Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626995149 - Original comment: ** |
Wikispaces>TallKite **Imported revision 627091177 - 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- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-01 18:52:12 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>627091177</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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(a',b') = [k',s'] = (-11k'+19s', 7k'-12s') | (a',b') = [k',s'] = (-11k'+19s', 7k'-12s') | ||
a'·b - a·b' works out to be k·s' - k'·s, and we have GCD ((k·s' - k'·s)·m/b, b'·n/b) | a'·b - a·b' works out to be k·s' - k'·s, and we have GCD ((k·s' - k'·s)·m/b, b'·n/b) | ||
If s is a multiple of n (E is an A1) and s' is a multiple of n, let s = x·n and s' = y·n | If s is a multiple of n (happens when E is an A1) and s' is a multiple of n, let s = x·n and s' = y·n | ||
GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b') | GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b') | ||
Thus every such interval is split, e.g. half-5th splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. | Thus every such interval is split, e.g. half-5th splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones. | ||
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Assume | Assume the pergen is a false double, and there's a comma C = (u,v,w) that splits both P8 and (a,b) appropriately. Can we prove r = 1? | ||
2.3.7 and (22,-5,-5) = P8/5 | 2.3.7 and (22,-5,-5) = P8/5 | ||
GCD (u,v,w) = 1 | GCD (u,v,w) = 1 | ||
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finish proofs<br /> | finish proofs<br /> | ||
link from: ups and downs page, Kite Giedraitis page, MOS scale names page,<br /> | link from: ups and downs page, Kite Giedraitis page, MOS scale names page,<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:101:&lt;h3&gt; --><h3 id="toc22"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a><!-- ws:end:WikiTextHeadingRule:101 -->Notaion guide PDF</h3> | <!-- ws:start:WikiTextHeadingRule:101:&lt;h3&gt; --><h3 id="toc22"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a><!-- ws:end:WikiTextHeadingRule:101 -->Notaion guide PDF</h3> | ||
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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. 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:7113: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:7113 --><br /> | ||
(screenshot)<br /> | (screenshot)<br /> | ||
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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:7114: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:7114 --><br /> | ||
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Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. Screenshots of the first 38 pergens:<br /> | Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. Screenshots of the first 38 pergens:<br /> | ||
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(a',b') = [k',s'] = (-11k'+19s', 7k'-12s')<br /> | (a',b') = [k',s'] = (-11k'+19s', 7k'-12s')<br /> | ||
a'·b - a·b' works out to be k·s' - k'·s, and we have GCD ((k·s' - k'·s)·m/b, b'·n/b)<br /> | a'·b - a·b' works out to be k·s' - k'·s, and we have GCD ((k·s' - k'·s)·m/b, b'·n/b)<br /> | ||
If s is a multiple of n (E is an A1) and s' is a multiple of n, let s = x·n and s' = y·n<br /> | If s is a multiple of n (happens when E is an A1) and s' is a multiple of n, let s = x·n and s' = y·n<br /> | ||
GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')<br /> | GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')<br /> | ||
Thus every such interval is split, e.g. half-5th splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones.<br /> | Thus every such interval is split, e.g. half-5th splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones.<br /> | ||
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Assume | Assume the pergen is a false double, and there's a comma C = (u,v,w) that splits both P8 and (a,b) appropriately. Can we prove r = 1?<br /> | ||
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2.3.7 and (22,-5,-5) = P8/5<br /> | 2.3.7 and (22,-5,-5) = P8/5<br /> | ||
GCD (u,v,w) = 1<br /> | GCD (u,v,w) = 1<br /> | ||