Pergen names: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-22 00:01:15 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-22 00:06:52 UTC</tt>.

: The original revision id was <tt>~~622159545~~</tt>.

: The original revision id was <tt>622159615</tt>.

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

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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vvA1,

^^\\d2,

vv\\M2 ||= C //= Db~~,//~~

vv\\M2 ||= C``//`` = Db

//C^^ = C#//

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

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\\m2,

vv\\A1 ||= C^^ = B#

C //= Db~~,//~~

C``//`` = Db

//C^^// = C# ||= C - F#v=Gb^ - C,

C - D/=Eb\ - F,

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||= {P8/2, P4/3} ||= ||= ||= ||= ||= ||= 14, 22, 28*, 30* ||

||= {P8/2, P5/3} ||= P8/2 = v3AA4 = ^3dd5

P5/3 = vvA2 = ^4dd3 ||= ^6d32 ||= C^6 ``=`` B#3 || C - Fxv3=Gbb^3 CC - D#vv - Fb^^ - G ||= ||= 16, 20*, 26, 30* ||

P5/3 = vvA2 = ^4dd3 ||= ^6d32 ||= C^6 ``=`` B#3 || C - Fxv3=Gbb^3 C

C - D#vv - Fb^^ - G ||= ||= 16, 20*, 26, 30* ||

||= " ||= P8/2 = vA4 = ^d5

P5/3 = /M2 = \\m3 ||= ^^d2,

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

C/// = Db ||= C - F#v=Gb^ - C

C``///`` = Db ||= C - F#v=Gb^ - C

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

||= {P8/2, P11/3} ||= ||= ||= ||= ||= ||= ||

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||~ fifths ||~ ||~ ||~ ||~ ||~ ||~ ||

||= {P8/5, P5} ||= P8/5 = ||= m2 ||= ||= ||= ||= ||

||= {P8, P5/5} ||= P5/5 = ^m2 = v4A31 ||= ^5d32 ||= C^5 ``=`` B#3 || C - Db^ - Ebb^ -- E#vv - F#v - G ||= ||= ||

||= {P8, P5/5} ||= P5/5 = ^m2 = v4A31 ||= ^5d32 ||= C^5 ``=`` B#3 || C - Db^ - Ebb^ -

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

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

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

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

Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) 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¢. 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. Thus __**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 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.

//Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) 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¢. 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. Thus __**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 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.//

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

//[//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 enharmonic interval, the following table shows in what parts of this range the interval should be upped or 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>

<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 N = 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 N parts, and if N' &gt; N, 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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vv\\M2

</td>

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

C^^ = C#

C^^ = D

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</td>

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

C ~~= Db,~~<~~/em~~>

C// = Db

C^^ = C#

</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>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>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>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>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>C - Fxv3=Gbb^3 CC - D#vv - Fb^^ - G

<td>C - Fxv3=Gbb^3 C

C - D#vv - Fb^^ - G

</td>

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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>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>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>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>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>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>C - Db^ - Ebb^ -- E#vv - F#v - G

<td>C - Db^ - Ebb^ -

- E#vv - F#v - G

</td>

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

[Question: how to find all possible pergens?]

Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) 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¢. 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. Thus 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 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.

Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) 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¢. 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. Thus 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 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.

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

[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 enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.

@@ 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 00:01:15 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-22 00:06:52 UTC</tt>.<br>
-: The original revision id was <tt>622159545</tt>.<br>
+: The original revision id was <tt>622159615</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 158: / Line 158: @@
 vvA1,
 ^^\\d2,
-vv\\M2 ||= C //= Db,//
+vv\\M2 ||= C``//`` = Db
 //C^^ = C#//
 //C^^// = D ||= C - D/=Eb\ - F,
@@ Line 172: / Line 172: @@
 \\m2,
 vv\\A1 ||= C^^ = B#
-C //= Db,//
+C``//`` = Db
 //C^^// = C# ||= C - F#v=Gb^ - C,
 C - D/=Eb\ - F,
@@ Line 202: / Line 202: @@
 ||= {P8/2, P4/3} ||=   ||=   ||=   ||=   ||=   ||= 14, 22, 28*, 30* ||
 ||= {P8/2, P5/3} ||= P8/2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;AA4 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd5
-P5/3 = vvA2 = ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||= ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; || 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; CC - D#vv - Fb^^ - G ||=   ||= 16, 20*, 26, 30* ||
+P5/3 = vvA2 = ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||= ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; || C - F&lt;span style="vertical-align: super;"&gt;x&lt;/span&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Gbb^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; C
+C - D#vv - Fb^^ - G ||=   ||= 16, 20*, 26, 30* ||
 ||= " ||= P8/2 = vA4 = ^d5
 P5/3 = /M2 = \\m3 ||= ^^d2,
 \\\m2 ||= C^^ = B#
-C/// = Db ||= C - F#v=Gb^ - C
+C``///`` = Db ||= C - F#v=Gb^ - C
 C - /D - \F - G ||=   ||= " ||
 ||= {P8/2, P11/3} ||=   ||=   ||=   ||=   ||=   ||=   ||
@@ Line 223: / Line 224: @@
 ||~ fifths ||~   ||~   ||~   ||~   ||~   ||~   ||
 ||= {P8/5, P5} ||= P8/5 = ||= m2 ||=   ||=   ||=   ||=   ||
-||= {P8, P5/5} ||= P5/5 = ^m2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;1 ||= ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt; ``=`` B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; || C - Db^ - Ebb^ -- E#vv - F#v - G ||=   ||=   ||
+||= {P8, P5/5} ||= P5/5 = ^m2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;1 ||= ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt; ``=`` B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; || C - Db^ - Ebb^ -
+- 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?//]
+[Question: how to find all possible pergens?//]//
-Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) 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¢. 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. Thus __**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 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.
+//Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) 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¢. 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. Thus __**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 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.//
-[//Question: What to do if the edo's 5th falls in the sweet spot? Example?//]
+//[//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 enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.
@@ Line 272: / Line 274: @@
 (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:19:&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:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&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:22:&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:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&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 522: / Line 524: @@
 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:15:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:15 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 For any comma containing primes 2 and 3, let N = 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 N parts, and if N' &amp;gt; N, 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 808: / Line 810: @@
 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:17:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&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 1,001: / Line 1,003: @@
 vv\\M2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C &lt;em&gt;= Db,&lt;/em&gt;&lt;br /&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;em&gt;C^^ = C#&lt;/em&gt;&lt;br /&gt;
 &lt;em&gt;C^^&lt;/em&gt; = D&lt;br /&gt;
@@ Line 1,030: / Line 1,032: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C^^ = B#&lt;br /&gt;
-C &lt;em&gt;= Db,&lt;/em&gt;&lt;br /&gt;
+C&lt;!-- ws:start:WikiTextRawRule:02:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:02 --&gt; = Db&lt;br /&gt;
 &lt;em&gt;C^^&lt;/em&gt; = C#&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,093: / Line 1,095: @@
          &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:01:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:01 --&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:03:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:03 --&gt; B#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;C - Ev - Ab^ - C&lt;br /&gt;
@@ Line 1,110: / Line 1,112: @@
          &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:02:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:02 --&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:04:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:04 --&gt; &lt;/span&gt;C#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;C - Dv - Eb^ - F&lt;br /&gt;
@@ Line 1,127: / Line 1,129: @@
          &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:03:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:03 --&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:05:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:05 --&gt; Db&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - D^ - Fv - G&lt;br /&gt;
@@ Line 1,144: / Line 1,146: @@
          &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:04:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:04 --&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:06:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:06 --&gt; B##&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;C - F#v - Cb^ - F&lt;br /&gt;
@@ Line 1,160: / Line 1,162: @@
          &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:05:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:05 --&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:07:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:07 --&gt; D&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;C F^ Cv F&lt;br /&gt;
@@ Line 1,225: / Line 1,227: @@
          &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:06:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:06 --&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:08:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:08 --&gt; B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&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; CC - D#vv - Fb^^ - G&lt;br /&gt;
+         &lt;td&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;
+C - D#vv - Fb^^ - G&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 1,244: / Line 1,247: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C^^ = B#&lt;br /&gt;
-C&lt;em&gt;/ = Db&lt;br /&gt;
+C&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 - F#v=Gb^ - C&lt;br /&gt;
@@ Line 1,309: / Line 1,312: @@
          &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:07:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:07 --&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:010:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:010 --&gt; B#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;C Ebv Gbvv A^ C&lt;br /&gt;
@@ Line 1,326: / Line 1,329: @@
          &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:08:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:08 --&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:011:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:011 --&gt; B##&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;C Db^ Ebb^^ Ev F&lt;br /&gt;
@@ Line 1,342: / Line 1,345: @@
          &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:09:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:09 --&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:012:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:012 --&gt; C#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;C Dv Evv F^ G^&lt;br /&gt;
@@ Line 1,358: / Line 1,361: @@
          &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:010:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:010 --&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:013:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:013 --&gt; Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;C E^ G#^^ Dbv F&lt;br /&gt;
@@ Line 1,374: / Line 1,377: @@
          &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:011:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:011 --&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:014:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:014 --&gt; Db&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;C Fv Bbvv D^ G&lt;br /&gt;
@@ Line 1,486: / Line 1,489: @@
          &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: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 style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:015:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:015 --&gt; B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;C - Db^ - Ebb^ -- E#vv - F#v - G&lt;br /&gt;
+         &lt;td&gt;C - Db^ - Ebb^ -&lt;br /&gt;
+- E#vv - F#v - G&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 1,514: / Line 1,518: @@
 &lt;br /&gt;
-[&lt;/em&gt;Question: how to find all possible pergens?&lt;em&gt;]&lt;br /&gt;
+[Question: how to find all possible pergens?&lt;em&gt;]&lt;/em&gt;&lt;br /&gt;
 &lt;br /&gt;
-Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) 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¢. 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. Thus &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 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;
+&lt;em&gt;Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) 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¢. 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. Thus &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 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;/em&gt;&lt;br /&gt;
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-[&lt;/em&gt;Question: What to do if the edo's 5th falls in the sweet spot? Example?//]&lt;br /&gt;
+&lt;em&gt;[&lt;/em&gt;Question: What to do if the edo's 5th falls in the sweet spot? Example?//]&lt;br /&gt;
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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 enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.&lt;br /&gt;