Pergen names

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**Pergen** (pronounced "peer-gen") names are a way of identifying rank-2 and rank-3 regular temperaments by their periods and generators. They are somewhat JI-agnostic in that they don't use higher primes. Rank-2 names only refer to the first two primes in the prime subgroup, and rank-3 names only refer to the first three primes.

If a rank-2 temperament uses the primes 2 and 3 in its comma(s), then the period can be expressed as the octave 2/1, or some fraction of an octave. The generator can be expressed as a 3-limit interval, or some fraction of one. The fraction is always of the form 1/N, in other words, the octave or the 3-limit interval is **split** into N parts. A 3-limit interval which is split into multiple generators is called a **multi-gen**.

For example,the srutal temperament splits the octave in two, and its pergen name is half-octave. The name is written {P8/2, P5}. Curly brackets are used because the name is a set of intervals. The dicot temperament splits the fifth in two, and is called half-fifth, written {P8, P5/2}. Porcupine is third-fourth, etc. Semaphore, which means "semi-fourth", is already sort of a pergen name. 

In a sense, pergen names are categories, because many temperaments will have the same pergen name. This has the advantage of reducing the hundreds (thousands?) of temperament names to perhaps a few dozen categories. It also focuses on the melodic properties of the temperament.

The largest category contains all commas of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime (e.g. 81/80 or 64/63). The period is the octave, and the generator is the fifth. Such temperaments are called fifth-based. The 4th is also a generator, and in fact every temperament has at least one alternate generator. To avoid ambiguity, the generator and the multi-gen are chosen to minimize the amount of splitting of the multi-gen, and as a tie-breaker, to minimize the size in cents of the multi-gen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent. 

For example, srutal could be {P8/2, M2/2}, but P5 is preferred because it is unsplit. Or it could be {P8/2, P12}, but P5 is preferred because it is smaller. Or it could be {P8/2, P4}, but P5 is always preferred over P4. Note that P5/2 is __not__ preferred over P4/2. For example, decimal is {P8/2, P4/2}, not {P8/2, P5/2}.

||||~ pergen name ||||||||~ examples ||
||~ written ||~ spoken ||~ comma(s) ||~ name ||||~ color name ||
||= {P8, P5} ||= fifth-based ||= 81/80 ||= meantone ||= green ||= gT ||
||= " ||= " ||= 64/63 ||= archy ||= red ||= rT ||
||= " ||= " ||= (-14,8,0,0,1) ||= schismic ||= large yellow ||= LyT ||
||= " ||= " ||= 81/80 & 126/125 ||= septimal meantone ||= green and bluish-blue ||= g&bg<span style="vertical-align: super;">3</span>T ||
||= {P8/2, P5} ||= half-octave ||= (11, -4, -2) ||= srutal ||= small deep green ||= sggT ||
||= " ||= " ||= 81/80 & 50/49 ||= injera ||= deep reddish and green ||= rryy&gT ||
||= {P8, P5/2} ||= half-fifth ||= 25/24 ||= dicot ||= deep yellow ||= yyT ||
||= " ||= " ||= (-1,5,0,0,-2) ||= mohajira ||= deep amber ||= aaT ||
||= {P8, P4/2} ||= half-fourth ||= 49/48 ||= semaphore ||= deep blue ||= bbT ||
||= {P8, P4/3} ||= third-fourth ||= 250/243 ||= porcupine ||= triple yellow ||= y<span style="vertical-align: super;">3</span>T ||
||= {P8, P11/3} ||= third-eleventh ||= (12,-1,0,0,-3) ||= small triple amber ||= small triple amber ||= sa<span style="vertical-align: super;">3</span>T ||
||= {P8/2, P4/2} ||= half-octave, half-fourth ||= 25/24 & 49/48 ||= decimal ||= deep yellow and deep blue ||= yy&bbT ||
||= {P8/4, P5} ||= quarter-octave ||= (3,4,-4) ||= diminished ||= quadruple green ||= g<span style="vertical-align: super;">4</span>T ||
For the 2.3... prime subgroup, the color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc.

For quadruple colors, the 3-limit multi-gen may be the major 2nd 9/8. For hextuple, it may be the minor 3rd 32/27. These intervals may also be voiced wider, as 3/1, 9/4, etc. To avoid cumbersome degree names like 16th or 18th, for degrees above 11, the widening is indicated with a "W". Thus 3/1 = WP5, 9/2 = WWM2, etc. Thus magic is {P8, WP5/5} = fifth-wide-fifth.

For non-standard prime groups, the period uses the first prime only, and the multi-gen uses the first two primes only. [[Kite's color notation|Color notation ]]is used to indicate primes higher than 3. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third.

Rank-3 pergen names have three intervals, any of which may be split. The unsplit 2.3.5... subgroup is {P8, P5, y3} = fifth-and-third-based. 2.3.5.7 with 50/49 tempered out is {P8/2, P5, y3}.

Rank-4 temperaments can be named similarly. Rank-1 temperaments could have pergen names, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.

Original HTML content:

<html><head><title>pergen names</title></head><body><strong>Pergen</strong> (pronounced &quot;peer-gen&quot;) names are a way of identifying rank-2 and rank-3 regular temperaments by their periods and generators. They are somewhat JI-agnostic in that they don't use higher primes. Rank-2 names only refer to the first two primes in the prime subgroup, and rank-3 names only refer to the first three primes.<br />
<br />
If a rank-2 temperament uses the primes 2 and 3 in its comma(s), then the period can be expressed as the octave 2/1, or some fraction of an octave. The generator can be expressed as a 3-limit interval, or some fraction of one. The fraction is always of the form 1/N, in other words, the octave or the 3-limit interval is <strong>split</strong> into N parts. A 3-limit interval which is split into multiple generators is called a <strong>multi-gen</strong>.<br />
<br />
For example,the srutal temperament splits the octave in two, and its pergen name is half-octave. The name is written {P8/2, P5}. Curly brackets are used because the name is a set of intervals. The dicot temperament splits the fifth in two, and is called half-fifth, written {P8, P5/2}. Porcupine is third-fourth, etc. Semaphore, which means &quot;semi-fourth&quot;, is already sort of a pergen name. <br />
<br />
In a sense, pergen names are categories, because many temperaments will have the same pergen name. This has the advantage of reducing the hundreds (thousands?) of temperament names to perhaps a few dozen categories. It also focuses on the melodic properties of the temperament.<br />
<br />
The largest category contains all commas of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime (e.g. 81/80 or 64/63). The period is the octave, and the generator is the fifth. Such temperaments are called fifth-based. The 4th is also a generator, and in fact every temperament has at least one alternate generator. To avoid ambiguity, the generator and the multi-gen are chosen to minimize the amount of splitting of the multi-gen, and as a tie-breaker, to minimize the size in cents of the multi-gen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent. <br />
<br />
For example, srutal could be {P8/2, M2/2}, but P5 is preferred because it is unsplit. Or it could be {P8/2, P12}, but P5 is preferred because it is smaller. Or it could be {P8/2, P4}, but P5 is always preferred over P4. Note that P5/2 is <u>not</u> preferred over P4/2. For example, decimal is {P8/2, P4/2}, not {P8/2, P5/2}.<br />
<br />


<table class="wiki_table">
    <tr>
        <th colspan="2">pergen name<br />
</th>
        <th colspan="4">examples<br />
</th>
    </tr>
    <tr>
        <th>written<br />
</th>
        <th>spoken<br />
</th>
        <th>comma(s)<br />
</th>
        <th>name<br />
</th>
        <th colspan="2">color name<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">{P8, P5}<br />
</td>
        <td style="text-align: center;">fifth-based<br />
</td>
        <td style="text-align: center;">81/80<br />
</td>
        <td style="text-align: center;">meantone<br />
</td>
        <td style="text-align: center;">green<br />
</td>
        <td style="text-align: center;">gT<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">64/63<br />
</td>
        <td style="text-align: center;">archy<br />
</td>
        <td style="text-align: center;">red<br />
</td>
        <td style="text-align: center;">rT<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">(-14,8,0,0,1)<br />
</td>
        <td style="text-align: center;">schismic<br />
</td>
        <td style="text-align: center;">large yellow<br />
</td>
        <td style="text-align: center;">LyT<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">81/80 &amp; 126/125<br />
</td>
        <td style="text-align: center;">septimal meantone<br />
</td>
        <td style="text-align: center;">green and bluish-blue<br />
</td>
        <td style="text-align: center;">g&amp;bg<span style="vertical-align: super;">3</span>T<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">{P8/2, P5}<br />
</td>
        <td style="text-align: center;">half-octave<br />
</td>
        <td style="text-align: center;">(11, -4, -2)<br />
</td>
        <td style="text-align: center;">srutal<br />
</td>
        <td style="text-align: center;">small deep green<br />
</td>
        <td style="text-align: center;">sggT<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">81/80 &amp; 50/49<br />
</td>
        <td style="text-align: center;">injera<br />
</td>
        <td style="text-align: center;">deep reddish and green<br />
</td>
        <td style="text-align: center;">rryy&amp;gT<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">{P8, P5/2}<br />
</td>
        <td style="text-align: center;">half-fifth<br />
</td>
        <td style="text-align: center;">25/24<br />
</td>
        <td style="text-align: center;">dicot<br />
</td>
        <td style="text-align: center;">deep yellow<br />
</td>
        <td style="text-align: center;">yyT<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">&quot;<br />
</td>
        <td style="text-align: center;">(-1,5,0,0,-2)<br />
</td>
        <td style="text-align: center;">mohajira<br />
</td>
        <td style="text-align: center;">deep amber<br />
</td>
        <td style="text-align: center;">aaT<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">{P8, P4/2}<br />
</td>
        <td style="text-align: center;">half-fourth<br />
</td>
        <td style="text-align: center;">49/48<br />
</td>
        <td style="text-align: center;">semaphore<br />
</td>
        <td style="text-align: center;">deep blue<br />
</td>
        <td style="text-align: center;">bbT<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">{P8, P4/3}<br />
</td>
        <td style="text-align: center;">third-fourth<br />
</td>
        <td style="text-align: center;">250/243<br />
</td>
        <td style="text-align: center;">porcupine<br />
</td>
        <td style="text-align: center;">triple yellow<br />
</td>
        <td style="text-align: center;">y<span style="vertical-align: super;">3</span>T<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">{P8, P11/3}<br />
</td>
        <td style="text-align: center;">third-eleventh<br />
</td>
        <td style="text-align: center;">(12,-1,0,0,-3)<br />
</td>
        <td style="text-align: center;">small triple amber<br />
</td>
        <td style="text-align: center;">small triple amber<br />
</td>
        <td style="text-align: center;">sa<span style="vertical-align: super;">3</span>T<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">{P8/2, P4/2}<br />
</td>
        <td style="text-align: center;">half-octave, half-fourth<br />
</td>
        <td style="text-align: center;">25/24 &amp; 49/48<br />
</td>
        <td style="text-align: center;">decimal<br />
</td>
        <td style="text-align: center;">deep yellow and deep blue<br />
</td>
        <td style="text-align: center;">yy&amp;bbT<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">{P8/4, P5}<br />
</td>
        <td style="text-align: center;">quarter-octave<br />
</td>
        <td style="text-align: center;">(3,4,-4)<br />
</td>
        <td style="text-align: center;">diminished<br />
</td>
        <td style="text-align: center;">quadruple green<br />
</td>
        <td style="text-align: center;">g<span style="vertical-align: super;">4</span>T<br />
</td>
    </tr>
</table>

For the 2.3... prime subgroup, the color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc.<br />
<br />
For quadruple colors, the 3-limit multi-gen may be the major 2nd 9/8. For hextuple, it may be the minor 3rd 32/27. These intervals may also be voiced wider, as 3/1, 9/4, etc. To avoid cumbersome degree names like 16th or 18th, for degrees above 11, the widening is indicated with a &quot;W&quot;. Thus 3/1 = WP5, 9/2 = WWM2, etc. Thus magic is {P8, WP5/5} = fifth-wide-fifth.<br />
<br />
For non-standard prime groups, the period uses the first prime only, and the multi-gen uses the first two primes only. <a class="wiki_link" href="/Kite%27s%20color%20notation">Color notation </a>is used to indicate primes higher than 3. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third.<br />
<br />
Rank-3 pergen names have three intervals, any of which may be split. The unsplit 2.3.5... subgroup is {P8, P5, y3} = fifth-and-third-based. 2.3.5.7 with 50/49 tempered out is {P8/2, P5, y3}.<br />
<br />
Rank-4 temperaments can be named similarly. Rank-1 temperaments could have pergen names, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.</body></html>