Omnitetrachordality

A scale is omnitetrachordal if any mode of the scale (that is, any particular octave span of the infinite scale) can be expressed as two identical sequences of steps ("tetrachords") each spanning 4/3, plus a 9/8 that may or may not be divided into smaller steps. The definition can of course be generalized to intervals of quasi-equivalence other than 4/3, but the original version is with 4/3.

This definition could be difficult to understand, so take the 5L+2s diatonic scale as an example. This scale has 7 notes and 7 different modes, so we should check each one.

C D E F G A B C = (C D E F) + 9/8 + (G A B C) (both tetrachords are LLs)

D E F G A B C D = (D E F G) + 9/8 + (A B C D) (both tetrachords are LsL)

E F G A B C D E = (E F G A) + 9/8 + (B C D E) (both tetrachords are sLL)

F G A B C D E F = 9/8 + (G A B C) + (C D E F) (both tetrachords are LLs)

G A B C D E F G = 9/8 + (A B C D) + (D E F G) (both tetrachords are LsL) or alternatively (G A B C) + (C D E F) + 9/8 (both tetrachords are LLs)

A B C D E F G A = 9/8 + (B C D E) + (E F G A) (both tetrachords are sLL) or alternatively (A B C D) + (D E F G) + 9/8 (both tetrachords are LsL)

B C D E F G A B = (B C D E) + (E F G A) + 9/8 (both tetrachords are sLL)

Since each mode can be expressed as two tetrachords each spanning 4/3 and a leftover 9/8 (some in more than one way), the diatonic scale is omnitetrachordal.

If you understand MOS scales well it should be clear that any MOS of a temperament in which the period represents 2/1 and the generator represents 4/3 (including meantone, mavila, superpyth, schismatic, etc.) will be omnitetrachordal. However, these are not the only possible omnitetrachordal scales. For an example of a different kind of omnitetrachordal scale, take the MODMOS of the 2L+8s scale (in pajara for example) with the pattern LsssLsssss.

(Lsss)(Lsss)(ss)

(sssL)(ss)(sssL)

(ssLs)(ss)(ssLs)

(sLss)(ss)(sLss)

(Lsss)(ss)(Lsss)

(ss)(sssL)(sssL)

(ss)(ssLs)(ssLs)

(ss)(sLss)(sLss) OR (sssL)(sssL)(ss)

(ss)(Lsss)(Lsss) OR (ssLs)(ssLs)(ss)

(sLss)(sLss)(ss)

In this case, each 4/3 is spanned by a 5-note scale segment rather than a 4-note one, so they are more properly called "pentachords". This is why this specific MODMOS of pajara was named the "pentachordal decatonic scale" by Paul Erlich (who is believed to have originated the concept of omnitetrachordality, circa 2002). However, the property is still called "omnitetrachordality" (unless someone proposes a better name and it sticks).

See also Gallery of omnitetrachordal scales.