Rank 3 scale: Difference between revisions

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'''Conjecture:''' Only the middle interval classes of an SN of odd cardinality may have a variety of 5, and no SN of even cardinality has has 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. MET-24 has three sizes of 2nd and 24th, 2 sizes of 3rd, 5th, 7th, …, 23rd etc. (the ~~Pythagorean~~ 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 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.

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

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 '''Conjecture:''' Only the middle interval classes of an SN of odd cardinality may have a variety of 5, and no SN of even cardinality has has 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. MET-24 has three sizes of 2nd and 24th, 2 sizes of 3rd, 5th, 7th, …, 23rd etc. (the Pythagorean 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 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.
 '''Theorem:''' The mean variety of scales generated by a single incidence of a third generator at the top or bottom of each step of a WF, with cardinality ''N'', is equal to (3''N''-4)/(''N''-1)