Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 625206059 - Original comment: ** |
Wikispaces>TallKite **Imported revision 625221961 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-22 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-22 20:23:25 UTC</tt>.<br> | ||
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How many edos support a given pergen? In general, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime. | How many edos support a given pergen? In general, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime. | ||
Given an edo, a period, and a generator, what is the pergen? There is more than one right answer. For 12edo, if P = 12\12 and G = 1\12 (one semitone), it could be either (P8, P4/5) or (P8, P5/7). | Given an edo, a period, and a generator, what is the pergen? There is more than one right answer. For 12edo, if P = 12\12 and G = 1\12 (one semitone), it could be either (P8, P4/5) or (P8, P5/7). | ||
For N-edo, if P = p\N and G = g\N, ... | For N-edo, if P = p\N and G = g\N, ... | ||
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Therefore assume m > |b| and unreduce | Therefore assume m > |b| and unreduce | ||
(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) | (P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) | ||
Simplify by dividing by b to get (P8/m, ( | Let p = m/b and q = n/b, p and q are coprime, |p| > 1 and |q| > 1 | ||
Simplify by dividing by b to get (P8/m, (q - ap, -m) / qm) = (P8/m, (a',b')/n') | |||
Can b' be reduced by simplifying further? | |||
No, because GCD (a', b') = GCD (q - ap, -bp) = r | |||
Since a and b are coprime, GCD (-ap, -bp) = p | |||
Since GCD (p, q) = 1 | |||
b' = -m, therefore m = |b'|, and the unreduced pergen is explicitly false | b' = -m, therefore m = |b'|, and the unreduced pergen is explicitly false | ||
Therefore the original pergen is a false double | Therefore the original pergen is a false double | ||
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(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n') | (P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n') | ||
Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b | Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b | ||
|b'| = m, so the unreduced pergen is explicitly false, and the test works | |b'| = m, so the unreduced pergen is explicitly false, and the test works | ||
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How many edos support a given pergen? In general, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.<br /> | How many edos support a given pergen? In general, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.<br /> | ||
<br /> | <br /> | ||
Given an edo, a period, and a generator, what is the pergen? There is more than one right answer. For 12edo, if P = 12\12 and G = 1\12 (one semitone), it could be either (P8, P4/5) or (P8, P5/7). <br /> | Given an edo, a period, and a generator, what is the pergen? There is more than one right answer. For 12edo, if P = 12\12 and G = 1\12 (one semitone), it could be either (P8, P4/5) or (P8, P5/7).<br /> | ||
<br /> | <br /> | ||
For N-edo, if P = p\N and G = g\N, ...<br /> | For N-edo, if P = p\N and G = g\N, ...<br /> | ||
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Therefore assume m &gt; |b| and unreduce<br /> | Therefore assume m &gt; |b| and unreduce<br /> | ||
(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn)<br /> | (P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn)<br /> | ||
Simplify by dividing by b to get (P8/m, ( | Let p = m/b and q = n/b, p and q are coprime, |p| &gt; 1 and |q| &gt; 1<br /> | ||
Simplify by dividing by b to get (P8/m, (q - ap, -m) / qm) = (P8/m, (a',b')/n')<br /> | |||
Can b' be reduced by simplifying further?<br /> | |||
No, because GCD (a', b') = GCD (q - ap, -bp) = r<br /> | |||
Since a and b are coprime, GCD (-ap, -bp) = p<br /> | |||
Since GCD (p, q) = 1<br /> | |||
b' = -m, therefore m = |b'|, and the unreduced pergen is explicitly false<br /> | b' = -m, therefore m = |b'|, and the unreduced pergen is explicitly false<br /> | ||
Therefore the original pergen is a false double<br /> | Therefore the original pergen is a false double<br /> | ||
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(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')<br /> | (P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')<br /> | ||
Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b<br /> | Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b<br /> | ||
|b'| = m, so the unreduced pergen is explicitly false, and the test works<br /> | |b'| = m, so the unreduced pergen is explicitly false, and the test works<br /> | ||
<br /> | <br /> | ||