Omnitetrachordality: Difference between revisions
m →Strongly Omnitetrachordal Examples: changed year of origin of the concept from 2002 and 1993 after advice from Paul |
Fixed up my theorems and proofs by changin 'any word' to 'any word spanning an approximation of... |
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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 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 | 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 and B | '''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. | '''Proof:''' An OTC scale comprises two iterations of one word - the tetrachord, and one iteration of another - the disjunction. | ||
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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. | 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 | 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 Scales|MOS Cradle Scale]] with parent scale ababa, or any [[SN scale]] generated from ababa, is strongly omnitetrachordal. | Any [[MOS Cradle Scales|MOS Cradle Scale]] with parent scale ababa, or any [[SN scale]] generated from ababa, is strongly omnitetrachordal. | ||