Rank-3 scale: Difference between revisions

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

It follows from the proof of [[omnitetrachordality]] that any SN scale (or any [[MOS Cradle|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. 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 instance 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 instance 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 generated from an OTC scale is OTC, and any MOS Cradle scale generated from a strongly OTC scale is strongly OTC.

It follows from the proof of [[omnitetrachordality]] that any SN scale (or any [[MOS Cradle|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. 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 instance 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 instance of a new or the existing smallest step is added to the top or bottom of every instance of any larger step size or subset of larger stop sizes. Even more broadly, any MOS Cradle scales generated from an OTC scale is OTC, and any MOS Cradle scale generated from a strongly OTC scale is strongly OTC.

SN scales, OTC scales and MOS Cradle scales can have any rank.

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 == SN scales, OTC scales, and MOS Cradle Scales ==
-It follows from the proof of [[omnitetrachordality]] that any SN scale (or any [[MOS Cradle|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. 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 instance 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 instance 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 generated from an OTC scale is OTC, and any MOS Cradle scale generated from a strongly OTC scale is strongly OTC.
+It follows from the proof of [[omnitetrachordality]] that any SN scale (or any [[MOS Cradle|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. 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 instance 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 instance of a new or the existing smallest step is added to the top or bottom of every instance of any larger step size or subset of larger stop sizes. Even more broadly, any MOS Cradle scales generated from an OTC scale is OTC, and any MOS Cradle scale generated from a strongly OTC scale is strongly OTC.
 SN scales, OTC scales and MOS Cradle scales can have any rank.