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An (incomplete) list of omnitetrachordal (OTC) scales.



Scales with two step sizes

(not necessarily MOS or DE)
These are mostly tempered (irrational) scales. For a few patterns (usually where the generator is close to a perfect 4/3 or 3/2), a Pythagorean (3-limit JI) version with only two step sizes is possible.

In some cases, multiple OTC patterns with the same number of large and small steps exist.

Follow the links for more detailed info on each scale!

3 tones

2L+s (the simplest possible OTC scale; s=9/8, L=4/3)

4 tones

(no OTC scales possible)

5 tones

2L+3s - sLsLs (MOS)
3L+2s - LsLsL (MOS)

6 tones

2L+4s - LsssLs

7 tones

2L+5s - sLsssLs (MOS)
5L+2s - LsLLLsL (MOS)

8 tones

2L+6s - LssssLss

9 tones

2L+7s - LsssssLss

10 tones

2L+8s - LsssssLsss
3L+7s - LsssLsLsss
5L+5s - LsLsLsLsLs (MOS)
7L+3s - sLLLsLLLsL
8L+2s - LLsLLLsLLL

11 tones

2L+9s - LssssssLsss

12 tones

2L+10s - LsssLsssssss (4+4+4), sssLssssLsss (5+2+5)
5L+7s - ssLsLssLsLsL (MOS)
7L+5s - LLsLsLLsLsLs (MOS)
10L+2s - LLLsLLLLsLLL

13 tones

2L+11s - LsssssssLssss
7L+6s - sLLLssLssLLLs

14 tones

2L+12s - LssssssssLssss
5L+9s - sssLsLsssLsLsL
7L+7s - LsLsLsLsLsLsLs (MOS)
12L+2s - LLLLLLLsLLLLLs

15 tones

2L+13s - LssssssssLsssss
7L+8s - LsLLssLsLLssLss, LssLLsLssLLsLss
12L+3s - LsLLLLLsLLLLLsL

16 tones

2L+14s - LsssssssssLsssss

17 tones

2L+15s - LsssssssssLssssss (7+3+7), LssssssssssLsssss (6+5+6)
5L+12s - sLsssLssLssLsssLs (MOS)
7L+10s - sLsLsLssLssLsLsLs, sLssLLssLssLLssLs
10L+7s - LsLsLsLLsLLsLsLsL, LsLLssLLsLLssLLsL
12L+5s - LsLLLsLLsLLsLLLsL (MOS)

18 tones

2L+16s - LssssssssssLssssss
7L+11s - LLsssLsLLsssLsssLs, LsssLLsLsssLLsLsss

19 tones

2L+17s - LsssssssssssLssssss
5L+14s - sLssssLssLssLssssLs
7L+12s - sLssLsLssLssLsLssLs (MOS)
10L+9s - LsLLsssLLsLLsssLLsL
12L+7s - LsLLsLsLLsLLsLsLLsL (MOS)
14L+5s - LLLLsLLsLLLLsLLsLLs
17L+2s - LLLLLsLLLLLLLsLLLLL

20 tones

2L+18s - LsssssssssssLsssssss
7L+13s - sssLsssLssLLsssLssLL, sssLsssLsLsLsssLsLsL, sssLsssLLssLsssLLssL
12L+8s - sLLssLLsLLLssLLsLLLs, sLLssLLLsLLssLLLsLLs, sLsLsLLLsLsLsLLLsLsL

21 tones

2L+19s - LssssssssssssLsssssss
5L+16s - sLsssssLssLssLsssssLs
7L+14s - ssssLssssLsLLssssLsLL, ssssLssssLLsLssssLLsL

22 tones

2L+20s - LssssssssssssLssssssss (9+4+9), LsssssssssssssLsssssss (8+6+8)
5L+17s - ssssLsssLssssLsssLsssL (MOS)
7L+15s - LssLsLsssLssLsLsssLsss, LsssLsLssLsssLsLssLsss, LsssLLsssLsssLLsssLsss
10L+12s - LsLsLsLsLsLssLsLsLsLss
12L+10s - sLsLsLsLsLsLLsLsLsLsLL, sLLsLsLLssLLssLLsLsLLs, sLLssLLLssLLssLLLssLLs
15L+7s - sLLLssLLLsLLLssLLLsLLL
17L+5s - sLLLsLLLsLLLLsLLLsLLLL (MOS)

23 tones

2L+21s - LsssssssssssssLssssssss
7L+16s - ssssLssssLssLLssssLssLL, ssLsLsLssssLssssLsLsLss, ssssLssssLLssLssssLLssL

24 tones

2L+22s - LssssssssssssssLssssssss
5L+19s - sssssLsssLsssssLsssLsssL
7L+17s - sssLsssLssLssLsssLssLssL, sssLsssLsssLsLsssLsssLsL
10L+14s - LssLLssssLLssLLssssLLssL, sssLsLsLsLsssLsLsLsLsLsL
12L+12s - LsLsLsLsLsLsLsLsLsLsLsLs (MOS), ssLsLLssLLssLsLLssLLssLL, ssLLsLssLLssLLsLssLLssLL
14L+10s - LssLLLLssLLssLLssLLLLssL, LLsLsLsLsLsLsLLLsLsLsLsL
15L+9s - LLsssLLLsLLLsLLLsssLLLsL
17L+7s - LLLsLLsLLsLLLsLLsLLsLLLs, LLLsLsLLLsLLLsLLLsLsLLLs
19L+5s - LLLsLLLLLsLLLsLLLLLsLLLs

>24 tones

Many larger OTC L+s scales are believed to exist, but due to exponentially increasing computation time, these have not yet been studied in detail. Scales of the form 2L+ns are known to exist up to size 53, and assumed to exist for any larger size.

L/s matrix


1
2
3
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6
7
8
9
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11
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15
16
17
18
19
20
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24
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2


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3

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4

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5

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x

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6

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7

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x

x

x


x

x


x

x








8

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x




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9

x


x




x




x










10

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x




x

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11

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12

x


x

x


x

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13

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14

x


x

x


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15

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16

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17

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x


















18

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19

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20

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21

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22

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23

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24

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s
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   



Scales with 3 step sizes




Definitions and formulas


"Dual" refers to the "inverse" of a L+s scale pattern, where every L is replaced by s, and vice versa. For example, sLssL and LsLLs are duals. If a scale is OTC, its dual is often OTC as well, but not always!

"Perfect" means that values for L and s exist such that L > s and that every mode of the scale will contain a perfect (just) 3/2 or 4/3 (or both). (See also Eigenmonzo subgroup.)

In this case the value P is given, where P = L/s. For a perfect scale, P > 1. Note that if a scale "a" is perfect (Pa = L/s), its dual "b" will have the value Pb = s/L = 1/Pa, and therefore must be imperfect (if Pa > 1, then Pb < 1 ).

In some cases, P may be less than zero. I'm not yet sure what this means :)

The value of P is calculated as follows:

a = the number of L steps per 2/1
b = the number of s steps per 2/1
c = the number of L steps per 4/3
d = the number of s steps per 4/3
x = log2(4/3) = ~0.41504 octaves = ~498.045 cents

aL+bs = log2(2/1) = 1
cL+ds = x

s = (c-ax)/(bc-ad)
L = (x-ds)/c
P = L/s

Note that the same procedure could be used to calculate the L/s ratio necessary to give any other just interval, such as 5/4, 11/8, etc.

Q is calculated similarly to P, but indicates a limit of sorts -- a point on the L/s continuum beyond which the omnitetrachordality of the scale can be considered to 'break down' in some way.

Consider for example the OTC 2L+8s pajara MODMOS, LsssssLsss -- at P = L/s = 1.885, L+3s forms a just 4/3. As L/s increases, L gets larger and s smaller; at Q = L/s = 4.827, a just 4/3 is not L+3s, but L+2.5s. Past this point, L+2s will therefore be closer to a just 4/3 than L+3s:
P_Q_C_LsssssLsss.png

For some scales, Q will not exist. For others, a second Q may exist that is less than P, placing a lower bound on the L/s ratio as well as an upper one.

We can also consider a point C, where the number of steps "crosses" the just 4/3 entirely - in the example above, this corresponds to 4/3 = L+2s. Q is then halfway between P and C, i.e.
C = 2Q-P
Q = (C+P)/2
P = 2Q-C

...which leads to the curious result that, although P is undefined when calculated by the normal method (due to division by zero) for a scale such as blackwood[10] (LsLsLsLsLs), C and Q can be calculated, and thus a value of P = -1 can be found anyway, even though it seems not to be of any use :)

"L/s range": For any L+s scale pattern, the ratio L/s may range from 1 (L=s, in which case the scale is (L+s)edo ) to ~infinity (s=0, in which case the scale is (L)edo ). A note such as "full L/s range is good" simply means that the approximation of 3/2 or 4/3 is reasonable across the entire range; no other assessment of the scale's "goodness" is intended.