Kite's thoughts on pergens: Difference between revisions

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A **pergen** (pronounced "peer-gen") is a way of identifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.

If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. ~~The fraction is~~ always of the form 1/N, thus the octave and/or the 3-limit interval is **split** into N parts. The interval which is split into multiple generators is the **multigen**. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.

If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is **split** into N parts. The interval which is split into multiple generators is the **multigen**. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.

For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The dicot temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore, a pun on "semi-fourth", is of course half-fourth.

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||= pythagorean ||= (P8, P5) ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= --- ||

||= meantone ||= (P8, P5) ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= --- ||

||= srutal ||= (P8/2, P5) ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = d2 ||

||= srutal ||= (P8/2, P5) ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = ^^d2 ||

||= semaphore ||= (P8, P4/2) ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = vvm2 ||

||= decimal ||= (P8/2, P4/2) ||= rank-2 ||= double-pair ||= rank-4 ||= 2 ||= E = vvd2, E' = \\m2 = ^^\\A1 ||

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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 = ``//``d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as 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\.

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)-2 · (64/63)2 · (19,-12)-1. This can be rewritten as vv1 + ``//``1 - d2 = vv``//``-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus 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 directly 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)-2 · (64/63)2 · (19,-12)-1. This can be rewritten as vv1 + ``//``1 - d2 = vv``//``-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.

This 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.

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Unlike the previous examples, Demeter's 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, vm3/2). It could also be called (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^^, very awkward! 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.

There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 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 gen2 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 there will always be a 3-limit comma small enough to 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 gen2'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:

||~ unsplit ||~ ^1 = 81/80 ||~~~ spoken name~~ ||~ ^1 = 64/63 ~~||~ spoken name~~ ||

||~ __pergen number__ ||||||||~ __prime subgroup__ ||

||~ unsplit ||||~ 2.3.5 (^1 = 81/80) ||||~ 2.3.7 (^1 = 64/63) ||

||= 1 ||= (P8, P5, ^1) ||= rank-3 unsplit ||= same ||= same ||

||~ half-splits ||~ ||~ ||~ ||~ ||

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==Notating non-8ve and no-5ths pergens==

In Blackwood-like pergens, the 5th is present but not independent. In ~~non~~-~~5th~~ pergens, the 5th is not present, and the prime subgroup doesn't contain 3.

In Blackwood-like pergens, the 5th is present but not independent. In no-5ths pergens, the 5th is not present, and the prime subgroup doesn't contain 3.

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.

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But in non-8ve and no-5ths pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and ~~non~~-~~5th~~ pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a __huge__ number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen ~~except Blackwood-like ones~~ can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.

||~ __pergen number__ ||||||||||||~ __prime subgroup__ ||

||~ unsplit ||~ 2.3 ||~ 2.5 (M3 = 5/4) ||~ 2.7 (M2 = 8/7) ||~ 3.5 (M6 = 5/3) ||~ 3.7 (M3 = 9/7) ||~ 5.7 (WWM3 = 5/1, d5 = 7/5) ||

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||= 14 ||= (P8/2, P11/3) ||= (P8/2, M10/3) ||= (P8/2, M9/3) ||= (P12/2, WWM3/3) ||= (P12/2, WM7/3) ||= (M9, WWm7/3)* ||

||= 15 ||= (P8/3, P4/3) ||= (P8/3, M3/3) ||= (P8/3, M2/3) ||= (P12/3, M6/3) ||= (P12/3, P4)* ||= (WWM3/3, d5/3) ||

For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.

For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and period-inverting if it's more than half of p/1.

Blackwood-like pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (WWM3/5, d5) can optionally be replaced too.

||~ pergen number ||~ 2.3 ||~ 2.5 ||~ 2.7 ||~ 3.5 ||~ 3.7 ||~ 5.7 ||

||= 33 ||= (P8/5, P5) ||= (P8/5, ^1) ||= (P8/5, ^1) ||= (P12/5, M6) ||= (P12/5, M3) ||= (WWM3/5, ^1) ||

Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the 1st 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.

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===Notation guide PDF===

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.

This PDF is a rank-2 notation guide that shows the full lattice for the first 32 pergens, up through the quarter-splits block. It includes alternate enharmonics for many pergens.

http://~~www~~.tallkite.com/misc_files/pergens.pdf

[[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf|tallkite.com/misc_files/notation guide for rank-2 pergens.pdf]]

Screenshots of the first 2 pages:

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A pergen (pronounced &quot;peer-gen&quot;) is a way of identifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.

If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. ~~The fraction is~~ always of the form 1/N, thus the octave and/or the 3-limit interval is split into N parts. The interval which is split into multiple generators is the multigen. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.

If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is split into N parts. The interval which is split into multiple generators is the multigen. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.

For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The dicot temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore, a pun on &quot;semi-fourth&quot;, is of course half-fourth.

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<td style="text-align: center;">1

</td>

<td style="text-align: center;">E = d2

<td style="text-align: center;">E = ^^d2

</td>

</tr>

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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 = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as 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\.

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)-2 · (64/63)2 · (19,-12)-1. This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus 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 directly 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)-2 · (64/63)2 · (19,-12)-1. This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.

This 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.

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Unlike the previous examples, Demeter's 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, vm3/2). It could also be called (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^^, very awkward! 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.

There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 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 &lt; 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 gen2 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 there will always be a 3-limit comma small enough to 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 &lt; 3, 5/4 is preferred.

If ^1 = 81/80, possible half-split gen2'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.

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<tr>

<th>~~unsplit~~

<th>pergen number

</th>

<th>~~^1 = 81~~/80

<th colspan="4">prime subgroup

</th>

<th>~~spoken name~~

</tr>

<tr>

<th>unsplit

</th>

<th>^1 = 64/63

<th colspan="2">2.3.5 (^1 = 81/80)

</th>

<th>~~spoken name~~

<th colspan="2">2.3.7 (^1 = 64/63)

</th>

</tr>

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<h2 id="toc18"><a name="Further Discussion-Notating non-8ve and no-5ths pergens"></a>Notating non-8ve and no-5ths pergens</h2>

In Blackwood-like pergens, the 5th is present but not independent. In ~~non~~-~~5th~~ pergens, the 5th is not present, and the prime subgroup doesn't contain 3.

In Blackwood-like pergens, the 5th is present but not independent. In no-5ths pergens, the 5th is not present, and the prime subgroup doesn't contain 3.

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.

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But in non-8ve and no-5ths pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and ~~non~~-~~5th~~ pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a huge number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a huge number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen ~~except Blackwood-like ones~~ can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.

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</table>

For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.

For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and period-inverting if it's more than half of p/1.

Blackwood-like pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (WWM3/5, d5) can optionally be replaced too.

<tr>

<th>pergen number

</th>

<th>2.3

</th>

<th>2.5

</th>

<th>2.7

</th>

<th>3.5

</th>

<th>3.7

</th>

<th>5.7

</th>

</tr>

<tr>

<td style="text-align: center;">33

</td>

<td style="text-align: center;">(P8/5, P5)

</td>

<td style="text-align: center;">(P8/5, ^1)

</td>

<td style="text-align: center;">(P8/5, ^1)

</td>

<td style="text-align: center;">(P12/5, M6)

</td>

<td style="text-align: center;">(P12/5, M3)

</td>

<td style="text-align: center;">(WWM3/5, ^1)

</td>

</tr>

</table>

Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the 1st 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.

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Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.

<img src="/file/view/pergen%20squares.png/627986281/pergen%20squares.png" alt="pergen squares.png" title="pergen squares.png" />

<img src="/file/view/pergen%20squares.png/627986281/pergen%20squares.png" alt="pergen squares.png" title="pergen squares.png" />

A similar chart could be made for all rank-3 pergens, using pergen cubes.

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<h3 id="toc24"><a name="Further Discussion-Supplemental materials-Notation guide PDF"></a>Notation guide PDF</h3>

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.

This PDF is a rank-2 notation guide that shows the full lattice for the first 32 pergens, up through the quarter-splits block. It includes alternate enharmonics for many pergens.

~~~~<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:8179 --~~>

<a class="wiki_link_ext" href="http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf" rel="nofollow">tallkite.com/misc_files/notation guide for rank-2 pergens.pdf</a>

Screenshots of the first 2 pages:

<img src="/file/view/pergens%201.png/627986243/704x948/pergens%201.png" alt="pergens 1.png" title="pergens 1.png" style="height: 948px; width: 704px;" />

<img src="/file/view/pergens%201.png/627986243/704x948/pergens%201.png" alt="pergens 1.png" title="pergens 1.png" style="height: 948px; width: 704px;" />

<img src="/file/view/pergens%202.png/627986267/704x760/pergens%202.png" alt="pergens 2.png" title="pergens 2.png" style="height: 760px; width: 704px;" />

<img src="/file/view/pergens%202.png/627986267/704x760/pergens%202.png" alt="pergens 2.png" title="pergens 2.png" style="height: 760px; width: 704px;" />

<h3 id="toc25"> </h3>

<h3 id="toc26"><a name="Further Discussion-Supplemental materials-alt-pergenLister"></a>alt-pergenLister</h3>

Alt-pergenLister lists out thousands of rank-2 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 or edo pair. Written in Jesusonic, runs inside Reaper.

<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>

<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>

The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.

Screenshots of the first 69 pergens:

<img src="/file/view/alt-pergenLister%201.png/627986177/800x427/alt-pergenLister%201.png" alt="alt-pergenLister 1.png" title="alt-pergenLister 1.png" style="height: 427px; width: 800px;" />

<img src="/file/view/alt-pergenLister%201.png/627986177/800x427/alt-pergenLister%201.png" alt="alt-pergenLister 1.png" title="alt-pergenLister 1.png" style="height: 427px; width: 800px;" />

<img src="http://xenharmonic.wikispaces.com/file/view/alt-pergenLister%202.png/627986179/800x455/alt-pergenLister%202.png" alt="alt-pergenLister 2.png" title="alt-pergenLister 2.png" style="height: 455px; width: 800px;" />

<img src="http://xenharmonic.wikispaces.com/file/view/alt-pergenLister%202.png/627986179/800x455/alt-pergenLister%202.png" alt="alt-pergenLister 2.png" title="alt-pergenLister 2.png" style="height: 455px; width: 800px;" />

The first 29 pergens supported by 12edo:

<img src="/file/view/alt-pergenLister%2012edo.png/627986189/800x449/alt-pergenLister%2012edo.png" alt="alt-pergenLister 12edo.png" title="alt-pergenLister 12edo.png" style="height: 449px; width: 800px;" />

<img src="/file/view/alt-pergenLister%2012edo.png/627986189/800x449/alt-pergenLister%2012edo.png" alt="alt-pergenLister 12edo.png" title="alt-pergenLister 12edo.png" style="height: 449px; width: 800px;" />

Some of the pergens supported by 15edo. A red asterisk means partial support.

<img src="/file/view/alt-pergenLister%2015edo.png/627986197/800x493/alt-pergenLister%2015edo.png" alt="alt-pergenLister 15edo.png" title="alt-pergenLister 15edo.png" style="height: 493px; width: 800px;" />

<img src="/file/view/alt-pergenLister%2015edo.png/627986197/800x493/alt-pergenLister%2015edo.png" alt="alt-pergenLister 15edo.png" title="alt-pergenLister 15edo.png" style="height: 493px; width: 800px;" />

Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.

<img src="/file/view/alt-pergenLister%2019edo.png/627986203/800x459/alt-pergenLister%2019edo.png" alt="alt-pergenLister 19edo.png" title="alt-pergenLister 19edo.png" style="height: 459px; width: 800px;" />

<img src="/file/view/alt-pergenLister%2019edo.png/627986203/800x459/alt-pergenLister%2019edo.png" alt="alt-pergenLister 19edo.png" title="alt-pergenLister 19edo.png" style="height: 459px; width: 800px;" />

Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:

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 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>2018-03-31 06:27:07 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-04-15 03:00:50 UTC</tt>.<br>
-: The original revision id was <tt>628165867</tt>.<br>
+: The original revision id was <tt>628679225</tt>.<br>
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 A **pergen** (pronounced "peer-gen") is a way of identifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.
-If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is **split** into N parts. The interval which is split into multiple generators is the **multigen**. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.
+If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is **split** into N parts. The interval which is split into multiple generators is the **multigen**. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.
 For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The dicot temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore, a pun on "semi-fourth", is of course half-fourth.
@@ Line 577: / Line 577: @@
 ||= pythagorean ||= (P8, P5) ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= --- ||
 ||= meantone ||= (P8, P5) ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= --- ||
-||= srutal ||= (P8/2, P5) ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = d2 ||
+||= srutal ||= (P8/2, P5) ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = ^^d2 ||
 ||= semaphore ||= (P8, P4/2) ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = vvm2 ||
 ||= decimal ||= (P8/2, P4/2) ||= rank-2 ||= double-pair ||= rank-4 ||= 2 ||= E = vvd2, E' = \\m2 = ^^\\A1 ||
@@ Line 604: / Line 604: @@
 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 (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as 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\.
-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)&lt;span style="vertical-align: super;"&gt;-2&lt;/span&gt; · (64/63)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; · (19,-12)&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;. This can be rewritten as vv1 + ``//``1 - d2 = vv``//``-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus 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 directly 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)&lt;span style="vertical-align: super;"&gt;-2&lt;/span&gt; · (64/63)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; · (19,-12)&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;. This can be rewritten as vv1 + ``//``1 - d2 = vv``//``-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.
 This 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.
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 Unlike the previous examples, Demeter's 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, vm3/2). It could also be called (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^^, very awkward! 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.
-There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 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 &lt; 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 gen2 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 there will always be a 3-limit comma small enough to 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 &lt; 3, 5/4 is preferred.
 If ^1 = 81/80, possible half-split gen2'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:
-||~ unsplit ||~ ^1 = 81/80 ||~ spoken name ||~ ^1 = 64/63 ||~ spoken name ||
+||~ __pergen number__ ||||||||~ __prime subgroup__ ||
+||~ unsplit ||||~ 2.3.5 (^1 = 81/80) ||||~ 2.3.7 (^1 = 64/63) ||
 ||= 1 ||= (P8, P5, ^1) ||= rank-3 unsplit ||= same ||= same ||
 ||~ half-splits ||~   ||~   ||~   ||~   ||
@@ Line 667: / Line 668: @@
 ==Notating non-8ve and no-5ths pergens==
-In Blackwood-like pergens, the 5th is present but not independent. In non-5th pergens, the 5th is not present, and the prime subgroup doesn't contain 3.
+In Blackwood-like pergens, the 5th is present but not independent. In no-5ths pergens, the 5th is not present, and the prime subgroup doesn't contain 3.
 In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.
@@ Line 673: / Line 674: @@
 But in non-8ve and no-5ths pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.
-Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.
+Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a __huge__ number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.
-Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.
+Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.
 ||~ __pergen number__ ||||||||||||~ __prime subgroup__ ||
 ||~ unsplit ||~ 2.3 ||~ 2.5 (M3 = 5/4) ||~ 2.7 (M2 = 8/7) ||~ 3.5 (M6 = 5/3) ||~ 3.7 (M3 = 9/7) ||~ 5.7 (WWM3 = 5/1, d5 = 7/5) ||
@@ Line 695: / Line 696: @@
 ||= 14 ||= (P8/2, P11/3) ||= (P8/2, M10/3) ||= (P8/2, M9/3) ||= (P12/2, WWM3/3) ||= (P12/2, WM7/3) ||= (M9, WWm7/3)* ||
 ||= 15 ||= (P8/3, P4/3) ||= (P8/3, M3/3) ||= (P8/3, M2/3) ||= (P12/3, M6/3) ||= (P12/3, P4)* ||= (WWM3/3, d5/3) ||
-For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.
+For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and period-inverting if it's more than half of p/1.
+Blackwood-like pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (WWM3/5, d5) can optionally be replaced too.
+||~ pergen number ||~ 2.3 ||~ 2.5 ||~ 2.7 ||~ 3.5 ||~ 3.7 ||~ 5.7 ||
+||= 33 ||= (P8/5, P5) ||= (P8/5, ^1) ||= (P8/5, ^1) ||= (P12/5, M6) ||= (P12/5, M3) ||= (WWM3/5, ^1) ||
 Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the 1st 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.
@@ Line 968: / Line 973: @@
 ===Notation guide PDF===
-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.
+This PDF is a rank-2 notation guide that shows the full lattice for the first 32 pergens, up through the quarter-splits block. It includes alternate enharmonics for many pergens.
-http://www.tallkite.com/misc_files/pergens.pdf
+[[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf|tallkite.com/misc_files/notation guide for rank-2 pergens.pdf]]
 Screenshots of the first 2 pages:
@@ Line 1,199: / Line 1,204: @@
 A &lt;strong&gt;pergen&lt;/strong&gt; (pronounced &amp;quot;peer-gen&amp;quot;) is a way of identifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.&lt;br /&gt;
 &lt;br /&gt;
-If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is &lt;strong&gt;split&lt;/strong&gt; into N parts. The interval which is split into multiple generators is the &lt;strong&gt;multigen&lt;/strong&gt;. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.&lt;br /&gt;
+If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is &lt;strong&gt;split&lt;/strong&gt; into N parts. The interval which is split into multiple generators is the &lt;strong&gt;multigen&lt;/strong&gt;. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.&lt;br /&gt;
 &lt;br /&gt;
 For example, the srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The dicot temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore, a pun on &amp;quot;semi-fourth&amp;quot;, is of course half-fourth.&lt;br /&gt;
@@ Line 3,367: / Line 3,372: @@
          &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;E = d2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;E = ^^d2&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 3,626: / Line 3,631: @@
 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 = &lt;!-- ws:start:WikiTextRawRule:060:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:060 --&gt;d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as 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\.&lt;br /&gt;
 &lt;br /&gt;
-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)&lt;span style="vertical-align: super;"&gt;-2&lt;/span&gt; · (64/63)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; · (19,-12)&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;. This can be rewritten as vv1 + &lt;!-- ws:start:WikiTextRawRule:061:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:061 --&gt;1 - d2 = vv&lt;!-- ws:start:WikiTextRawRule:062:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:062 --&gt;-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.&lt;br /&gt;
+With this standardization, the enharmonic can be derived directly 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)&lt;span style="vertical-align: super;"&gt;-2&lt;/span&gt; · (64/63)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; · (19,-12)&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;. This can be rewritten as vv1 + &lt;!-- ws:start:WikiTextRawRule:061:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:061 --&gt;1 - d2 = vv&lt;!-- ws:start:WikiTextRawRule:062:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:062 --&gt;-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.&lt;br /&gt;
 &lt;br /&gt;
 This 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.&lt;br /&gt;
@@ Line 3,632: / Line 3,637: @@
 Unlike the previous examples, Demeter's 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, vm3/2). It could also be called (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^^, very awkward! 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.&lt;br /&gt;
 &lt;br /&gt;
-There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 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 &lt;strong&gt;DOL&lt;/strong&gt; (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &amp;lt; 3, 5/4 is preferred.&lt;br /&gt;
+There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 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 there will always be a 3-limit comma small enough to 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 &lt;strong&gt;DOL&lt;/strong&gt; (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &amp;lt; 3, 5/4 is preferred.&lt;br /&gt;
 &lt;br /&gt;
 If ^1 = 81/80, possible half-split gen2'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.&lt;br /&gt;
@@ Line 3,641: / Line 3,646: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;th&gt;unsplit&lt;br /&gt;
+         &lt;th&gt;&lt;u&gt;pergen number&lt;/u&gt;&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;^1 = 81/80&lt;br /&gt;
+         &lt;th colspan="4"&gt;&lt;u&gt;prime subgroup&lt;/u&gt;&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;spoken name&lt;br /&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+         &lt;th&gt;unsplit&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;^1 = 64/63&lt;br /&gt;
+         &lt;th colspan="2"&gt;2.3.5 (^1 = 81/80)&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;spoken name&lt;br /&gt;
+         &lt;th colspan="2"&gt;2.3.7 (^1 = 64/63)&lt;br /&gt;
 &lt;/th&gt;
      &lt;/tr&gt;
@@ Line 4,144: / Line 4,151: @@
 &lt;!-- ws:start:WikiTextHeadingRule:99:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Further Discussion-Notating non-8ve and no-5ths pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:99 --&gt;Notating non-8ve and no-5ths pergens&lt;/h2&gt;
   &lt;br /&gt;
-In Blackwood-like pergens, the 5th is present but not independent. In non-5th pergens, the 5th is not present, and the prime subgroup doesn't contain 3.&lt;br /&gt;
+In Blackwood-like pergens, the 5th is present but not independent. In no-5ths pergens, the 5th is not present, and the prime subgroup doesn't contain 3.&lt;br /&gt;
 &lt;br /&gt;
 In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.&lt;br /&gt;
@@ Line 4,150: / Line 4,157: @@
 But in non-8ve and no-5ths pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.&lt;br /&gt;
 &lt;br /&gt;
-Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a &lt;u&gt;huge&lt;/u&gt; number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.&lt;br /&gt;
+Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a &lt;u&gt;huge&lt;/u&gt; number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.&lt;br /&gt;
 &lt;br /&gt;
-Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.&lt;br /&gt;
+Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.&lt;br /&gt;
@@ Line 4,452: / Line 4,459: @@
 &lt;/table&gt;
-For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.&lt;br /&gt;
+For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and period-inverting if it's more than half of p/1.&lt;br /&gt;
+&lt;br /&gt;
+Blackwood-like pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (WWM3/5, d5) can optionally be replaced too.&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;pergen number&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2.3&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2.5&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2.7&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;3.5&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;3.7&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;5.7&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;33&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/5, P5)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/5, ^1)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/5, ^1)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P12/5, M6)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P12/5, M3)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(WWM3/5, ^1)&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
 &lt;br /&gt;
 Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the 1st 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.&lt;br /&gt;
@@ Line 4,496: / Line 4,542: @@
 &lt;br /&gt;
 Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:4926:&amp;lt;img src=&amp;quot;/file/view/pergen%20squares.png/627986281/pergen%20squares.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/pergen%20squares.png/627986281/pergen%20squares.png" alt="pergen squares.png" title="pergen squares.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4926 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:4962:&amp;lt;img src=&amp;quot;/file/view/pergen%20squares.png/627986281/pergen%20squares.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/pergen%20squares.png/627986281/pergen%20squares.png" alt="pergen squares.png" title="pergen squares.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4962 --&gt;&lt;br /&gt;
 A similar chart could be made for all rank-3 pergens, using pergen cubes.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 6,579: / Line 6,625: @@
 &lt;!-- ws:start:WikiTextHeadingRule:111:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc24"&gt;&lt;a name="Further Discussion-Supplemental materials-Notation guide PDF"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:111 --&gt;Notation guide PDF&lt;/h3&gt;
   &lt;br /&gt;
-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.&lt;br /&gt;
+This PDF is a rank-2 notation guide that shows the full lattice for the first 32 pergens, up through the quarter-splits block. It includes alternate enharmonics for many pergens.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:8179:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:8179 --&gt;&lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf" rel="nofollow"&gt;tallkite.com/misc_files/notation guide for rank-2 pergens.pdf&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
 Screenshots of the first 2 pages:&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:4927:&amp;lt;img src=&amp;quot;/file/view/pergens%201.png/627986243/704x948/pergens%201.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 948px; width: 704px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/pergens%201.png/627986243/704x948/pergens%201.png" alt="pergens 1.png" title="pergens 1.png" style="height: 948px; width: 704px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4927 --&gt;&lt;br /&gt;
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 Alt-pergenLister lists out thousands of rank-2 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 or edo pair. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
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 The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.&lt;br /&gt;
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 Screenshots of the first 69 pergens:&lt;br /&gt;
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 The first 29 pergens supported by 12edo:&lt;br /&gt;
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 Some of the pergens supported by 15edo. A red asterisk means partial support.&lt;br /&gt;
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 Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.&lt;br /&gt;
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 Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:&lt;br /&gt;