Kite's thoughts on pergens: Difference between revisions

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For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.

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A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.

=__Further Discussion__=  

==Extremely large multigens==  

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For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.&lt;br /&gt;

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A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.&lt;br /&gt;

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So far, the largest multigen has been a 12th. As the multigen fractions get larger, the multigen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &amp;quot;W&amp;quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen can be P4, P5, P11, P12, WWP4 or WWP5.&lt;br /&gt;

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If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a &lt;strong&gt;single-split&lt;/strong&gt; pergen. If it has two fractions, it's a &lt;strong&gt;double-split&lt;/strong&gt; pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called &lt;strong&gt;single-pair&lt;/strong&gt; notation because it adds only a single pair of accidentals to conventional notation. &lt;strong&gt;Double-pair&lt;/strong&gt; notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs can be freely exchanged with highs/lows.&lt;br /&gt;

A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.&lt;br /&gt;

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To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P and P8. If P is 6/5, the comma is 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P - P8 = (6/5)&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P = (2/1) · (7/6)&lt;span style="vertical-align: super;"&gt;-4&lt;/span&gt;. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - P4 = (10/9)^3 ÷ (4/3) = 250/243.&lt;br /&gt;

There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.&lt;br /&gt;

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In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64.&lt;br /&gt;

We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.&lt;br /&gt;

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There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:&lt;br /&gt;

This is a lot of math, but it only needs to be done once for each pergen.&lt;br /&gt;

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Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;A2, which is an improvement but still awkward. The period is ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m3 and the generator is v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2.&lt;br /&gt;

Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese is (P8, P11/3), with G = 7/5 = d5. E = 3·d5 - P11 = descending dd3.&lt;br /&gt;

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One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.&lt;br /&gt;

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Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt; page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.&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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Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).&lt;br /&gt;

However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.&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.&lt;br /&gt;

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This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. 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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