Pergen names
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**Pergen** (pronounced "peer-gen") sets 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 use the first two primes in the prime subgroup, and rank-3 names only use 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 is called half-octave. The set is written {P8/2, P5}. The dicot temperament splits the fifth in two, and is called half-fifth, written {P8, P5/2}. Porcupine is third-fourth, {P8, P4/3}. Semaphore, which means "semi-fourth", is of course {P8, P4/2}.
Many temperaments will share the same pergen set. This has the advantage of reducing the hundreds (thousands?) of temperament names to perhaps a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. Both srutal and injera sound the same, although they temper out different commas.
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: {P8, P5}. Such temperaments are called fifth-based.
Every temperament has at least one alternate generator. More, if the octave is split. To avoid ambiguity, the generator is 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 ||||||||~ example temperaments ||
||~ 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 example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone (9/8 is a whole tone). For hextuple colors, the multi-gen 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 sets 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} = half-octave fifth-and-third.
Rank-4 temperaments have pergen sets of four intervals. Rank-1 temperaments could have pergen sets of one, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.
To find a temperament's pergen set, first find the PGM, the period generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each generator, counting the period as a special type of generator. Choose your generators so that all entries below the diagonal are zero. You can use x31.com's temperament finder to find this matrix. Next make a square matrix by discarding columns for the higher primes. Then invert the matrix to get the monzos for each generator. Add/subtract periods from the generator to get alternate generators. Add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.
Example: 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. x31.com gives us this matrix:
||~ ||~ 2/1 ||~ 3/1 ||~ 5/1 ||~ 7/1 ||
||~ period ||= 1 ||= 1 ||= 1 ||= 2 ||
||~ gen1 ||= 0 ||= 2 ||= 1 ||= 1 ||
||~ gen2 ||= 0 ||= 0 ||= 2 ||= 1 ||
Thus 2/1 = P, 3/1 = P + 2 G1, 5/1 = P + G1 + 2 G2, and 7/1 = 2 P + G1 + G2.
Discard the last column to make a square matrix:
||~ ||~ 2/1 ||~ 3/1 ||~ 5/1 ||
||~ period ||= 1 ||= 1 ||= 1 ||
||~ gen1 ||= 0 ||= 2 ||= 1 ||
||~ gen2 ||= 0 ||= 0 ||= 2 ||
Invert it. "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.
||~ ||~ period ||~ gen1 ||~ gen2 ||~ ||
||~ 2/1 ||= 4 ||= -2 ||= -1 || ||
||~ 3/1 ||= 0 ||= 2 ||= -1 || ||
||~ 5/1 ||= 0 ||= 0 ||= 2 || /4 ||
Thus the period = (4, 0, 0)/4 = (1, 0, 0) = 2/1, gen1 = (-2, 2, 0)/4 = (-1, 1, 0)/2 = P5/2, and gen2 = (-1, -1, 2)/4 = [25/6]/4.
Next, search for alternate generators... (to be continued)Original HTML content:
<html><head><title>pergen names</title></head><body><strong>Pergen</strong> (pronounced "peer-gen") sets 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 use the first two primes in the prime subgroup, and rank-3 names only use 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 is called half-octave. The set is written {P8/2, P5}. The dicot temperament splits the fifth in two, and is called half-fifth, written {P8, P5/2}. Porcupine is third-fourth, {P8, P4/3}. Semaphore, which means "semi-fourth", is of course {P8, P4/2}.<br />
<br />
Many temperaments will share the same pergen set. This has the advantage of reducing the hundreds (thousands?) of temperament names to perhaps a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. Both srutal and injera sound the same, although they temper out different commas.<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: {P8, P5}. Such temperaments are called fifth-based.<br />
<br />
Every temperament has at least one alternate generator. More, if the octave is split. To avoid ambiguity, the generator is 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">example temperaments<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;">"<br />
</td>
<td style="text-align: center;">"<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;">"<br />
</td>
<td style="text-align: center;">"<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;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">81/80 & 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&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;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">81/80 & 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&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;">"<br />
</td>
<td style="text-align: center;">"<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 & 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&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. For quadruple colors, the 3-limit multi-gen may be the major 2nd 9/8. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone (9/8 is a whole tone). For hextuple colors, the multi-gen 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.<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 sets 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} = half-octave fifth-and-third.<br />
<br />
Rank-4 temperaments have pergen sets of four intervals. Rank-1 temperaments could have pergen sets of one, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.<br />
<br />
To find a temperament's pergen set, first find the PGM, the period generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each generator, counting the period as a special type of generator. Choose your generators so that all entries below the diagonal are zero. You can use x31.com's temperament finder to find this matrix. Next make a square matrix by discarding columns for the higher primes. Then invert the matrix to get the monzos for each generator. Add/subtract periods from the generator to get alternate generators. Add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.<br />
<br />
Example: 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. x31.com gives us this matrix:<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>2/1<br />
</th>
<th>3/1<br />
</th>
<th>5/1<br />
</th>
<th>7/1<br />
</th>
</tr>
<tr>
<th>period<br />
</th>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">2<br />
</td>
</tr>
<tr>
<th>gen1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
<tr>
<th>gen2<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
</table>
Thus 2/1 = P, 3/1 = P + 2 G1, 5/1 = P + G1 + 2 G2, and 7/1 = 2 P + G1 + G2.<br />
<br />
Discard the last column to make a square matrix:<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>2/1<br />
</th>
<th>3/1<br />
</th>
<th>5/1<br />
</th>
</tr>
<tr>
<th>period<br />
</th>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
<tr>
<th>gen1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
<tr>
<th>gen2<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
</tr>
</table>
Invert it. "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>period<br />
</th>
<th>gen1<br />
</th>
<th>gen2<br />
</th>
<th><br />
</th>
</tr>
<tr>
<th>2/1<br />
</th>
<td style="text-align: center;">4<br />
</td>
<td style="text-align: center;">-2<br />
</td>
<td style="text-align: center;">-1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<th>3/1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">-1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<th>5/1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td>/4<br />
</td>
</tr>
</table>
Thus the period = (4, 0, 0)/4 = (1, 0, 0) = 2/1, gen1 = (-2, 2, 0)/4 = (-1, 1, 0)/2 = P5/2, and gen2 = (-1, -1, 2)/4 = [25/6]/4.<br />
<br />
Next, search for alternate generators... (to be continued)</body></html>