Rank 3 scale: Difference between revisions

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'''Proof:'''

Since there are three step sizes, a,b, and c, interval class ''N'' has variety 3

Since there are three step sizes, a, b, and c, interval class ''N'' has variety 3

Scale segments of length 1≤length≤''N''/2-1 comprise either all a’s, all a’s but single b, or all a’s but for a single c, and therefore interval classes of length 1≤length≤''N''/2-1 have variety 3. Interval classes of length ''N''/2+1≤length≤''N''-1 also have variety 3 by symmetry (given that scale segments of length ''N''/2+1≤length≤''N''-1 are the complement of scale segments of length 1≤length≤''N''/2-1.

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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)

'''Theorem:''' The mean variety of scales X 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)

'''Proof:'''

Every second step of ~~the scale~~ gives the WF scale ~~the 3-SN~~ scale is generated from. We can this scale W.

Every second step of X gives the WF scale X scale is generated from. We can this scale 'W'.

Call the small and large steps of ~~the WF scale~~ 'S' and 'L', respectively, the size of the new generator 'G', where G<S<L, and the period of the scale 'P'. There are 3 sizes of second (interval class 1), G, S-G, and L-G~~, and~~ three sizes of the largest interval class of the scale, interval class ''N''-1, i.e., the difference between P and the 3 sizes of second.

Call the small and large steps of W 'S' and 'L', respectively, the size of the new generator 'G', where G<S<L, and the period of the scale 'P'. There are 3 sizes of second (interval class 1): G, S-G, and L-G. It follows immediately that there are also three sizes of the largest interval class of the scale, interval class ''N''-1, i.e., the difference between P and the 3 sizes of second.

For a scale of cardinality ''N'', ~~The WF scale it is generated by~~ has cardinality ''N''/2, so we have ''N''/2-1 interval classes with 2 step sizes.

For a scale X of cardinality ''N'', W has cardinality ''N''/2, so we have ''N''/2-1 interval classes with 2 step sizes.

For interval class 1+2C, for 1 ≤C ≤ (''N''/2)-2, from the 2 sizes A and A+L-S of interval class 2C, may be added the steps G, S-G, or L-G, leading to the possible interval sizes A+G, A+S-G, A+L-G, A+L-S+G, A+L-S+S-G=A+L-G, and A+L-S+L-G. However, since A<A+L-S, if we have both A+S-G, and A+L-S+L-G, then, after adding G, the next step of the scale, to both, to get to an interval class of W, we have step sizes differing by 2S-2L, and W would not be WF, and so we can have only one of these, reducing our set of possible interval sizes to 4.

Then the total number of specific intervals in ~~the scale~~ 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)

Then the total number of specific intervals in E 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, have mean variety < 3.

@@ Line 89: / Line 89: @@
 '''Proof:'''
-Since there are three step sizes, a,b, and c, interval class ''N'' has variety 3
+Since there are three step sizes, a, b, and c, interval class ''N'' has variety 3
 Scale segments of length 1≤length≤''N''/2-1 comprise either all a’s, all a’s but single b, or all a’s but for a single c, and therefore interval classes of length 1≤length≤''N''/2-1 have variety 3. Interval classes of length ''N''/2+1≤length≤''N''-1 also have variety 3 by symmetry (given that scale segments of length ''N''/2+1≤length≤''N''-1 are the complement of scale segments of length 1≤length≤''N''/2-1.
@@ Line 105: / Line 105: @@
 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)
+'''Theorem:''' The mean variety of scales X 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)
 '''Proof:'''
-Every second step of the scale gives the WF scale the 3-SN scale is generated from. We can this scale W.
+Every second step of X gives the WF scale X scale is generated from. We can this scale 'W'.
-Call the small and large steps of the WF scale 'S' and 'L', respectively, the size of the new generator 'G', where G<S<L, and the period of the scale 'P'. There are 3 sizes of second (interval class 1), G, S-G, and L-G, and three sizes of the largest interval class of the scale, interval class ''N''-1, i.e., the difference between P and the 3 sizes of second.
+Call the small and large steps of W 'S' and 'L', respectively, the size of the new generator 'G', where G<S<L, and the period of the scale 'P'. There are 3 sizes of second (interval class 1): G, S-G, and L-G. It follows immediately that there are also three sizes of the largest interval class of the scale, interval class ''N''-1, i.e., the difference between P and the 3 sizes of second.
-For a scale of cardinality ''N'', The WF scale it is generated by has cardinality ''N''/2, so we have ''N''/2-1 interval classes with 2 step sizes.
+For a scale X of cardinality ''N'', W has cardinality ''N''/2, so we have ''N''/2-1 interval classes with 2 step sizes.
 For interval class 1+2C, for 1 ≤C ≤ (''N''/2)-2, from the 2 sizes A and A+L-S of interval class 2C, may be added the steps G, S-G, or L-G, leading to the possible interval sizes A+G, A+S-G, A+L-G, A+L-S+G, A+L-S+S-G=A+L-G, and A+L-S+L-G. However, since A<A+L-S, if we have both A+S-G, and A+L-S+L-G, then, after adding G, the next step of the scale, to both, to get to an interval class of W, we have step sizes differing by 2S-2L, and W would not be WF, and so we can have only one of these, reducing our set of possible interval sizes to 4.
-Then the total number of specific intervals in the scale 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)
+Then the total number of specific intervals in E 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, have mean variety < 3.