Pergen names: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-22 01:34:58 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-22 02:23:58 UTC</tt>.

: The original revision id was <tt>~~622161181~~</tt>.

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P5/2 = ^m3 = vM3

P8/2 = v/A4 = ^\d5

``= ^/P4 = v\P5`` ||= \\m2,

``=`` ^/P4 ``=`` v\P5 ||= \\m2,

vvA1,

^^\\d2,

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[//Question: What to do if the edo's 5th falls in the sweet spot? Example?//]

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic, thus the edo implies the enharmonic.

||||~ bare enharmonic interval ||~ implied edo ||~ edo's 5th ||~ upping range ||~ downing range ||~ if 5th is just ||

||||~ bare enharmonic interval ||~ implied edo ||~ edo's 5th ||~ upping range ||~ downing range ||~ if the 5th is just ||

||= M2 ||= C - D ||= 2-edo ||= 600¢ ||= none ||= all ||= downed ||

||= m3 ||= C - Eb ||= 3-edo ||= 800¢ ||= none ||= all ||= downed ||

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[//Question: how many entries does this table realistically need?//]

As a ~~corollary~~, 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.

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.

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. __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.

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For {P8/M, multi-gen/N}, there are two conditions on the enharmonic. If T is the 3-factor of the multi-gen, 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 and two ~~implied edos~~, ~~and~~

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. In addition, ups/downs can be exchanged for highs/lows, making 6 possibilities.

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

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 ^6d32, which implies 26-edo. Most of the other edos aren't practical. 10 and 20 imply the m3. 22-edo suggests a triple-dim 4th. The octave genchain would be C - E#^3=Abbv3 - C. Seeing the same pitch represented as both E and A is rather disconcerting. For this reason, enharmonics that are unisons or 2nds are preferred.

~~For {P8~~, ~~multi~~-~~gen~~/N}, the ~~implied~~ edo = ~~Ny ± 1~~

~~For {P8/M, multi~~-~~gen~~/~~N}, the implied edo~~ = Mx = ~~Ny ± T, where T is the~~ 3-~~factor of~~ the ~~multi-gen~~,

or~~, the 1st implied edo = Mx and the 2nd implied edo = Ny ± 1~~

[//Question: if the edo is 14, is the enharmonic 2 A1's = AA1?//]

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 in the main table.

Even if a pergen with two fractions __can__ be notated with a single accidental pair, a second pair may be preferred. Again, analogous to 22-edo, a notation that causes familiar chords to be misspelled is not very welcoming.

[//Example?//]

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

For {**P8, multi-gen/N}**, the implied edo = Ny ± 1

For **{P8/M, multi-gen/N}**, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,

or, the 8ve's implied edo = Mx and the multi-gen's implied edo = Ny ± T

The main table lists all possible notations for each pergen, following the above rules. 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 in the main table.

[//Question: how to find the notation for multi-comma tempers?//]

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(to be continued)</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 names</title></head><body><a href="#Definition">Definition</a> | <a href="#Derivation">Derivation</a> | <a href="#Applications">Applications</a>

<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 names</title></head><body><a href="#Definition">Definition</a> | <a href="#Derivation">Derivation</a> | <a href="#Applications">Applications</a>

<h1 id="toc0"><a name="Definition"></a>Definition</h1>

<h1 id="toc0"><a name="Definition"></a>Definition</h1>

A pergen (pronounced &quot;peer-gen&quot;) is a way of identifying a rank-2 or rank-3 regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.

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The preferred pergen for untempered just intonation is the octave, the fifth, and a list of commas, each containing only one higher prime: {P8, P5, 81/80, 64/63, ...}. These commas are called notational commas. They determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is universal agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th. Note that the choice of 11's notational comma affects the mapping of other 11-limit commas.

<h1 id="toc1"><a name="Derivation"></a>Derivation</h1>

<h1 id="toc1"><a name="Derivation"></a>Derivation</h1>

For any comma containing primes 2 and 3, let M = the GCD of all its prime factors other than the 2-factor, and let N = the GCD of all its higher-prime factors. The comma will split the octave into M parts, and if N &gt; M, it will split some other 3-limit interval into N parts. Therefore a comma with only two factors, one of which is the 2-factor, always splits the octave (unless the other factor is ±1, e.g. 32/31). And a comma with only one higher-prime factor will always split something, unless that factor is ±1.

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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 multi-gen, the multi-gen2. A double half-fifth is a fifth = (-1, 1, 0), and this gives us (-4, 0, 2)/2 = 7/4. The fraction disappears, the multi-gen 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-fifths makes 64/63 = r1. The pergen is {P8, P5/2, r1} = half-fifth with red. 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.

<h1 id="toc2"><a name="Applications"></a>Applications</h1>

<h1 id="toc2"><a name="Applications"></a>Applications</h1>

Pergens allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>. And certain rank-2 temperaments require another additional pair. One possibility is highs and lows, written / and \. v\D is down-low D, and /P5 is high-five. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.

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P5/2 = ^m3 = vM3

P8/2 = v/A4 = ^\d5

= ~~^/P4 = v\P5~~

= ^/P4 = v\P5

</td>

<td style="text-align: center;">\\m2,

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vv\\M2

</td>

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

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

C^^ = C#

C^^// = D

C^^// = D

</td>

<td style="text-align: center;">C - D/=Eb\ - F,

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</td>

<td style="text-align: center;">C^^ = B#

C// = Db

C// = Db

C^^// = C#

C^^// = C#

</td>

<td style="text-align: center;">C - F#v=Gb^ - C,

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</td>

<td style="text-align: center;">C^^ = B#

C// = C#

C// = C#

C^^\\ = B

</td>

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<td style="text-align: center;">^3d2

</td>

<td style="text-align: center;">C^3 = B#

<td style="text-align: center;">C^3 = B#

</td>

<td style="text-align: center;">C - Ev - Ab^ - C

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<td style="text-align: center;">v3A1

</td>

<td style="text-align: center;">C^3 = C#

<td style="text-align: center;">C^3 = C#

</td>

<td style="text-align: center;">C - Dv - Eb^ - F

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<td style="text-align: center;">v3m2

</td>

<td style="text-align: center;">C^3 = Db

<td style="text-align: center;">C^3 = Db

</td>

<td style="text-align: center;">C - D^ - Fv - G

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<td style="text-align: center;">^3dd2

</td>

<td style="text-align: center;">C^3 = B##

<td style="text-align: center;">C^3 = B##

</td>

<td style="text-align: center;">C - F#v - Cb^ - F

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<td style="text-align: center;">v3M2

</td>

<td style="text-align: center;">C^3 = D

<td style="text-align: center;">C^3 = D

</td>

<td style="text-align: center;">C F^ Cv F

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<td style="text-align: center;">^6d32

</td>

<td style="text-align: center;">C^6 = B#3

<td style="text-align: center;">C^6 = B#3

</td>

<td style="text-align: center;">C - Fxv3=Gbb^3 C

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</td>

<td style="text-align: center;">C^^ = B#

C/// = Db

C/// = Db

</td>

<td style="text-align: center;">C - F#v=Gb^ - C

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<td style="text-align: center;">^4d2

</td>

<td style="text-align: center;">C^4 = B#

<td style="text-align: center;">C^4 = B#

</td>

<td style="text-align: center;">C Ebv Gbvv A^ C

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<td style="text-align: center;">^4dd2

</td>

<td style="text-align: center;">C^4 = B##

<td style="text-align: center;">C^4 = B##

</td>

<td style="text-align: center;">C Db^ Ebb^^ Ev F

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<td style="text-align: center;">v4A1

</td>

<td style="text-align: center;">C^4 = C#

<td style="text-align: center;">C^4 = C#

</td>

<td style="text-align: center;">C Dv Evv F^ G

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<td style="text-align: center;">v4dd3

</td>

<td style="text-align: center;">C^4 = Eb3

<td style="text-align: center;">C^4 = Eb3

</td>

<td style="text-align: center;">C E^ G#^^ Dbv F

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<td style="text-align: center;">v4m2

</td>

<td style="text-align: center;">C^4 = Db

<td style="text-align: center;">C^4 = Db

</td>

<td style="text-align: center;">C Fv Bbvv D^ G

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<td style="text-align: center;">^5d32

</td>

<td style="text-align: center;">C^5 = B#3

<td style="text-align: center;">C^5 = B#3

</td>

<td style="text-align: center;">C - Db^ - Ebb^ -

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[Question: What to do if the edo's 5th falls in the sweet spot? Example?]

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic, thus the edo implies the enharmonic.

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<th>downing range

</th>

<th>if 5th is just

<th>if the 5th is just

</th>

</tr>

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[Question: how many entries does this table realistically need?]

As a ~~corollary~~, 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.

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.

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. 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.

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For {P8/M, multi-gen/N}, there are two conditions on the enharmonic. If T is the 3-factor of the multi-gen, 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 and two ~~implied~~ edos, and

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. In addition, ups/downs can be exchanged for highs/lows, making 6 possibilities.

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 ^6d32, which implies 26-edo. Most of the other edos aren't practical. 10 and 20 imply the m3. 22-edo suggests a triple-dim 4th. The octave genchain would be C - E#^3=Abbv3 - C. Seeing the same pitch represented as both E and A is rather disconcerting. For this reason, enharmonics that are unisons or 2nds are preferred.

[Question: if the edo is 14, is the enharmonic 2 A1's = AA1?]

Even if a pergen with two fractions can be notated with a single accidental pair, a second pair may be preferred. Again, analogous to 22-edo, a notation that causes familiar chords to be misspelled is not very welcoming.

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

[Example?]

~~For {P8, multi-gen/N}, the implied edo = Ny ± 1~~<~~br /~~>

~~For {P8/M, multi-gen/N}, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,~~

~~or, the 1st implied edo = Mx and the 2nd implied edo = Ny ± 1~~

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

For {P8, multi-gen/N}, the implied edo = Ny ± 1

For {P8/M, multi-gen/N}, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,

or, the 8ve's implied edo = Mx and the multi-gen's implied edo = Ny ± T

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 in the main table.

The main table lists all possible notations for each pergen, following the above rules. 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 in the main table.

[Question: how to find the notation for multi-comma tempers?]

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-22 01:34:58 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-22 02:23:58 UTC</tt>.<br>
-: The original revision id was <tt>622161181</tt>.<br>
+: The original revision id was <tt>622162431</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 157: / Line 157: @@
 P5/2 = ^m3 = vM3
 P8/2 = v/A4 = ^\d5
-``= ^/P4 = v\P5`` ||= \\m2,
+``=`` ^/P4 ``=`` v\P5 ||= \\m2,
 vvA1,
 ^^\\d2,
@@ Line 236: / Line 236: @@
 [//Question: What to do if the edo's 5th falls in the sweet spot? Example?//]
-Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic.
+Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic, thus the edo implies the enharmonic.
-||||~ bare enharmonic interval ||~ implied edo ||~ edo's 5th ||~ upping range ||~ downing range ||~ if 5th is just ||
+||||~ bare enharmonic interval ||~ implied edo ||~ edo's 5th ||~ upping range ||~ downing range ||~ if the 5th is just ||
 ||= M2 ||= C - D ||= 2-edo ||= 600¢ ||= none ||= all ||= downed ||
 ||= m3 ||= C - Eb ||= 3-edo ||= 800¢ ||= none ||= all ||= downed ||
@@ Line 252: / Line 252: @@
 [//Question: how many entries does this table realistically need?//]
-As a corollary, 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.
+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.
 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. __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.
@@ Line 265: / Line 265: @@
 For {P8/M, multi-gen/N}, there are two conditions on the enharmonic. If T is the 3-factor of the multi-gen, 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 and two implied edos, and
+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. In addition, ups/downs can be exchanged for highs/lows, making 6 possibilities.
-For {P8/M, P5}, the implied edo = Mx
+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 triple-dim 4th. The octave genchain 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. Seeing the same pitch represented as both E and A is rather disconcerting. For this reason, enharmonics that are unisons or 2nds are preferred.
-For {P8, multi-gen/N}, the implied edo = Ny ± 1
-For {P8/M, multi-gen/N}, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,
-or, the 1st implied edo = Mx and the 2nd implied edo = Ny ± 1
+[//Question: if the edo is 14, is the enharmonic 2 A1's = AA1?//]
-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 in the main table.
+Even if a pergen with two fractions __can__ be notated with a single accidental pair, a second pair may be preferred. Again, analogous to 22-edo, a notation that causes familiar chords to be misspelled is not very welcoming.
+[//Example?//]
+For **{P8/M, P5}**, the implied edo = Mx
+For {**P8, multi-gen/N}**, the implied edo = Ny ± 1
+For **{P8/M, multi-gen/N}**, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,
+or, the 8ve's implied edo = Mx and the multi-gen's implied edo = Ny ± T
+The main table lists all possible notations for each pergen, following the above rules. 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 in the main table.
 [//Question: how to find the notation for multi-comma tempers?//]
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 (to be continued)</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">&lt;html&gt;&lt;head&gt;&lt;title&gt;pergen names&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:25:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;
+<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;pergen names&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:26:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt; | &lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:19:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 A &lt;strong&gt;pergen&lt;/strong&gt; (pronounced &amp;quot;peer-gen&amp;quot;) is a way of identifying a rank-2 or rank-3 regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.&lt;br /&gt;
@@ Line 530: / Line 537: @@
 The preferred pergen for untempered just intonation is the octave, the fifth, and a list of commas, each containing only one higher prime: {P8, P5, 81/80, 64/63, ...}. These commas are called &lt;strong&gt;notational commas&lt;/strong&gt;. They determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is universal agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th. Note that the choice of 11's notational comma affects the mapping of other 11-limit commas.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:21:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:21 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&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 its prime factors other than the 2-factor, and let N = the GCD of all its higher-prime factors. The comma will split the octave into M parts, and if N &amp;gt; M, it will split some other 3-limit interval into N parts. Therefore a comma with only two factors, one of which is the 2-factor, always splits the octave (unless the other factor is ±1, e.g. 32/31). And a comma with only one higher-prime factor will always split something, unless that factor is ±1.&lt;br /&gt;
@@ Line 816: / Line 823: @@
 Again, period = P8 and gen1 = P5/2. Gen2 = (-3, -1, 2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multi-gen, the multi-gen2. A double half-fifth is a fifth = (-1, 1, 0), and this gives us (-4, 0, 2)/2 = 7/4. The fraction disappears, the multi-gen 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-fifths makes 64/63 = r1. The pergen is {P8, P5/2, r1} = half-fifth with red. 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:23:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:23 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 Pergens allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;. And certain rank-2 temperaments require another additional pair. One possibility is highs and lows, written / and \. v\D is down-low D, and /P5 is high-five. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.&lt;br /&gt;
@@ Line 1,004: / Line 1,011: @@
 P5/2 = ^m3 = vM3&lt;br /&gt;
 P8/2 = v/A4 = ^\d5&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRawRule:00:``= ^/P4 = v\P5`` --&gt;= ^/P4 = v\P5&lt;!-- ws:end:WikiTextRawRule:00 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRawRule:00:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:00 --&gt; ^/P4 &lt;!-- ws:start:WikiTextRawRule:01:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:01 --&gt; v\P5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;\\m2,&lt;br /&gt;
@@ Line 1,011: / Line 1,018: @@
 vv\\M2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C&lt;!-- ws:start:WikiTextRawRule:01:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:01 --&gt; = Db&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C&lt;!-- ws:start:WikiTextRawRule:02:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:02 --&gt; = Db&lt;br /&gt;
 C^^ = C#&lt;br /&gt;
-C^^&lt;!-- ws:start:WikiTextRawRule:02:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:02 --&gt; = D&lt;br /&gt;
+C^^&lt;!-- ws:start:WikiTextRawRule:03:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:03 --&gt; = D&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - D/=Eb\ - F,&lt;br /&gt;
@@ Line 1,040: / Line 1,047: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C^^ = B#&lt;br /&gt;
-C&lt;!-- ws:start:WikiTextRawRule:03:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:03 --&gt; = Db&lt;br /&gt;
+C&lt;!-- ws:start:WikiTextRawRule:04:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:04 --&gt; = Db&lt;br /&gt;
-C^^&lt;!-- ws:start:WikiTextRawRule:04:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:04 --&gt; = C#&lt;br /&gt;
+C^^&lt;!-- ws:start:WikiTextRawRule:05:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:05 --&gt; = C#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - F#v=Gb^ - C,&lt;br /&gt;
@@ Line 1,066: / Line 1,073: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C^^ = B#&lt;br /&gt;
-C&lt;!-- ws:start:WikiTextRawRule:05:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:05 --&gt; = C#&lt;br /&gt;
+C&lt;!-- ws:start:WikiTextRawRule:06:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:06 --&gt; = C#&lt;br /&gt;
 C^^\\ = B&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,103: / Line 1,110: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;3&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;3 &lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:06:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:06 --&gt; B#&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:07:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:07 --&gt; B#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - Ev - Ab^ - C&lt;br /&gt;
@@ Line 1,120: / Line 1,127: @@
          &lt;td style="text-align: center;"&gt;v&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;!-- ws:start:WikiTextRawRule:07:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:07 --&gt; &lt;/span&gt;C#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3 &lt;!-- ws:start:WikiTextRawRule:08:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:08 --&gt; &lt;/span&gt;C#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - Dv - Eb^ - F&lt;br /&gt;
@@ Line 1,137: / Line 1,144: @@
          &lt;td style="text-align: center;"&gt;v&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:08:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:08 --&gt; Db&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:09:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:09 --&gt; Db&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - D^ - Fv - G&lt;br /&gt;
@@ Line 1,154: / Line 1,161: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;3&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;3&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:09:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:09 --&gt; B##&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:010:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:010 --&gt; B##&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - F#v - Cb^ - F&lt;br /&gt;
@@ Line 1,170: / Line 1,177: @@
          &lt;td style="text-align: center;"&gt;v&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:010:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:010 --&gt; D&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:011:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:011 --&gt; D&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C F^ Cv F&lt;br /&gt;
@@ Line 1,235: / Line 1,242: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:011:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:011 --&gt; B#&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;6&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:012:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:012 --&gt; B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;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&lt;br /&gt;
@@ Line 1,255: / Line 1,262: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C^^ = B#&lt;br /&gt;
-C&lt;!-- ws:start:WikiTextRawRule:012:``///`` --&gt;///&lt;!-- ws:end:WikiTextRawRule:012 --&gt; = Db&lt;br /&gt;
+C&lt;!-- ws:start:WikiTextRawRule:013:``///`` --&gt;///&lt;!-- ws:end:WikiTextRawRule:013 --&gt; = Db&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - F#v=Gb^ - C&lt;br /&gt;
@@ Line 1,320: / Line 1,327: @@
          &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:013:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:013 --&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:014:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:014 --&gt; B#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C Ebv Gbvv A^ C&lt;br /&gt;
@@ Line 1,337: / Line 1,344: @@
          &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:014:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:014 --&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:015:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:015 --&gt; B##&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C Db^ Ebb^^ Ev F&lt;br /&gt;
@@ Line 1,353: / Line 1,360: @@
          &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:015:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:015 --&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:016:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:016 --&gt; C#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C Dv Evv F^ G&lt;br /&gt;
@@ Line 1,369: / Line 1,376: @@
          &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:016:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:016 --&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:017:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:017 --&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;C E^ G#^^ Dbv F&lt;br /&gt;
@@ Line 1,385: / Line 1,392: @@
          &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:017:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:017 --&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:018:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:018 --&gt; Db&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C Fv Bbvv D^ G&lt;br /&gt;
@@ Line 1,497: / Line 1,504: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:018:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:018 --&gt; B#&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;5&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:019:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:019 --&gt; B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - Db^ - Ebb^ -&lt;br /&gt;
@@ Line 1,532: / Line 1,539: @@
 [&lt;em&gt;Question: What to do if the edo's 5th falls in the sweet spot? Example?&lt;/em&gt;]&lt;br /&gt;
 &lt;br /&gt;
-Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic.&lt;br /&gt;
+Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic, thus the edo implies the enharmonic.&lt;br /&gt;
@@ Line 1,547: / Line 1,554: @@
          &lt;th&gt;downing range&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;if 5th is just&lt;br /&gt;
+         &lt;th&gt;if the 5th is just&lt;br /&gt;
 &lt;/th&gt;
      &lt;/tr&gt;
@@ Line 1,731: / Line 1,738: @@
 [&lt;em&gt;Question: how many entries does this table realistically need?&lt;/em&gt;]&lt;br /&gt;
 &lt;br /&gt;
-As a corollary, 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;
+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;
 &lt;br /&gt;
 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. &lt;u&gt;The enharmonic interval's number of ups or downs is equal to the height&lt;/u&gt;. 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;br /&gt;
@@ Line 1,744: / Line 1,751: @@
 &lt;br /&gt;
 For {P8/M, multi-gen/N}, there are two conditions on the enharmonic. If T is the 3-factor of the multi-gen, 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.&lt;br /&gt;
-Therefore there must be two accidental pairs, each and two implied edos, and &lt;br /&gt;
+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. In addition, ups/downs can be exchanged for highs/lows, making 6 possibilities.&lt;br /&gt;
+&lt;br /&gt;
+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 triple-dim 4th. The octave genchain 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. Seeing the same pitch represented as both E and A is rather disconcerting. For this reason, enharmonics that are unisons or 2nds are preferred.&lt;br /&gt;
+&lt;br /&gt;
+[&lt;em&gt;Question: if the edo is 14, is the enharmonic 2 A1's = AA1?&lt;/em&gt;]&lt;br /&gt;
+&lt;br /&gt;
+Even if a pergen with two fractions &lt;u&gt;can&lt;/u&gt; be notated with a single accidental pair, a second pair may be preferred. Again, analogous to 22-edo, a notation that causes familiar chords to be misspelled is not very welcoming.&lt;br /&gt;
 &lt;br /&gt;
-For {P8/M, P5}, the implied edo = Mx&lt;br /&gt;
+[&lt;em&gt;Example?&lt;/em&gt;]&lt;br /&gt;
-For {P8, multi-gen/N}, the implied edo = Ny ± 1&lt;br /&gt;
-For {P8/M, multi-gen/N}, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,&lt;br /&gt;
-or, the 1st implied edo = Mx and the 2nd implied edo = Ny ± 1&lt;br /&gt;
 &lt;br /&gt;
+For &lt;strong&gt;{P8/M, P5}&lt;/strong&gt;, the implied edo = Mx&lt;br /&gt;
+For {&lt;strong&gt;P8, multi-gen/N}&lt;/strong&gt;, the implied edo = Ny ± 1&lt;br /&gt;
+For &lt;strong&gt;{P8/M, multi-gen/N}&lt;/strong&gt;, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,&lt;br /&gt;
+or, the 8ve's implied edo = Mx and the multi-gen's implied edo = Ny ± T&lt;br /&gt;
 &lt;br /&gt;
-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 in the main table.&lt;br /&gt;
+The main table lists all possible notations for each pergen, following the above rules. 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 in the main table.&lt;br /&gt;
 &lt;br /&gt;
 [&lt;em&gt;Question: how to find the notation for multi-comma tempers?&lt;/em&gt;]&lt;br /&gt;