Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 624957075 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624962099 - 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>2018-01-16 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-16 21:35:20 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>624962099</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 | 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). | ||
G = | G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz | ||
<span style="display: block; text-align: center;">**<span style="font-size: 110%;">The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, ( | <span style="display: block; text-align: center;">**<span style="font-size: 110%;">The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| <= x</span>** | ||
</span> | </span> | ||
A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen. | A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen. | ||
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eighth-comma hedgehog: # = 157¢, ^ = 49¢, / = 52¢ (eighth-comma = 1/8 of 250/243) | eighth-comma hedgehog: # = 157¢, ^ = 49¢, / = 52¢ (eighth-comma = 1/8 of 250/243) | ||
Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are different. | Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different. | ||
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==Misc notes== | ==Misc notes== | ||
A square mapping [(x, y), (0, z)] creates the pergen (P8/x, ( | Given: | ||
If z = 1, let | 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 | If z = 1, let i = y - x, and the pergen = (P8/x, P5) | ||
A pergen (P8/m, (a,b)/n) arises from a square mapping [(m, m-am/b), (0, n/b)] | If z = -1, let i = 2x - y, and the pergen = (P8/x, P4) = (P8/x, P5) | ||
The GCD is always > 0: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3 | Therefore if |z| = 1, n = 1 | ||
Since P12 = y·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)] | |||
The GCD is always > 0: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3 | |||
To prove: inverse of unreducing is also unreducing | To prove: inverse of unreducing is also unreducing | ||
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To prove: n is always a multiple of b, and unless n = 1, n > |b| | To prove: n is always a multiple of b, and unless n = 1, n > |b| | ||
b = x/k and n = xz/k, where k = sign (z) | b = x/k and n = xz/k, where k = sign (z) · GCD (iz-y, x) | ||
n = zb | n = zb = |z|·|b| | ||
Therefore multigens like M9/3 or M3/5 never occur, because they always reduce to something simpler | |||
(P8/m, M9/3) = ??? | |||
Therefore a and b must be coprime, otherwise M/n could be simplified by GCD (a,b) | |||
To prove: (a,b)/n splits P8 into b periods | |||
M - b·P5 = (a,b) - (-b,b) = (a+b,0) = (a+b)·P8 | |||
Because n is a multiple of b, M/b is a multiple of M/n = G | |||
M/b = (n/b)·M/n = (n/b)·G | |||
(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 | |||
c·(a+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/b = (c+d,-c) + c·(n/b)·G | |||
Therefore P8 is split into b periods | |||
Therefore, since b = x and n = xz, (a,b) splits P8 into |b| periods | Therefore, since b = x and n = xz, (a,b) splits P8 into |b| periods | ||
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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? | ||
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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 | ||
Can b' be reduced by simplifying further? | |||
No, because GCD (a', b', n') = GCD (n/b - am/b, -m, mn/b) | |||
GCD (b', n') = m | |||
GCD (n/b, m) = 1 | |||
|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 | ||
(P8/3, P4/2) --> (P8/3, M6/6) = (-4,3)/6 | |||
M6/6 --> (18,-9)/18 = (2,-1)/2 = P4/2 | |||
--> (6,-3)/6 = m10/6 | |||
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 | 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 /> | ||
G = | G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz<br /> | ||
<br /> | <br /> | ||
<span style="display: block; text-align: center;"><strong><span style="font-size: 110%;">The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, ( | <span style="display: block; text-align: center;"><strong><span style="font-size: 110%;">The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| &lt;= x</span></strong><br /> | ||
</span><br /> | </span><br /> | ||
A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.<br /> | A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.<br /> | ||
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49¢, / = 52¢ (eighth-comma = 1/8 of 250/243)<br /> | 49¢, / = 52¢ (eighth-comma = 1/8 of 250/243)<br /> | ||
<br /> | <br /> | ||
Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are different.<br /> | Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.<br /> | ||
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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:3984: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:3984 --><br /> | ||
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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:3985: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:3985 --><br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:91:&lt;h2&gt; --><h2 id="toc25"><a name="x157¢, ^-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:91 -->Misc notes</h2> | <!-- ws:start:WikiTextHeadingRule:91:&lt;h2&gt; --><h2 id="toc25"><a name="x157¢, ^-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:91 -->Misc notes</h2> | ||
<br /> | <br /> | ||
A square mapping [(x, y), (0, z)] creates the pergen (P8/x, ( | Given:<br /> | ||
If z = 1, let | 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 /> | ||
If z = -1, let | If z = 1, let i = y - x, and the pergen = (P8/x, P5)<br /> | ||
A pergen (P8/m, (a,b)/n) arises from a square mapping [(m, m-am/b), (0, n/b)] | If z = -1, let i = 2x - y, and the pergen = (P8/x, P4) = (P8/x, P5)<br /> | ||
The GCD is always &gt; 0: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3 | Therefore if |z| = 1, n = 1<br /> | ||
<br /> | Since P12 = y·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 /> | |||
The GCD is always &gt; 0: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3<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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<br /> | <br /> | ||
To prove: n is always a multiple of b, and unless n = 1, n &gt; |b|<br /> | To prove: n is always a multiple of b, and unless n = 1, n &gt; |b|<br /> | ||
b = x/k and n = xz/k, where k = sign (z) | b = x/k and n = xz/k, where k = sign (z) · GCD (iz-y, x)<br /> | ||
n = zb<br /> | n = zb = |z|·|b|<br /> | ||
Therefore multigens like M9/3 or M3/5 never occur, because they always reduce to something simpler<br /> | |||
(P8/m, M9/3) = ???<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 splits P8 into b periods<br /> | |||
M - b·P5 = (a,b) - (-b,b) = (a+b,0) = (a+b)·P8<br /> | |||
Because n is a multiple of b, M/b is a multiple of M/n = 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 /> | |||
Let c and d be the bezout pair of a+b and b such that c·(a+b) + d·b = 1<br /> | |||
c·(a+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/b = (c+d,-c) + c·(n/b)·G<br /> | |||
Therefore P8 is split into b periods<br /> | |||
<br /> | <br /> | ||
Therefore, since b = x and n = xz, (a,b) splits P8 into |b| periods<br /> | Therefore, since b = x and n = xz, (a,b) splits P8 into |b| periods<br /> | ||
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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 /> | |||
<br /> | <br /> | ||
<br /> | <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 /> | ||
Can b' be reduced by simplifying further?<br /> | |||
No, because GCD (a', b', n') = GCD (n/b - am/b, -m, mn/b)<br /> | |||
GCD (b', n') = m<br /> | |||
GCD (n/b, m) = 1<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 /> | ||
(P8/3, P4/2) --&gt; (P8/3, M6/6) = (-4,3)/6<br /> | |||
M6/6 --&gt; (18,-9)/18 = (2,-1)/2 = P4/2<br /> | |||
--&gt; (6,-3)/6 = m10/6<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 /> | ||