Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 627970489 - Original comment: ** |
Wikispaces>TallKite **Imported revision 627980589 - 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>2018-03-24 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-24 16:28:16 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>627980589</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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Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4. | Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4. | ||
==Pergens and EDOs | ==Pergens and EDOs== | ||
Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored. | Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored. | ||
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__**EDO-pair notation**__ | __**EDO-pair notation**__ | ||
Just as a pair of edos and a prime subgroup 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'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16 | Just as a pair of edos and a prime subgroup 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'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16. | ||
For each edo, find the nearest edomapping (also known as the patent val) for the 2.3 subgroup. | For each edo, find the nearest **edomapping** (also known as the patent val) for the 2.3 subgroup. If the edo has a "b" wart, e.g. 13b-edo, use the second-nearest edomapping. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If d = m, -m or 0, the generator is the 5th, and the pergen is simply (P8/m, P5). | ||
For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4. | For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4. | ||
If |d| ≠ m, P5 is not the generator, and we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree, | If |d| ≠ m or 0, P5 is not the generator, and we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree, let g/g' be the smallest ancestor of this ratio. The generator G maps to both g\N and g'\N'. The two edos come closest to coinciding here. | ||
For example, for 7-edo and 17-edo, m = 1 but d = 2. The smallest ancestor of 7/17 is 2/5. | For example, for 7-edo and 17-edo, m = 1 but d = 2, so G ≠ P5. The smallest ancestor of 7/17 is 2/5. G maps to both 2\7 and 5\17, which are both neutral 3rds. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths. 2\7 = 343¢ and 5\17 = 353¢, and their difference is only 1\N", where N" = LCM (N, N'). This 10¢ difference is the least difference between any 7edo note and any 17edo note, except that the two neutral 6ths differ by the same 10¢, and of course some note pairs coincide exactly, such as 0\7 and 0\17. | ||
Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, use the generator we found from the scale tree, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this new vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for. | Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found from the scale tree, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this new vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for. If the comma is (a, b, c), then a·P8 + b·P5 + c·G = 0, and G = (b-a, -b) / c. | ||
For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2). | For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2). | ||
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To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24. 11/9 also works, it yields 243/242. | To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24. 11/9 also works, it yields 243/242. | ||
If the | If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a Blackwood-like pergen. | ||
The closer two edos are in the scale tree, the simpler the pergen they make: | Here are some example edo pairs. The closer two edos are in the scale tree, the simpler the pergen they make: | ||
||~ ||~ 12-edo ||~ 13b-edo ||~ 14-edo ||~ 15-edo ||~ 16-edo ||~ 17-edo ||~ 18b-edo ||~ 19-edo ||~ 20-edo || | ||~ ||~ 12-edo ||~ 13b-edo ||~ 14-edo ||~ 15-edo ||~ 16-edo ||~ 17-edo ||~ 18b-edo ||~ 19-edo ||~ 20-edo || | ||
||~ 13b-edo ||= (P8, P5/7) ||= ||= ||= ||= ||= ||= ||= ||= || | ||~ 13b-edo ||= (P8, P5/7) ||= ||= ||= ||= ||= ||= ||= ||= || | ||
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||~ 18b-edo ||= (P8/6, P5) ||= (P8, P12/4) ||= (P8/2, P4/2) ||= (P8/3, P12/4) ||= (P8/2, P5) ||= (P8, P5/10) ||= ||= ||= || | ||~ 18b-edo ||= (P8/6, P5) ||= (P8, P12/4) ||= (P8/2, P4/2) ||= (P8/3, P12/4) ||= (P8/2, P5) ||= (P8, P5/10) ||= ||= ||= || | ||
||~ 19-edo ||= (P8, P5) ||= (P8, P12/10) ||= (P8, P4/2) ||= (P8, P12/6) ||= (P8, P12/5) ||= (P8, P11/3) ||= (P8, P4/8) ||= ||= || | ||~ 19-edo ||= (P8, P5) ||= (P8, P12/10) ||= (P8, P4/2) ||= (P8, P12/6) ||= (P8, P12/5) ||= (P8, P11/3) ||= (P8, P4/8) ||= ||= || | ||
||~ 20-edo ||= (P8/4, P5) ||= (P8, WWP4/16) ||= (P8/2, P5/4) ||= (P8/5, | ||~ 20-edo ||= (P8/4, P5) ||= (P8, WWP4/16) ||= (P8/2, P5/4) ||= (P8/5, ^1) ||= (P8/4, P5/3) ||= (P8, P11/4) ||= (P8/2, P4/8) ||= (P8, P4/8) ||= || | ||
||~ 21-edo ||= (P8/3, P5) ||= (P8, W<span style="vertical-align: super;">3</span>P4/9) ||= (P8/7, | ||~ 21-edo ||= (P8/3, P5) ||= (P8, W<span style="vertical-align: super;">3</span>P4/9) ||= (P8/7, ^1) ||= (P8/3, P4/3) ||= (P8, P5/3) ||= (P8, P11/6) ||= (P8/3, P5/2) ||= (P8, P11/3) ||= (P8, P5/12) || | ||
||~ 22-edo ||= (P8/2, P5) ||= (P8, W<span style="vertical-align: super;">3</span>P4/15) ||= (P8/2, P4/3) ||= (P8, P4/3) ||= (P8/2, P12/5) ||= (P8, P5) ||= (P8/2, P12/7) ||= (P8, P12/5) ||= (P8/2, M2/4) || | ||~ 22-edo ||= (P8/2, P5) ||= (P8, W<span style="vertical-align: super;">3</span>P4/15) ||= (P8/2, P4/3) ||= (P8, P4/3) ||= (P8/2, P12/5) ||= (P8, P5) ||= (P8/2, P12/7) ||= (P8, P12/5) ||= (P8/2, M2/4) || | ||
||~ 23-edo ||= (P8, P4/5) ||= (P8, WWP4/8) ||= (P8, P4/2) ||= (P8, P12/12) ||= (P8, P5) ||= (P8, P12/9) ||= (P8, P12/4) ||= (P8, P12/6) ||= (P8, W<span style="vertical-align: super;">5</span>P5/16) || | ||~ 23-edo ||= (P8, P4/5) ||= (P8, WWP4/8) ||= (P8, P4/2) ||= (P8, P12/12) ||= (P8, P5) ||= (P8, P12/9) ||= (P8, P12/4) ||= (P8, P12/6) ||= (P8, W<span style="vertical-align: super;">5</span>P5/16) || | ||
||~ 24-edo ||= (P8/12, | ||~ 24-edo ||= (P8/12, ^1) ||= (P8, W<span style="vertical-align: super;">6</span>P4/14) ||= (P8/2, P4/2) ||= (P8/3, P4/2) ||= (P8/8, P5) ||= (P8, P5/2) ||= (P8/6, P4/2) ||= (P8, P4/2) ||= (P8/4, P4/2) || | ||
A specific pergen can be converted to an edo pair by finding the range of 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 among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2). | A specific pergen can be converted to an edo pair by finding the range of 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 among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2). | ||
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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 | ||
repost alt-pergenlister code | |||
to do: | to do: | ||
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<!-- ws:end:WikiTextTocRule:136 --><!-- ws:start:WikiTextTocRule:137: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating tunings with an arbitrary generator">Notating tunings with an arbitrary generator</a></div> | <!-- ws:end:WikiTextTocRule:136 --><!-- ws:start:WikiTextTocRule:137: --><div style="margin-left: 2em;"><a href="#Further Discussion-Notating tunings with an arbitrary generator">Notating tunings with an arbitrary generator</a></div> | ||
<!-- ws:end:WikiTextTocRule:137 --><!-- ws:start:WikiTextTocRule:138: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div> | <!-- ws:end:WikiTextTocRule:137 --><!-- ws:start:WikiTextTocRule:138: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div> | ||
<!-- ws:end:WikiTextTocRule:138 --><!-- ws:start:WikiTextTocRule:139: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs | <!-- ws:end:WikiTextTocRule:138 --><!-- ws:start:WikiTextTocRule:139: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule:139 --><!-- ws:start:WikiTextTocRule:140: --><div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials*">Supplemental materials*</a></div> | <!-- ws:end:WikiTextTocRule:139 --><!-- ws:start:WikiTextTocRule:140: --><div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials*">Supplemental materials*</a></div> | ||
<!-- ws:end:WikiTextTocRule:140 --><!-- ws:start:WikiTextTocRule:141: --><div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-Notaion guide PDF">Notaion guide PDF</a></div> | <!-- ws:end:WikiTextTocRule:140 --><!-- ws:start:WikiTextTocRule:141: --><div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-Notaion guide PDF">Notaion guide PDF</a></div> | ||
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Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4.<br /> | Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:104:&lt;h2&gt; --><h2 id="toc22"><a name="Further Discussion-Pergens and EDOs | <!-- ws:start:WikiTextHeadingRule:104:&lt;h2&gt; --><h2 id="toc22"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:104 -->Pergens and EDOs</h2> | ||
<br /> | <br /> | ||
Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.<br /> | Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.<br /> | ||
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<u><strong>EDO-pair notation</strong></u><br /> | <u><strong>EDO-pair notation</strong></u><br /> | ||
<br /> | <br /> | ||
Just as a pair of edos and a prime subgroup 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'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16 | Just as a pair of edos and a prime subgroup 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'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16.<br /> | ||
<br /> | <br /> | ||
For each edo, find the nearest edomapping (also known as the patent val) for the 2.3 subgroup. | For each edo, find the nearest <strong>edomapping</strong> (also known as the patent val) for the 2.3 subgroup. If the edo has a &quot;b&quot; wart, e.g. 13b-edo, use the second-nearest edomapping. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If d = m, -m or 0, the generator is the 5th, and the pergen is simply (P8/m, P5).<br /> | ||
<br /> | <br /> | ||
For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.<br /> | For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.<br /> | ||
<br /> | <br /> | ||
If |d| ≠ m, P5 is not the generator, and we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree, | If |d| ≠ m or 0, P5 is not the generator, and we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree, let g/g' be the smallest ancestor of this ratio. The generator G maps to both g\N and g'\N'. The two edos come closest to coinciding here.<br /> | ||
<br /> | <br /> | ||
For example, for 7-edo and 17-edo, m = 1 but d = 2. The smallest ancestor of 7/17 is 2/5. | For example, for 7-edo and 17-edo, m = 1 but d = 2, so G ≠ P5. The smallest ancestor of 7/17 is 2/5. G maps to both 2\7 and 5\17, which are both neutral 3rds. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths. 2\7 = 343¢ and 5\17 = 353¢, and their difference is only 1\N&quot;, where N&quot; = LCM (N, N'). This 10¢ difference is the least difference between any 7edo note and any 17edo note, except that the two neutral 6ths differ by the same 10¢, and of course some note pairs coincide exactly, such as 0\7 and 0\17.<br /> | ||
<br /> | <br /> | ||
Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, use the generator we found from the scale tree, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this new vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for.<br /> | Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found from the scale tree, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this new vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for. If the comma is (a, b, c), then a·P8 + b·P5 + c·G = 0, and G = (b-a, -b) / c.<br /> | ||
<br /> | <br /> | ||
For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2).<br /> | For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2).<br /> | ||
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To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24. 11/9 also works, it yields 243/242.<br /> | To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24. 11/9 also works, it yields 243/242.<br /> | ||
<br /> | <br /> | ||
If the | If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a Blackwood-like pergen. <br /> | ||
<br /> | <br /> | ||
The closer two edos are in the scale tree, the simpler the pergen they make:<br /> | Here are some example edo pairs. The closer two edos are in the scale tree, the simpler the pergen they make:<br /> | ||
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<td style="text-align: center;">(P8/2, P5/4)<br /> | <td style="text-align: center;">(P8/2, P5/4)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">(P8/5, | <td style="text-align: center;">(P8/5, ^1)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">(P8/4, P5/3)<br /> | <td style="text-align: center;">(P8/4, P5/3)<br /> | ||
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<td style="text-align: center;">(P8, W<span style="vertical-align: super;">3</span>P4/9)<br /> | <td style="text-align: center;">(P8, W<span style="vertical-align: super;">3</span>P4/9)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">(P8/7, | <td style="text-align: center;">(P8/7, ^1)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">(P8/3, P4/3)<br /> | <td style="text-align: center;">(P8/3, P4/3)<br /> | ||
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<th>24-edo<br /> | <th>24-edo<br /> | ||
</th> | </th> | ||
<td style="text-align: center;">(P8/12, | <td style="text-align: center;">(P8/12, ^1)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">(P8, W<span style="vertical-align: super;">6</span>P4/14)<br /> | <td style="text-align: center;">(P8, W<span style="vertical-align: super;">6</span>P4/14)<br /> | ||
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A specific pergen can be converted to an edo pair by finding the range of 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 among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2).<br /> | A specific pergen can be converted to an edo pair by finding the range of 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 among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2).<br /> | ||
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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 /> | ||
repost alt-pergenlister code<br /> | |||
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to do:<br /> | to do:<br /> | ||
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make a glossary of all bolded terms?<br /> | make a glossary of all bolded terms?<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:108:&lt;h3&gt; --><h3 id="toc24"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a><!-- ws:end:WikiTextHeadingRule:108 -->Notaion guide PDF</h3> | <!-- ws:start:WikiTextHeadingRule:108:&lt;h3&gt; --><h3 id="toc24"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a><!-- ws:end:WikiTextHeadingRule:108 -->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 /> | ||
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(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 /> | ||
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The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.<br /> | The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.<br /> | ||