Rank 3 scale: Difference between revisions
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== Theorems, Proofs and Conjectures on 3-SN scales == | == 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:''' | '''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 | 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 | 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 SN scales have max variety > 5. | ||
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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. | 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 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) | '''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:''' | '''Proof:''' | ||
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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. | 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 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) | 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. | ||
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'''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:''' 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. | ||