Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626543807 - Original comment: ** |
Wikispaces>TallKite **Imported revision 626544099 - 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>2018-02-17 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-17 03:41:35 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>626544099</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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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. | ||
This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, information for playing the score. Examples: | This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, information for playing the score. Examples: | ||
15-edo: # ``=`` 240¢, ^ ``=`` 80¢ (^ = 1/3 #) | 15-edo: # ``=`` 240¢, ^ ``=`` 80¢ (^ = 1/3 #) | ||
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</span><span style="display: block; text-align: center;">P1 -- /m2 -- ``//``d3=\\A2 -- \M3 -- P4 | </span><span style="display: block; text-align: center;">P1 -- /m2 -- ``//``d3=\\A2 -- \M3 -- P4 | ||
</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb``//``=D#\\ -- E\ -- F</span> | </span><span style="display: block; text-align: center;">C -- Db/ -- Ebb``//``=D#\\ -- E\ -- F</span> | ||
Because the enharmonic found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans | Because the enharmonic found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. | ||
To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see [[pergen#Applications-Tipping%20points|tipping points]] above. Add n**·**count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period). | To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see [[pergen#Applications-Tipping%20points|tipping points]] above. Add n**·**count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period). | ||
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Double-pair notation has two enharmonics, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two enharmonics can be either A1, m2 or d2. One can choose to use whichever two enharmonics best avoid tipping. | Double-pair notation has two enharmonics, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two enharmonics can be either A1, m2 or d2. One can choose to use whichever two enharmonics best avoid tipping. | ||
An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation. | An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an E of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation. | ||
Another "tippy" temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. | |||
==Notating unsplit pergens== | ==Notating unsplit pergens== | ||
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||= breedsmic ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 || | ||= breedsmic ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 || | ||
||= 7-limit JI ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= --- || | ||= 7-limit JI ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= --- || | ||
A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors. | A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors. | ||
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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.) | ||
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A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided. | A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided. | ||
||~ MOS scale ||||~ primary example ||~ secondary examples || | ||~ MOS scale ||||~ primary example ||~ secondary examples || | ||
||~ Pentatonic ||~ ||~ ||~ || | ||~ Pentatonic ||~ ||~ ||~ || | ||
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||= 9L 1s ||= (P8, P4/2) [10] ||= quarter-4th decatonic ||< || | ||= 9L 1s ||= (P8, P4/2) [10] ||= quarter-4th decatonic ||< || | ||
The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators | The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen. | ||
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, (5/4)/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, (5/4)/2), where 5·G = 7/4. | ||
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==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, but | 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. | ||
Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset. | Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset. | ||
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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 subtracting periods and inverting. | 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. | ||
||~ edos ||~ octave split ||~ period ||~ generator(s) ||~ 5th's | ||~ 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 & 12 ||= 1 ||= 7\7 = 12\12 ||= 3\7 = 5\12 ||= 4\7 = 7\12 ||= unsplit ||= (P8, WWP5/6) || | ||
||= 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 | 3\8 = 4\12, **3\8 = 5\12** ||= 5\8 = 7\12 ||= quarter-8ve ||= || | ||
||= 9 & 12 ||= 3 ||= 3\9 = 4\12 ||= 1\9 = 1\12, 2\9 = 3\12, 4\9 = 5\12 ||= 5 | ||= 9 & 12 ||= 3 ||= 3\9 = 4\12 ||= 1\9 = 1\12, 2\9 = 3\12, **4\9 = 5\12** ||= 5\9 = 7\12 ||= third-8ve ||= || | ||
||= 10 & 12 ||= 2 ||= 5\10 = 6\12 ||= 1\10 = 1\12, 4\10 = 5\12 ||= 6 | ||= 10 & 12 ||= 2 ||= 5\10 = 6\12 ||= 1\10 = 1\12, **4\10 = 5\12** ||= 6\10 = 7\12 ||= half-8ve ||= || | ||
||= 11 & 12 ||= 1 ||= 11\11 = 12\12 ||= 1\11 = 1\12 ||= 6 | ||= 11 & 12 ||= 1 ||= 11\11 = 12\12 ||= 1\11 = 1\12 ||= 6\11 = 7\12 ||= fifth-4th ||= || | ||
||= 12 & 13 ||= 1 ||= 12\12 = 13\13 ||= 1\12 = 1\13 ||= 7 | ||= 12 & 13 ||= 1 ||= 12\12 = 13\13 ||= 1\12 = 1\13 ||= 7\12 = 8\13 ||= fifth-4th ||= || | ||
||= 12 & 13b ||= 1 ||= 12\12 = 13\13 ||= 1\12 = 1\13 ||= 7 | ||= 12 & 13b ||= 1 ||= 12\12 = 13\13 ||= 1\12 = 1\13 ||= 7\12 = 7\13 ||= seventh-5th ||= || | ||
||= 12 & 15 ||= 3 ||= 4\12 = 5\15 ||= 1\12 = 1\15, 3\12 = 4\15, 5\12 = 6\15 ||= 7 | ||= 12 & 15 ||= 3 ||= 4\12 = 5\15 ||= 1\12 = 1\15, 3\12 = 4\15, **5\12 = 6\15** ||= 7\12 = 9\15 ||= third-8ve ||= || | ||
||= 12 & 17 ||= 1 ||= 12\12 = 17\17 ||= 5\12 = 7\17 ||= 7 | ||= 12 & 17 ||= 1 ||= 12\12 = 17\17 ||= 5\12 = 7\17 ||= 7\12 = 10\17 ||= unsplit ||= || | ||
||= 15 & 17 ||= 1 ||= 15\15 = 17\17 ||= 7\15 = 8\17 ||= 9 | ||= 15 & 17 ||= 1 ||= 15\15 = 17\17 ||= 7\15 = 8\17 ||= 9\15 = 10\17 ||= third-11th ||= || | ||
||= 22 & 24 ||= 2 ||= 11\22 = 12\24 ||= 1\22= 1\24, 10\22 = 11\24 ||= 13 | ||= 22 & 24 ||= 2 ||= 11\22 = 12\24 ||= **1\22= 1\24**, 10\22 = 11\24 ||= 13\22 = 14\24 ||= 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. | 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 among others 2\7, 3\10 and 5\17. Any two of those three edos defines half-5th. | ||
==Supplemental materials*== | ==Supplemental materials*== | ||
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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 | ||
add a mapping commas section somewhere/ | |||
===Notaion guide PDF=== | ===Notaion guide PDF=== | ||
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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 /> | ||
<br /> | <br /> | ||
This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, information for playing the score. Examples:<br /> | This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, information for playing the score. Examples:<br /> | ||
<br /> | <br /> | ||
15-edo: # <!-- ws:start:WikiTextRawRule:036:``=`` -->=<!-- ws:end:WikiTextRawRule:036 --> 240¢, ^ <!-- ws:start:WikiTextRawRule:037:``=`` -->=<!-- ws:end:WikiTextRawRule:037 --> 80¢ (^ = 1/3 #)<br /> | 15-edo: # <!-- ws:start:WikiTextRawRule:036:``=`` -->=<!-- ws:end:WikiTextRawRule:036 --> 240¢, ^ <!-- ws:start:WikiTextRawRule:037:``=`` -->=<!-- ws:end:WikiTextRawRule:037 --> 80¢ (^ = 1/3 #)<br /> | ||
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</span><span style="display: block; text-align: center;">P1 -- /m2 -- <!-- ws:start:WikiTextRawRule:050:``//`` -->//<!-- ws:end:WikiTextRawRule:050 -->d3=\\A2 -- \M3 -- P4<br /> | </span><span style="display: block; text-align: center;">P1 -- /m2 -- <!-- ws:start:WikiTextRawRule:050:``//`` -->//<!-- ws:end:WikiTextRawRule:050 -->d3=\\A2 -- \M3 -- P4<br /> | ||
</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb<!-- ws:start:WikiTextRawRule:051:``//`` -->//<!-- ws:end:WikiTextRawRule:051 -->=D#\\ -- E\ -- F</span><br /> | </span><span style="display: block; text-align: center;">C -- Db/ -- Ebb<!-- ws:start:WikiTextRawRule:051:``//`` -->//<!-- ws:end:WikiTextRawRule:051 -->=D#\\ -- E\ -- F</span><br /> | ||
Because the enharmonic found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans | Because the enharmonic found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic.<br /> | ||
<br /> | <br /> | ||
To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see <a class="wiki_link" href="/pergen#Applications-Tipping%20points">tipping points</a> above. Add n<strong>·</strong>count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period).<br /> | To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see <a class="wiki_link" href="/pergen#Applications-Tipping%20points">tipping points</a> above. Add n<strong>·</strong>count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period).<br /> | ||
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Double-pair notation has two enharmonics, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two enharmonics can be either A1, m2 or d2. One can choose to use whichever two enharmonics best avoid tipping.<br /> | Double-pair notation has two enharmonics, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two enharmonics can be either A1, m2 or d2. One can choose to use whichever two enharmonics best avoid tipping.<br /> | ||
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An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.<br /> | An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an E of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.<br /> | ||
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Another &quot;tippy&quot; temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:87:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Notating unsplit pergens"></a><!-- ws:end:WikiTextHeadingRule:87 -->Notating unsplit pergens</h2> | <!-- ws:start:WikiTextHeadingRule:87:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Notating unsplit pergens"></a><!-- ws:end:WikiTextHeadingRule:87 -->Notating unsplit pergens</h2> | ||
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</table> | </table> | ||
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A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &quot;superfalse&quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors.<br /> | A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &quot;superfalse&quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors.<br /> | ||
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</table> | </table> | ||
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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 /> | ||
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A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided.<br /> | A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided.<br /> | ||
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The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators | The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.<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, (5/4)/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, (5/4)/2), where 5·G = 7/4.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:97:&lt;h2&gt; --><h2 id="toc20"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:97 -->Pergens and EDOs</h2> | <!-- ws:start:WikiTextHeadingRule:97:&lt;h2&gt; --><h2 id="toc20"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:97 -->Pergens and EDOs</h2> | ||
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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, but | 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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Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.<br /> | Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.<br /> | ||
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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 /> | ||
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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 &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. | 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 /> | ||
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<th>generator(s)<br /> | <th>generator(s)<br /> | ||
</th> | </th> | ||
<th>5th's | <th>5th's keyspans<br /> | ||
</th> | </th> | ||
<th>pergen<br /> | <th>pergen<br /> | ||
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<td style="text-align: center;">3\7 = 5\12<br /> | <td style="text-align: center;">3\7 = 5\12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">4 | <td style="text-align: center;">4\7 = 7\12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">unsplit<br /> | <td style="text-align: center;">unsplit<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(P8, WWP5/6)<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</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, <strong>3\8 = 5\12</strong><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">5 | <td style="text-align: center;">5\8 = 7\12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">quarter-8ve<br /> | <td style="text-align: center;">quarter-8ve<br /> | ||
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<td style="text-align: center;">3\9 = 4\12<br /> | <td style="text-align: center;">3\9 = 4\12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">1\9 = 1\12, 2\9 = 3\12, 4\9 = 5\12<br /> | <td style="text-align: center;">1\9 = 1\12, 2\9 = 3\12, <strong>4\9 = 5\12</strong><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">5 | <td style="text-align: center;">5\9 = 7\12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third-8ve<br /> | <td style="text-align: center;">third-8ve<br /> | ||
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<td style="text-align: center;">5\10 = 6\12<br /> | <td style="text-align: center;">5\10 = 6\12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">1\10 = 1\12, 4\10 = 5\12<br /> | <td style="text-align: center;">1\10 = 1\12, <strong>4\10 = 5\12</strong><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">6 | <td style="text-align: center;">6\10 = 7\12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">half-8ve<br /> | <td style="text-align: center;">half-8ve<br /> | ||
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<td style="text-align: center;">1\11 = 1\12<br /> | <td style="text-align: center;">1\11 = 1\12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">6 | <td style="text-align: center;">6\11 = 7\12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">fifth-4th<br /> | <td style="text-align: center;">fifth-4th<br /> | ||
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<td style="text-align: center;">1\12 = 1\13<br /> | <td style="text-align: center;">1\12 = 1\13<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">7 | <td style="text-align: center;">7\12 = 8\13<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">fifth-4th<br /> | <td style="text-align: center;">fifth-4th<br /> | ||
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<td style="text-align: center;">1\12 = 1\13<br /> | <td style="text-align: center;">1\12 = 1\13<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">7 | <td style="text-align: center;">7\12 = 7\13<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">seventh-5th<br /> | <td style="text-align: center;">seventh-5th<br /> | ||
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<td style="text-align: center;">4\12 = 5\15<br /> | <td style="text-align: center;">4\12 = 5\15<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">1\12 = 1\15, 3\12 = 4\15, 5\12 = 6\15<br /> | <td style="text-align: center;">1\12 = 1\15, 3\12 = 4\15, <strong>5\12 = 6\15</strong><br /> | ||
</td> | </td> | ||
<td style="text-align: center;">7 | <td style="text-align: center;">7\12 = 9\15<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third-8ve<br /> | <td style="text-align: center;">third-8ve<br /> | ||
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<td style="text-align: center;">5\12 = 7\17<br /> | <td style="text-align: center;">5\12 = 7\17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">7 | <td style="text-align: center;">7\12 = 10\17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">unsplit<br /> | <td style="text-align: center;">unsplit<br /> | ||
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<td style="text-align: center;">7\15 = 8\17<br /> | <td style="text-align: center;">7\15 = 8\17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">9 | <td style="text-align: center;">9\15 = 10\17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third-11th<br /> | <td style="text-align: center;">third-11th<br /> | ||
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<td style="text-align: center;">11\22 = 12\24<br /> | <td style="text-align: center;">11\22 = 12\24<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">1\22= 1\24, 10\22 = 11\24<br /> | <td style="text-align: center;"><strong>1\22= 1\24</strong>, 10\22 = 11\24<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">13 | <td style="text-align: center;">13\22 = 14\24<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">half-8ve quarter-tone<br /> | <td style="text-align: center;">half-8ve quarter-tone<br /> | ||
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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 /> | 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 among others 2\7, 3\10 and 5\17. Any two of those three edos defines half-5th.<br /> | ||
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<!-- 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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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 /> | ||
add a mapping commas section somewhere/<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:6458: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:6458 --><br /> | ||
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<!-- 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:6459: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:6459 --><br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||