Pergen names: Difference between revisions

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-18 02:01:59 UTC</tt>.<br>

: The original revision id was <tt>621910225</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">**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.

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&lt;span style="vertical-align: super;"&gt;x &lt;/span&gt;3&lt;span style="vertical-align: super;"&gt;y &lt;/span&gt;P or 2&lt;span style="vertical-align: super;"&gt;x &lt;/span&gt;3&lt;span style="vertical-align: super;"&gt;y &lt;/span&gt;P&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;, 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.  

||||~ pergen name ||||||||~ examples ||

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.

<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;strong&gt;Pergen&lt;/strong&gt; (pronounced &amp;quot;peer-gen&amp;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.&lt;br /&gt;

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 &amp;quot;semi-fourth&amp;quot;, is already sort of a pergen name. &lt;br /&gt;

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.&lt;br /&gt;

The largest category contains all commas of the form 2&lt;span style="vertical-align: super;"&gt;x &lt;/span&gt;3&lt;span style="vertical-align: super;"&gt;y &lt;/span&gt;P or 2&lt;span style="vertical-align: super;"&gt;x &lt;/span&gt;3&lt;span style="vertical-align: super;"&gt;y &lt;/span&gt;P&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;, 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. &lt;br /&gt;

         &lt;th colspan="4"&gt;examples&lt;br /&gt;

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 &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;