Rank-3 scale: Difference between revisions

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== Theorems, Proofs and Conjectures on 3-SN scales ==

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

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

'''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 for a 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.

Scale segments of length 1 ≤ length ≤ ''N''/2-1 comprise either all ''a''’s, all ''a''’s but for a 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.

Finally, scale segments of length ''N''/2 contain all a’s but for one b, or all ~~a’s~~ but for one c, and so interval class ''N''/2 has variety 2.

Finally, scale segments of length ''N''/2 contain all a’s but for one ''b'', or all ''a''’s but for one ''c'', and so interval class ''N''/2 has variety 2.

The total variety of the scale is then 2+(''N''-2)*3 = 3''N''-4, and the mean variety of the scale is (3''N''-4) / (''N''-1).

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

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

'''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.

'''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.

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.

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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 instance 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).

'''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).

'''Conjecture:''' Scales with 2 instances of a generator added to a WF scale have mean 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.

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

== List of named rank-3 scales ==

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 == Theorems, Proofs and Conjectures on 3-SN scales ==
-'''Theorem:''' Scales of the form a...ba...c have mean variety (3''N''-4) / (''N''-1).
+'''Theorem:''' Scales of the form ''a...ba...c'' have mean variety (3''N''-4) / (''N''-1).
 '''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 for a 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.
+Scale segments of length 1 ≤ length ≤ ''N''/2-1 comprise either all ''a''’s, all ''a''’s but for a 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.
-Finally, scale segments of length ''N''/2 contain all a’s but for one b, or all a’s but for one c, and so interval class ''N''/2 has variety 2.
+Finally, scale segments of length ''N''/2 contain all a’s but for one ''b'', or all ''a''’s but for one ''c'', and so interval class ''N''/2 has variety 2.
 The total variety of the scale is then 2+(''N''-2)*3 = 3''N''-4, and the mean variety of the scale is (3''N''-4) / (''N''-1).
-'''Conjecture:''' No SN scales have max variety > 5.
+'''Conjecture:''' No 3-SN scales have max variety > 5.
-'''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.
+'''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.
 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.
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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 instance 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).
+'''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).
 '''Conjecture:''' Scales with 2 instances of a generator added to a WF scale have mean 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.
+'''Conjecture:''' ''abacaba'' and ''aabaabaac'' are the only SN scales with mean variety = 3.
 == List of named rank-3 scales ==