Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-~~07 22~~:33:53 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-08 02:47:32 UTC</tt>.

: The original revision id was <tt>~~626117063~~</tt>.

: The original revision id was <tt>626121387</tt>.

: The revision comment was: <tt></tt>

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For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6⋅G - m3. The comma splits both the octave and the fifth.

This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2⋅P8 - 4⋅P4) / (2⋅4) = (2⋅M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus ~~true~~ doubles require commas of at least 7-~~limit~~, whereas false doubles require only 5-limit.

This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2⋅P8 - 4⋅P4) / (2⋅4) = (2⋅M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus __true doubles require commas of at least 7-limit__, whereas false doubles require only 5-limit.

A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.

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Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.

Bracketed equivalences can also be used for innate-comma chords, aka essentially tempered chords. For example, injera's dom7b5 chord, tuned with 5/4, 7/5 and 7/4, is C,v7(vd5). It might be voiced C Gbv Bbv E. The Gbv - E interval is an awkward ^A6, even though it's the same interval as C - Bbv. Spelling Gbv as F#^ makes a more natural vm7 interval. But that would change the Gbv - Bbv = M3 interval to F#^ - Bbv, an awkward vvd4. To indicate all this, the chord could be thought of as C,v7(vd5=^A4). Likewise, pajara's dom7b5 chord can be C7([^d5=vA4]v3).

==Tipping points and sweet spots==

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==Notating rank-3 pergens==

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. ~~Every~~ additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:

||~ tuning ||~ tuning's rank ||~ notation ||~ notation's rank ||~ # of enharmonics needed ||~ enharmonics ||

||= 12-edo ||= rank-1 ||= conventional ||= rank-2 ||= 1 ||= E = d2 ||

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||= marvel ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- ||

||= breedsmic ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 ||

||= 7-limit JI ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= ---- ||

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more.

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All the previous rank-3 examples had the 2nd generator be a mapping comma, represented by an up. However, this isn't always possible.

||~ 7-limit temperament ||~ comma ||~ pergen ||~ spoken pergen ||~ period ||~ gen1 ||~ gen2 ||~ enharmonic ||

||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= unsplit with ups ||= P8 ||= P5 ||= ^1 = 81/80 ||= ~~none~~ ||

||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= unsplit with ups ||= P8 ||= P5 ||= ^1 = 81/80 ||= --- ||

||= deep reddish ||= 50/49 ||= (P8/2, P5, ^1) ||= half-8ve with ups ||= /d5 = 7/5 ||= P5 ||= ^1 = 81/80 ||= ``//``d2 ||

||= triple bluish ||= 1029/1000 ||= (P8, P11/3, ^1) ||= third-11th with ups ||= P8 ||= \d5 = 7/5 ||= ^1 = 81/80 ||= ``\\\``dd3 ||

||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= half-5th with ups ||= P8 ||= \d4 = 49/40 ||= ^1 = 64/63 ||= ``\\``dd3 ||

||= demeter ||= 686/675 ||= (P8, P5, ~~vm3~~/2) ||= half-~~downminor~~-3rd ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^\\dd3 ||

||= demeter ||= 686/675 ||= (P8, P5, \m3/2) ||= half-lowminor-3rd ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 ||

Demeter equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's, and 7/6 into two 15/14's. Its pergen is (P8, P5, ~~vm3~~/2), not (P8, P5, y3/3) or (P8, P5, vM3/3), in order to minimize the splitting fraction.

Demeter equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's, and 7/6 into two 15/14's. Its pergen is (P8, P5, \m3/2), not (P8, P5, y3/3) or (P8, P5, vM3/3), in order to minimize the splitting fraction. Then why not (P8, P5, v/A1)? Because gen2 has 2 kinds of accidentals? Or maybe (P8, P5, v\5/5)?

fifth-comma: c = 9.5¢, P5 = 700¢ + c, vM3 = 400¢ + d, d = -10¢

~~gen2 = vM3/3 = [4,2,-1,0]/3 = [1,0,-1,1] = v/A1, E = vM3 - 3gen2 = [1,2,2,-3] = ^^\\\dd3~~

~~gen2 = vM3/3 = [1,1,0,1] = /m2, E = [-1,1,1,3] = ^///dd2, C^/// = B##, C --~~

gen2 = \m3/2 = [3,2,0,-1]/2 = [1,1,1,0] = ^m2, E = [1,0,-2,-1] = vv\A1

gen2 = vM3/3 = [4,2,-1,0]/3 = [1,1,0,1] = /m2, E = [-1,1,1,3] = ^///dd2, C^/// = B##, C --

gen2 = \m3/2 = [3,2,0,-1]/2 = [1,1,1,0] = ^m2 = v\M2, E = [1,0,-2,-1] = vv\A1

gen2 = \m3/2 = [3,2,0,-1]/2 = [1,0,-1,1] = v/A1, E = [1,2,2,-3] = ^^\\\dd3

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A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5)/2 or (25/16)/4.

Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair.

Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair. Removing the 3-limit comma makes a rank-3 pergen.

Blackwood 5edo+y perchain: D E=F G A B=C D (E = m2), genchain: ... Gb^^ Bb^ D F#v A#vv... (^1 = 81/80, no E)

12edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E)

17edo+y perchain: C C^ Dv D... (E = vm2), genchain: same as blackwood, but with / and \

17edo+y perchain: C C^ Dv D... (E = dd3, E' = vm2 = vvA1), genchain: same as blackwood, but with / and \

==Pergens and MOS scales==

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For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This is explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6⋅G - m3. The comma splits both the octave and the fifth.

This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2⋅P8 - 4⋅P4) / (2⋅4) = (2⋅M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit.

This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2⋅P8 - 4⋅P4) / (2⋅4) = (2⋅M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit.

A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.

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Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.

~~ ~~

Bracketed equivalences can also be used for innate-comma chords, aka essentially tempered chords. For example, injera's dom7b5 chord, tuned with 5/4, 7/5 and 7/4, is C,v7(vd5). It might be voiced C Gbv Bbv E. The Gbv - E interval is an awkward ^A6, even though it's the same interval as C - Bbv. Spelling Gbv as F#^ makes a more natural vm7 interval. But that would change the Gbv - Bbv = M3 interval to F#^ - Bbv, an awkward vvd4. To indicate all this, the chord could be thought of as C,v7(vd5=^A4). Likewise, pajara's dom7b5 chord can be C7([^d5=vA4]v3).

<h2 id="toc14"><a name="Further Discussion-Tipping points and sweet spots"></a>Tipping points and sweet spots</h2>

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<h2 id="toc16"><a name="Further Discussion-Notating rank-3 pergens"></a>Notating rank-3 pergens</h2>

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. ~~Every~~ additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:

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</td>

<td style="text-align: center;">E = \\dd3

</td>

</tr>

<tr>

<td style="text-align: center;">7-limit JI

</td>

<td style="text-align: center;">rank-4

</td>

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

</td>

<td style="text-align: center;">rank-4

</td>

<td style="text-align: center;">0

</td>

</td>

</tr>

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<td style="text-align: center;">^1 = 81/80

</td>

<td style="text-align: center;">~~none~~

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

</td>

</tr>

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<td style="text-align: center;">686/675

</td>

<td style="text-align: center;">(P8, P5, ~~vm3~~/2)

<td style="text-align: center;">(P8, P5, \m3/2)

</td>

<td style="text-align: center;">half-~~downminor~~-3rd

<td style="text-align: center;">half-lowminor-3rd

</td>

<td style="text-align: center;">P8

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<td style="text-align: center;">v/A1 = 15/14

</td>

<td style="text-align: center;">^\\dd3

<td style="text-align: center;">^^\\\dd3

</td>

</tr>

</table>

Demeter equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's, and 7/6 into two 15/14's. Its pergen is (P8, P5, ~~vm3~~/2), not (P8, P5, y3/3) or (P8, P5, vM3/3), in order to minimize the splitting fraction.

Demeter equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's, and 7/6 into two 15/14's. Its pergen is (P8, P5, \m3/2), not (P8, P5, y3/3) or (P8, P5, vM3/3), in order to minimize the splitting fraction. Then why not (P8, P5, v/A1)? Because gen2 has 2 kinds of accidentals? Or maybe (P8, P5, v\5/5)?

fifth-comma: c = 9.5¢, P5 = 700¢ + c, vM3 = 400¢ + d, d = -10¢

~~gen2 = vM3/3 = [4,2,-1,0]/3 = [1,0,-1,1] = v/A1, E = vM3 - 3gen2 = [1,2,2,-3] = ^^\\\dd3 ~~

~~gen2 = vM3/3 = [1,1,0,1] = /m2, E = [-1,1,1,3] = ^/dd2, C^/ = B##, C -- ~~

gen2 = \m3/2 = [3,2,0,-1]/2 = [1,1,1,0] = ^m2, E = [1,0,-2,-1] = vv\A1

gen2 = vM3/3 = [4,2,-1,0]/3 = [1,1,0,1] = /m2, E = [-1,1,1,3] = ^/dd2, C^/ = B##, C --

gen2 = \m3/2 = [3,2,0,-1]/2 = [1,1,1,0] = ^m2 = v\M2, E = [1,0,-2,-1] = vv\A1

gen2 = \m3/2 = [3,2,0,-1]/2 = [1,0,-1,1] = v/A1, E = [1,2,2,-3] = ^^\\\dd3

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A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5)/2 or (25/16)/4.

Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair.

Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair. Removing the 3-limit comma makes a rank-3 pergen.

Blackwood 5edo+y perchain: D E=F G A B=C D (E = m2), genchain: ... Gb^^ Bb^ D F#v A#vv... (^1 = 81/80, no E)

12edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E)

17edo+y perchain: C C^ Dv D... (E = vm2), genchain: same as blackwood, but with / and \

17edo+y perchain: C C^ Dv D... (E = dd3, E' = vm2 = vvA1), genchain: same as blackwood, but with / and \

<h2 id="toc18"><a name="Further Discussion-Pergens and MOS scales"></a>Pergens and MOS scales</h2>

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This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.

<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>

<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>

Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.

<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>

Screenshot of the first 38 pergens:

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" />

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" />

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:

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-07 22:33:53 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-08 02:47:32 UTC</tt>.<br>
-: The original revision id was <tt>626117063</tt>.<br>
+: The original revision id was <tt>626121387</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 362: / Line 362: @@
 For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P5) / (3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - m3. The comma splits both the octave and the fifth.
-This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P4) / (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;4) = (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit.
+This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P4) / (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;4) = (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus __true doubles require commas of at least 7-limit__, whereas false doubles require only 5-limit.
 A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.
@@ Line 483: / Line 483: @@
 Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.
-Bracketed equivalences can also be used for innate-comma chords, aka essentially tempered chords. For example, injera's dom7b5 chord, tuned with 5/4, 7/5 and 7/4, is C,v7(vd5). It might be voiced C Gbv Bbv E. The Gbv - E interval is an awkward ^A6, even though it's the same interval as C - Bbv. Spelling Gbv as F#^ makes a more natural vm7 interval. But that would change the Gbv - Bbv = M3 interval to F#^ - Bbv, an awkward vvd4. To indicate all this, the chord could be thought of as C,v7(vd5=^A4). Likewise, pajara's dom7b5 chord can be C7([^d5=vA4]v3).
 ==Tipping points and sweet spots==
@@ Line 524: / Line 522: @@
 ==Notating rank-3 pergens==
-Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Every additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:
+Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:
 ||~ tuning ||~ tuning's rank ||~ notation ||~ notation's rank ||~ # of enharmonics needed ||~ enharmonics ||
 ||= 12-edo ||= rank-1 ||= conventional ||= rank-2 ||= 1 ||= E = d2 ||
@@ Line 537: / Line 535: @@
 ||= marvel ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- ||
 ||= breedsmic ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 ||
+||= 7-limit JI ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= ---- ||
 A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more.
@@ Line 542: / Line 541: @@
 All the previous rank-3 examples had the 2nd generator be a mapping comma, represented by an up. However, this isn't always possible.
 ||~ 7-limit temperament ||~ comma ||~ pergen ||~ spoken pergen ||~ period ||~ gen1 ||~ gen2 ||~ enharmonic ||
-||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= unsplit with ups ||= P8 ||= P5 ||= ^1 = 81/80 ||= none ||
+||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= unsplit with ups ||= P8 ||= P5 ||= ^1 = 81/80 ||= --- ||
 ||= deep reddish ||= 50/49 ||= (P8/2, P5, ^1) ||= half-8ve with ups ||= /d5 = 7/5 ||= P5 ||= ^1 = 81/80 ||= ``//``d2 ||
 ||= triple bluish ||= 1029/1000 ||= (P8, P11/3, ^1) ||= third-11th with ups ||= P8 ||= \d5 = 7/5 ||= ^1 = 81/80 ||= ``\\\``dd3 ||
 ||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= half-5th with ups ||= P8 ||= \d4 = 49/40 ||= ^1 = 64/63 ||= ``\\``dd3 ||
-||= demeter ||= 686/675 ||= (P8, P5, vm3/2) ||= half-downminor-3rd ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^\\dd3 ||
+||= demeter ||= 686/675 ||= (P8, P5, \m3/2) ||= half-lowminor-3rd ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 ||
-Demeter equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's, and 7/6 into two 15/14's. Its pergen is (P8, P5, vm3/2), not (P8, P5, y3/3) or (P8, P5, vM3/3), in order to minimize the splitting fraction.
+Demeter equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's, and 7/6 into two 15/14's. Its pergen is (P8, P5, \m3/2), not (P8, P5, y3/3) or (P8, P5, vM3/3), in order to minimize the splitting fraction. Then why not (P8, P5, v/A1)? Because gen2 has 2 kinds of accidentals? Or maybe (P8, P5, v\5/5)?
 fifth-comma: c = 9.5¢, P5 = 700¢ + c, vM3 = 400¢ + d, d = -10¢
-gen2 = vM3/3 = [4,2,-1,0]/3 = [1,0,-1,1] = v/A1, E = vM3 - 3gen2 = [1,2,2,-3] = ^^\\\dd3
-gen2 = vM3/3 = [1,1,0,1] = /m2, E = [-1,1,1,3] = ^///dd2, C^/// = B##, C --
-gen2 = \m3/2 = [3,2,0,-1]/2 = [1,1,1,0] = ^m2, E = [1,0,-2,-1] = vv\A1
+gen2 = vM3/3 = [4,2,-1,0]/3 = [1,1,0,1] = /m2, E = [-1,1,1,3] = ^///dd2, C^/// = B##, C --
+gen2 = \m3/2 = [3,2,0,-1]/2 = [1,1,1,0] = ^m2 = v\M2, E = [1,0,-2,-1] = vv\A1
+gen2 = \m3/2 = [3,2,0,-1]/2 = [1,0,-1,1] = v/A1, E = [1,2,2,-3] = ^^\\\dd3
@@ Line 561: / Line 561: @@
 A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5)/2 or (25/16)/4.
-Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair.
+Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair. Removing the 3-limit comma makes a rank-3 pergen.
 Blackwood 5edo+y perchain: D E=F G A B=C D (E = m2), genchain: ... Gb^^ Bb^ D F#v A#vv... (^1 = 81/80, no E)
 edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E)
-edo+y perchain: C C^ Dv D... (E = vm2), genchain: same as blackwood, but with / and \
+edo+y perchain: C C^ Dv D... (E = dd3, E' = vm2 = vvA1), genchain: same as blackwood, but with / and \
 ==Pergens and MOS scales==
@@ Line 2,336: / Line 2,336: @@
 For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P5) / (3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This &lt;u&gt;is&lt;/u&gt; explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - m3. The comma splits both the octave and the fifth.&lt;br /&gt;
 &lt;br /&gt;
-This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P4) / (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;4) = (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit.&lt;br /&gt;
+This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P4) / (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;4) = (2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus &lt;u&gt;true doubles require commas of at least 7-limit&lt;/u&gt;, whereas false doubles require only 5-limit.&lt;br /&gt;
 &lt;br /&gt;
 A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.&lt;br /&gt;
@@ Line 2,448: / Line 2,448: @@
 &lt;br /&gt;
 Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.&lt;br /&gt;
-&lt;br /&gt;
-Bracketed equivalences can also be used for innate-comma chords, aka essentially tempered chords. For example, injera's dom7b5 chord, tuned with 5/4, 7/5 and 7/4, is C,v7(vd5). It might be voiced C Gbv Bbv E. The Gbv - E interval is an awkward ^A6, even though it's the same interval as C - Bbv. Spelling Gbv as F#^ makes a more natural vm7 interval. But that would change the Gbv - Bbv = M3 interval to F#^ - Bbv, an awkward vvd4. To indicate all this, the chord could be thought of as C,v7(vd5=^A4). Likewise, pajara's dom7b5 chord can be C7([^d5=vA4]v3).&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:83:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Tipping points and sweet spots"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:83 --&gt;Tipping points and sweet spots&lt;/h2&gt;
@@ Line 2,670: / Line 2,668: @@
 &lt;!-- ws:start:WikiTextHeadingRule:87:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Further Discussion-Notating rank-3 pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:87 --&gt;Notating rank-3 pergens&lt;/h2&gt;
   &lt;br /&gt;
-Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Every additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:&lt;br /&gt;
+Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:&lt;br /&gt;
@@ Line 2,839: / Line 2,837: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;E = \\dd3&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;7-limit JI&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;double-pair&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;hr /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,883: / Line 2,895: @@
          &lt;td style="text-align: center;"&gt;^1 = 81/80&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;none&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;---&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,945: / Line 2,957: @@
          &lt;td style="text-align: center;"&gt;686/675&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P5, vm3/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5, \m3/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;half-downminor-3rd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;half-lowminor-3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
@@ Line 2,955: / Line 2,967: @@
          &lt;td style="text-align: center;"&gt;v/A1 = 15/14&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;^\\dd3&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;^^\\\dd3&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
 &lt;/table&gt;
-Demeter equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's, and 7/6 into two 15/14's. Its pergen is (P8, P5, vm3/2), not (P8, P5, y3/3) or (P8, P5, vM3/3), in order to minimize the splitting fraction. &lt;br /&gt;
+Demeter equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's, and 7/6 into two 15/14's. Its pergen is (P8, P5, \m3/2), not (P8, P5, y3/3) or (P8, P5, vM3/3), in order to minimize the splitting fraction. Then why not (P8, P5, v/A1)? Because gen2 has 2 kinds of accidentals? Or maybe (P8, P5, v\5/5)?&lt;br /&gt;
+&lt;br /&gt;
 fifth-comma: c = 9.5¢, P5 = 700¢ + c, vM3 = 400¢ + d, d = -10¢&lt;br /&gt;
-gen2 = vM3/3 = [4,2,-1,0]/3 = [1,0,-1,1] = v/A1, E = vM3 - 3gen2 = [1,2,2,-3] = ^^\\\dd3&lt;br /&gt;
-gen2 = vM3/3 = [1,1,0,1] = /m2, E = [-1,1,1,3] = ^&lt;em&gt;/dd2, C^&lt;/em&gt;/ = B##, C --&lt;br /&gt;
 &lt;br /&gt;
-gen2 = \m3/2 = [3,2,0,-1]/2 = [1,1,1,0] = ^m2, E = [1,0,-2,-1] = vv\A1&lt;br /&gt;
+gen2 = vM3/3 = [4,2,-1,0]/3 = [1,1,0,1] = /m2, E = [-1,1,1,3] = ^&lt;em&gt;/dd2, C^&lt;/em&gt;/ = B##, C --&lt;br /&gt;
+gen2 = \m3/2 = [3,2,0,-1]/2 = [1,1,1,0] = ^m2 = v\M2, E = [1,0,-2,-1] = vv\A1&lt;br /&gt;
 &lt;br /&gt;
+gen2 = \m3/2 = [3,2,0,-1]/2 = [1,0,-1,1] = v/A1, E = [1,2,2,-3] = ^^\\\dd3&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
@@ Line 2,974: / Line 2,987: @@
 A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5)/2 or (25/16)/4.&lt;br /&gt;
 &lt;br /&gt;
-Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair.&lt;br /&gt;
+Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair. Removing the 3-limit comma makes a rank-3 pergen.&lt;br /&gt;
 &lt;br /&gt;
 Blackwood 5edo+y perchain: D E=F G A B=C D (E = m2), genchain: ... Gb^^ Bb^ D F#v A#vv... (^1 = 81/80, no E)&lt;br /&gt;
 edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E)&lt;br /&gt;
-edo+y perchain: C C^ Dv D... (E = vm2), genchain: same as blackwood, but with / and \&lt;br /&gt;
+edo+y perchain: C C^ Dv D... (E = dd3, E' = vm2 = vvA1), genchain: same as blackwood, but with / and \&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:91:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:91 --&gt;Pergens and MOS scales&lt;/h2&gt;
@@ Line 4,398: / Line 4,411: @@
   &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;
-&lt;!-- ws:start:WikiTextUrlRule:5479: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:5479 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:5498: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:5498 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:5480:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:5480 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:5499:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:5499 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Screenshot of the first 38 pergens:&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:3215:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 460px; width: 704px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3215 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:3230:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 460px; width: 704px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3230 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 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;