Omnitetrachordality: Difference between revisions

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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|4/3]], plus a [[9/8|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.

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|4/3]], plus a [[9/8|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|5L+2s]] diatonic scale as an example. This scale has 7 notes and 7 different modes, so we should check each one.

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-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|4/3]], plus a [[9/8|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.
+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|4/3]], plus a [[9/8|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|5L+2s]] diatonic scale as an example. This scale has 7 notes and 7 different modes, so we should check each one.