Omnitetrachordality

From Xenharmonic Wiki
Jump to: navigation, search

A scale is weakly 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 (a "disjunction"). The definition can of course be generalized to intervals of quasi-equivalence other than 4/3, but the original version is with 4/3.

If, for each mode of the scale, the entire pattern of scale steps within the disjunction also occurs within the tetrachord, the scale is strongly omnitetrachordal (Paul Erlich's original definition of omnitetrachordal). For example, the scale "abc abc ac" is weakly omnitetrachordal but not strongly omnitetrachordal, because the disjunction "ac" does not occur in the tetrachords "abc."

There are other related definitions in between the above two, that all differ by the restrictions placed on the disjunction, for example:

  • That the entire disjunction must simply occur elsewhere in the scale, not within an individual tetrachord (so "aa aba aba" is allowed, since the "aa" occurs between the two tetrachords, but "abc abc ac" not allowed)
  • That only each individual step size in the disjunction must occur elsewhere in the scale (so "abc abc ac" is now allowed, but not "ab ab ac")

Below we will look at strongly omnitetrachordal scales.

Strongly Omnitetrachordal Examples

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.