53 (except 4), and probably exists for any larger scale size as well. With the exception of the 3-, 5-, and 7-note scales, these patterns are not MOS.

In every case so far studied, the strings of 's' steps come in a small group and a large group. Let x be the number of 's' steps in the small group, and y the number of 's' steps in the large group. Then:

  • 9/8 = (y-x)s
  • 4/3 = L+xs
  • 3/2 = L+ys
  • If either x or y is odd, the scale has a symmetric mode (ex. sLsLs).
  • If both x and y are odd, there are two symmetric modes (ex. sLssssLs and ssLssLss).
  • If both x and y are even, no symmetric mode exists (ex. LsLsss).

As scale size increases:

  • y/x appears to converge on a value around 0.6 .
  • For odd size scales, the generator is large (at least 4/3 = 498 cents), and appears to approach sqrt(2) = 600 cents.
  • For even size scales, the period is 1/2 octave, and the generator is small, and appears to approach 1/1 = 0 cents.

Note the patterns in values of P:

  • P(3) = P(5)+1 = P(7)+2
  • P(6) = P(8)+1 = P(10)+2 = P(12a)+3
  • P(9) = P(11)+1 = P(13)+2 = P(15)+3 = P(17)+4
  • P(12b) = P(14)+1 = P(16)+2 = P(18)+3 = P(20)+4 = P(22)+5
  • P(19) = P(21)+1 = P(23)+2 = P(25)+3 = P(27)+4 = P(29)+5
  • etc.

These groups appear to share generators and temperaments.

For some scale sizes (12, 17, 22, 29, 34, 41, 46, 53, and possibly others), more than one class of OTC scale is possible! In such cases, the two will have different tetrachordal divisions (i.e. representing 4/3 with different numbers of steps). One version will have P very near 1, and the other will have a much higher value of P. Note that these scale sizes tend to correspond to EDOs that represent 4/3 and 3/2 with relatively high accuracy.

size 2L+_s P pattern
3 2L+s 2.4424745962 L s L (MOS)
5 2L+3s 1.4424745962 Ls s Ls (MOS)
6 2L+4s 3.8849491924 Ls ss Ls
7 2L+5s 0.4424745962 Lss s Lss (MOS)
8 2L+6s 2.8849491924 Lss ss Lss
9 2L+7s 5.3274237885 Lss sss Lss
10 2L+8s 1.8849491924 Lsss ss Lsss
11 2L+9s 4.3274237885 Lsss sss Lsss
12a 2L+10s 0.8849491924 Lssss ss Lssss
12b 2L+10s 6.7698983847 Lsss ssss Lsss
13 2L+11s 3.3274237885 Lssss sss Lssss
14 2L+12s 5.7698983847 Lssss ssss Lssss
15 2L+13s 2.3274237885 Lsssss sss Lsssss
16 2L+14s 4.7698983847 Lsssss ssss Lsssss
17a 2L+15s 1.3274237885 Lssssss sss Lssssss
17b 2L+15s 7.2123729809 Lsssss sssss Lsssss
18 2L+16s 3.7698983847 Lssssss ssss Lssssss
19 2L+17s 6.2123729809 Lssssss sssss Lssssss
20 2L+18s 2.7698983847 Lsssssss ssss Lsssssss
21 2L+19s 5.2123729809 Lsssssss sssss Lsssssss
22a 2L+20s 1.7698983847 Lssssssss ssss Lssssssss
22b 2L+20s 7.6548475771 Lsssssss ssssss Lsssssss
23 2L+21s 4.2123729809 Lssssssss sssss Lssssssss
24 2L+22s 6.6548475771 Lssssssss ssssss Lssssssss
25 2L+23s 3.2123729809 Lsssssssss sssss Lsssssssss
26 2L+24s 5.6548475771 Lsssssssss ssssss Lsssssssss
27 2L+25s 2.2123729809 Lssssssssss sssss Lssssssssss
28 2L+26s 4.6548475771 Lssssssssss ssssss Lssssssssss
29a 2L+27s 1.2123729809 Lsssssssssss sssss Lsssssssssss
29b 2L+27s 7.0973221733 Lssssssssss sssssss Lssssssssss
30 2L+28s 3.6548475771 Lsssssssssss ssssss Lsssssssssss
31 2L+29s 6.0973221733 Lsssssssssss sssssss Lsssssssssss
32 2L+30s 2.6548475771 Lssssssssssss ssssss Lssssssssssss
33 2L+31s 5.0973221733 Lssssssssssss sssssss Lssssssssssss
34a 2L+32s 1.6548475771 Lsssssssssssss ssssss Lsssssssssssss
34b 2L+32s 6.5397967694 Lssssssssssss ssssssss Lssssssssssss
35 2L+33s 4.0973221733 Lsssssssssssss sssssss Lsssssssssssss
36 2L+34s 6.5397967694 Lsssssssssssss ssssssss Lsssssssssssss
37 2L+35s 3.0973221733 Lssssssssssssss sssssss Lssssssssssssss
38 2L+36s 5.5397967694 Lssssssssssssss ssssssss Lssssssssssssss
39 2L+37s 2.0973221733 Lsssssssssssssss sssssss Lsssssssssssssss
40 2L+38s 4.5397967694 Lsssssssssssssss ssssssss Lsssssssssssssss
41a 2L+39s 1.0973221733 Lssssssssssssssss sssssss Lssssssssssssssss
41b 2L+39s 6.9822713656 Lsssssssssssssss sssssssss Lsssssssssssssss
42 2L+40s 3.5397967694 Lssssssssssssssss ssssssss Lssssssssssssssss
43 2L+41s 5.9822713656 Lssssssssssssssss sssssssss Lssssssssssssssss
44 2L+42s 2.5397967694 Lsssssssssssssssss ssssssss Lsssssssssssssssss
45 2L+43s 4.9822713656 Lsssssssssssssssss sssssssss Lsssssssssssssssss
46a 2L+44s 1.5397967694 Lssssssssssssssssss ssssssss Lssssssssssssssssss
46b 2L+44s 7.4247459618 Lsssssssssssssssss ssssssssss Lsssssssssssssssss
47 2L+45s 3.9822713656 Lssssssssssssssssss sssssssss Lssssssssssssssssss
48 2L+46s 6.4247459618 Lssssssssssssssssss ssssssssss Lssssssssssssssssss
49 2L+47s 2.9822713656 Lsssssssssssssssssss sssssssss Lsssssssssssssssssss
50 2L+48s 5.4247459618 Lsssssssssssssssssss ssssssssss Lsssssssssssssssssss
51 2L+49s 1.9822713656 Lssssssssssssssssssss sssssssss Lssssssssssssssssssss
52 2L+50s 4.4247459618 Lssssssssssssssssssss ssssssssss Lssssssssssssssssssss
53a 2L+51s 0.9822713656 Lsssssssssssssssssssss sssssssss Lsssssssssssssssssssss
53b 2L+51s 6.8672205580 Lssssssssssssssssssss sssssssssss Lssssssssssssssssssss

etc.

See also