Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to ~~quarter~~-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.

The enharmonic interval, or more briefly the **enharmonic**, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.

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C - G implies ...Eb Bb F C G D A E B F# C#...

C - Eb^=Ev - G implies ...F -- Ab^=Av -- C -- Eb^=Ev -- G -- Bb^=Bv -- D -- F^=F#v -- A -- C^=C#v -- E...

If the octave is split, the table has a **perchain** ("peer-chain", chain of periods) that shows the octave: In C -- F#v=Gb^ -- C~~, the last C is an octave above the first one.~~

If the octave is split, the table has a **perchain** ("peer-chain", chain of periods) that shows the octave: C -- F#v=Gb^ -- C

The table lists several possible notations for each pergen. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination.

~~(table is unfinished)~~

||~ pergen ||~ enharmonic

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third-8ve ||= ^<span style="vertical-align: super;">3</span>d2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented ||

||= (P8, P4/3)

third-4th ||= v<span style="vertical-align: super;">3</span>A1 ||= C^<span style="vertical-align: super;">3 ``=`` </span>C# ||= P4/3 = ^^m2 ~~= vM2~~ ||= C - Dv - Eb^ - F ||= porcupine ||

third-4th ||= v<span style="vertical-align: super;">3</span>A1 ||= C^<span style="vertical-align: super;">3 ``=`` </span>C# ||= P4/3 = vM2 = ^^m2 ||= C - Dv - Eb^ - F ||= porcupine ||

||= (P8, P5/3)

third-5th ||= v<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric ||

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third-8ve, half-4th ||= v<span style="vertical-align: super;">6</span>A2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D# ||= P8/3 = ^^m3 = v<span style="vertical-align: super;">4</span>A4

P4/2 = ^<span style="vertical-align: super;">3</span>m2 = v<span style="vertical-align: super;">3</span>M3 ||= C - Eb^^ - Avv - C

C - Db^<span style="vertical-align: super;">3</span>=Ev<span style="vertical-align: super;">3</span> - F ||= ||

C - Db^<span style="vertical-align: super;">3</span>=Ev<span style="vertical-align: super;">3</span> - F ||= sixfold jade ||

||= " ||= ^<span style="vertical-align: super;">3</span>d2,

\\m2 ||= C^<span style="vertical-align: super;">3</span> ``=`` B#

C``//`` = Db ||= P8/3 = vM3 = ^^d4

P4/2 = /M2 = \m3 ||= C - Ev - Ab^ - C

C - D/=Eb\ - F ||= ||

C - D/=Eb\ - F ||= 128/125 & 49/48 ||

||= (P8/3, P5/2)

third-8ve, half-5th ||= ^<span style="vertical-align: super;">3</span>d2

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C``//`` = C# ||= P8/3 = vM3 = ^^d4

P5/2 = /m3 = \M3 ||= C - Ev - Ab/ - C

C - Eb/=E\ - G ||= ||

C - Eb/=E\ - G ||= small sixfold blue ||

||= (P8/2, P4/3)

half-8ve, third-4th ||= ^^d2

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C/<span style="vertical-align: super;">3</span> ``=`` C# ||= P8/2 = vA4 = ^d5

P4/3 = \M2 = ``//``m2 ||= C - F#v=Gb^ - C

C - D\ - Eb/ - F ||= ||

C - D\ - Eb/ - F ||= large sixfold red ||

||= (P8/2, P5/3)

half-8ve,

third-5th ||= ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2 ||= C^<span style="vertical-align: super;">6</span> ``=`` B#<span style="vertical-align: super;">3</span> ||= P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5

P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3 ||= C - F<span style="vertical-align: super;">x</span>v<span style="vertical-align: super;">3</span>=Gbb^<span style="vertical-align: super;">3</span> C

C - D#vv - Fb^^ - G ||= ||

C - D#vv - Fb^^ - G ||= large sixfold yellow ||

||= " ||= ^^d2,

\\\m2 ||= C^^ = B#

C``///`` = Db ||= P8/2 = vA4 = ^d5

P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C

C - /D - \F - G ||= ||

C - /D - \F - G ||= 50/49 & 1029/1024 ||

||= (P8/2, P11/3)

half-8ve,

third-11th ||= v<span style="vertical-align: super;">6</span>M2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D ||= P8/2 = ^<span style="vertical-align: super;">3</span>4 = v<span style="vertical-align: super;">3</span>5

P11/3 = ^^4 = v<span style="vertical-align: super;">4</span>5 ||= C - F^<span style="vertical-align: super;">3</span>=Gv<span style="vertical-align: super;">3</span> - C

C - F^^ - Cvv - F ||= ||

C - F^^ - Cvv - F ||= large sixfold jade ||

||= (P8/3, P4/3)

third-

everything ||= ||= ||= ||= ||= ||

everything ||= v<span style="vertical-align: super;">3</span>d2,

\<span style="vertical-align: super;">3</span>A1 ||= C^<span style="vertical-align: super;">3 </span> ``=`` Dbb

C/3 ``=`` C# ||= P8/3 = ^M3 = vvd4

P4/3 = \M2 = ``//``m2

P5/3 = v/M2 ||= C - E^ - Abv - C

C - D\ - Eb/ - F

C - Dv/ - F^\ - G ||= 250/243 & 729/686

^1 = 64/63

/1 = 81/80 ||

||= " ||= ^<span style="vertical-align: super;">3</span>d2,

\<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` B#

C/<span style="vertical-align: super;">3</span> ``=`` Db ||= P8/3 = vM3 = ^^d4

P5/3 = /M2 = \\m3

P4/3 = v\M2 ||= C - Ev - Ab^ - C

C - D/ - F\ - G

C - Dv\ - Eb^/ - F ||= 128/125 & 1029/1024

^1 = 81/80

/1 = 64/63 ||

||= " ||= v<span style="vertical-align: super;">3</span>A1,

\<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` C#

C/3 ``=`` Db ||= P4/3 = vM2 = ^^m2

P5/3 = /M2 = \\m3

P8/3 = v/M3 ||= C - Dv - Eb^ - F

C - D/ - F\ - G

C - Ev/ - Ab^\ - C ||= 250/243 & 1029/1024

^1 = 81/80

/1 = 64/63 ||

||~ quarters ||~ ||~ ||~ ||~ ||~ ||

||= (P8/4, P5) ||= ^<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B# ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= C Ebv Gbvv=F#^^ A^ C ||= diminished,

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Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at http://xenharmonic.wikispaces.com/pergen+names

__**Extra stuff:**__

Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:

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c = -g

d = -r

~~__**Extra paragraphs:**__~~

~~Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).~~

~~For {P8/M, multigen/N}, an octave = M periods ± some number of enharmonics, and a multigen = N generators ± some number of enharmonics.~~

~~The first is based on the enharmonic's degree, which can be deduced from the pergen as follows:~~

The octave spans 7 steps. If the octave is split into M periods, each spanning x steps, x is roughly 7/M. The enharmonic, or some multiple of it, must span |Mx - 7| steps. Likewise, if the multigen is split into N generators, each spanning y steps, and S = the multigen's degree - 1, then y is roughly S/N, and the enharmonic spans |Ny - S| steps.

The enharmonic's degree depends on exactly how 7/M or S/N is rounded off, and alternate degrees are possible. For example, third-11th has S/N = 10/3 = 3 or 4, implying a generator that's a 4th or a 5th, and an enharmonic that's a 2nd or a 3rd. The lower degree is generally preferred. However, for single-comma temperaments, the enharmonic interval should be the same degree as the comma. So sometimes larger degrees are preferred.

For {P8/M, multigen/N}, there are two conditions on the enharmonic's degree, which may be mutually exclusive. If so, two unconventional accidental pairs (e.g. ups/downs and highs/lows) must be used, and each accidental pair has its own enharmonic interval.

~~For **{P8/M, P5}**, the enharmonic's degree = |M * round (7/M) - 7 + a*M| + 1 (a = 0, or 1 or -1 for the alternate enharmonics)~~

~~For {**P8, multigen/N}**, the enharmonic's degree = |N * round ((S/N) - S ± N| + 1, where S = the multigen's degree - 1~~

~~For **{P8/M, multigen/N}**, the enharmonic's degree = |M * round (7/M) - 7 ± M| + 1 = |N * round ((S/N) - S ± S| + 1~~

~~or, the 8ve's enharmonic = |M * round (7/M) - 7 ± M| + 1 and the multigen's enharmonic = |N * round ((S/N) - S ± N| +1~~

~~The 2nd restriction is based on the implied edo. The possible edos, and thus the possible enharmonics, can be deduced from the pergen.~~

For {P8/M, P5}, the octave is split into M periods. If the period has a 3-exponent of x, then the enharmonic interval's 3-exponent is Mx, and the implied edo is |Mx|. Thus half-8ve implies an even-numbered edo.

For {P8, multigen/N}, since the octave is unsplit, the only possible multigens are some voicing of the 5th, and the multigen's 3-exponent is ±1. If the generator has a 3-exponent of y, the enharmonic's 3-exponent is Ny ± 1, and the implied edo is |Ny ± 1|. Thus half-fourth and half-fifth both imply an odd-numbered edo.

For {P8/M, multigen/N}, there are two conditions on the enharmonic. If T is the 3-exponent of the multigen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd. Therefore there must be two accidental pairs, each with its own enharmonic interval. In the main table, this pergen is notated with both ups/downs and highs/lows. Since the 8ve and 4th are split, the 5th is too. Each interval has its own genchain. One of these is notated with ups/downs, another with highs/lows, and the third with both. The 3 possible ways of allocating the two accidental pairs are all listed. Furthermore, ups/downs can be exchanged for highs/lows.

For {P8/2, P5/3}, the edo = 2x = 3y ± 1. The edo must be even, thus y must be odd. Possible edos are 2, 4, 8, 10, 14, 16, 20, 22, 26, 28... The main table has ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2, which implies 26-edo. Most of the other edos aren't practical. 10 and 20 imply the m3. 22-edo suggests a d<span style="vertical-align: super;">3</span>4. The perchain would be C - E#^<span style="vertical-align: super;">3</span>=Abbv<span style="vertical-align: super;">3</span> - C.

~~For **{P8/M, P5}**, the implied edo = Mx~~

~~For {**P8, multigen/N}**, the implied edo = Ny ± 1 (the multigen is some voicing of the 5th)~~

~~For **{P8/M, multigen/N}**, the implied edo = Mx = Ny ± T, where T is the 3-exponent of the multigen,~~

~~or, the 8ve's implied edo = Mx and the multigen's implied edo = Ny ± F~~

As a side note, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.

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The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><h1 id="toc0"> </h1>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><h1 id="toc0"> </h1>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#toc0"> </a></div>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#toc0"> </a></div>

<div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

<div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

<div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>

<div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>

<div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>

<div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>

<div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>

<div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div>

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

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

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div>

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

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

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

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

<div style="margin-left: 2em;"><a href="#Further Discussion-Combining pergens">Combining pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Combining pergens">Combining pergens</a></div>

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

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

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

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

<div style="margin-left: 2em;"><a href="#Further Discussion-Misc notes">Misc notes</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Misc notes">Misc notes</a></div>

</div>

</div>

<h1 id="toc1"><a name="Definition"></a><u><strong>Definition</strong></u></h1>

<h1 id="toc1"><a name="Definition"></a><u><strong>Definition</strong></u></h1>

<br />

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Untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).<br />

<br />

<h1 id="toc2"><a name="Derivation"></a><u>Derivation</u></h1>

<h1 id="toc2"><a name="Derivation"></a><u>Derivation</u></h1>

<br />

For any comma containing primes 2 and 3, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents. The comma will split the octave into m parts, and if n &gt; m, it will split some 3-limit interval into n parts.<br />

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Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-5ths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-5th with red. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, gg7/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.<br />

<br />

<h1 id="toc3"><a name="Applications"></a><u>Applications</u></h1>

<h1 id="toc3"><a name="Applications"></a><u>Applications</u></h1>

<br />

One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit, and might not actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.<br />

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Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.<br />

<br />

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to ~~quarter~~-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.<br />

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.<br />

<br />

The enharmonic interval, or more briefly the <strong>enharmonic</strong>, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.<br />

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C - G implies ...Eb Bb F C G D A E B F# C#...<br />

C - Eb^=Ev - G implies ...F -- Ab^=Av -- C -- Eb^=Ev -- G -- Bb^=Bv -- D -- F^=F#v -- A -- C^=C#v -- E...<br />

If the octave is split, the table has a <strong>perchain</strong> (&quot;peer-chain&quot;, chain of periods) that shows the octave: In C -- F#v=Gb^ -- C~~, the last C is an octave above the first one.<br />~~

If the octave is split, the table has a <strong>perchain</strong> (&quot;peer-chain&quot;, chain of periods) that shows the octave: C -- F#v=Gb^ -- C<br />

~~<br />~~

The table lists several possible notations for each pergen. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination.<br />

~~<br />~~

~~(table is unfinished)~~<br />

<br />

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</td>

</td>

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C - Db^<span style="vertical-align: super;">3</span>=Ev<span style="vertical-align: super;">3</span> - F<br />

</td>

<td style="text-align: center;">sixfold jade<br />

</td>

</tr>

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C - D/=Eb\ - F<br />

</td>

</td>

</tr>

Line 1,539:

Line 1,521:

C - Eb/=E\ - G<br />

</td>

<td style="text-align: center;">small sixfold blue<br />

</td>

</tr>

Line 1,558:

Line 1,540:

C - D\ - Eb/ - F<br />

</td>

<td style="text-align: center;">large sixfold red<br />

</td>

</tr>

Line 1,576:

Line 1,558:

C - D#vv - Fb^^ - G<br />

</td>

<td style="text-align: center;">large sixfold yellow<br />

</td>

</tr>

Line 1,594:

Line 1,576:

C - /D - \F - G<br />

</td>

</td>

</tr>

Line 1,612:

Line 1,594:

C - F^^ - Cvv - F<br />

</td>

<td style="text-align: center;">large sixfold jade<br />

</td>

</tr>

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Line 1,602:

everything<br />

</td>

\<span style="vertical-align: super;">3</span>A1<br />

</td>

C/3 = C#<br />

</td>

P4/3 = \M2 = //m2<br />

P5/3 = v/M2<br />

</td>

C - D\ - Eb/ - F<br />

C - Dv/ - F^\ - G<br />

</td>

^1 = 64/63<br />

/1 = 81/80<br />

</td>

</tr>

<tr>

</td>

\<span style="vertical-align: super;">3</span>m2<br />

</td>

C/<span style="vertical-align: super;">3</span> = Db<br />

</td>

P5/3 = /M2 = \\m3<br />

P4/3 = v\M2<br />

</td>

C - D/ - F\ - G<br />

C - Dv\ - Eb^/ - F<br />

</td>

^1 = 81/80<br />

/1 = 64/63<br />

</td>

</tr>

<tr>

</td>

\<span style="vertical-align: super;">3</span>m2<br />

</td>

C/3 = Db<br />

</td>

P5/3 = /M2 = \\m3<br />

P8/3 = v/M3<br />

</td>

C - D/ - F\ - G<br />

C - Ev/ - Ab^\ - C<br />

</td>

^1 = 81/80<br />

/1 = 64/63<br />

</td>

</tr>

Line 1,650:

Line 1,684:

</td>

</td>

Line 1,665:

Line 1,699:

</td>

</td>

Line 1,679:

Line 1,713:

</td>

</td>

Line 1,693:

Line 1,727:

</td>

</td>

Line 1,707:

Line 1,741:

</td>

</td>

Line 2,134:

Line 2,168:

<br />

<h1 id="toc4"><a name="Further Discussion"></a><u>Further Discussion</u></h1>

<h1 id="toc4"><a name="Further Discussion"></a><u>Further Discussion</u></h1>

<br />

<h2 id="toc5"><a name="Further Discussion-Extremely large multigens"></a>Extremely large multigens</h2>

<h2 id="toc5"><a name="Further Discussion-Extremely large multigens"></a>Extremely large multigens</h2>

<br />

So far, the largest multigen has been a 12th. As the multigen fractions get larger, the multigen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen can be P4, P5, P11, P12, WWP4 or WWP5.<br />

<br />

<h2 id="toc6"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

<h2 id="toc6"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

<br />

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a <strong>single-split</strong> pergen. If it has two fractions, it's a <strong>double-split</strong> pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called <strong>single-pair</strong> notation because it adds only a single pair of accidentals to conventional notation. <strong>Double-pair</strong> notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs can be freely exchanged with highs/lows.<br />

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Line 2,182:

A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.<br />

<br />

<h2 id="toc7"><a name="Further Discussion-Finding an example temperament"></a>Finding an example temperament</h2>

<h2 id="toc7"><a name="Further Discussion-Finding an example temperament"></a>Finding an example temperament</h2>

<br />

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4<span class="nowrap">⋅</span>P and P8. If P is 6/5, the comma is 4<span class="nowrap">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4<span class="nowrap">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3<span class="nowrap">⋅</span>G - P4 = (10/9)^3 ÷ (4/3) = 250/243.<br />

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There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.<br />

<br />

<h2 id="toc8"><a name="Further Discussion-Ratio and cents of the accidentals"></a>Ratio and cents of the accidentals</h2>

<h2 id="toc8"><a name="Further Discussion-Ratio and cents of the accidentals"></a>Ratio and cents of the accidentals</h2>

<br />

In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64.<br />

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We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢ and ^1 = (A1)/2. But A1 = 100¢ + 7c, so ^1 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.<br />

<br />

<h2 id="toc9"><a name="Further Discussion-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

<h2 id="toc9"><a name="Further Discussion-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

<br />

There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:<br />

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A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3).<br />

<br />

Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = v<span style="vertical-align: super;">4</span>//ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:<br />

Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = v<span style="vertical-align: super;">4</span>//ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:<br />

One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v<span style="vertical-align: super;">12</span>m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /<span style="vertical-align: super;">3</span>d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling.<br />

Enharmonic = v<span style="vertical-align: super;">12</span>m3 = /<span style="vertical-align: super;">3</span>d2 = v<span style="vertical-align: super;">4</span>/m2 = v<span style="vertical-align: super;">4</span>\\A1. Period = \M3 = v<span style="vertical-align: super;">4</span>4 = //d4. Generator = ^\M3 = v<span style="vertical-align: super;">3</span>4 = ^//d4.<br />

Enharmonic = v<span style="vertical-align: super;">12</span>m3 = /<span style="vertical-align: super;">3</span>d2 = v<span style="vertical-align: super;">4</span>/m2 = v<span style="vertical-align: super;">4</span>\\A1. Period = \M3 = v<span style="vertical-align: super;">4</span>4 = //d4. Generator = ^\M3 = v<span style="vertical-align: super;">3</span>4 = ^//d4.<br />

<br />

This is a lot of math, but it only needs to be done once for each pergen.<br />

<br />

<h2 id="toc10"><a name="Further Discussion-Alternate enharmonics"></a>Alternate enharmonics</h2>

<h2 id="toc10"><a name="Further Discussion-Alternate enharmonics"></a>Alternate enharmonics</h2>

<br />

Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2.<br />

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Line 2,256:

</span><span style="display: block; text-align: center;">C -- Ev -- Ab^ -- C<br />

</span><span style="display: block; text-align: center;">P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4<br />

</span><span style="display: block; text-align: center;">P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4<br />

</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb//=D#\\ -- E\ -- F</span><br />

</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb//=D#\\ -- E\ -- F</span><br />

Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2. Thus the vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.<br />

<br />

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This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, tend to have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. So alternate enharmonics are needed sometimes.<br />

<br />

<h2 id="toc11"><a name="Further Discussion-Alternate keyspans and stepspans"></a>Alternate keyspans and stepspans</h2>

<h2 id="toc11"><a name="Further Discussion-Alternate keyspans and stepspans"></a>Alternate keyspans and stepspans</h2>

<br />

One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.<br />

<br />

<h2 id="toc12"><a name="Further Discussion-Chord names and scale names"></a>Chord names and scale names</h2>

<h2 id="toc12"><a name="Further Discussion-Chord names and scale names"></a>Chord names and scale names</h2>

<br />

Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.<br />

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<br />

<h2 id="toc13"><a name="Further Discussion-Combining pergens"></a>Combining pergens</h2>

<h2 id="toc13"><a name="Further Discussion-Combining pergens"></a>Combining pergens</h2>

<br />

Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).<br />

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However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.<br />

<br />

<h2 id="toc14"><a name="Further Discussion-Pergens and EDOs"></a>Pergens and EDOs</h2>

<h2 id="toc14"><a name="Further Discussion-Pergens and EDOs"></a>Pergens and EDOs</h2>

<br />

Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br />

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<br />

<h2 id="toc15"><a name="Further Discussion-Supplemental materials"></a>Supplemental materials</h2>

<h2 id="toc15"><a name="Further Discussion-Supplemental materials"></a>Supplemental materials</h2>

<br />

This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-something block.<br />

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(links coming soon)<br />

<br />

<h2 id="toc16"><a name="Further Discussion-Misc notes"></a>Misc notes</h2>

<h2 id="toc16"><a name="Further Discussion-Misc notes"></a>Misc notes</h2>

<br />

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a><br />

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a><br />

<br />

<u><strong>Extra stuff:</strong></u><br />

<br />

Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:<br />

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Line 3,094:

c = -g<br />

d = -r<br />

~~<br />~~

~~<u><strong>Extra paragraphs:</strong></u><br />~~

~~<br />~~

~~Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).<br />~~

~~<br />~~

~~For {P8/M, multigen/N}, an octave = M periods ± some number of enharmonics, and a multigen = N generators ± some number of enharmonics.<br />~~

~~<br />~~

~~The first is based on the enharmonic's degree, which can be deduced from the pergen as follows:<br />~~

~~<br />~~

The octave spans 7 steps. If the octave is split into M periods, each spanning x steps, x is roughly 7/M. The enharmonic, or some multiple of it, must span |Mx - 7| steps. Likewise, if the multigen is split into N generators, each spanning y steps, and S = the multigen's degree - 1, then y is roughly S/N, and the enharmonic spans |Ny - S| steps.<br />

~~<br />~~

The enharmonic's degree depends on exactly how 7/M or S/N is rounded off, and alternate degrees are possible. For example, third-11th has S/N = 10/3 = 3 or 4, implying a generator that's a 4th or a 5th, and an enharmonic that's a 2nd or a 3rd. The lower degree is generally preferred. However, for single-comma temperaments, the enharmonic interval should be the same degree as the comma. So sometimes larger degrees are preferred.<br />

~~<br />~~

For {P8/M, multigen/N}, there are two conditions on the enharmonic's degree, which may be mutually exclusive. If so, two unconventional accidental pairs (e.g. ups/downs and highs/lows) must be used, and each accidental pair has its own enharmonic interval.<br />

~~<br />~~

~~For <strong>{P8/M, P5}</strong>, the enharmonic's degree = |M * round (7/M) - 7 + a*M| + 1 (a = 0, or 1 or -1 for the alternate enharmonics)<br />~~

~~For {<strong>P8, multigen/N}</strong>, the enharmonic's degree = |N * round ((S/N) - S ± N| + 1, where S = the multigen's degree - 1<br />~~

~~For <strong>{P8/M, multigen/N}</strong>, the enharmonic's degree = |M * round (7/M) - 7 ± M| + 1 = |N * round ((S/N) - S ± S| + 1<br />~~

~~or, the 8ve's enharmonic = |M * round (7/M) - 7 ± M| + 1 and the multigen's enharmonic = |N * round ((S/N) - S ± N| +1<br />~~

~~<br />~~

~~The 2nd restriction is based on the implied edo. The possible edos, and thus the possible enharmonics, can be deduced from the pergen.<br />~~

~~<br />~~

For {P8/M, P5}, the octave is split into M periods. If the period has a 3-exponent of x, then the enharmonic interval's 3-exponent is Mx, and the implied edo is |Mx|. Thus half-8ve implies an even-numbered edo.<br />

~~<br />~~

For {P8, multigen/N}, since the octave is unsplit, the only possible multigens are some voicing of the 5th, and the multigen's 3-exponent is ±1. If the generator has a 3-exponent of y, the enharmonic's 3-exponent is Ny ± 1, and the implied edo is |Ny ± 1|. Thus half-fourth and half-fifth both imply an odd-numbered edo.<br />

~~<br />~~

For {P8/M, multigen/N}, there are two conditions on the enharmonic. If T is the 3-exponent of the multigen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd. Therefore there must be two accidental pairs, each with its own enharmonic interval. In the main table, this pergen is notated with both ups/downs and highs/lows. Since the 8ve and 4th are split, the 5th is too. Each interval has its own genchain. One of these is notated with ups/downs, another with highs/lows, and the third with both. The 3 possible ways of allocating the two accidental pairs are all listed. Furthermore, ups/downs can be exchanged for highs/lows.<br />

~~<br />~~

For {P8/2, P5/3}, the edo = 2x = 3y ± 1. The edo must be even, thus y must be odd. Possible edos are 2, 4, 8, 10, 14, 16, 20, 22, 26, 28... The main table has ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2, which implies 26-edo. Most of the other edos aren't practical. 10 and 20 imply the m3. 22-edo suggests a d<span style="vertical-align: super;">3</span>4. The perchain would be C - E#^<span style="vertical-align: super;">3</span>=Abbv<span style="vertical-align: super;">3</span> - C.<br />

~~<br />~~

~~For <strong>{P8/M, P5}</strong>, the implied edo = Mx<br />~~

~~For {<strong>P8, multigen/N}</strong>, the implied edo = Ny ± 1 (the multigen is some voicing of the 5th)<br />~~

~~For <strong>{P8/M, multigen/N}</strong>, the implied edo = Mx = Ny ± T, where T is the 3-exponent of the multigen,<br />~~

~~or, the 8ve's implied edo = Mx and the multigen's implied edo = Ny ± F<br />~~

~~<br />~~

<br />

As a side note, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.<br />

<br />

The LCM of the pergen's two splitting fractions is called the <strong>height</strong> of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</body></html></pre></div>

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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:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-13 00:57:51 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-13 02:34:39 UTC</tt>.<br>
-: The original revision id was <tt>624812535</tt>.<br>
+: The original revision id was <tt>624813333</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 127: / Line 127: @@
 Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.
-Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to quarter-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.
+Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.
 The enharmonic interval, or more briefly the **enharmonic**, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.
@@ Line 134: / Line 134: @@
 C - G implies ...Eb Bb F C G D A E B F# C#...
 C - Eb^=Ev - G implies ...F -- Ab^=Av -- C -- Eb^=Ev -- G -- Bb^=Bv -- D -- F^=F#v -- A -- C^=C#v -- E...
-If the octave is split, the table has a **perchain** ("peer-chain", chain of periods) that shows the octave: In C -- F#v=Gb^ -- C, the last C is an octave above the first one.
+If the octave is split, the table has a **perchain** ("peer-chain", chain of periods) that shows the octave: C -- F#v=Gb^ -- C
-The table lists several possible notations for each pergen. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination.
-(table is unfinished)
 ||~ pergen ||~ enharmonic
@@ Line 207: / Line 203: @@
 third-8ve ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented ||
 ||= (P8, P4/3)
-third-4th ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;3 ``=`` &lt;/span&gt;C# ||= P4/3 = ^^m2 = vM2 ||= C - Dv - Eb^ - F ||= porcupine ||
+third-4th ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;3 ``=`` &lt;/span&gt;C# ||= P4/3 = vM2 = ^^m2 ||= C - Dv - Eb^ - F ||= porcupine ||
 ||= (P8, P5/3)
 third-5th ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric ||
@@ Line 216: / Line 212: @@
 third-8ve, half-4th ||= v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;A2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` D# ||= P8/3 = ^^m3 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A4
 P4/2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M3 ||= C - Eb^^ - Avv - C
-C - Db^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Ev&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - F ||=   ||
+C - Db^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Ev&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - F ||= sixfold jade ||
 ||= " ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2,
 \\m2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` B#
 C``//`` = Db ||= P8/3 = vM3 = ^^d4
 P4/2 = /M2 = \m3 ||= C - Ev - Ab^ - C
-C - D/=Eb\ - F ||=   ||
+C - D/=Eb\ - F ||= 128/125 &amp; 49/48 ||
 ||= (P8/3, P5/2)
 third-8ve, half-5th ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2
@@ Line 227: / Line 223: @@
 C``//`` = C# ||= P8/3 = vM3 = ^^d4
 P5/2 = /m3 = \M3 ||= C - Ev - Ab/ - C
-C - Eb/=E\ - G ||=   ||
+C - Eb/=E\ - G ||= small sixfold blue ||
 ||= (P8/2, P4/3)
 half-8ve, third-4th ||= ^^d2
@@ Line 233: / Line 229: @@
 C/&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` C# ||= P8/2 = vA4 = ^d5
 P4/3 = \M2 = ``//``m2 ||= C - F#v=Gb^ - C
-C - D\ - Eb/ - F ||=   ||
+C - D\ - Eb/ - F ||= large sixfold red ||
 ||= (P8/2, P5/3)
 half-8ve,
 third-5th ||= ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= P8/2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;AA4 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd5
 P5/3 = vvA2 = ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||= C - F&lt;span style="vertical-align: super;"&gt;x&lt;/span&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Gbb^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; C
-C - D#vv - Fb^^ - G ||=   ||
+C - D#vv - Fb^^ - G ||= large sixfold yellow ||
 ||= " ||= ^^d2,
 \\\m2 ||= C^^ = B#
 C``///`` = Db ||= P8/2 = vA4 = ^d5
 P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C
-C - /D - \F - G ||=   ||
+C - /D - \F - G ||= 50/49 &amp; 1029/1024 ||
 ||= (P8/2, P11/3)
 half-8ve,
 third-11th ||= v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` D ||= P8/2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;5
 P11/3 = ^^4 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;5 ||= C - F^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Gv&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - C
-C - F^^ - Cvv - F ||=   ||
+C - F^^ - Cvv - F ||= large sixfold jade ||
 ||= (P8/3, P4/3)
 third-
-everything ||=   ||=   ||=   ||=   ||=   ||
+everything ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2,
+\&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` Dbb
+C/3 ``=`` C# ||= P8/3 = ^M3 = vvd4
+P4/3 = \M2 = ``//``m2
+P5/3 = v/M2 ||= C - E^ - Abv - C
+C - D\ - Eb/ - F
+C - Dv/ - F^\ - G ||= 250/243 &amp; 729/686
+^1 = 64/63
+/1 = 81/80 ||
+||= " ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2,
+\&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` B#
+C/&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` Db ||= P8/3 = vM3 = ^^d4
+P5/3 = /M2 = \\m3
+P4/3 = v\M2 ||= C - Ev - Ab^ - C
+C - D/ - F\ - G
+C - Dv\ - Eb^/ - F ||= 128/125 &amp; 1029/1024
+^1 = 81/80
+/1 = 64/63 ||
+||= " ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1,
+\&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` C#
+C/3 ``=`` Db ||= P4/3 = vM2 = ^^m2
+P5/3 = /M2 = \\m3
+P8/3 = v/M3 ||= C - Dv - Eb^ - F
+C - D/ - F\ - G
+C - Ev/ - Ab^\ - C ||= 250/243 &amp; 1029/1024
+^1 = 81/80
+/1 = 64/63 ||
 ||~ quarters ||~   ||~   ||~   ||~   ||~   ||
 ||= (P8/4, P5) ||= ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;d2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ``=`` B# ||= P8/4 = vm3 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A2 ||= C Ebv Gbvv=F#^^ A^ C ||= diminished,
@@ Line 537: / Line 559: @@
 Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at http://xenharmonic.wikispaces.com/pergen+names
+__**Extra stuff:**__
 Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:
@@ Line 548: / Line 576: @@
 c = -g
 d = -r
-__**Extra paragraphs:**__
-Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).
-For {P8/M, multigen/N}, an octave = M periods ± some number of enharmonics, and a multigen = N generators ± some number of enharmonics.
-The first is based on the enharmonic's degree, which can be deduced from the pergen as follows:
-The octave spans 7 steps. If the octave is split into M periods, each spanning x steps, x is roughly 7/M. The enharmonic, or some multiple of it, must span |Mx - 7| steps. Likewise, if the multigen is split into N generators, each spanning y steps, and S = the multigen's degree - 1, then y is roughly S/N, and the enharmonic spans |Ny - S| steps.
-The enharmonic's degree depends on exactly how 7/M or S/N is rounded off, and alternate degrees are possible. For example, third-11th has S/N = 10/3 = 3 or 4, implying a generator that's a 4th or a 5th, and an enharmonic that's a 2nd or a 3rd. The lower degree is generally preferred. However, for single-comma temperaments, the enharmonic interval should be the same degree as the comma. So sometimes larger degrees are preferred.
-For {P8/M, multigen/N}, there are two conditions on the enharmonic's degree, which may be mutually exclusive. If so, two unconventional accidental pairs (e.g. ups/downs and highs/lows) must be used, and each accidental pair has its own enharmonic interval.
-For **{P8/M, P5}**, the enharmonic's degree = |M * round (7/M) - 7 + a*M| + 1 (a = 0, or 1 or -1 for the alternate enharmonics)
-For {**P8, multigen/N}**, the enharmonic's degree = |N * round ((S/N) - S ± N| + 1, where S = the multigen's degree - 1
-For **{P8/M, multigen/N}**, the enharmonic's degree = |M * round (7/M) - 7 ± M| + 1 = |N * round ((S/N) - S ± S| + 1
-or, the 8ve's enharmonic = |M * round (7/M) - 7 ± M| + 1 and the multigen's enharmonic = |N * round ((S/N) - S ± N| +1
-The 2nd restriction is based on the implied edo. The possible edos, and thus the possible enharmonics, can be deduced from the pergen.
-For {P8/M, P5}, the octave is split into M periods. If the period has a 3-exponent of x, then the enharmonic interval's 3-exponent is Mx, and the implied edo is |Mx|. Thus half-8ve implies an even-numbered edo.
-For {P8, multigen/N}, since the octave is unsplit, the only possible multigens are some voicing of the 5th, and the multigen's 3-exponent is ±1. If the generator has a 3-exponent of y, the enharmonic's 3-exponent is Ny ± 1, and the implied edo is |Ny ± 1|. Thus half-fourth and half-fifth both imply an odd-numbered edo.
-For {P8/M, multigen/N}, there are two conditions on the enharmonic. If T is the 3-exponent of the multigen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd. Therefore there must be two accidental pairs, each with its own enharmonic interval. In the main table, this pergen is notated with both ups/downs and highs/lows. Since the 8ve and 4th are split, the 5th is too. Each interval has its own genchain. One of these is notated with ups/downs, another with highs/lows, and the third with both. The 3 possible ways of allocating the two accidental pairs are all listed. Furthermore, ups/downs can be exchanged for highs/lows.
-For {P8/2, P5/3}, the edo = 2x = 3y ± 1. The edo must be even, thus y must be odd. Possible edos are 2, 4, 8, 10, 14, 16, 20, 22, 26, 28... The main table has ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2, which implies 26-edo. Most of the other edos aren't practical. 10 and 20 imply the m3. 22-edo suggests a d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4. The perchain would be C - E#^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Abbv&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - C.
-For **{P8/M, P5}**, the implied edo = Mx
-For {**P8, multigen/N}**, the implied edo = Ny ± 1 (the multigen is some voicing of the 5th)
-For **{P8/M, multigen/N}**, the implied edo = Mx = Ny ± T, where T is the 3-exponent of the multigen,
-or, the 8ve's implied edo = Mx and the multigen's implied edo = Ny ± F
 As a side note, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.
@@ Line 595: / Line 581: @@
 The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</pre></div>
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-&lt;!-- ws:end:WikiTextTocRule:77 --&gt;&lt;!-- ws:start:WikiTextTocRule:78: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding a notation for a pergen"&gt;Finding a notation for a pergen&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:84 --&gt;&lt;!-- ws:start:WikiTextTocRule:85: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding a notation for a pergen"&gt;Finding a notation for a pergen&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:78 --&gt;&lt;!-- ws:start:WikiTextTocRule:79: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate enharmonics"&gt;Alternate enharmonics&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:85 --&gt;&lt;!-- ws:start:WikiTextTocRule:86: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate enharmonics"&gt;Alternate enharmonics&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:79 --&gt;&lt;!-- ws:start:WikiTextTocRule:80: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate keyspans and stepspans"&gt;Alternate keyspans and stepspans&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:86 --&gt;&lt;!-- ws:start:WikiTextTocRule:87: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate keyspans and stepspans"&gt;Alternate keyspans and stepspans&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:80 --&gt;&lt;!-- ws:start:WikiTextTocRule:81: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Chord names and scale names"&gt;Chord names and scale names&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:87 --&gt;&lt;!-- ws:start:WikiTextTocRule:88: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Chord names and scale names"&gt;Chord names and scale names&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:81 --&gt;&lt;!-- ws:start:WikiTextTocRule:82: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Combining pergens"&gt;Combining pergens&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:88 --&gt;&lt;!-- ws:start:WikiTextTocRule:89: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Combining pergens"&gt;Combining pergens&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:82 --&gt;&lt;!-- ws:start:WikiTextTocRule:83: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and EDOs"&gt;Pergens and EDOs&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:89 --&gt;&lt;!-- ws:start:WikiTextTocRule:90: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and EDOs"&gt;Pergens and EDOs&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:83 --&gt;&lt;!-- ws:start:WikiTextTocRule:84: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Supplemental materials"&gt;Supplemental materials&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:90 --&gt;&lt;!-- ws:start:WikiTextTocRule:91: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Supplemental materials"&gt;Supplemental materials&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:84 --&gt;&lt;!-- ws:start:WikiTextTocRule:85: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Misc notes"&gt;Misc notes&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:91 --&gt;&lt;!-- ws:start:WikiTextTocRule:92: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Misc notes"&gt;Misc notes&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:85 --&gt;&lt;!-- ws:start:WikiTextTocRule:86: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:92 --&gt;&lt;!-- ws:start:WikiTextTocRule:93: --&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:86 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:end:WikiTextTocRule:93 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:43:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:43 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 &lt;br /&gt;
@@ Line 864: / Line 850: @@
 Untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:45:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:45 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 For any comma containing primes 2 and 3, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents. The comma will split the octave into m parts, and if n &amp;gt; m, it will split some 3-limit interval into n parts.&lt;br /&gt;
@@ Line 1,149: / Line 1,135: @@
 Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-5ths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-5th with red. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, gg7/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &amp;gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:47:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:47 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit, and might not actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.&lt;br /&gt;
@@ Line 1,165: / Line 1,151: @@
 Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.&lt;br /&gt;
 &lt;br /&gt;
-Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to quarter-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.&lt;br /&gt;
+Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.&lt;br /&gt;
 &lt;br /&gt;
 The enharmonic interval, or more briefly the &lt;strong&gt;enharmonic&lt;/strong&gt;, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.&lt;br /&gt;
@@ Line 1,172: / Line 1,158: @@
 C - G implies ...Eb Bb F C G D A E B F# C#...&lt;br /&gt;
 C - Eb^=Ev - G implies ...F -- Ab^=Av -- C -- Eb^=Ev -- G -- Bb^=Bv -- D -- F^=F#v -- A -- C^=C#v -- E...&lt;br /&gt;
-If the octave is split, the table has a &lt;strong&gt;perchain&lt;/strong&gt; (&amp;quot;peer-chain&amp;quot;, chain of periods) that shows the octave: In C -- F#v=Gb^ -- C, the last C is an octave above the first one.&lt;br /&gt;
+If the octave is split, the table has a &lt;strong&gt;perchain&lt;/strong&gt; (&amp;quot;peer-chain&amp;quot;, chain of periods) that shows the octave: C -- F#v=Gb^ -- C&lt;br /&gt;
-&lt;br /&gt;
-The table lists several possible notations for each pergen. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination.&lt;br /&gt;
-&lt;br /&gt;
-(table is unfinished)&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,437: / Line 1,419: @@
          &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3 &lt;!-- ws:start:WikiTextRawRule:09:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:09 --&gt; &lt;/span&gt;C#&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;P4/3 = ^^m2 = vM2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;P4/3 = vM2 = ^^m2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - Dv - Eb^ - F&lt;br /&gt;
@@ Line 1,502: / Line 1,484: @@
 C - Db^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Ev&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;sixfold jade&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,520: / Line 1,502: @@
 C - D/=Eb\ - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;128/125 &amp;amp; 49/48&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,539: / Line 1,521: @@
 C - Eb/=E\ - G&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;small sixfold blue&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,558: / Line 1,540: @@
 C - D\ - Eb/ - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;large sixfold red&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,576: / Line 1,558: @@
 C - D#vv - Fb^^ - G&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;large sixfold yellow&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,594: / Line 1,576: @@
 C - /D - \F - G&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;50/49 &amp;amp; 1029/1024&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,612: / Line 1,594: @@
 C - F^^ - Cvv - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;large sixfold jade&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,620: / Line 1,602: @@
 everything&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2,&lt;br /&gt;
+\&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:024:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:024 --&gt; Dbb&lt;br /&gt;
+C/3 &lt;!-- ws:start:WikiTextRawRule:025:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:025 --&gt; C#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P8/3 = ^M3 = vvd4&lt;br /&gt;
+P4/3 = \M2 = &lt;!-- ws:start:WikiTextRawRule:026:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:026 --&gt;m2&lt;br /&gt;
+P5/3 = v/M2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C - E^ - Abv - C&lt;br /&gt;
+C - D\ - Eb/ - F&lt;br /&gt;
+C - Dv/ - F^\ - G&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;250/243 &amp;amp; 729/686&lt;br /&gt;
+^1 = 64/63&lt;br /&gt;
+/1 = 81/80&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2,&lt;br /&gt;
+\&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:027:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:027 --&gt; B#&lt;br /&gt;
+C/&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:028:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:028 --&gt; Db&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P8/3 = vM3 = ^^d4&lt;br /&gt;
+P5/3 = /M2 = \\m3&lt;br /&gt;
+P4/3 = v\M2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C - Ev - Ab^ - C&lt;br /&gt;
+C - D/ - F\ - G&lt;br /&gt;
+C - Dv\ - Eb^/ - F&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;128/125 &amp;amp; 1029/1024&lt;br /&gt;
+^1 = 81/80&lt;br /&gt;
+/1 = 64/63&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1,&lt;br /&gt;
+\&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:029:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:029 --&gt; C#&lt;br /&gt;
+C/3 &lt;!-- ws:start:WikiTextRawRule:030:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:030 --&gt; Db&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;P4/3 = vM2 = ^^m2&lt;br /&gt;
+P5/3 = /M2 = \\m3&lt;br /&gt;
+P8/3 = v/M3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C - Dv - Eb^ - F&lt;br /&gt;
+C - D/ - F\ - G&lt;br /&gt;
+C - Ev/ - Ab^\ - C&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;250/243 &amp;amp; 1029/1024&lt;br /&gt;
+^1 = 81/80&lt;br /&gt;
+/1 = 64/63&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,650: / Line 1,684: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;d2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:024:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:024 --&gt; B#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:031:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:031 --&gt; B#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;P8/4 = vm3 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A2&lt;br /&gt;
@@ Line 1,665: / Line 1,699: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:025:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:025 --&gt; B##&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:032:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:032 --&gt; B##&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;P4/4 = ^m2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;AA1&lt;br /&gt;
@@ Line 1,679: / Line 1,713: @@
          &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:026:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:026 --&gt; C#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:033:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:033 --&gt; C#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;P5/4 = vM2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2&lt;br /&gt;
@@ Line 1,693: / Line 1,727: @@
          &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:027:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:027 --&gt; Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:034:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:034 --&gt; Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;P11/4 = ^M3 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd5&lt;br /&gt;
@@ Line 1,707: / Line 1,741: @@
          &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:028:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:028 --&gt; Db&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:035:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:035 --&gt; Db&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;P12/4 = v4 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M3&lt;br /&gt;
@@ Line 2,134: / Line 2,168: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Further Discussion"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;&lt;u&gt;Further Discussion&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:49:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Further Discussion"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:49 --&gt;&lt;u&gt;Further Discussion&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Further Discussion-Extremely large multigens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;Extremely large multigens&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:51:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Further Discussion-Extremely large multigens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:51 --&gt;Extremely large multigens&lt;/h2&gt;
   &lt;br /&gt;
 So far, the largest multigen has been a 12th. As the multigen fractions get larger, the multigen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &amp;quot;W&amp;quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen can be P4, P5, P11, P12, WWP4 or WWP5.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Further Discussion-Singles and doubles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;Singles and doubles&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:53:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Further Discussion-Singles and doubles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:53 --&gt;Singles and doubles&lt;/h2&gt;
   &lt;br /&gt;
 If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a &lt;strong&gt;single-split&lt;/strong&gt; pergen. If it has two fractions, it's a &lt;strong&gt;double-split&lt;/strong&gt; pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called &lt;strong&gt;single-pair&lt;/strong&gt; notation because it adds only a single pair of accidentals to conventional notation. &lt;strong&gt;Double-pair&lt;/strong&gt; notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs can be freely exchanged with highs/lows.&lt;br /&gt;
@@ Line 2,148: / Line 2,182: @@
 A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Further Discussion-Finding an example temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;Finding an example temperament&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:55:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Further Discussion-Finding an example temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:55 --&gt;Finding an example temperament&lt;/h2&gt;
   &lt;br /&gt;
 To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P and P8. If P is 6/5, the comma is 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P - P8 = (6/5)&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P = (2/1) · (7/6)&lt;span style="vertical-align: super;"&gt;-4&lt;/span&gt;. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - P4 = (10/9)^3 ÷ (4/3) = 250/243.&lt;br /&gt;
@@ Line 2,170: / Line 2,204: @@
 There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Further Discussion-Ratio and cents of the accidentals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;Ratio and cents of the accidentals&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:57:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Further Discussion-Ratio and cents of the accidentals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:57 --&gt;Ratio and cents of the accidentals&lt;/h2&gt;
   &lt;br /&gt;
 In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64.&lt;br /&gt;
@@ Line 2,176: / Line 2,210: @@
 We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢ and ^1 = (A1)/2. But A1 = 100¢ + 7c, so ^1 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Further Discussion-Finding a notation for a pergen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;Finding a notation for a pergen&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:59:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Further Discussion-Finding a notation for a pergen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:59 --&gt;Finding a notation for a pergen&lt;/h2&gt;
   &lt;br /&gt;
 There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:&lt;br /&gt;
@@ Line 2,203: / Line 2,237: @@
 A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3).&lt;br /&gt;
 &lt;br /&gt;
-Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\d3 = 2·vv\m2, and E - E' = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;&lt;!-- ws:start:WikiTextRawRule:029:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:029 --&gt;ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:&lt;br /&gt;
+Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\d3 = 2·vv\m2, and E - E' = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;&lt;!-- ws:start:WikiTextRawRule:036:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:036 --&gt;ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:&lt;br /&gt;
 &lt;span style="display: block; text-align: center;"&gt;P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- E^=Fv\ -- Ab/=A\ -- C^/=Dbv -- F&lt;/span&gt;&lt;br /&gt;
 One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling.&lt;br /&gt;
-Enharmonic = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3 = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;/m2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\A1. Period = \M3 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;4 = &lt;!-- ws:start:WikiTextRawRule:030:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:030 --&gt;d4. Generator = ^\M3 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4 = ^&lt;!-- ws:start:WikiTextRawRule:031:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:031 --&gt;d4.&lt;br /&gt;
+Enharmonic = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3 = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;/m2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\A1. Period = \M3 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;4 = &lt;!-- ws:start:WikiTextRawRule:037:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:037 --&gt;d4. Generator = ^\M3 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4 = ^&lt;!-- ws:start:WikiTextRawRule:038:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:038 --&gt;d4.&lt;br /&gt;
 &lt;br /&gt;
 &lt;span style="display: block; text-align: center;"&gt;P1 — \M3 — \\A5=/m6 — P8&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C — E\ — Ab/ — C&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;8=v/m9 — F&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C — E^\ — Ab^^/=Avv\ — Dbv/ — F&lt;/span&gt;&lt;br /&gt;
 This is a lot of math, but it only needs to be done once for each pergen.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Further Discussion-Alternate enharmonics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;Alternate enharmonics&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:61:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Further Discussion-Alternate enharmonics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:61 --&gt;Alternate enharmonics&lt;/h2&gt;
   &lt;br /&gt;
 Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;A2, which is an improvement but still awkward. The period is ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m3 and the generator is v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2.&lt;br /&gt;
@@ Line 2,222: / Line 2,256: @@
 &lt;span style="display: block; text-align: center;"&gt;P1 -- vM3 -- ^m6 -- P8&lt;br /&gt;
 &lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Ev -- Ab^ -- C&lt;br /&gt;
-&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 -- /m2 -- &lt;!-- ws:start:WikiTextRawRule:032:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:032 --&gt;d3=\\A2 -- \M3 -- P4&lt;br /&gt;
+&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 -- /m2 -- &lt;!-- ws:start:WikiTextRawRule:039:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:039 --&gt;d3=\\A2 -- \M3 -- P4&lt;br /&gt;
-&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Db/ -- Ebb&lt;!-- ws:start:WikiTextRawRule:033:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:033 --&gt;=D#\\ -- E\ -- F&lt;/span&gt;&lt;br /&gt;
+&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Db/ -- Ebb&lt;!-- ws:start:WikiTextRawRule:040:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:040 --&gt;=D#\\ -- E\ -- F&lt;/span&gt;&lt;br /&gt;
 Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2. Thus the vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 2,232: / Line 2,266: @@
 This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, tend to have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. So alternate enharmonics are needed sometimes.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Further Discussion-Alternate keyspans and stepspans"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;Alternate keyspans and stepspans&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:63:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Further Discussion-Alternate keyspans and stepspans"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:63 --&gt;Alternate keyspans and stepspans&lt;/h2&gt;
   &lt;br /&gt;
 One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Further Discussion-Chord names and scale names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;Chord names and scale names&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:65:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Further Discussion-Chord names and scale names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:65 --&gt;Chord names and scale names&lt;/h2&gt;
   &lt;br /&gt;
 Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt; page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.&lt;br /&gt;
@@ Line 2,716: / Line 2,750: @@
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 Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).&lt;br /&gt;
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 However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Pergens and EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;Pergens and EDOs&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:69:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Pergens and EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:69 --&gt;Pergens and EDOs&lt;/h2&gt;
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 Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.&lt;br /&gt;
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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-something block.&lt;br /&gt;
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-Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at &lt;!-- ws:start:WikiTextUrlRule:3788:http://xenharmonic.wikispaces.com/pergen+names --&gt;&lt;a href="http://xenharmonic.wikispaces.com/pergen+names"&gt;http://xenharmonic.wikispaces.com/pergen+names&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:3788 --&gt;&lt;br /&gt;
+Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at &lt;!-- ws:start:WikiTextUrlRule:3829:http://xenharmonic.wikispaces.com/pergen+names --&gt;&lt;a href="http://xenharmonic.wikispaces.com/pergen+names"&gt;http://xenharmonic.wikispaces.com/pergen+names&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:3829 --&gt;&lt;br /&gt;
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 Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:&lt;br /&gt;
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 d = -r&lt;br /&gt;
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-&lt;u&gt;&lt;strong&gt;Extra paragraphs:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
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-Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).&lt;br /&gt;
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-For {P8/M, multigen/N}, an octave = M periods ± some number of enharmonics, and a multigen = N generators ± some number of enharmonics.&lt;br /&gt;
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-The first is based on the enharmonic's degree, which can be deduced from the pergen as follows:&lt;br /&gt;
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-The octave spans 7 steps. If the octave is split into M periods, each spanning x steps, x is roughly 7/M. The enharmonic, or some multiple of it, must span |Mx - 7| steps. Likewise, if the multigen is split into N generators, each spanning y steps, and S = the multigen's degree - 1, then y is roughly S/N, and the enharmonic spans |Ny - S| steps.&lt;br /&gt;
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-The enharmonic's degree depends on exactly how 7/M or S/N is rounded off, and alternate degrees are possible. For example, third-11th has S/N = 10/3 = 3 or 4, implying a generator that's a 4th or a 5th, and an enharmonic that's a 2nd or a 3rd. The lower degree is generally preferred. However, for single-comma temperaments, the enharmonic interval should be the same degree as the comma. So sometimes larger degrees are preferred.&lt;br /&gt;
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-For {P8/M, multigen/N}, there are two conditions on the enharmonic's degree, which may be mutually exclusive. If so, two unconventional accidental pairs (e.g. ups/downs and highs/lows) must be used, and each accidental pair has its own enharmonic interval.&lt;br /&gt;
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-For &lt;strong&gt;{P8/M, P5}&lt;/strong&gt;, the enharmonic's degree = |M * round (7/M) - 7 + a*M| + 1 (a = 0, or 1 or -1 for the alternate enharmonics)&lt;br /&gt;
-For {&lt;strong&gt;P8, multigen/N}&lt;/strong&gt;, the enharmonic's degree = |N * round ((S/N) - S ± N| + 1, where S = the multigen's degree - 1&lt;br /&gt;
-For &lt;strong&gt;{P8/M, multigen/N}&lt;/strong&gt;, the enharmonic's degree = |M * round (7/M) - 7 ± M| + 1 = |N * round ((S/N) - S ± S| + 1&lt;br /&gt;
-or, the 8ve's enharmonic = |M * round (7/M) - 7 ± M| + 1 and the multigen's enharmonic = |N * round ((S/N) - S ± N| +1&lt;br /&gt;
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-The 2nd restriction is based on the implied edo. The possible edos, and thus the possible enharmonics, can be deduced from the pergen.&lt;br /&gt;
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-For {P8/M, P5}, the octave is split into M periods. If the period has a 3-exponent of x, then the enharmonic interval's 3-exponent is Mx, and the implied edo is |Mx|. Thus half-8ve implies an even-numbered edo.&lt;br /&gt;
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-For {P8, multigen/N}, since the octave is unsplit, the only possible multigens are some voicing of the 5th, and the multigen's 3-exponent is ±1. If the generator has a 3-exponent of y, the enharmonic's 3-exponent is Ny ± 1, and the implied edo is |Ny ± 1|. Thus half-fourth and half-fifth both imply an odd-numbered edo.&lt;br /&gt;
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-For {P8/M, multigen/N}, there are two conditions on the enharmonic. If T is the 3-exponent of the multigen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd. Therefore there must be two accidental pairs, each with its own enharmonic interval. In the main table, this pergen is notated with both ups/downs and highs/lows. Since the 8ve and 4th are split, the 5th is too. Each interval has its own genchain. One of these is notated with ups/downs, another with highs/lows, and the third with both. The 3 possible ways of allocating the two accidental pairs are all listed. Furthermore, ups/downs can be exchanged for highs/lows.&lt;br /&gt;
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-For {P8/2, P5/3}, the edo = 2x = 3y ± 1. The edo must be even, thus y must be odd. Possible edos are 2, 4, 8, 10, 14, 16, 20, 22, 26, 28... The main table has ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2, which implies 26-edo. Most of the other edos aren't practical. 10 and 20 imply the m3. 22-edo suggests a d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4. The perchain would be C - E#^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Abbv&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - C.&lt;br /&gt;
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-For &lt;strong&gt;{P8/M, P5}&lt;/strong&gt;, the implied edo = Mx&lt;br /&gt;
-For {&lt;strong&gt;P8, multigen/N}&lt;/strong&gt;, the implied edo = Ny ± 1 (the multigen is some voicing of the 5th)&lt;br /&gt;
-For &lt;strong&gt;{P8/M, multigen/N}&lt;/strong&gt;, the implied edo = Mx = Ny ± T, where T is the 3-exponent of the multigen,&lt;br /&gt;
-or, the 8ve's implied edo = Mx and the multigen's implied edo = Ny ± F&lt;br /&gt;
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 As a side note, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.&lt;br /&gt;
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 The LCM of the pergen's two splitting fractions is called the &lt;strong&gt;height&lt;/strong&gt; of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.&lt;/body&gt;&lt;/html&gt;</pre></div>