Rank 3 scale: Difference between revisions

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The only SN scales that are MV3, and have mean variety < 3 are a…ba…c

It follows from the proof of [[omnitetrachordality]] that any SN scale (or any [[MOS Cradle Scales|MOS Cradle Scale]]) generated from an approximation of the Pythagorean trichord 4/3 4/3 9/8 is omnitetrachordal, and any SN scale generated from an approximation of the Pythagorean pentatonic is strongly ~~omnitetrachordal~~.

=== SN scales, OTC scales, and MOS Cradle Scales ===

It follows from the proof of [[omnitetrachordality]] that any SN scale (or any [[MOS Cradle Scales|MOS Cradle Scale]]) generated from an approximation of the Pythagorean trichord 4/3 4/3 9/8 is omnitetrachordal (OTC), and any SN scale generated from an approximation of the Pythagorean pentatonic is strongly OTC. SN scales not generated from an approximation of the Pythagorean trichord may also be OTC, including the scale (2/1, 3/2, 5/4: 225/224)[19] with step signature 10L+2M+7s (LsLsLMLsLsLsLMLsLsL), mapped to (135/128~21/20, 25/24~28/27, 64/63~50/49), for which 9/8 ~ LsL, and 32/27 ~ sLMLs. All SN scales generated from OTC scales are OTC, and all SN scales generated from strongly OTC scales are strongly OTC. Further, the episturmium morphism that generates SN scales (in which an iteration of a new or the existing smallest step is added to the top or bottom of every larger step, see [[SN scales]]) can be applied to generate larger OTC and strongly OTC scales from OTC and strongly OTC scales, or a more general morphism, in which an iteration of a new or the existing smallest step is added to the top or bottom of every instance of any larger step size. Even more broadly, any [[MOS Cradle Scales|MOS Cradle Scale]] generated from an OTC scale is OTC, and any MOS Cradle Scale generated from a strongly OTC scale is strongly OTC.

=== Theorem, Proofs and Conjectures on 3-SN scales ===

'''Theorem:''' Scales of the form a...ba...c have mean variety (3''N''-4)/(''N''-1)

@@ Line 83: / Line 83: @@
 The only SN scales that are MV3, and have mean variety < 3 are a…ba…c
-It follows from the proof of [[omnitetrachordality]] that any SN scale (or any [[MOS Cradle Scales|MOS Cradle Scale]]) generated from an approximation of the Pythagorean trichord 4/3 4/3 9/8 is omnitetrachordal, and any SN scale generated from an approximation of the Pythagorean pentatonic is strongly omnitetrachordal.
+=== SN scales, OTC scales, and MOS Cradle Scales ===
+It follows from the proof of [[omnitetrachordality]] that any SN scale (or any [[MOS Cradle Scales|MOS Cradle Scale]]) generated from an approximation of the Pythagorean trichord 4/3 4/3 9/8 is omnitetrachordal (OTC), and any SN scale generated from an approximation of the Pythagorean pentatonic is strongly OTC. SN scales not generated from an approximation of the Pythagorean trichord may also be OTC, including the scale (2/1, 3/2, 5/4: 225/224)[19] with step signature 10L+2M+7s (LsLsLMLsLsLsLMLsLsL), mapped to (135/128~21/20, 25/24~28/27, 64/63~50/49), for which 9/8 ~ LsL, and 32/27 ~ sLMLs. All SN scales generated from OTC scales are OTC, and all SN scales generated from strongly OTC scales are strongly OTC. Further, the episturmium morphism that generates SN scales (in which an iteration of a new or the existing smallest step is added to the top or bottom of every larger step, see [[SN scales]]) can be applied to generate larger OTC and strongly OTC scales from OTC and strongly OTC scales, or a more general morphism, in which an iteration of a new or the existing smallest step is added to the top or bottom of every instance of any larger step size. Even more broadly, any [[MOS Cradle Scales|MOS Cradle Scale]] generated from an OTC scale is OTC, and any MOS Cradle Scale generated from a strongly OTC scale is strongly OTC.
+=== Theorem, Proofs and Conjectures on 3-SN scales ===
 '''Theorem:''' Scales of the form a...ba...c have mean variety (3''N''-4)/(''N''-1)