Pergen names
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=__**Definition**__=
A **pergen** set (pronounced "peer-gen") 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, but avoided if possible.
If a rank-2 temperament uses the primes 2 and 3 in its comma(s), then the period can be expressed as the octave 2/1, or some fraction of an octave. The generator can usually be expressed as a 3-limit interval, or some fraction of one. The fraction is always of the form 1/N, in other words, the octave or the 3-limit interval is **split** into N parts. An interval which is split into multiple generators is called a **multi-gen**.
For example, the srutal temperament splits the octave in two, and is called half-octave. The set is written {P8/2, P5}. The dicot temperament splits the fifth in two, and is called half-fifth, written {P8, P5/2}. Porcupine is third-fourth, {P8, P4/3}. Semaphore, which means "semi-fourth", is of course {P8, P4/2}.
Many temperaments will share the same pergen set. This has the advantage of reducing the 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. In addition, the pergen set tells us how to notate the temperament using [[Ups and Downs Notation|ups and downs]].
The largest category contains all commas of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime (e.g. 81/80 or 64/63). The period is the octave, and the generator is the fifth: {P8, P5}. Such temperaments are called fifth-based.
Every temperament has at least one alternate generator. More, if the octave is split. To avoid ambiguity, the generator is 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.
For example, srutal could be {P8/2, M2/2}, but P5 is preferred because it is unsplit. Or it could be {P8/2, P12}, but P5 is preferred because it is smaller. Or it could be {P8/2, P4}, but P5 is always preferred over P4. Note that P5/2 is __not__ preferred over P4/2. For example, decimal is {P8/2, P4/2}, not {P8/2, P5/2}.
||||~ pergen set ||||||||~ example temperaments ||
||~ written ||~ spoken ||~ comma(s) ||~ name ||||~ color name ||
||= {P8, P5} ||= fifth-based ||= 81/80 ||= meantone ||= green ||= gT ||
||= " ||= " ||= 64/63 ||= archy ||= red ||= rT ||
||= " ||= " ||= (-14,8,0,0,1) ||= schismic ||= large yellow ||= LyT ||
||= " ||= " ||= 81/80 & 126/125 ||= septimal meantone ||= green and bluish-blue ||= g&bg<span style="vertical-align: super;">3</span>T ||
||= {P8/2, P5} ||= half-octave ||= (11, -4, -2) ||= srutal ||= small deep green ||= sggT ||
||= " ||= " ||= 81/80 & 50/49 ||= injera ||= deep reddish and green ||= rryy&gT ||
||= {P8, P5/2} ||= half-fifth ||= 25/24 ||= dicot ||= deep yellow ||= yyT ||
||= " ||= " ||= (-1,5,0,0,-2) ||= mohajira ||= deep amber ||= aaT ||
||= {P8, P4/2} ||= half-fourth ||= 49/48 ||= semaphore ||= deep blue ||= bbT ||
||= {P8, P4/3} ||= third-fourth ||= 250/243 ||= porcupine ||= triple yellow ||= y<span style="vertical-align: super;">3</span>T ||
||= {P8, P11/3} ||= third-eleventh ||= (12,-1,0,0,-3) ||= small triple amber ||= small triple amber ||= sa<span style="vertical-align: super;">3</span>T ||
||= {P8/2, P4/2} ||= half-octave, half-fourth ||= 25/24 & 49/48 ||= decimal ||= deep yellow and deep blue ||= yy&bbT ||
||= {P8/4, P5} ||= quarter-octave ||= (3,4,-4) ||= diminished ||= quadruple green ||= g<span style="vertical-align: super;">4</span>T ||
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.
For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. [[Kite's color notation|Color notation ]]is used to indicate primes higher than 3. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third (y3 = 5/4).
Rank-3 pergen sets have three intervals, period, gen1 and gen2, any of which may be split. The unsplit 2.3.5... subgroup's set could be {P8, P5, y3}. However, to avoid higher primes, and more closely mimic conventional notation, it's better to reduce gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with an up symbol, and we have {P8, P5, ^1} = fifth-based with ups. However, if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and colors must be used for gen2.
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} = third-tone with ups.
A rank-4 temperament has a pergen set of four intervals. A rank-1 temperament could have a pergen set of one, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.
Finally, the pergen set for JI is best thought of as the octave, the fifth, and a set of commas: {P8, P5, 81/80, 64/63, ...}. The choice of commas for certain primes, notably 11 and 13, is somewhat arbitrary.
=__Derivation__=
In a multi-comma temperament, it's possible that one comma will contain only the 1st and 2nd primes. The 2nd prime is directly related to the 1st prime. If this happens, the multi-gen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.
For example, 2.3.5.7 with commas 256/243 and 225/224. The 1st comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multi-gen must use primes 2 and 5. In this case, the pergen set is {P8/5, y3}, the same as Blackwood.
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 > 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.
For rank-2, we can compute the pergen set right from the PGM [(x y), (0, z)]. For a period P and a generator G:
P8 = xP and WP5 = yP + zG
P = P8/x
G = [WP5 - y(P8/x)] / z = [-yP8 + xWP5]/xz = (-y, x) / xz
To G, add n periods, which are P8/x:
G = (-y, x) / xz + nP8/x = (nz - y, x) / xz
n ranges from -x (subtracting a full octave) to +x (adding a full octave).
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&limit=7|x31.com]] gives us this matrix:
||~ ||~ 2/1 ||~ 3/1 ||~ 5/1 ||~ 7/1 ||
||~ period ||= 1 ||= 1 ||= 1 ||= 2 ||
||~ gen1 ||= 0 ||= 2 ||= 1 ||= 1 ||
||~ gen2 ||= 0 ||= 0 ||= 2 ||= 1 ||
Thus 2/1 = P, 3/1 = P + 2 G1, 5/1 = P + G1 + 2 G2, and 7/1 = 2 P + G1 + G2. Discard the last column, to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:
||~ ||~ 2/1 ||~ 3/1 ||~ 5/1 ||
||~ period ||= 1 ||= 1 ||= 1 ||
||~ gen1 ||= 0 ||= 2 ||= 1 ||
||~ gen2 ||= 0 ||= 0 ||= 2 ||
Use an [[http://wims.unice.fr/wims/wims.cgi?session=GF84B8C7BF.1&lang=en&cmd=reply&module=tool%2Flinear%2Fmatmult.en&matA=1+1+1%0D%0A0+2+1%0D%0A0+0+2&matB=&show=A%5E-1|online tool]] to invert it. "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.
||~ ||~ period ||~ gen1 ||~ gen2 ||~ ||
||~ 2/1 ||= 4 ||= -2 ||= -1 || ||
||~ 3/1 ||= 0 ||= 2 ||= -1 || ||
||~ 5/1 ||= 0 ||= 0 ||= 2 || /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) = WWyy1/4.
Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multi-gen P5 is split in half, one multi-gen equals two gens, and adding an octave to the gen adds a __double__ octave to the multi-gen. The alternate gens are WWP5/2 and P11/2, both of which are larger, so the best gen1 is P5/2.
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.
Alternatively, we could discard the 3rd column and keep the 4th one:
||~ ||~ 2/1 ||~ 3/1 ||~ 7/1 ||
||~ period ||= 1 ||= 1 ||= 2 ||
||~ gen1 ||= 0 ||= 2 ||= 1 ||
||~ gen2 ||= 0 ||= 0 ||= 1 ||
This inverts to this matrix:
||~ ||~ period ||~ gen1 ||~ gen2 ||~ ||
||~ 2/1 ||= 2 ||= -1 ||= -3 || ||
||~ 3/1 ||= 0 ||= 1 ||= -1 || ||
||~ 7/1 ||= 0 ||= 0 ||= 2 || /2 ||
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, but only if the smaller prime is > 3.
=__Applications__=
Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair. Highs and lows are written / and \. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this contrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.
Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is fifth-based and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled C Ev G.
Not all possible combinations of periods and generators are unique pergens. {P8, WWP5/2} is actually {P8, P5/2}. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. This table lists all unique rank-2 pergens, ordered by the size of the largest splitting factor.
The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.
(table is under construction)
||~ pergen set ||~ valid range
of the 5th ||~ equivalences ||~ enharmonic
interval ||~ genchain(s) ||~ example
temperament ||~ compatible edos
(12-31 only) ||
||= {P8, P5} ||= 600-720¢ ||= none ||= none ||= C - G ||= meantone ||= 12, 16, 19, 23, 26 ||
||= {P8/2, P5} ||= 700-720¢ ||= P8/2 = vA4 = ^d5 ||= ^^d2 ||= C - F#v=Gb^ - C ||= srutal ||= 12, 20, 22, 24, 30 ||
||= " ||= 600-700¢ ||= P8/2 = ^A4 = vd5 ||= vvd2 ||= C - F#^=Gbv - C ||= ||= 12, 14, 16, 18b, 26 ||
||= " ||= 600-720¢ ||= P8/2 = ^P4 = vP5 ||= vvM2 ||= C - F^=Gv - C ||= (is this needed?) ||= ||
||= {P8, P5/2} ||= 4\7 - 720¢ ||= P5/2 = ^m3 = vM3 ||= vvA1 ||= C - Eb^=Ev - G ||= mohajira ||= 14, 17, 20, 21, 24
27, 28, 30, 31 ||
||= " ||= 600¢ - 4\7 ||= P5/2 = ^M3 = vm3 ||= vvd1 ||= ||= ||= 14, 18b, 21, 28 ||
||= {P8, P4/2} ||= ||= P4/2 = ^M2 = vm3 ||= vvm2 ||= D - E^=Fv - G ||= semaphore ||= 22 ||
||= " ||= ||= P4/2 = vA2 = ^d3 ||= ^^dd2 ||= ||= ||= ||
||= " ||= ||= P4/2 = ^A2 = vd3 ||= vvdd2 ||= ||= ||= ||
||= {P8, P11/2} ||= ||= P11/2 = ^m6 = vM6 ||= vvA1 ||= ||= ||= ||
||= ||= ||= ||= ||= ||= ||= ||
||= {P8, P12/2} ||= ||= P12/2 = ^M6 = vm7 ||= vvm2 ||= ||= magic ||= ||
||= ||= ||= ||= ||= ||= ||= ||
|| || || || || || || ||
||= {P8/2, P4/2} ||= ||= P8/2 = vA4 = ^d5,
P4/2 = /M2 = \m3 ||= ^^d2,
\\m2 ||= C - F#v=Gb^ - C,
C - D/=Eb\ - F ||= bb&aaT ||= 22 ||
||= {P8/2, P11/2} ||= ||= ||= ||= ||= ||= ||
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||= {P8/2, P12/2} ||= ||= ||= ||= ||= ||= ||
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(to be continued)Original HTML content:
<html><head><title>pergen names</title></head><body><!-- ws:start:WikiTextTocRule:6:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Derivation">Derivation</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Applications">Applications</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: -->
<!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 --><u><strong>Definition</strong></u></h1>
<br />
A <strong>pergen</strong> set (pronounced "peer-gen") 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, but avoided if possible.<br />
<br />
If a rank-2 temperament uses the primes 2 and 3 in its comma(s), then the period can be expressed as the octave 2/1, or some fraction of an octave. The generator can usually be expressed as a 3-limit interval, or some fraction of one. The fraction is always of the form 1/N, in other words, the octave or the 3-limit interval is <strong>split</strong> into N parts. An interval which is split into multiple generators is called a <strong>multi-gen</strong>.<br />
<br />
For example, the srutal temperament splits the octave in two, and is called half-octave. The set is written {P8/2, P5}. The dicot temperament splits the fifth in two, and is called half-fifth, written {P8, P5/2}. Porcupine is third-fourth, {P8, P4/3}. Semaphore, which means "semi-fourth", is of course {P8, P4/2}.<br />
<br />
Many temperaments will share the same pergen set. This has the advantage of reducing the 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. In addition, the pergen set tells us how to notate the temperament using <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>.<br />
<br />
The largest category contains all commas of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime (e.g. 81/80 or 64/63). The period is the octave, and the generator is the fifth: {P8, P5}. Such temperaments are called fifth-based.<br />
<br />
Every temperament has at least one alternate generator. More, if the octave is split. To avoid ambiguity, the generator is 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.<br />
<br />
For example, srutal could be {P8/2, M2/2}, but P5 is preferred because it is unsplit. Or it could be {P8/2, P12}, but P5 is preferred because it is smaller. Or it could be {P8/2, P4}, but P5 is always preferred over P4. Note that P5/2 is <u>not</u> preferred over P4/2. For example, decimal is {P8/2, P4/2}, not {P8/2, P5/2}.<br />
<br />
<table class="wiki_table">
<tr>
<th colspan="2">pergen set<br />
</th>
<th colspan="4">example temperaments<br />
</th>
</tr>
<tr>
<th>written<br />
</th>
<th>spoken<br />
</th>
<th>comma(s)<br />
</th>
<th>name<br />
</th>
<th colspan="2">color name<br />
</th>
</tr>
<tr>
<td style="text-align: center;">{P8, P5}<br />
</td>
<td style="text-align: center;">fifth-based<br />
</td>
<td style="text-align: center;">81/80<br />
</td>
<td style="text-align: center;">meantone<br />
</td>
<td style="text-align: center;">green<br />
</td>
<td style="text-align: center;">gT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">64/63<br />
</td>
<td style="text-align: center;">archy<br />
</td>
<td style="text-align: center;">red<br />
</td>
<td style="text-align: center;">rT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">(-14,8,0,0,1)<br />
</td>
<td style="text-align: center;">schismic<br />
</td>
<td style="text-align: center;">large yellow<br />
</td>
<td style="text-align: center;">LyT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">81/80 & 126/125<br />
</td>
<td style="text-align: center;">septimal meantone<br />
</td>
<td style="text-align: center;">green and bluish-blue<br />
</td>
<td style="text-align: center;">g&bg<span style="vertical-align: super;">3</span>T<br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8/2, P5}<br />
</td>
<td style="text-align: center;">half-octave<br />
</td>
<td style="text-align: center;">(11, -4, -2)<br />
</td>
<td style="text-align: center;">srutal<br />
</td>
<td style="text-align: center;">small deep green<br />
</td>
<td style="text-align: center;">sggT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">81/80 & 50/49<br />
</td>
<td style="text-align: center;">injera<br />
</td>
<td style="text-align: center;">deep reddish and green<br />
</td>
<td style="text-align: center;">rryy&gT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8, P5/2}<br />
</td>
<td style="text-align: center;">half-fifth<br />
</td>
<td style="text-align: center;">25/24<br />
</td>
<td style="text-align: center;">dicot<br />
</td>
<td style="text-align: center;">deep yellow<br />
</td>
<td style="text-align: center;">yyT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">(-1,5,0,0,-2)<br />
</td>
<td style="text-align: center;">mohajira<br />
</td>
<td style="text-align: center;">deep amber<br />
</td>
<td style="text-align: center;">aaT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8, P4/2}<br />
</td>
<td style="text-align: center;">half-fourth<br />
</td>
<td style="text-align: center;">49/48<br />
</td>
<td style="text-align: center;">semaphore<br />
</td>
<td style="text-align: center;">deep blue<br />
</td>
<td style="text-align: center;">bbT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8, P4/3}<br />
</td>
<td style="text-align: center;">third-fourth<br />
</td>
<td style="text-align: center;">250/243<br />
</td>
<td style="text-align: center;">porcupine<br />
</td>
<td style="text-align: center;">triple yellow<br />
</td>
<td style="text-align: center;">y<span style="vertical-align: super;">3</span>T<br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8, P11/3}<br />
</td>
<td style="text-align: center;">third-eleventh<br />
</td>
<td style="text-align: center;">(12,-1,0,0,-3)<br />
</td>
<td style="text-align: center;">small triple amber<br />
</td>
<td style="text-align: center;">small triple amber<br />
</td>
<td style="text-align: center;">sa<span style="vertical-align: super;">3</span>T<br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8/2, P4/2}<br />
</td>
<td style="text-align: center;">half-octave, half-fourth<br />
</td>
<td style="text-align: center;">25/24 & 49/48<br />
</td>
<td style="text-align: center;">decimal<br />
</td>
<td style="text-align: center;">deep yellow and deep blue<br />
</td>
<td style="text-align: center;">yy&bbT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8/4, P5}<br />
</td>
<td style="text-align: center;">quarter-octave<br />
</td>
<td style="text-align: center;">(3,4,-4)<br />
</td>
<td style="text-align: center;">diminished<br />
</td>
<td style="text-align: center;">quadruple green<br />
</td>
<td style="text-align: center;">g<span style="vertical-align: super;">4</span>T<br />
</td>
</tr>
</table>
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.<br />
<br />
For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. <a class="wiki_link" href="/Kite%27s%20color%20notation">Color notation </a>is used to indicate primes higher than 3. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third (y3 = 5/4).<br />
<br />
Rank-3 pergen sets have three intervals, period, gen1 and gen2, any of which may be split. The unsplit 2.3.5... subgroup's set could be {P8, P5, y3}. However, to avoid higher primes, and more closely mimic conventional notation, it's better to reduce gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with an up symbol, and we have {P8, P5, ^1} = fifth-based with ups. However, if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and colors must be used for gen2.<br />
<br />
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} = third-tone with ups.<br />
<br />
A rank-4 temperament has a pergen set of four intervals. A rank-1 temperament could have a pergen set of one, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.<br />
<br />
Finally, the pergen set for JI is best thought of as the octave, the fifth, and a set of commas: {P8, P5, 81/80, 64/63, ...}. The choice of commas for certain primes, notably 11 and 13, is somewhat arbitrary.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:2 --><u>Derivation</u></h1>
<br />
In a multi-comma temperament, it's possible that one comma will contain only the 1st and 2nd primes. The 2nd prime is directly related to the 1st prime. If this happens, the multi-gen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.<br />
<br />
For example, 2.3.5.7 with commas 256/243 and 225/224. The 1st comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multi-gen must use primes 2 and 5. In this case, the pergen set is {P8/5, y3}, the same as Blackwood.<br />
<br />
To find a temperament's pergen set, first find the <strong>PGM</strong>, 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 <a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank">x31eq.com/temper/uv.html</a> 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 > 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.<br />
<br />
For rank-2, we can compute the pergen set right from the PGM [(x y), (0, z)]. For a period P and a generator G:<br />
<br />
P8 = xP and WP5 = yP + zG<br />
P = P8/x<br />
G = [WP5 - y(P8/x)] / z = [-yP8 + xWP5]/xz = (-y, x) / xz<br />
<br />
To G, add n periods, which are P8/x:<br />
G = (-y, x) / xz + nP8/x = (nz - y, x) / xz<br />
n ranges from -x (subtracting a full octave) to +x (adding a full octave).<br />
<br />
Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7" rel="nofollow">x31.com</a> gives us this matrix:<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>2/1<br />
</th>
<th>3/1<br />
</th>
<th>5/1<br />
</th>
<th>7/1<br />
</th>
</tr>
<tr>
<th>period<br />
</th>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">2<br />
</td>
</tr>
<tr>
<th>gen1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
<tr>
<th>gen2<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
</table>
Thus 2/1 = P, 3/1 = P + 2 G1, 5/1 = P + G1 + 2 G2, and 7/1 = 2 P + G1 + G2. Discard the last column, to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>2/1<br />
</th>
<th>3/1<br />
</th>
<th>5/1<br />
</th>
</tr>
<tr>
<th>period<br />
</th>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
<tr>
<th>gen1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
<tr>
<th>gen2<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
</tr>
</table>
Use an <a class="wiki_link_ext" href="http://wims.unice.fr/wims/wims.cgi?session=GF84B8C7BF.1&lang=en&cmd=reply&module=tool%2Flinear%2Fmatmult.en&matA=1+1+1%0D%0A0+2+1%0D%0A0+0+2&matB=&show=A%5E-1" rel="nofollow">online tool</a> to invert it. "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>period<br />
</th>
<th>gen1<br />
</th>
<th>gen2<br />
</th>
<th><br />
</th>
</tr>
<tr>
<th>2/1<br />
</th>
<td style="text-align: center;">4<br />
</td>
<td style="text-align: center;">-2<br />
</td>
<td style="text-align: center;">-1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<th>3/1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">-1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<th>5/1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td>/4<br />
</td>
</tr>
</table>
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) = WWyy1/4.<br />
<br />
Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multi-gen P5 is split in half, one multi-gen equals two gens, and adding an octave to the gen adds a <u>double</u> octave to the multi-gen. The alternate gens are WWP5/2 and P11/2, both of which are larger, so the best gen1 is P5/2.<br />
<br />
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.<br />
<br />
Alternatively, we could discard the 3rd column and keep the 4th one:<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>2/1<br />
</th>
<th>3/1<br />
</th>
<th>7/1<br />
</th>
</tr>
<tr>
<th>period<br />
</th>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">2<br />
</td>
</tr>
<tr>
<th>gen1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
<tr>
<th>gen2<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
</table>
This inverts to this matrix:<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>period<br />
</th>
<th>gen1<br />
</th>
<th>gen2<br />
</th>
<th><br />
</th>
</tr>
<tr>
<th>2/1<br />
</th>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">-1<br />
</td>
<td style="text-align: center;">-3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<th>3/1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">-1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<th>7/1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td>/2<br />
</td>
</tr>
</table>
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, but only if the smaller prime is > 3.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:4 --><u>Applications</u></h1>
<br />
Pergen sets allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all fifth-based temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, ups and downs. And certain rank-2 temperaments require another additional pair. Highs and lows are written / and \. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this contrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.<br />
<br />
Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is fifth-based and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled C Ev G.<br />
<br />
Not all possible combinations of periods and generators are unique pergens. {P8, WWP5/2} is actually {P8, P5/2}. {P8/2, P5/2} is actually {P8/2, P4/2}. There is no {P8, M2/2}, and {P8/2, M2/2} is actually {P8/2, P5}. This table lists all unique rank-2 pergens, ordered by the size of the largest splitting factor.<br />
<br />
The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.<br />
<br />
(table is under construction)<br />
<br />
<table class="wiki_table">
<tr>
<th>pergen set<br />
</th>
<th>valid range<br />
of the 5th<br />
</th>
<th>equivalences<br />
</th>
<th>enharmonic<br />
interval<br />
</th>
<th>genchain(s)<br />
</th>
<th>example<br />
temperament<br />
</th>
<th>compatible edos <br />
(12-31 only)<br />
</th>
</tr>
<tr>
<td style="text-align: center;">{P8, P5}<br />
</td>
<td style="text-align: center;">600-720¢<br />
</td>
<td style="text-align: center;">none<br />
</td>
<td style="text-align: center;">none<br />
</td>
<td style="text-align: center;">C - G<br />
</td>
<td style="text-align: center;">meantone<br />
</td>
<td style="text-align: center;">12, 16, 19, 23, 26<br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8/2, P5}<br />
</td>
<td style="text-align: center;">700-720¢<br />
</td>
<td style="text-align: center;">P8/2 = vA4 = ^d5<br />
</td>
<td style="text-align: center;">^^d2<br />
</td>
<td style="text-align: center;">C - F#v=Gb^ - C<br />
</td>
<td style="text-align: center;">srutal<br />
</td>
<td style="text-align: center;">12, 20, 22, 24, 30<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">600-700¢<br />
</td>
<td style="text-align: center;">P8/2 = ^A4 = vd5<br />
</td>
<td style="text-align: center;">vvd2<br />
</td>
<td style="text-align: center;">C - F#^=Gbv - C<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">12, 14, 16, 18b, 26<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">600-720¢<br />
</td>
<td style="text-align: center;">P8/2 = ^P4 = vP5<br />
</td>
<td style="text-align: center;">vvM2<br />
</td>
<td style="text-align: center;">C - F^=Gv - C<br />
</td>
<td style="text-align: center;">(is this needed?)<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8, P5/2}<br />
</td>
<td style="text-align: center;">4\7 - 720¢<br />
</td>
<td style="text-align: center;">P5/2 = ^m3 = vM3<br />
</td>
<td style="text-align: center;">vvA1<br />
</td>
<td style="text-align: center;">C - Eb^=Ev - G<br />
</td>
<td style="text-align: center;">mohajira<br />
</td>
<td style="text-align: center;">14, 17, 20, 21, 24 <br />
27, 28, 30, 31<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">600¢ - 4\7<br />
</td>
<td style="text-align: center;">P5/2 = ^M3 = vm3<br />
</td>
<td style="text-align: center;">vvd1<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">14, 18b, 21, 28<br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8, P4/2}<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">P4/2 = ^M2 = vm3<br />
</td>
<td style="text-align: center;">vvm2<br />
</td>
<td style="text-align: center;">D - E^=Fv - G<br />
</td>
<td style="text-align: center;">semaphore<br />
</td>
<td style="text-align: center;">22<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">P4/2 = vA2 = ^d3<br />
</td>
<td style="text-align: center;">^^dd2<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">P4/2 = ^A2 = vd3<br />
</td>
<td style="text-align: center;">vvdd2<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8, P11/2}<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">P11/2 = ^m6 = vM6<br />
</td>
<td style="text-align: center;">vvA1<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8, P12/2}<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">P12/2 = ^M6 = vm7<br />
</td>
<td style="text-align: center;">vvm2<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">magic<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
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<tr>
<td style="text-align: center;">{P8/2, P4/2}<br />
</td>
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</td>
<td style="text-align: center;">P8/2 = vA4 = ^d5,<br />
P4/2 = /M2 = \m3<br />
</td>
<td style="text-align: center;">^^d2,<br />
\\m2<br />
</td>
<td style="text-align: center;">C - F#v=Gb^ - C,<br />
C - D/=Eb\ - F<br />
</td>
<td style="text-align: center;">bb&aaT<br />
</td>
<td style="text-align: center;">22<br />
</td>
</tr>
<tr>
<td style="text-align: center;">{P8/2, P11/2}<br />
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</tr>
<tr>
<td style="text-align: center;">{P8/2, P12/2}<br />
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(to be continued)</body></html>