Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626694485 - Original comment: ** |
Wikispaces>TallKite **Imported revision 626696627 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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>2018-02-21 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-21 05:37:09 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>626696627</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> | ||
| Line 786: | Line 786: | ||
See the screenshots in the next section for examples of which pergens are supported by a specific edo. | See the screenshots in the next section for examples of which pergens are supported by a specific edo. | ||
Just as a pair of edos can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo & N'-edo, m = GCD (N,N'). P = (N/m)\N = (N'/m)\N'. If the bezout pair of (N/m,N'/m) is (g,g'), G = |g'|\N= |g|\N'. Alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking sometimes creates a 2nd pergen. | Just as a pair of edos can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo & N'-edo, m = GCD (N,N'). P = (N/m)\N = (N'/m)\N'. If the bezout pair of (N/m,N'/m) is (g,g'), G = |g'|\N= |g|\N'. The bezout pair is also either ancestor of N/N' in the scale tree. Alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking sometimes creates a 2nd pergen. | ||
||~ edos ||~ octave split ||~ period ||~ generator(s) ||~ 5th's keyspans ||~ pergen ||~ 2nd pergen || | ||~ edos ||~ octave split ||~ period ||~ generator(s) ||~ 5th's keyspans ||~ pergen ||~ 2nd pergen || | ||
||= 7 & 12 ||= 1 ||= 7\7 = 12\12 ||= 3\7 = 5\12 ||= 4\7 = 7\12 ||= unsplit ||= (P8, WWP5/6) || | ||= 7 & 12 ||= 1 ||= 7\7 = 12\12 ||= 3\7 = 5\12 ||= 4\7 = 7\12 ||= unsplit ||= (P8, WWP5/6) || | ||
| Line 849: | Line 849: | ||
==Various proofs (unfinished)== | ==Various proofs (unfinished)== | ||
The interval P8/2 has a "ratio" of the square root of 2 | The interval P8/2 has a "ratio" of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the **pergen matrix** [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping. | ||
Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise both (a,b) and n could be reduced by GCD (a,b). Since -am/b is an integer, it follows that m must be a multiple of |b| as well. Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|)= |b| · r. | Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise both (a,b) and n could be reduced by GCD (a,b). Since -am/b is an integer, it follows that m must be a multiple of |b| as well. Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|)= |b| · r. | ||
| Line 862: | Line 862: | ||
If m = 1 and n = 1, we have GCD (a', b') = the "normal" split. | If m = 1 and n = 1, we have GCD (a', b') = the "normal" split. | ||
====== ====== | |||
Assume it's a false double, and there's a comma (u,v,w) that splits both P8 and (a,b) appropriately | Assume it's a false double, and there's a comma (u,v,w) that splits both P8 and (a,b) appropriately | ||
| Line 873: | Line 874: | ||
Assume GCD (x,y) = 1 | Assume GCD (x,y) = 1 | ||
To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into | To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts | ||
if r > 1, it's a true double | if r > 1, it's a true double | ||
a·P8 = (a,0) = (a,b) - (0,b) = M - b·P12 | a·P8 = (a,0) = (a,b) - (0,b) = M - b·P12 | ||
M = n·G = qrb·G | M = n·G = qrb·G | ||
a·P8 = qrb·G - b·P12 = b·(qr·G - P12) | a·P8 = qrb·G - b·P12 = b·(qr·G - P12) | ||
Let c and d be the bezout pair of a and b, with c·a + d·b = 1 | Let c and d be the bezout pair of a and b, with c·a + d·b = 1 | ||
If |b| = 1, c = | If |b| = 1, let c = 1 and d = ±a, to avoid c = 0 | ||
ca·P8 = cb·(qr·G - P12) | ca·P8 = cb·(qr·G - P12) | ||
(1 - d·b)·P8 = c·b·(qr·G - P12) | (1 - d·b)·P8 = c·b·(qr·G - P12) | ||
P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) | P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G | ||
| Line 1,017: | Line 1,013: | ||
<!-- ws:end:WikiTextTocRule:137 --><!-- ws:start:WikiTextTocRule:138: --><div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-pergenLister app">pergenLister app</a></div> | <!-- ws:end:WikiTextTocRule:137 --><!-- ws:start:WikiTextTocRule:138: --><div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-pergenLister app">pergenLister app</a></div> | ||
<!-- ws:end:WikiTextTocRule:138 --><!-- ws:start:WikiTextTocRule:139: --><div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div> | <!-- ws:end:WikiTextTocRule:138 --><!-- ws:start:WikiTextTocRule:139: --><div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div> | ||
<!-- ws:end:WikiTextTocRule:139 --><!-- ws:start:WikiTextTocRule:140: --><div style="margin-left: 6em;"><a href="# | <!-- ws:end:WikiTextTocRule:139 --><!-- ws:start:WikiTextTocRule:140: --><div style="margin-left: 6em;"><a href="#toc26"> </a></div> | ||
<!-- ws:end:WikiTextTocRule:140 --><!-- ws:start:WikiTextTocRule:141: --><div style="margin-left: 2em;"><a href="#Further Discussion-Miscellaneous Notes">Miscellaneous Notes</a></div> | <!-- ws:end:WikiTextTocRule:140 --><!-- ws:start:WikiTextTocRule:141: --><div style="margin-left: 2em;"><a href="#Further Discussion-Miscellaneous Notes">Miscellaneous Notes</a></div> | ||
<!-- ws:end:WikiTextTocRule:141 --><!-- ws:start:WikiTextTocRule:142: --></div> | <!-- ws:end:WikiTextTocRule:141 --><!-- ws:start:WikiTextTocRule:142: --></div> | ||
| Line 5,386: | Line 5,382: | ||
See the screenshots in the next section for examples of which pergens are supported by a specific edo.<br /> | See the screenshots in the next section for examples of which pergens are supported by a specific edo.<br /> | ||
<br /> | <br /> | ||
Just as a pair of edos can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo &amp; N'-edo, m = GCD (N,N'). P = (N/m)\N = (N'/m)\N'. If the bezout pair of (N/m,N'/m) is (g,g'), G = |g'|\N= |g|\N'. Alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking sometimes creates a 2nd pergen.<br /> | Just as a pair of edos can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo &amp; N'-edo, m = GCD (N,N'). P = (N/m)\N = (N'/m)\N'. If the bezout pair of (N/m,N'/m) is (g,g'), G = |g'|\N= |g|\N'. The bezout pair is also either ancestor of N/N' in the scale tree. Alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking sometimes creates a 2nd pergen.<br /> | ||
| Line 5,598: | Line 5,594: | ||
<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. It includes alternate enharmonics for many pergens.<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. It includes alternate enharmonics for many pergens.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:7017: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:7017 --><br /> | ||
(screenshot)<br /> | (screenshot)<br /> | ||
<br /> | <br /> | ||
| Line 5,608: | Line 5,604: | ||
<br /> | <br /> | ||
Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. 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. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. 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:7018:http://www.tallkite.com/misc_files/alt-pergenLister.zip --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a><!-- ws:end:WikiTextUrlRule:7018 --><br /> | ||
<br /> | <br /> | ||
Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
| Line 5,627: | Line 5,623: | ||
<!-- ws:start:WikiTextHeadingRule:107:&lt;h2&gt; --><h2 id="toc25"><a name="Further Discussion-Various proofs (unfinished)"></a><!-- ws:end:WikiTextHeadingRule:107 -->Various proofs (unfinished)</h2> | <!-- ws:start:WikiTextHeadingRule:107:&lt;h2&gt; --><h2 id="toc25"><a name="Further Discussion-Various proofs (unfinished)"></a><!-- ws:end:WikiTextHeadingRule:107 -->Various proofs (unfinished)</h2> | ||
<br /> | <br /> | ||
The interval P8/2 has a &quot;ratio&quot; of the square root of 2 | The interval P8/2 has a &quot;ratio&quot; of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the <strong>pergen matrix</strong> [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.<br /> | ||
<br /> | <br /> | ||
Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise both (a,b) and n could be reduced by GCD (a,b). Since -am/b is an integer, it follows that m must be a multiple of |b| as well. Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|)= |b| · r.<br /> | Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise both (a,b) and n could be reduced by GCD (a,b). Since -am/b is an integer, it follows that m must be a multiple of |b| as well. Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|)= |b| · r.<br /> | ||
| Line 5,640: | Line 5,636: | ||
If m = 1 and n = 1, we have GCD (a', b') = the &quot;normal&quot; split.<br /> | If m = 1 and n = 1, we have GCD (a', b') = the &quot;normal&quot; split.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:109:&lt;h6&gt; --><h6 id="toc26" | <!-- ws:start:WikiTextHeadingRule:109:&lt;h6&gt; --><h6 id="toc26"><!-- ws:end:WikiTextHeadingRule:109 --> </h6> | ||
<br /> | |||
<br /> | <br /> | ||
Assume it's a false double, and there's a comma (u,v,w) that splits both P8 and (a,b) appropriately<br /> | Assume it's a false double, and there's a comma (u,v,w) that splits both P8 and (a,b) appropriately<br /> | ||
| Line 5,651: | Line 5,648: | ||
Assume GCD (x,y) = 1<br /> | Assume GCD (x,y) = 1<br /> | ||
<br /> | <br /> | ||
To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into | To prove: if r = 1, it's a false double, and (a,b)/n splits P8 into m parts<br /> | ||
if r &gt; 1, it's a true double<br /> | if r &gt; 1, it's a true double<br /> | ||
a·P8 = (a,0) = (a,b) - (0,b) = M - b·P12<br /> | a·P8 = (a,0) = (a,b) - (0,b) = M - b·P12<br /> | ||
M = n·G = qrb·G<br /> | M = n·G = qrb·G<br /> | ||
a·P8 = qrb·G - b·P12 = b·(qr·G - P12)<br /> | a·P8 = qrb·G - b·P12 = b·(qr·G - P12)<br /> | ||
Let c and d be the bezout pair of a and b, with c·a + d·b = 1<br /> | Let c and d be the bezout pair of a and b, with c·a + d·b = 1<br /> | ||
If |b| = 1, c = | If |b| = 1, let c = 1 and d = ±a, to avoid c = 0<br /> | ||
ca·P8 = cb·(qr·G - P12)<br /> | ca·P8 = cb·(qr·G - P12)<br /> | ||
(1 - d·b)·P8 = c·b·(qr·G - P12)<br /> | (1 - d·b)·P8 = c·b·(qr·G - P12)<br /> | ||
P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) | P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G<br /> | ||
<br /> | |||
<br /> | <br /> | ||
<br /> | <br /> | ||