Rank 3 scale: Difference between revisions

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Pairwise-DE scales have MV3. Pairwise-DE scales that are not PWF are not trivalent; and one of the DE scales / MOS scales found by equating a pair of steps of such scales is a Multi-MOS, which is DE / MV2, but does not demonstrate Myhill's property.

PWF and pairwise-DE scales include the same number of ~~incidences~~ of steps of 2 of the 3 different step sizes, apart from abacaba.

PWF and pairwise-DE scales include the same number of instances of steps of 2 of the 3 different step sizes, apart from abacaba.

The scale abacaba is the only PWF / PDE / PMOS scale, and the only trivalent scale that is also symmetric.

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=== 3-SN scales ===

The scales a...ba...c and abacaba are [[SN scales]], which are symmetric, and can be uniquely defined by a signature.

SN scales are generated iteratively by placing an instance of a new or the existing smallest step at the top or bottom of every larger step.

SN scales include MOS scales. MOS scales are the rank-2 SN scales, or 2-SN scales.

3-SN scales are generated from MOS scales, and 4-SN scales are generated from 3-SN scales, etc.

The only SN scale that is trivalent is abacaba

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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 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.

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 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|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 ===

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'''Conjecture:''' Only the two interval classes of an SN of odd cardinality may have a variety of 5, and no SN of even cardinality has max variety > 4.

Many SN scales have max variety 4 and mean variety < 3, including all scales for which the number of ~~incidences~~ of one step size is equal to the sum of the numbers of ~~incidences~~ of the remaining 2 step sizes. It follows that such scales are of even cardinality (possess an even number of notes), and can be generated by adding to any WF scale an ~~incidence~~ of a third generator, smaller than the small step of the WF scale, above or below each step of the WF scale. A reasonably well-known example of this is MET-24, which can be generated by adding an ~~incidence~~ of a generator around 57c above or below each step of the Pythagorean chromatic scale (and then tempering). MET-24 has three sizes of 2nd and 24th, 2 sizes of 3rd, 5th, 7th, …, 23rd etc. (the parapythagorean chromatic scale), and 4 sizes of 4th, 6th, 8th, …, 22nd.

Many SN scales have max variety 4 and mean variety < 3, including all scales for which the number of instances of one step size is equal to the sum of the numbers of instances of the remaining 2 step sizes. It follows that such scales are of even cardinality (possess an even number of notes), and can be generated by adding to any WF scale an instance of a third generator, smaller than the small step of the WF scale, above or below each step of the WF scale. A reasonably well-known example of this is MET-24, which can be generated by adding an instance of a generator around 57c above or below each step of the Pythagorean chromatic scale (and then tempering). MET-24 has three sizes of 2nd and 24th, 2 sizes of 3rd, 5th, 7th, …, 23rd etc. (the parapythagorean chromatic scale), and 4 sizes of 4th, 6th, 8th, …, 22nd.

'''Theorem:''' The mean variety of scales X with cardinality ''N,'' generated by a single ~~incidence~~ of a third generator G at the top or bottom of each step of a WF scale W is equal to (3''N''-4)/(''N''-1)

'''Theorem:''' The mean variety of scales X with cardinality ''N,'' generated by a single instance of a third generator G at the top or bottom of each step of a WF scale W is equal to (3''N''-4)/(''N''-1)

'''Proof:'''

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Then the total number of specific intervals in X is (''N''/2-1)*2 + 2*3 + (''N''/2-2)*4 = 6+''N''-2+2''N''-8 = 3''N''-4, and the mean variety = (3''N''-4)/(''N''-1)

'''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 scale, have mean variety < 3.

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

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

@@ Line 64: / Line 64: @@
 Pairwise-DE scales have MV3. Pairwise-DE scales that are not PWF are not trivalent; and one of the DE scales / MOS scales found by equating a pair of steps of such scales is a Multi-MOS, which is DE / MV2, but does not demonstrate Myhill's property.
-PWF and pairwise-DE scales include the same number of incidences of steps of 2 of the 3 different step sizes, apart from abacaba.
+PWF and pairwise-DE scales include the same number of instances of steps of 2 of the 3 different step sizes, apart from abacaba.
 The scale abacaba is the only PWF / PDE / PMOS scale, and the only trivalent scale that is also symmetric.
@@ Line 74: / Line 74: @@
 === 3-SN scales ===
 The scales a...ba...c and abacaba are [[SN scales]], which are symmetric, and can be uniquely defined by a signature.
+SN scales are generated iteratively by placing an instance of a new or the existing smallest step at the top or bottom of every larger step.
 SN scales include MOS scales. MOS scales are the rank-2 SN scales, or 2-SN scales.
+-SN scales are generated from MOS scales, and 4-SN scales are generated from 3-SN scales, etc.
 The only SN scale that is trivalent is abacaba
@@ Line 84: / Line 88: @@
 === 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.
+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 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|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 ===
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 '''Conjecture:''' Only the two interval classes of an SN of odd cardinality may have a variety of 5, and no SN of even cardinality has max variety > 4.
-Many SN scales have max variety 4 and mean variety < 3, including all scales for which the number of incidences of one step size is equal to the sum of the numbers of incidences of the remaining 2 step sizes. It follows that such scales are of even cardinality (possess an even number of notes), and can be generated by adding to any WF scale an incidence of a third generator, smaller than the small step of the WF scale, above or below each step of the WF scale. A reasonably well-known example of this is MET-24, which can be generated by adding an incidence of a generator around 57c above or below each step of the Pythagorean chromatic scale (and then tempering). MET-24 has three sizes of 2nd and 24th, 2 sizes of 3rd, 5th, 7th, …, 23rd etc. (the parapythagorean chromatic scale), and 4 sizes of 4th, 6th, 8th, …, 22nd.
+Many SN scales have max variety 4 and mean variety < 3, including all scales for which the number of instances of one step size is equal to the sum of the numbers of instances of the remaining 2 step sizes. It follows that such scales are of even cardinality (possess an even number of notes), and can be generated by adding to any WF scale an instance of a third generator, smaller than the small step of the WF scale, above or below each step of the WF scale. A reasonably well-known example of this is MET-24, which can be generated by adding an instance of a generator around 57c above or below each step of the Pythagorean chromatic scale (and then tempering). MET-24 has three sizes of 2nd and 24th, 2 sizes of 3rd, 5th, 7th, …, 23rd etc. (the parapythagorean chromatic scale), and 4 sizes of 4th, 6th, 8th, …, 22nd.
-'''Theorem:''' The mean variety of scales X with cardinality ''N,'' generated by a single incidence of a third generator G at the top or bottom of each step of a WF scale W is equal to (3''N''-4)/(''N''-1)
+'''Theorem:''' The mean variety of scales X with cardinality ''N,'' generated by a single instance of a third generator G at the top or bottom of each step of a WF scale W is equal to (3''N''-4)/(''N''-1)
 '''Proof:'''
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 Then the total number of specific intervals in X is (''N''/2-1)*2 + 2*3 + (''N''/2-2)*4 = 6+''N''-2+2''N''-8 = 3''N''-4, and the mean variety = (3''N''-4)/(''N''-1)
-'''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 scale, have mean variety < 3.
+'''Conjecture:''' SN scales only of the form a…ba…c, or generated by a single instance of a third gen at the top or bottom of each step of a WF scale, have mean variety < 3.
 '''Conjecture:''' Scales of the form a...ba...ba...c have mean variety ((''N''/3-1)*(2*3+4) + 2*2) / (''N''-1) =(10''N''/3-6) / (''N''-1).