Omnitetrachordality: Difference between revisions

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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.

An [[period|octave-repeating]] [[scale]] is '''weakly omnitetrachordal''' if any [[mode]] of the scale (that is, any particular octave span of the 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"). Although the property is called ''omnitetrachordality'', it does not require exactly four notes (three steps) in the sequence. 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".

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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|contents=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.}}

{{Theorem|contents=All strongly omnitetrachordal scales are of the form ABABA, where A is any [[word]] (an abstract scale pattern, without regard to relative step sizes) spanning an approximation of 9/8 and B is any word spanning an approximation of 32/27.}}

{{Proof|contents=An OTC scale comprises two iterations of one word - the tetrachord, and one iteration of another - the disjunction.

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-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.
+An [[period|octave-repeating]] [[scale]] is '''weakly omnitetrachordal''' if any [[mode]] of the scale (that is, any particular octave span of the 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"). Although the property is called ''omnitetrachordality'', it does not require exactly four notes (three steps) in the sequence. 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".
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 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|contents=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.}}
+{{Theorem|contents=All strongly omnitetrachordal scales are of the form ABABA, where A is any [[word]] (an abstract scale pattern, without regard to relative step sizes) spanning an approximation of 9/8 and B is any word spanning an approximation of 32/27.}}
 {{Proof|contents=An OTC scale comprises two iterations of one word - the tetrachord, and one iteration of another - the disjunction.