Gallery of omnitetrachordal scales

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

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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	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	L
1		x
2			x		x			x		x		x					x					x
3		x					x					x
4		x
5		x			x		x					x		x			x		x
6		x					x
7		x	x		x		x			x		x			x		x
8		x					x					x
9		x			x					x					x
10		x					x					x		x
11		x					x
12		x			x		x			x		x
13		x					x
14		x			x		x			x
15		x					x
16		x			x		x
17		x			x		x
18		x
19		x			x
20		x
21		x
22		x
23		x
24		x
s

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Scales with 3 step sizes

• 22-tone Indian 'sruti' scale

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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.