Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 624962099 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624970849 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-17 04:20:44 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>624970849</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz | 3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz | ||
M's 3-limit monzo is (-y, x). To get alternate generators, add i periods to G, with i ranging from -x (subtracting a full octave) to +x (adding a full octave). | M's 3-limit monzo is (-y, x), or (y, -x) if z is negative. To get alternate generators, add i periods to G, with i ranging from -x (subtracting a full octave) to +x (adding a full octave). | ||
G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz | G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz | ||
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Given: | Given: | ||
A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x > 0, z ≠ 0, and |i| <= x | A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x > 0, z ≠ 0, and |i| <= x | ||
If z = 1, let i = y - x, and the pergen = (P8/x, P5) | If z = 1, let i = y - x, and the pergen = (P8/x, P5) | ||
If z = -1, let i = 2x - y, and the pergen = (P8/x, P4) = (P8/x, P5) | If z = -1, let i = 2x - y, and the pergen = (P8/x, P4) = (P8/x, P5) | ||
Therefore if |z| = 1, n = 1 | Therefore if |z| = 1, n = 1 | ||
Since P12 = | Since P5 = P12 - P8 = (y-x)·P + z·G, n = GCD (y,z) | ||
A pergen (P8/m, (a,b)/n) arises from a square mapping [(m, m-am/b), (0, n/b)] | A pergen (P8/m, (a,b)/n) arises from a square mapping [(m, m-am/b), (0, n/b)] | ||
m = x | |||
ka = iz - y | |||
kb = x | |||
kn = xz | |||
k = GCD (iz - y, x) | |||
m = kb, therefore k = m/b | |||
x = m | |||
z = kn/x = kn/kb = n/b | |||
y = iz - ka = iz - am/b | |||
P8 = mP, P12 = mP - (am/b)P + (n/b)·(G + iP) | |||
(a,b) = amP + bmP - amP + nG + inP = bmP + inP + nG | |||
(P8/m, (i·n/b - m + am/b, m) / mn/b), M/n = (i·n/m - b + a, b) / n), i = m·(b/n) | |||
To prove: inverse of unreducing is also unreducing | To prove: inverse of unreducing is also unreducing | ||
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(P8/m, (a',b')/n') unreduced is (P8/m, (n'-a'm, -b'm) / mn') = (P8/m, (mn-(n-am)m, bmm) / mmn) = (P8/m, (a,b)/n) | (P8/m, (a',b')/n') unreduced is (P8/m, (n'-a'm, -b'm) / mn') = (P8/m, (mn-(n-am)m, bmm) / mmn) = (P8/m, (a,b)/n) | ||
To prove: n is always a multiple of b, and | |||
To prove: n is always a multiple of b, and n = |b| if and only if n = 1 | |||
b = x/k and n = xz/k, where k = sign (z) · GCD (iz-y, x) | b = x/k and n = xz/k, where k = sign (z) · GCD (iz-y, x) | ||
The GCD is defined here as always positive: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3 | |||
n = zb = |z|·|b| | n = zb = |z|·|b| | ||
Therefore multigens like M9/3 or M3/5 never occur, because they always reduce to something simpler | Therefore multigens like M9/3 or M3/5 never occur, because they always reduce to something simpler | ||
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Therefore a and b must be coprime, otherwise M/n could be simplified by GCD (a,b) | Therefore a and b must be coprime, otherwise M/n could be simplified by GCD (a,b) | ||
To prove: (a,b)/n | To prove: test for explicitly false | ||
If m = |b|, is the pergen explicitly false? | |||
Because n is a multiple of b, | Does (a,b)/n split P8 into |b| periods? | ||
(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5 | |||
Because n is a multiple of b, n/b is an integer | |||
M/b = (n/b)·M/n = (n/b)·G | M/b = (n/b)·M/n = (n/b)·G | ||
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5) | (a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5) | ||
Let c and d be the bezout pair of a+b and b such that c·(a+b) + d·b = 1 | Let c and d be the bezout pair of a+b and b such that c·(a+b) + d·b = 1 | ||
Since the pergen is a double-split, m > 1, therefore |b| > 1, therefore c ≠ 0 | |||
c·(a+b)·P8 = c·b·((n/b)·G - P5) | c·(a+b)·P8 = c·b·((n/b)·G - P5) | ||
(1 - d·b)·P8 = c·b·((n/b)·G - P5) | (1 - d·b)·P8 = c·b·((n/b)·G - P5) | ||
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5) | P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5) | ||
P8/b = ( | P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G) | ||
Therefore P8 is split | Therefore P8 is split into |b| periods | ||
Therefore if m = |b|, the pergen is explicitly false | Therefore if m = |b|, the pergen is explicitly false | ||
To prove: true/false test | To prove: true/false test | ||
If GCD (m,n) = b, is the pergen a false double? | If GCD (m,n) = |b|, is the pergen a false double? | ||
If m = |b|, the pergen is explicitly false | |||
Therefore assume m > |b| and unreduce | |||
(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, M'/n') | |||
Simplify by dividing M' and n' by b to get (P8/m, (n/b - a(m/b), -m) / m(n/b)) | |||
b' = -m, therefore m = |b'|, and the unreduced pergen is explicitly false | |||
Therefore the original pergen is a false double | |||
To prove: alternate true/false test | To prove: alternate true/false test | ||
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GCD (b', n') = m | GCD (b', n') = m | ||
GCD (n/b, m) = 1 | GCD (n/b, m) = 1 | ||
GCD ( | |||
|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 | ||
Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. | Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. | ||
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3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz<br /> | 3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz<br /> | ||
<br /> | <br /> | ||
M's 3-limit monzo is (-y, x). To get alternate generators, add i periods to G, with i ranging from -x (subtracting a full octave) to +x (adding a full octave). <br /> | M's 3-limit monzo is (-y, x), or (y, -x) if z is negative. To get alternate generators, add i periods to G, with i ranging from -x (subtracting a full octave) to +x (adding a full octave). <br /> | ||
G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz<br /> | G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block.<br /> | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:4003:http://www.tallkite.com/misc_files/pergens.pdf --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><!-- ws:end:WikiTextUrlRule:4003 --><br /> | ||
<br /> | <br /> | ||
Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:4004:http://www.tallkite.com/misc_files/alt-pergensLister.zip --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergensLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergensLister.zip</a><!-- ws:end:WikiTextUrlRule:4004 --><br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
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Given:<br /> | Given:<br /> | ||
A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x &gt; 0, z ≠ 0, and |i| &lt;= x<br /> | A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x &gt; 0, z ≠ 0, and |i| &lt;= x<br /> | ||
<br /> | |||
If z = 1, let i = y - x, and the pergen = (P8/x, P5)<br /> | If z = 1, let i = y - x, and the pergen = (P8/x, P5)<br /> | ||
If z = -1, let i = 2x - y, and the pergen = (P8/x, P4) = (P8/x, P5)<br /> | If z = -1, let i = 2x - y, and the pergen = (P8/x, P4) = (P8/x, P5)<br /> | ||
Therefore if |z| = 1, n = 1<br /> | Therefore if |z| = 1, n = 1<br /> | ||
Since P12 = | Since P5 = P12 - P8 = (y-x)·P + z·G, n = GCD (y,z)<br /> | ||
A pergen (P8/m, (a,b)/n) arises from a square mapping [(m, m-am/b), (0, n/b)]<br /> | A pergen (P8/m, (a,b)/n) arises from a square mapping [(m, m-am/b), (0, n/b)]<br /> | ||
<br /> | |||
m = x<br /> | |||
ka = iz - y<br /> | |||
kb = x<br /> | |||
kn = xz<br /> | |||
k = GCD (iz - y, x)<br /> | |||
m = kb, therefore k = m/b<br /> | |||
<br /> | |||
x = m<br /> | |||
z = kn/x = kn/kb = n/b<br /> | |||
y = iz - ka = iz - am/b<br /> | |||
<br /> | |||
P8 = mP, P12 = mP - (am/b)P + (n/b)·(G + iP)<br /> | |||
(a,b) = amP + bmP - amP + nG + inP = bmP + inP + nG<br /> | |||
(P8/m, (i·n/b - m + am/b, m) / mn/b), M/n = (i·n/m - b + a, b) / n), i = m·(b/n)<br /> | |||
<br /> | |||
<br /> | <br /> | ||
To prove: inverse of unreducing is also unreducing<br /> | To prove: inverse of unreducing is also unreducing<br /> | ||
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(P8/m, (a',b')/n') unreduced is (P8/m, (n'-a'm, -b'm) / mn') = (P8/m, (mn-(n-am)m, bmm) / mmn) = (P8/m, (a,b)/n)<br /> | (P8/m, (a',b')/n') unreduced is (P8/m, (n'-a'm, -b'm) / mn') = (P8/m, (mn-(n-am)m, bmm) / mmn) = (P8/m, (a,b)/n)<br /> | ||
<br /> | <br /> | ||
To prove: n is always a multiple of b, and | <br /> | ||
To prove: n is always a multiple of b, and n = |b| if and only if n = 1<br /> | |||
b = x/k and n = xz/k, where k = sign (z) · GCD (iz-y, x)<br /> | b = x/k and n = xz/k, where k = sign (z) · GCD (iz-y, x)<br /> | ||
The GCD is defined here as always positive: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3<br /> | |||
n = zb = |z|·|b|<br /> | n = zb = |z|·|b|<br /> | ||
Therefore multigens like M9/3 or M3/5 never occur, because they always reduce to something simpler<br /> | Therefore multigens like M9/3 or M3/5 never occur, because they always reduce to something simpler<br /> | ||
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Therefore a and b must be coprime, otherwise M/n could be simplified by GCD (a,b)<br /> | Therefore a and b must be coprime, otherwise M/n could be simplified by GCD (a,b)<br /> | ||
<br /> | <br /> | ||
To prove: (a,b)/n | To prove: test for explicitly false<br /> | ||
If m = |b|, is the pergen explicitly false?<br /> | |||
Because n is a multiple of b, | Does (a,b)/n split P8 into |b| periods?<br /> | ||
(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5<br /> | |||
Because n is a multiple of b, n/b is an integer<br /> | |||
M/b = (n/b)·M/n = (n/b)·G<br /> | M/b = (n/b)·M/n = (n/b)·G<br /> | ||
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)<br /> | (a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)<br /> | ||
Let c and d be the bezout pair of a+b and b such that c·(a+b) + d·b = 1<br /> | Let c and d be the bezout pair of a+b and b such that c·(a+b) + d·b = 1<br /> | ||
Since the pergen is a double-split, m &gt; 1, therefore |b| &gt; 1, therefore c ≠ 0<br /> | |||
c·(a+b)·P8 = c·b·((n/b)·G - P5)<br /> | c·(a+b)·P8 = c·b·((n/b)·G - P5)<br /> | ||
(1 - d·b)·P8 = c·b·((n/b)·G - P5)<br /> | (1 - d·b)·P8 = c·b·((n/b)·G - P5)<br /> | ||
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5)<br /> | P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5)<br /> | ||
P8/b = ( | P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)<br /> | ||
Therefore P8 is split | Therefore P8 is split into |b| periods<br /> | ||
Therefore if m = |b|, the pergen is explicitly false<br /> | Therefore if m = |b|, the pergen is explicitly false<br /> | ||
<br /> | <br /> | ||
To prove: true/false test<br /> | To prove: true/false test<br /> | ||
If GCD (m,n) = b, is the pergen a false double?<br /> | If GCD (m,n) = |b|, is the pergen a false double?<br /> | ||
<br /> | If m = |b|, the pergen is explicitly false<br /> | ||
<br /> | Therefore assume m &gt; |b| and unreduce<br /> | ||
(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, M'/n')<br /> | |||
Simplify by dividing M' and n' by b to get (P8/m, (n/b - a(m/b), -m) / m(n/b))<br /> | |||
b' = -m, therefore m = |b'|, and the unreduced pergen is explicitly false<br /> | |||
Therefore the original pergen is a false double<br /> | |||
<br /> | <br /> | ||
To prove: alternate true/false test<br /> | To prove: alternate true/false test<br /> | ||
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GCD (b', n') = m<br /> | GCD (b', n') = m<br /> | ||
GCD (n/b, m) = 1<br /> | GCD (n/b, m) = 1<br /> | ||
GCD (<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 /> | ||
Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.<br /> | Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.<br /> | ||