Rank-3 scale: Difference between revisions

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'''Conjecture:''' No 3-SN scales have max variety > 5.

'''Conjecture:''' Only the two interval classes of a 3-SNS of odd cardinality may have a variety of 5, and no SN of even cardinality has max variety > 4.

'''Conjecture:''' Only the two interval classes of a 3-SNS of odd cardinality may have a variety of 5, and no 3-SNS 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 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.

@@ Line 144: / Line 144: @@
 '''Conjecture:''' No 3-SN scales have max variety > 5.
-'''Conjecture:''' Only the two interval classes of a 3-SNS of odd cardinality may have a variety of 5, and no SN of even cardinality has max variety > 4.
+'''Conjecture:''' Only the two interval classes of a 3-SNS of odd cardinality may have a variety of 5, and no 3-SNS 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 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.