Kite's thoughts on pergens: Difference between revisions

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).

A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.

The **genchain** (chain of generators) in the table is only a short section of the full genchain.

If the octave is split, the table has a **perchain** ("peer-chain", chain of periods) that shows the octave: C -- F#v=Gb^ -- C

For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer).

==Singles and doubles==  

12edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E)

17edo+y perchain: C C^ Dv D... (E = dd3, E' = vm2 = vvA1), genchain: same as blackwood, but with / and \

==Pergens and MOS scales==  

||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* ||

||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 ||

This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.

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<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&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt; &lt;/h1&gt;

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The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).&lt;br /&gt;

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A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.&lt;br /&gt;

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Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.&lt;br /&gt;

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The &lt;strong&gt;genchain&lt;/strong&gt; (chain of generators) in the table is only a short section of the full genchain.&lt;br /&gt;

If the octave is split, the table has a &lt;strong&gt;perchain&lt;/strong&gt; (&amp;quot;peer-chain&amp;quot;, chain of periods) that shows the octave: C -- F#v=Gb^ -- C&lt;br /&gt;

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For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer).&lt;br /&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Further Discussion-Singles and doubles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;Singles and doubles&lt;/h2&gt;

17edo+y perchain: C C^ Dv D... (E = dd3, E' = vm2 = vvA1), genchain: same as blackwood, but with / and \&lt;br /&gt;

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Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.&lt;br /&gt;

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.&lt;br /&gt;

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Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.&lt;br /&gt;

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This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.&lt;br /&gt;

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Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;

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Screenshot of the first 38 pergens:&lt;br /&gt;

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Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:&lt;br /&gt;

The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.&lt;br /&gt;

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The interval P8/2 has a &amp;quot;ratio&amp;quot; of the square root of 2 = 2&lt;span style="vertical-align: super;"&gt;1/2&lt;/span&gt;, and its monzo can be written with fractions as (1/2, 0). Likewise, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the &lt;strong&gt;pergen matrix&lt;/strong&gt; [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.&lt;br /&gt;

Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.&lt;br /&gt;

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&lt;u&gt;&lt;strong&gt;Combining pergens&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;