Pergen names: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-19 20:07:01 UTC</tt>.<br>

: The original revision id was <tt>621981911</tt>.<br>

The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-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 12, the widening is indicated with one "W" per octave. Thus 27/8 = WM6, 9/2 = WWM2, etc. Thus magic is {P8, P12/5} = fifth-twelfth.

Examples: Marvel (2.3.5.7 and 225/224, or reddish yellow) is {P8, P5, ^1} = fifth-based with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is {P8/2, P5, g1} = half-octave with green.Triple bluish (1029/1000) is {P8, M2/3, ^1}.

 

 

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 period/generator. Choose your generators so that all entries below the diagonal are zero. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix. Next make a square matrix by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous example, to ensure that the diagonal has no zeros. Lower primes &gt; 3 may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the 1st generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.

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:

The 2nd multi-gen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be Wgg8/4 (Wgg8 = 96/25). A quadruple half-fifth is a double fifth is a M9. Subtracting that makes gen2 be gg7/4 (gg7 = 128/75). Subtracting M9 again, and inverting again, makes gen2 = (-9, 3, 2)/4 = Lyy3/4 (Lyy3 = 675/512). As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.

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. 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 set is {P8, P5/2, r1} = half-fifth with red. This is far better than {P8, P5/2, gg7/4}. The pergen set sometimes uses a larger prime in place of a smaller one, in order to avoid splitting gen2, IF both primes are &gt; 3.</pre></div>

The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-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 12, the widening is indicated with one &amp;quot;W&amp;quot; per octave. Thus 27/8 = WM6, 9/2 = WWM2, etc. Thus magic is {P8, P12/5} = fifth-twelfth.&lt;br /&gt;

Examples: Marvel (2.3.5.7 and 225/224, or reddish yellow) is {P8, P5, ^1} = fifth-based with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is {P8/2, P5, g1} = half-octave with green.Triple bluish (1029/1000) is {P8, M2/3, ^1}.&lt;br /&gt;

To find a temperament's pergen set, first find the &lt;strong&gt;PGM&lt;/strong&gt;, the period generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Choose your generators so that all entries below the diagonal are zero. Graham Breed's website has a temperament finder &lt;a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank"&gt;x31eq.com/temper/uv.html&lt;/a&gt; that will find such a matrix. Next make a square matrix by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous example, to ensure that the diagonal has no zeros. Lower primes &amp;gt; 3 may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the 1st generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, 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;

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;

The 2nd multi-gen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be Wgg8/4 (Wgg8 = 96/25). A quadruple half-fifth is a double fifth is a M9. Subtracting that makes gen2 be gg7/4 (gg7 = 128/75). Subtracting M9 again, and inverting again, makes gen2 = (-9, 3, 2)/4 = Lyy3/4 (Lyy3 = 675/512). As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.&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. 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 set is {P8, P5/2, r1} = half-fifth with red. This is far better than {P8, P5/2, gg7/4}. The pergen set sometimes uses a larger prime in place of a smaller one, in order to avoid splitting gen2, IF both primes are &amp;gt; 3.&lt;/body&gt;&lt;/html&gt;</pre></div>