OTC 2L ns
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.