Pergen names: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-18 16:25:50 UTC</tt>.<br>

: The original revision id was <tt>621930305</tt>.<br>

<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;white-space: pre-wrap ! important" class="old-revision-html">A **pergen** (pronounced "peer-gen") set 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. The pergen set is chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave.

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.

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.

Example: 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. x31.com gives us this matrix:

Discard the last column to make a square matrix:

Invert it. "/4" means that each entry is to be divided by the determinant of the last matrix, which is 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) ^ (1/4).

Next, search for alternate generators... (to be continued)</pre></div>

<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;A &lt;strong&gt;pergen&lt;/strong&gt; (pronounced &amp;quot;peer-gen&amp;quot;) set 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. The pergen set is chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave.&lt;br /&gt;

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.&lt;br /&gt;

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 &amp;quot;W&amp;quot;. Thus 3/1 = WP5, 9/2 = WWM2, etc. Thus magic is {P8, WP5/5} = fifth-wide-fifth.&lt;br /&gt;

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.&lt;br /&gt;

Example: 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. x31.com gives us this matrix:&lt;br /&gt;

Discard the last column to make a square matrix:&lt;br /&gt;

Invert it. &amp;quot;/4&amp;quot; means that each entry is to be divided by the determinant of the last matrix, which is 4.&lt;br /&gt;

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) ^ (1/4).&lt;br /&gt;

Next, search for alternate generators... (to be continued)&lt;/body&gt;&lt;/html&gt;</pre></div>