Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626886573 - Original comment: ** |
Wikispaces>TallKite **Imported revision 626889459 - 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>2018-02-26 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-26 05:38:33 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>626889459</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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==Ratio and cents of the accidentals== | ==Ratio and cents of the accidentals== | ||
Looking at the table in the Applications section, the up symbol equals only a few ratios. 81/80, 64/63 | The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, the up symbol often equals only a few ratios. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 11-limit temperaments use either 33/32 or 729/704. These **mapping commas** are used to map higher primes to 3-limit intervals, and are essential for notation. They also determine where a ratio "lands" on a keyboard. By definition they are a P1, and the only intervals that map to P1 are these commas and combinations of them. | ||
If a single-comma temperament uses double-pair notation, neither accidentals will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in lemba, where ^1 equals 64/63 minus 81/80. | |||
Sometimes the mapping comma needs to be inverted. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. See also blackwood-like pergens below. | |||
Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + c. Since the enharmonic = 0¢, we can derive the cents of the up symbol. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. | |||
In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the enharmonic is an A1. | In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the enharmonic is an A1. | ||
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||= 560-585¢ ||= P11/3 ||= ||= ||= ||= || | ||= 560-585¢ ||= P11/3 ||= ||= ||= ||= || | ||
||= 576-591¢ ||= WWP4/5 ||= 583-593¢ ||= WWWP4/7 ||= ||= || | ||= 576-591¢ ||= WWP4/5 ||= 583-593¢ ||= WWWP4/7 ||= ||= || | ||
The total range of possible generators is fairly well covered, providing notation options. In particular, if the tuning's generator is just over 720¢, instead of calling it a 5th, one can call its inverse a quarter-12th, to avoid descending 2nds. | The total range of possible generators is fairly well covered, providing notation options. In particular, if the tuning's generator is just over 720¢, instead of calling it a 5th, one can call its inverse a quarter-12th, to avoid descending 2nds. Nevertheless, there are gaps in the table, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation. | ||
Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen. | Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen. | ||
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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'. Let g\g' be the smaller-numbered ancestor of N\N' in the scale tree. G = g\N = g'\N'. If the octave is split, 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, or stacking an alternate generator, 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'. Let g\g' be the smaller-numbered ancestor of N\N' in the scale tree. G = g\N = g'\N'. If the octave is split, 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, or stacking an alternate generator, 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 ||= | ||= 7 & 12 ||= 1 ||= 7\7 = 12\12 ||= 3\7 = 5\12 ||= 4\7 = 7\12 ||= unsplit ||= || | ||
||= 8 & 12 ||= 4 ||= 2\8 = 3\12 ||= 1\8 = 1\12, 1\8 = 2\12, | ||= 8 & 12 ||= 4 ||= 2\8 = 3\12 ||= 1\8 = 1\12, 1\8 = 2\12, | ||
3\8 = 4\12, **3\8 = 5\12** ||= 5\8 = 7\12 ||= quarter-8ve ||= || | 3\8 = 4\12, **3\8 = 5\12** ||= 5\8 = 7\12 ||= quarter-8ve ||= || | ||
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needs more screenshots, including 12-edo's pergens and a page of the pdf | needs more screenshots, including 12-edo's pergens and a page of the pdf | ||
needs pergen squares picture | needs pergen squares picture | ||
fill in the 2nd pergens column above | fill in the 2nd pergens column above -- can one edo pair imply two pergens? | ||
to do: | to do: | ||
check references to mapping commas | |||
finish proofs | finish proofs | ||
link from: ups and downs page, Kite Giedraitis page, MOS scale names page, | link from: ups and downs page, Kite Giedraitis page, MOS scale names page, | ||
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<!-- ws:start:WikiTextHeadingRule:77:&lt;h2&gt; --><h2 id="toc10"><a name="Further Discussion-Ratio and cents of the accidentals"></a><!-- ws:end:WikiTextHeadingRule:77 -->Ratio and cents of the accidentals</h2> | <!-- ws:start:WikiTextHeadingRule:77:&lt;h2&gt; --><h2 id="toc10"><a name="Further Discussion-Ratio and cents of the accidentals"></a><!-- ws:end:WikiTextHeadingRule:77 -->Ratio and cents of the accidentals</h2> | ||
<br /> | <br /> | ||
Looking at the table in the Applications section, the up symbol equals only a few ratios. 81/80, 64/63 | The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, the up symbol often equals only a few ratios. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 11-limit temperaments use either 33/32 or 729/704. These <strong>mapping commas</strong> are used to map higher primes to 3-limit intervals, and are essential for notation. They also determine where a ratio &quot;lands&quot; on a keyboard. By definition they are a P1, and the only intervals that map to P1 are these commas and combinations of them.<br /> | ||
<br /> | <br /> | ||
If a single-comma temperament uses double-pair notation, neither accidentals will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in lemba, where ^1 equals 64/63 minus 81/80.<br /> | |||
<br /> | <br /> | ||
Sometimes the mapping comma needs to be inverted. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. See also blackwood-like pergens below.<br /> | |||
<br /> | |||
Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + c. Since the enharmonic = 0¢, we can derive the cents of the up symbol. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic.<br /> | |||
<br /> | <br /> | ||
In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the enharmonic is an A1.<br /> | In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the enharmonic is an A1.<br /> | ||
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</table> | </table> | ||
The total range of possible generators is fairly well covered, providing notation options. In particular, if the tuning's generator is just over 720¢, instead of calling it a 5th, one can call its inverse a quarter-12th, to avoid descending 2nds. | The total range of possible generators is fairly well covered, providing notation options. In particular, if the tuning's generator is just over 720¢, instead of calling it a 5th, one can call its inverse a quarter-12th, to avoid descending 2nds. Nevertheless, there are gaps in the table, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation.<br /> | ||
<br /> | <br /> | ||
Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.<br /> | Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.<br /> | ||
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<td style="text-align: center;">unsplit<br /> | <td style="text-align: center;">unsplit<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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needs more screenshots, including 12-edo's pergens and a page of the pdf<br /> | needs more screenshots, including 12-edo's pergens and a page of the pdf<br /> | ||
needs pergen squares picture<br /> | needs pergen squares picture<br /> | ||
fill in the 2nd pergens column above<br /> | fill in the 2nd pergens column above -- can one edo pair imply two pergens?<br /> | ||
<br /> | <br /> | ||
to do:<br /> | to do:<br /> | ||
check references to mapping commas<br /> | |||
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 /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:7104:http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments --><a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a><!-- ws:end:WikiTextUrlRule:7104 --><br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:7105:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --><a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a><!-- ws:end:WikiTextUrlRule:7105 --><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:7106: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:7106 --><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:7107: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:7107 --><br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||