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.

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.

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

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'''Conjecture:''' SN scales only of the form a…ba…c, or generated by a single incidence of a third gen at the top or bottom of each step of a WF, have mean variety < 3.

'''Conjecture:''' Scales of the form a...ba...ba...c have variety ((''N''/3-1)*(2*3+4) + 2*2) / (''N''-1) ~~= 2''N''-6+4''N''/3~~=(10''N''/3-6)/(''N''-1).

'''Conjecture:''' Scales of the form a...ba...ba...c have variety ((''N''/3-1)*(2*3+4) + 2*2) / (''N''-1) =(10''N''/3-6) / (''N''-1).

'''Conjecture:''' Scales with 2 instances of a gen added to a WF scale have variety (''N''/3-1)*2 + 4*3 + 2(''N''/3-2)*4 / (N-1) ~~= 2''N''/3-2+12+8''N''/3-16~~ = (10''N''/3-6)/(''N''-1)

'''Conjecture:''' Scales with 2 instances of a gen added to a WF scale have variety ((''N''/3-1)*2 + 4*3 + 2(''N''/3-2)*4) / (N-1) = (10''N''/3-6)/(''N''-1)

'''Conjecture:''' abacaba and aabaabaac are the only SN scales with mean variety = 3.

@@ Line 58: / Line 58: @@
 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.
+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.
 '''Theorem:''' Scales of the form a...ba...c have mean variety (3''N''-4)/(''N''-1)
@@ Line 96: / Line 96: @@
 '''Conjecture:''' SN scales only of the form a…ba…c, or generated by a single incidence of a third gen at the top or bottom of each step of a WF, have mean variety < 3.
-'''Conjecture:''' Scales of the form a...ba...ba...c have variety ((''N''/3-1)*(2*3+4) + 2*2) / (''N''-1) = 2''N''-6+4''N''/3=(10''N''/3-6)/(''N''-1).
+'''Conjecture:''' Scales of the form a...ba...ba...c have variety ((''N''/3-1)*(2*3+4) + 2*2) / (''N''-1) =(10''N''/3-6) / (''N''-1).
-'''Conjecture:''' Scales with 2 instances of a gen added to a WF scale have variety (''N''/3-1)*2 + 4*3 + 2(''N''/3-2)*4 / (N-1) = 2''N''/3-2+12+8''N''/3-16 = (10''N''/3-6)/(''N''-1)
+'''Conjecture:''' Scales with 2 instances of a gen added to a WF scale have variety ((''N''/3-1)*2 + 4*3 + 2(''N''/3-2)*4) / (N-1) = (10''N''/3-6)/(''N''-1)
 '''Conjecture:''' abacaba and aabaabaac are the only SN scales with mean variety = 3.

Rank 3 scale: Difference between revisions

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