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>2017-11-26 15:27:44 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-26 16:50:58 UTC</tt>.

: The original revision id was <tt>~~622435625~~</tt>.

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[//Question: does n ever need to range more widely?//]

**Rank-2 pergen from the ~~mapping~~: ~~P =~~ P8/x, ~~G =~~ (nz - y, x) / xz**

**Rank-2 pergen from the [(x, 0), (y, z)] matrix: {P8/x, (nz-y, x)/xz}**

Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7|x31.com]] gives us this matrix:

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||~ 3/1 ||= 0 ||= 1 ||= -1 || ||

||~ 7/1 ||= 0 ||= 0 ||= 2 || /2 ||

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 > 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.

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 > 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.

=__Applications__=

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One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names.

Also, 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, [[Ups and Downs Notation|ups and downs]]. 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.

Also, 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.

Most rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. 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.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But because the schisma does not map to a unison, this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result doesn't need ups and downs.

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}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting fraction.

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. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting fraction.

The enharmonic interval 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, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic interval.

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||= " ||= P8/2 = ^P4 = vP5 ||= vvM2 ||= C^^ = D ||= C - F^=Gv - C ||= 128/121,

^1 = 33/32 ||= " ||

|| " || P8/2 = vAA4 = ^dd5 || ^^d32 || C^^ = B#3 || C - F##v=Gbb^ - C || || " ||

||= " ||= P8/2 = vAA4 = ^dd5 ||= ^^d32 ||= C^^ = B#3 ||= C - F##v=Gbb^ - C ||= ||= " ||

|| " || || ^^d52 || C^^ = B#5 || || || ||

||= " ||= ||= ^^d52 ||= C^^ = B#5 ||= ||= ||= ||

||= {P8, P4/2} ||= P4/2 = ^M2 = vm3 ||= vvm2 ||= C^^ = Db ||= C - D^=Ebv - F ||= semaphore

^1 = 64/63 ||= 14, 15*, 18b*, 19, 20*,

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30* ||

||= " ||= P4/2 = vA2 = ^d3 ||= ^^dd2 ||= C^^ = B## ||= C - D#v=Ebb^ - F ||= ||= " ||

|| " || P4/2 = vAA2 = ^dd3 || ^^d42 || C^^ = B#4 || C - D##v=Eb3^ - F || || " ||

||= " ||= P4/2 = vAA2 = ^dd3 ||= ^^d42 ||= C^^ = B#4 ||= C - D##v=Eb3^ - F ||= ||= " ||

||= {P8, P5/2} ||= P5/2 = ^m3 = vM3 ||= vvA1 ||= C^^ = C# ||= C - Eb^=Ev - G ||= mohajira

^1 = 33/32 ||= 14*, 17, 18b, 20*, 21*,

24, 27, 28*, 30*, 31 ||

|| " || P5/2 = ^A2 = vd4 || vvdd3 || C^^ = Eb3 || || || ||

||= " ||= P5/2 = ^A2 = vd4 ||= vvdd3 ||= C^^ = Eb3 ||= ||= ||= ||

||= {P8/2, P4/2} ||= P4/2 = /M2 = \m3

P5/2 = ^m3 = vM3

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- E#vv - F#v - G ||= ||= ||

||= " ||= ||= v5dd3 ||= ||= ||= ||= ||

~~[//Question: how to find all possible pergens?//]~~

Removing the ups and downs from an enharmonic interval makes a "bare" conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the tipping point. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.

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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-exponent of the bare enharmonic, thus the edo implies the enharmonic.

||||~ bare enharmonic interval ||~ implied edo ||~ edo's 5th ||~ upping range ||~ downing range ||~ if the 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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(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> | <a href="#Explanations">Explanations</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> | <a href="#Explanations">Explanations</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, ...}. The higher prime's exponent in the monzo must be 1 or -1. These commas are called notational commas. They are not necessarily tempered out, they merely 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 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 other 3-limit interval into N parts. Therefore a comma with only two primes, one of which is 2, always splits the octave (unless the other prime's exponent is ±1, e.g. 32/31). And a comma with only one higher prime will always split something, unless that prime's exponent is ±1.

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[Question: does n ever need to range more widely?]

Rank-2 pergen from the ~~mapping~~: ~~P =~~ P8/x, ~~G =~~ (nz - y, x) / xz

Rank-2 pergen from the [(x, 0), (y, z)] matrix: {P8/x, (nz-y, x)/xz}

Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7" rel="nofollow">x31.com</a> gives us this matrix:

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</table>

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.

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>

One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names.

Also, 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.

Also, 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.

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.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But because the schisma does not map to a unison, this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result doesn't need ups and downs.

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}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting fraction.

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. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting fraction.

The enharmonic interval 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, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic interval.

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</tr>

<tr>

<td>&quot;

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

</td>

<td>P8/2 = vAA4 = ^dd5

<td style="text-align: center;">P8/2 = vAA4 = ^dd5

</td>

<td>^^d32

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

</td>

<td>C^^ = B#3

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

</td>

<td>C - F##v=Gbb^ - C

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

</td>

<td>

</td>

<td>&quot;

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

</td>

</tr>

<tr>

<td>&quot;

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

</td>

<td>

</td>

<td>^^d52

<td style="text-align: center;">^^d52

</td>

<td>C^^ = B#5

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

</td>

<td>

</td>

<td>

</td>

<td>

</td>

</tr>

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</tr>

<tr>

<td>&quot;

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

</td>

<td>P4/2 = vAA2 = ^dd3

<td style="text-align: center;">P4/2 = vAA2 = ^dd3

</td>

<td>^^d42

<td style="text-align: center;">^^d42

</td>

<td>C^^ = B#4

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

</td>

<td>C - D##v=Eb3^ - F

<td style="text-align: center;">C - D##v=Eb3^ - F

</td>

<td>

</td>

<td>&quot;

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

</td>

</tr>

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</tr>

<tr>

<td>&quot;

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

</td>

<td>P5/2 = ^A2 = vd4

<td style="text-align: center;">P5/2 = ^A2 = vd4

</td>

<td>vvdd3

<td style="text-align: center;">vvdd3

</td>

<td>C^^ = Eb3

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

</td>

<td>

</td>

<td>

</td>

<td>

</td>

</tr>

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</table>

~~ ~~

~~[Question: how to find all possible pergens?] ~~

Removing the ups and downs from an enharmonic interval makes a &quot;bare&quot; conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &quot;sweet spot&quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the tipping point. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore ups and downs may need to be swapped, depending on the size of the 5th in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.

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<tr>

<th colspan="2">bare enharmonic interval

<th colspan="2">bare enharmonic interval

</th>

<th>implied edo

<th>implied edo

</th>

<th>edo's 5th

</th>

<th>upping range

<th>upping range

</th>

<th>downing range

<th>downing range

</th>

<th>if the 5th is just

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<h1 id="toc3"><a name="Explanations"></a>Explanations</h1>

<h1 id="toc3"><a name="Explanations"></a>Explanations</h1>

Each enharmonic interval implies a different notation. If every pergen could use every enharmonic, there would be an overwhelming choice of notations! Fortunately, not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic.

@@ 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>2017-11-26 15:27:44 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-26 16:50:58 UTC</tt>.<br>
-: The original revision id was <tt>622435625</tt>.<br>
+: The original revision id was <tt>622441807</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 76: / Line 76: @@
 [//Question: does n ever need to range more widely?//]
-**Rank-2 pergen from the mapping: P = P8/x, G = (nz - y, x) / xz**
+**Rank-2 pergen from the [(x, 0), (y, z)] matrix: {P8/x, (nz-y, x)/xz}**
 Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7|x31.com]] gives us this matrix:
@@ Line 109: / Line 109: @@
 ||~ 3/1 ||= 0 ||= 1 ||= -1 ||   ||
 ||~ 7/1 ||= 0 ||= 0 ||= 2 || /2 ||
-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.
+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.
 =__Applications__=
@@ Line 115: / Line 115: @@
 One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names.
-Also, 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, [[Ups and Downs Notation|ups and downs]]. 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.
+Also, 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.
+Most rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. 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.
 Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But because the schisma does not map to a unison, this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result doesn't need ups and downs.
-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}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting fraction.
+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. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting fraction.
 The enharmonic interval 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, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic interval.
@@ Line 156: / Line 158: @@
 ||= " ||= P8/2 = ^P4 = vP5 ||= vvM2 ||= C^^ = D ||= C - F^=Gv - C ||= 128/121,
 ^1 = 33/32 ||= " ||
-|| " || P8/2 = vAA4 = ^dd5 || ^^d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 || C^^ = B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; || C - F##v=Gbb^ - C ||   || " ||
+||= " ||= P8/2 = vAA4 = ^dd5 ||= ^^d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^^ = B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= C - F##v=Gbb^ - C ||=   ||= " ||
-|| " ||   || ^^d&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;2 || C^^ = B#&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt; ||   ||   ||   ||
+||= " ||=   ||= ^^d&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;2 ||= C^^ = B#&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt; ||=   ||=   ||=   ||
 ||= {P8, P4/2} ||= P4/2 = ^M2 = vm3 ||= vvm2 ||= C^^ = Db ||= C - D^=Ebv - F ||= semaphore
 ^1 = 64/63 ||= 14, 15*, 18b*, 19, 20*,
@@ Line 163: / Line 165: @@
 * ||
 ||= " ||= P4/2 = vA2 = ^d3 ||= ^^dd2 ||= C^^ = B## ||= C - D#v=Ebb^ - F ||=   ||= " ||
-|| " || P4/2 = vAA2 = ^dd3 || ^^d&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;2 || C^^ = B#&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; || C - D##v=Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;^ - F ||   || " ||
+||= " ||= P4/2 = vAA2 = ^dd3 ||= ^^d&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;2 ||= C^^ = B#&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ||= C - D##v=Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;^ - F ||=   ||= " ||
 ||= {P8, P5/2} ||= P5/2 = ^m3 = vM3 ||= vvA1 ||= C^^ = C# ||= C - Eb^=Ev - G ||= mohajira
 ^1 = 33/32 ||= 14*, 17, 18b, 20*, 21*,
 , 27, 28*, 30*, 31 ||
-|| " || P5/2 = ^A2 = vd4 || vvdd3 || C^^ = Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||   ||   ||   ||
+||= " ||= P5/2 = ^A2 = vd4 ||= vvdd3 ||= C^^ = Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||=   ||=   ||=   ||
 ||= {P8/2, P4/2} ||= P4/2 = /M2 = \m3
 P5/2 = ^m3 = vM3
@@ Line 243: / Line 245: @@
 - E#vv - F#v - G ||=   ||=   ||
 ||= " ||=   ||= v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;dd3 ||=   ||=   ||=   ||=   ||
-[//Question: how to find all possible pergens?//]
 Removing the ups and downs from an enharmonic interval makes a "bare" conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the tipping point. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.
@@ Line 251: / Line 251: @@
 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-exponent of the bare enharmonic, thus the edo implies the enharmonic.
-||||~ bare enharmonic interval ||~ implied edo ||~ edo's 5th ||~ upping range ||~ downing range ||~ if the 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 319: / Line 319: @@
 (to be continued)</pre></div>
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-<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:28:&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:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt; | &lt;a href="#Explanations"&gt;Explanations&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&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:33:&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:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt; | &lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt; | &lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt; | &lt;a href="#Explanations"&gt;Explanations&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:33 --&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;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:25:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:25 --&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 583: / Line 583: @@
 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, ...}. The higher prime's exponent in the monzo must be 1 or -1. These commas are called &lt;strong&gt;notational commas&lt;/strong&gt;. They are not necessarily tempered out, they merely 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: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;!-- ws:start:WikiTextHeadingRule:27:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:27 --&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 other 3-limit interval into N parts. Therefore a comma with only two primes, one of which is 2, always splits the octave (unless the other prime's exponent is ±1, e.g. 32/31). And a comma with only one higher prime will always split something, unless that prime's exponent is ±1.&lt;br /&gt;
@@ Line 605: / Line 605: @@
 [&lt;em&gt;Question: does n ever need to range more widely?&lt;/em&gt;]&lt;br /&gt;
 &lt;br /&gt;
-&lt;strong&gt;Rank-2 pergen from the mapping: P = P8/x, G = (nz - y, x) / xz&lt;/strong&gt;&lt;br /&gt;
+&lt;strong&gt;Rank-2 pergen from the [(x, 0), (y, z)] matrix: {P8/x, (nz-y, x)/xz}&lt;/strong&gt;&lt;br /&gt;
 &lt;br /&gt;
 Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. &lt;a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;amp;limit=7" rel="nofollow"&gt;x31.com&lt;/a&gt; gives us this matrix:&lt;br /&gt;
@@ Line 867: / Line 867: @@
 &lt;/table&gt;
-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;
+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;
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+&lt;!-- ws:start:WikiTextHeadingRule:29:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:29 --&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.&lt;br /&gt;
 &lt;br /&gt;
-Also, 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;
+Also, 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. &lt;br /&gt;
+&lt;br /&gt;
+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;
 &lt;br /&gt;
 Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But because the schisma does not map to a unison, this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result doesn't need ups and downs.&lt;br /&gt;
 &lt;br /&gt;
-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}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting fraction.&lt;br /&gt;
+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. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting fraction.&lt;br /&gt;
 &lt;br /&gt;
 The enharmonic interval 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, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic interval.&lt;br /&gt;
@@ Line 1,008: / Line 1,010: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;P8/2 = vAA4 = ^dd5&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;P8/2 = vAA4 = ^dd5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;^^d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;^^d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;C^^ = B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^^ = B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;C - F##v=Gbb^ - C&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C - F##v=Gbb^ - C&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;^^d&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;^^d&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;C^^ = B#&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^^ = B#&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,075: / Line 1,077: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;P4/2 = vAA2 = ^dd3&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;P4/2 = vAA2 = ^dd3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;^^d&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;^^d&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;C^^ = B#&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^^ = B#&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;C - D##v=Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;^ - F&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C - D##v=Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;^ - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,109: / Line 1,111: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;P5/2 = ^A2 = vd4&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;P5/2 = ^A2 = vd4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;vvdd3&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;vvdd3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;C^^ = Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^^ = Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,651: / Line 1,653: @@
 &lt;/table&gt;
-&lt;br /&gt;
-[&lt;em&gt;Question: how to find all possible pergens?&lt;/em&gt;]&lt;br /&gt;
 &lt;br /&gt;
 Removing the ups and downs from an enharmonic interval makes a &amp;quot;bare&amp;quot; conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &amp;quot;sweet spot&amp;quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the tipping point. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore &lt;u&gt;&lt;strong&gt;ups and downs may need to be swapped, depending on the size of the 5th&lt;/strong&gt;&lt;/u&gt; in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.&lt;br /&gt;
@@ Line 1,663: / Line 1,663: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;th colspan="2"&gt;bare enharmonic interval&lt;br /&gt;
+         &lt;th colspan="2"&gt;bare enharmonic interval&lt;!-- ws:start:WikiTextRawRule:020:`` `` --&gt; &lt;!-- ws:end:WikiTextRawRule:020 --&gt;&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;implied edo&lt;br /&gt;
+         &lt;th&gt;implied edo&lt;!-- ws:start:WikiTextRawRule:021:`` `` --&gt; &lt;!-- ws:end:WikiTextRawRule:021 --&gt;&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;edo's 5th&lt;br /&gt;
+         &lt;th&gt;edo's 5th&lt;!-- ws:start:WikiTextRawRule:022:`` `` --&gt; &lt;!-- ws:end:WikiTextRawRule:022 --&gt;&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;upping range&lt;br /&gt;
+         &lt;th&gt;upping range&lt;!-- ws:start:WikiTextRawRule:023:`` `` --&gt; &lt;!-- ws:end:WikiTextRawRule:023 --&gt;&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;downing range&lt;br /&gt;
+         &lt;th&gt;downing range&lt;!-- ws:start:WikiTextRawRule:024:`` `` --&gt; &lt;!-- ws:end:WikiTextRawRule:024 --&gt;&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;if the 5th is just&lt;br /&gt;
@@ Line 1,858: / Line 1,858: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Explanations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Explanations&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:31:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Explanations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:31 --&gt;Explanations&lt;/h1&gt;
   &lt;br /&gt;
 Each enharmonic interval implies a different notation. If every pergen could use every enharmonic, there would be an overwhelming choice of notations! Fortunately, not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic.&lt;br /&gt;