Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626535901 - 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- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-17 02:46:14 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>626543807</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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||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= rank-3 half-5th ||= double-pair ||= P8 ||= v``//``A2 = 60/49 ||= /1 = 64/63 ||= ^^\<span style="vertical-align: super;">4</span>dd3 || | ||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= rank-3 half-5th ||= double-pair ||= P8 ||= v``//``A2 = 60/49 ||= /1 = 64/63 ||= ^^\<span style="vertical-align: super;">4</span>dd3 || | ||
||= demeter ||= 686/675 ||= (P8, P5, vM3/3) ||= third-downmajor-3rd ||= double-pair ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 || | ||= demeter ||= 686/675 ||= (P8, P5, vM3/3) ||= third-downmajor-3rd ||= double-pair ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 || | ||
There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = ``//``d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\. (In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.) | There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = ``//``d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\. (In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.) | ||
With this standardization, the enharmonic can be derived from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)<span style="vertical-align: super;">-2</span> · (64/63)<span style="vertical-align: super;">2</span> · (19,-12)<span style="vertical-align: super;">-1</span>. This can be rewritten as 50/49 = vv1 + ``//``1 - d2 = vv``//``-d2 = -^^\\d2. The comma is negative, but the enharmonic never is, therefore E = ^^\\d2. P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler. | With this standardization, the enharmonic can be derived from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)<span style="vertical-align: super;">-2</span> · (64/63)<span style="vertical-align: super;">2</span> · (19,-12)<span style="vertical-align: super;">-1</span>. This can be rewritten as 50/49 = vv1 + ``//``1 - d2 = vv``//``-d2 = -^^\\d2. The comma is negative, but the enharmonic never is, therefore E = ^^\\d2. P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler. | ||
The 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which is tempered to 0¢. However, a rank-3 temperament has two mapping commas, and neither is forced to be 0¢. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish doesn't tip. | The 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which is tempered to 0¢. However, a rank-3 temperament has two mapping commas, and neither is forced to be 0¢. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish doesn't tip. | ||
Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip. | Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip. | ||
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The next table lists all the ranges for all multigens up to seventh-splits. You can look up your generator in the first column and find a possible multigen. Use the octave inverse if G > 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns. | The next table lists all the ranges for all multigens up to seventh-splits. You can look up your generator in the first column and find a possible multigen. Use the octave inverse if G > 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns. | ||
||||~ | ||||~ __primary choice__ ||||||||~ __secondary choices__ || | ||
||~ generator ||~ possible multigen ||~ generator ||~ multigen ||~ generator ||~ multigen || | ||~ generator ||~ possible multigen ||~ generator ||~ multigen ||~ generator ||~ multigen || | ||
||= 23-60¢ ||= M2/4 (requires P8/2) ||= ||= ||= ||= || | ||= 23-60¢ ||= M2/4 (requires P8/2) ||= ||= ||= ||= || | ||
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Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk. | Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk. | ||
||||~ | ||||~ __pergen__ ||||||~ __MOS scales of 5-12 notes__ ||~ ||~ ||~ || | ||
||= (P8, P5) ||= unsplit ||= 5 = 2L 3s ||= 7 = 5L 2s |||| 12 = 7L 5s (or 5L 7s) || || || | ||= (P8, P5) ||= unsplit ||= 5 = 2L 3s ||= 7 = 5L 2s |||| 12 = 7L 5s (or 5L 7s) || || || | ||
||||~ halves ||~ ||~ ||~ ||~ ||~ ||~ || | ||||~ halves ||~ ||~ ||~ ||~ ||~ ||~ || | ||
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||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* || | ||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* || | ||
||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 || | ||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 || | ||
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 subtracting periods and inverting. The generator is stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. | 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 subtracting periods and inverting. The generator is stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. | ||
||~ edos ||~ octave split ||~ period ||~ generator(s) ||~ 5th's keyspan ||~ pergen ||~ 2nd pergen || | ||~ edos ||~ octave split ||~ period ||~ generator(s) ||~ 5th's keyspan ||~ pergen ||~ 2nd pergen || | ||
||= 7 & 12 ||= 1 ||= 7\7 = 12\12 ||= 3\7 = 5\12 ||= 4 & 7 ||= unsplit ||= || | ||= 7 & 12 ||= 1 ||= 7\7 = 12\12 ||= 3\7 = 5\12 ||= 4 & 7 ||= 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 & 7 ||= quarter-8ve ||= || | 3\8 = 4\12, 3\8 = 5\12 ||= 5 & 7 ||= quarter-8ve ||= || | ||
||= 9 & 12 ||= 3 ||= 3\9 = 4\12 ||= 1\9 = 1\12, 2\9 = 3\12, 4\9 = 5\12 ||= 5 & 7 ||= third-8ve ||= || | ||= 9 & 12 ||= 3 ||= 3\9 = 4\12 ||= 1\9 = 1\12, 2\9 = 3\12, 4\9 = 5\12 ||= 5 & 7 ||= third-8ve ||= || | ||
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||= 12 & 15 ||= 3 ||= 4\12 = 5\15 ||= 1\12 = 1\15, 3\12 = 4\15, 5\12 = 6\15 ||= 7 & 9 ||= third-8ve ||= || | ||= 12 & 15 ||= 3 ||= 4\12 = 5\15 ||= 1\12 = 1\15, 3\12 = 4\15, 5\12 = 6\15 ||= 7 & 9 ||= third-8ve ||= || | ||
||= 12 & 17 ||= 1 ||= 12\12 = 17\17 ||= 5\12 = 7\17 ||= 7 & 10 ||= unsplit ||= || | ||= 12 & 17 ||= 1 ||= 12\12 = 17\17 ||= 5\12 = 7\17 ||= 7 & 10 ||= unsplit ||= || | ||
||= | ||= 15 & 17 ||= 1 ||= 15\15 = 17\17 ||= 7\15 = 8\17 ||= 9 & 10 ||= third-11th ||= || | ||
||= 22 & 24 ||= 2 ||= 11\22 = 12\24 ||= 1\22= 1\24, 10\22 = 11\24 ||= 13 & 14 ||= half-8ve quarter-tone ||= || | ||= 22 & 24 ||= 2 ||= 11\22 = 12\24 ||= 1\22= 1\24, 10\22 = 11\24 ||= 13 & 14 ||= half-8ve quarter-tone ||= || | ||
A specific pergen can be converted to an edo pair by looking up its generator cents in the [[pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has 2\7, 3\10 and 5\17. Any two of those three edos defines half-5th. | |||
==Supplemental materials*== | ==Supplemental materials*== | ||
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</table> | </table> | ||
There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = <!-- ws:start:WikiTextRawRule:054:``//`` -->//<!-- ws:end:WikiTextRawRule:054 -->d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\. (In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.) <br /> | There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = <!-- ws:start:WikiTextRawRule:054:``//`` -->//<!-- ws:end:WikiTextRawRule:054 -->d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\. (In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)<br /> | ||
<br /> | <br /> | ||
With this standardization, the enharmonic can be derived from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)<span style="vertical-align: super;">-2</span> · (64/63)<span style="vertical-align: super;">2</span> · (19,-12)<span style="vertical-align: super;">-1</span>. This can be rewritten as 50/49 = vv1 + <!-- ws:start:WikiTextRawRule:055:``//`` -->//<!-- ws:end:WikiTextRawRule:055 -->1 - d2 = vv<!-- ws:start:WikiTextRawRule:056:``//`` -->//<!-- ws:end:WikiTextRawRule:056 -->-d2 = -^^\\d2. The comma is negative, but the enharmonic never is, therefore E = ^^\\d2. P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.<br /> | With this standardization, the enharmonic can be derived from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)<span style="vertical-align: super;">-2</span> · (64/63)<span style="vertical-align: super;">2</span> · (19,-12)<span style="vertical-align: super;">-1</span>. This can be rewritten as 50/49 = vv1 + <!-- ws:start:WikiTextRawRule:055:``//`` -->//<!-- ws:end:WikiTextRawRule:055 -->1 - d2 = vv<!-- ws:start:WikiTextRawRule:056:``//`` -->//<!-- ws:end:WikiTextRawRule:056 -->-d2 = -^^\\d2. The comma is negative, but the enharmonic never is, therefore E = ^^\\d2. P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.<br /> | ||
<br /> | <br /> | ||
The 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which is tempered to 0¢. However, a rank-3 temperament has two mapping commas, and neither is forced to be 0¢. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish doesn't tip. <br /> | The 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which is tempered to 0¢. However, a rank-3 temperament has two mapping commas, and neither is forced to be 0¢. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish doesn't tip.<br /> | ||
<br /> | <br /> | ||
Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.<br /> | Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.<br /> | ||
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<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<th colspan="2"> | <th colspan="2"><u>primary choice</u><br /> | ||
</ | |||
</th> | </th> | ||
<th colspan=" | <th colspan="4"><u>secondary choices</u><br /> | ||
</th> | </th> | ||
</tr> | </tr> | ||
| Line 3,523: | Line 3,523: | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<th colspan="2">pergen<br /> | <th colspan="2"><u>pergen</u><br /> | ||
</th> | </th> | ||
<th colspan=" | <th colspan="3"><u>MOS scales of 5-12 notes</u><br /> | ||
</th> | </th> | ||
<th><br /> | <th><br /> | ||
| Line 4,929: | Line 4,927: | ||
</table> | </table> | ||
<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 subtracting periods and inverting. The generator is stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen.<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 subtracting periods and inverting. The generator is stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen.<br /> | ||
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<td style="text-align: center;">2\8 = 3\12<br /> | <td style="text-align: center;">2\8 = 3\12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">1\8 = 1\12, 1\8 = 2\12<br /> | <td style="text-align: center;">1\8 = 1\12, 1\8 = 2\12,<br /> | ||
3\8 = 4\12, 3\8 = 5\12<br /> | 3\8 = 4\12, 3\8 = 5\12<br /> | ||
</td> | </td> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">15 &amp; 17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">1<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">15\15 = 17\17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">7\15 = 8\17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">9 &amp; 10<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">third-11th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<br /> | <br /> | ||
A specific pergen can be converted to an edo pair by looking up its generator cents in the <a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator">arbitrary generator</a> table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has 2\7, 3\10 and 5\17. Any two of those three edos defines half-5th.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:99:&lt;h2&gt; --><h2 id="toc21"><a name="Further Discussion-Supplemental materials*"></a><!-- ws:end:WikiTextHeadingRule:99 -->Supplemental materials*</h2> | <!-- ws:start:WikiTextHeadingRule:99:&lt;h2&gt; --><h2 id="toc21"><a name="Further Discussion-Supplemental materials*"></a><!-- ws:end:WikiTextHeadingRule:99 -->Supplemental materials*</h2> | ||
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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. 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:6454: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:6454 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:103:&lt;h3&gt; --><h3 id="toc23"><a name="Further Discussion-Supplemental materials*-Pergen squares pic"></a><!-- ws:end:WikiTextHeadingRule:103 -->Pergen squares pic</h3> | <!-- ws:start:WikiTextHeadingRule:103:&lt;h3&gt; --><h3 id="toc23"><a name="Further Discussion-Supplemental materials*-Pergen squares pic"></a><!-- ws:end:WikiTextHeadingRule:103 -->Pergen squares pic</h3> | ||
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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:6455: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:6455 --><br /> | ||
<br /> | <br /> | ||
Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
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Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | ||