# 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.