Omnitetrachordality: Difference between revisions

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* 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")

* 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 and B are any words.

'''Theorem:''' All strongly omnitetrachordal scales are of the form ABABA, where A and B are any words:

'''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, where A and B are any words.

Any [[MOS Cradle Scales|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.

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 * 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")
+* 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 and B are any words.
+'''Theorem:''' All strongly omnitetrachordal scales are of the form ABABA, where A and B are any words:
+'''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, where A and B are any words.
+Any [[MOS Cradle Scales|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.