Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 627189919 - 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- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-20 13:25:08 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>627818967</tt>.<br> | ||
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When there is more than one enharmonic, the first one can be combined with the 2nd to make an equivalent 2nd enharmonic. | When there is more than one enharmonic, the first one can be combined with the 2nd to make an equivalent 2nd enharmonic. | ||
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.) | 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. | 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. | ||
Quadruple-pair notation is needed for some true triples like (P8/2, P5/2, vM3/2). A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split. | Quadruple-pair notation is needed for some true triples like (P8/2, P5/2, vM3/2). A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split. | ||
Some examples of 7-limit rank-3 temperaments: | Some examples of 7-limit rank-3 temperaments: | ||
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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 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 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 vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. 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. | 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 vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. 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. | ||
Demeter is unusual in that its gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals. | Demeter is unusual in that its gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals. | ||
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There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from the 2nd generator to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For demeter, any combination of vm3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because any 3-limit comma can be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the **DOL** (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred. | There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from the 2nd generator to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For demeter, any combination of vm3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because any 3-limit comma can be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the **DOL** (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred. | ||
If ^1 = 81/80, possible half-split gens2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's. | If ^1 = 81/80, possible half-split gens2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's. | ||
All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens: | All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens: | ||
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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'. Let g/g' be the smaller-numbered ancestor of N/N' in the scale tree. G = g\N = g'\N'. If the octave is split, alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking, or stacking an alternate generator, sometimes creates a 2nd pergen. | Just as a pair of edos 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'). P = (N/m)\N = (N'/m)\N'. Let g/g' be the smaller-numbered ancestor of N/N' in the scale tree. G = g\N = g'\N'. If the octave is split, alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking, or stacking an alternate generator, sometimes creates a 2nd pergen. | ||
||~ edos ||~ octave split ||~ period ||~ generator(s) ||~ 5th's keyspans ||~ pergen ||~ 2nd pergen || | ||~ edos ||~ octave split ||~ period ||~ generator(s) ||~ 5th's keyspans ||~ pergen ||~ 2nd pergen || | ||
||= 7 & 12 ||= 1 ||= 7\7 = 12\12 ||= 3\7 = 5\12 ||= 4\7 = 7\12 ||= unsplit ||= || | ||= 7 & 12 ||= 1 ||= 7\7 = 12\12 ||= 3\7 = 5\12 ||= 4\7 = 7\12 ||= unsplit ||= || | ||
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||= 12 & 17 ||= 1 ||= 12\12 = 17\17 ||= 5\12 = 7\17 ||= 7\12 = 10\17 ||= unsplit ||= || | ||= 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 = 10\17 ||= third-11th ||= || | ||= 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 = 14\24 ||= half-8ve quarter-tone ||= || | ||= 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 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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When there is more than one enharmonic, the first one can be combined with the 2nd to make an equivalent 2nd enharmonic.<br /> | When there is more than one enharmonic, the first one can be combined with the 2nd to make an equivalent 2nd enharmonic.<br /> | ||
<br /> | <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.) <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.)<br /> | ||
<br /> | <br /> | ||
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 /> | 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 /> | ||
<br /> | <br /> | ||
Quadruple-pair notation is needed for some true triples like (P8/2, P5/2, vM3/2). A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split. <br /> | Quadruple-pair notation is needed for some true triples like (P8/2, P5/2, vM3/2). A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.<br /> | ||
<br /> | <br /> | ||
Some examples of 7-limit rank-3 temperaments:<br /> | Some examples of 7-limit rank-3 temperaments:<br /> | ||
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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 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 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 vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. 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. <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 vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. 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.<br /> | ||
<br /> | <br /> | ||
Demeter is unusual in that its gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.<br /> | Demeter is unusual in that its gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.<br /> | ||
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There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from the 2nd generator to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For demeter, any combination of vm3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because any 3-limit comma can be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the <strong>DOL</strong> (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred.<br /> | There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from the 2nd generator to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For demeter, any combination of vm3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because any 3-limit comma can be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the <strong>DOL</strong> (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred.<br /> | ||
<br /> | <br /> | ||
If ^1 = 81/80, possible half-split gens2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's. <br /> | If ^1 = 81/80, possible half-split gens2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.<br /> | ||
<br /> | <br /> | ||
All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens:<br /> | All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens:<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 /> | ||
<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'. Let g/g' be the smaller-numbered ancestor of N/N' in the scale tree. G = g\N = g'\N'. If the octave is split, alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking, or stacking an alternate generator, sometimes creates a 2nd pergen.<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'). P = (N/m)\N = (N'/m)\N'. Let g/g' be the smaller-numbered ancestor of N/N' in the scale tree. G = g\N = g'\N'. If the octave is split, alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking, or stacking an alternate generator, sometimes creates a 2nd pergen.<br /> | ||
| Line 6,712: | Line 6,712: | ||
<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;"><strong>1\22= 1\24</strong>, 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\22 = 14\24<br /> | <td style="text-align: center;">13\22 = 14\24<br /> | ||