Omnitetrachordality

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"

All omnitetrachordal scales are of the form ABA, where A is any word spanning an approximation of 4/3 and B is any word spanning an approximation of 9/8.

Theorem: All strongly omnitetrachordal scales are of the form ABABA, where A is any word spanning an approximation of 9/8 and B is any word spanning an approximation of 32/27.

Proof: An OTC scale comprises two iterations of one word - the tetrachord, and one iteration of another - the disjunction.

Any word may be written as a product of three words: ABC (where any of A, B, or C may be empty).

We write the tetrachord as a product of three words: ABC.

For a scale to be strongly OTC, the disjunction must be a factor of the tetrachord, so the disjunction must be either A, B, or C.

If the disjunction is A, the scale has the form ABCABCA in one of its modes. Writing BC as D, we have ADADA.

If the disjunction is B, then the scale is ABCABCB in one of its modes. In another one of its modes it is BCABCBA, which is not tetrachordal, therefore the scale is not OTC.

If the disjunction is C, then the scale is ABCABCC in one of its modes. Writing AB as D, we have DCDCC, or in another mode, CDCDC.

Clearly ADADA and CDCDC are equivalent in form to ABABA and therefore all strongly OTC scales have the form ABABA. For ABABA to be tetrachordal, it is also required for A to span an approximation of 9/8 and ABA to span an approximation of 4/3. Therefore B must span an approximation of 32/27.

Any MOS Cradle Scale with parent scale ababa, or any SN scale generated from ababa, is strongly omnitetrachordal.

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 1993). However, the property is still called "omnitetrachordality" (unless someone proposes a better name and it sticks).

See also Gallery of omnitetrachordal scales.