Kite's thoughts on pergens: Difference between revisions

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(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half.

The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).

The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the **color depth** is GCD (c,d...).

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[xenharmonic/Kite's color notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yellow, (a,b,-1) = green, (a,b,0,1) = blue, (a,b,0,-1) = red, (a,b,0,0,1) = jade, (a,b,0,0,-1) = amber, and (a,b) = white. Examples: 5/4 = y3 = yellow 3rd, 7/5 = bg5 = blue-green 5th or bluish 5th, etc.

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A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, ~~and~~ the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with either microtonal accidentals (P8, P5, ^1, /1,...) or colors (P8, P5, g1, r1,...).

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with either microtonal accidentals (P8, P5, ^1, /1,...) or colors (P8, P5, g1, r1,...).

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==Finding an example temperament==

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, the quadruple red temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, the quadruple red temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.

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==Notating unsplit pergens==

An unsplit pergen doesn't __require__ ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a comma that maps to P1, such as 81/80 or 64/63. For single-comma temperaments, the pergen is unsplit if and only if the vanishing comma's monzo has a final exponent of ±1.

An unsplit pergen doesn't __require__ ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a comma that maps to P1, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (the monzo has a final exponent of ±1).

The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate.

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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.

==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.

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Given an edo, a period, and a generator, what is the pergen? There is usually more than one right answer. For 10edo with P = 5\10 and G = 2\10, it could be either (P8/2, P4/2) or (P8/2, P5/3). Every coprime period/generator pair results in a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 11-edo and 23-edo could also be considered ambiguous.

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 6-edo, 11-edo and 23-edo could also be considered ambiguous.

||||~ pergen ||~ supporting edos (12-31 only) ||

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See the screenshots in the next section for examples of which pergens are supported by a specific edo.

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.

__**EDO-pair notation**__

||~ 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 ~~||=~~ ||

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, i.e. a quarter-8ve.

For each edo, find the nearest edomapping (also known as the patent val) for the 2.3 subgroup. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If |d| = m, 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, and 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.

Sometimes the second-nearest edomapping is preferred, more on this later.

If |d| ≠ m, we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree, find the two ancestors and the two descendants of this ratio. Choose one of the four (more on this below), and let this new ratio be g/g'. 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 ancestors and descendents of 7/17 are 2/5, 5/12, 9/22 and 12/29. Choose the smallest for now, 2/5. The generator maps to both 2\7 and 5\17. 2\7 is 343¢ and 5\17 is 353¢, both neutral 3rds. Their difference is ~10¢ = 1\119. (119 = LCM (7, 17). This is the smallest difference possible between 7edo's notes and 17edo's notes, except of course for 0\7 and 0\17. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths.

Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, we extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found earlier, 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 vector with either edomapping is zero. Treating this new vector as a monzo, 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.

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 neutral 3rds with a 5th. The pergen is obviously (P8, P5/2).

To verify the validity of this approach, one can find a specific 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 = 71¢. A smaller comma can be found with a ratio more in the center of the range, such as 11/9, which yeilds 242/243. Because the direction of the cross product vector depends on the order of the two vectors multiplied, the sign of the comma is arbitrary, and the comma may be a descending interval. In that case, invert the comma to make 243/242. Using unreduced edomappings gives the same result. If the basis is (2/1, 3/1, 5/1), we have (7, 11, 16) x (17, 27, 39) = (-3, -1, 2) = 25/24.

If the octave is split,

//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) ||~ 4th's keyspans ||~ 5th's keyspans ||~ pergen ||

||= 7 & 12 ||= 1 ||= 7\7 = 12\12 ||= 3\7 = 5\12 ||= 3\7 = 5\12 ||= 4\7 = 7\12 ||= unsplit ||

||= 8 & 12 ||= 4 ||= 2\8 = 3\12 ||= 1\8 = 1\12, 1\8 = 2\12,

3\8 = 4\12, **3\8 = 5\12** ||= 5\8 = 7\12 ||= quarter-8ve ~~||=~~ ||

3\8 = 4\12, **3\8 = 5\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 = 7\12 ||= third-8ve ~~||=~~ ||

||= 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 = 7\12 ||= half-8ve ~~||=~~ ||

||= 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 = 7\12 ||= fifth-4th ~~||=~~ ||

||= 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 = 8\13 ||= fifth-4th ~~||=~~ ||

||= 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 = 7\13 ||= seventh-5th ~~||=~~ ||

||= 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 = 9\15 ||= third-8ve ~~||=~~ ||

||= 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 = 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).

If the 8ve is split... [//how to find alternate generators?//]

Both 13edo and 18edo are best notated with their second-nearest edomappings, (13, 7) and (18,10). [//Deal with 6-edo and 8-edo as well here//]. For these edos, use the 2nd best, since the pergen is about notation.

==Supplemental materials*==

needs more screenshots, including 12-edo's pergens and a page of the pdf

~~needs pergen squares picture~~

fill in the 2nd pergens column above -- can one edo pair imply two pergens?

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<div style="margin-left: 2em;"><a href="#Further Discussion-Notating tunings with an arbitrary generator">Notating tunings with an arbitrary generator</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs*">Pergens and EDOs*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials*">Supplemental materials*</a></div>

<div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-Notaion guide PDF">Notaion guide PDF</a></div>

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(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half.

The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).

The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Kite%27s%20color%20notation">Color notation</a> (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yellow, (a,b,-1) = green, (a,b,0,1) = blue, (a,b,0,-1) = red, (a,b,0,0,1) = jade, (a,b,0,0,-1) = amber, and (a,b) = white. Examples: 5/4 = y3 = yellow 3rd, 7/5 = bg5 = blue-green 5th or bluish 5th, etc.

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A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, ~~and~~ the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with either microtonal accidentals (P8, P5, ^1, /1,...) or colors (P8, P5, g1, r1,...).

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with either microtonal accidentals (P8, P5, ^1, /1,...) or colors (P8, P5, g1, r1,...).

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<h2 id="toc9"><a name="Further Discussion-Finding an example temperament"></a>Finding an example temperament</h2>

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, the quadruple red temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, the quadruple red temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.

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<h2 id="toc15"><a name="Further Discussion-Notating unsplit pergens"></a>Notating unsplit pergens</h2>

An unsplit pergen doesn't require ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a comma that maps to P1, such as 81/80 or 64/63. For single-comma temperaments, the pergen is unsplit if and only if the vanishing comma's monzo has a final exponent of ±1.

An unsplit pergen doesn't require ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a comma that maps to P1, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (the monzo has a final exponent of ±1).

The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate.

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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.

<h2 id="toc22"><a name="Further Discussion-Pergens and EDOs"></a>Pergens and EDOs</h2>

<h2 id="toc22"><a name="Further Discussion-Pergens and EDOs*"></a>Pergens and EDOs*</h2>

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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Given an edo, a period, and a generator, what is the pergen? There is usually more than one right answer. For 10edo with P = 5\10 and G = 2\10, it could be either (P8/2, P4/2) or (P8/2, P5/3). Every coprime period/generator pair results in a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 11-edo and 23-edo could also be considered ambiguous.

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 6-edo, 11-edo and 23-edo could also be considered ambiguous.

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See the screenshots in the next section for examples of which pergens are supported by a specific edo.

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.

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 &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, i.e. a quarter-8ve.