Kite's thoughts on pergens: Difference between revisions

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3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz

G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, and the cents of any accidental pairs used, rounded off to the nearest integer. Ths gives the musician, who may not be well-versed in microtonal theory, the necessary information to play the score. Examples:

19-edo: # = 63¢

21-edo: ^ = 57¢

quarter-comma meantone: # = 76¢

fifth-comma meantone: # = 84¢

third-comma archy: # = 177¢

Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.

Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].

For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The enharmonic can also be found using gedras: xE = M - n&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G = P5 - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m3 = [7,4] - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[3,2] = [7,4] - [6,4] = [1,0] = A1.

The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</pre></div>

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In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the comma's monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).&lt;br /&gt;

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For any comma containing primes 2 and 3, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents. The comma will split the octave into m parts, and if n &amp;gt; m, it will split some 3-limit interval into n parts.&lt;br /&gt;

3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz&lt;br /&gt;

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G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz&lt;br /&gt;

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Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen 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-5ths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-5th with red. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, gg7/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &amp;gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.&lt;br /&gt;

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One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, because either it contains more than 2 primes, or because the multigen isn't actually a generator. For example, sensei, or semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen of (P8, (5/4)/2), but the tone isn't a generator.&lt;br /&gt;

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To list 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 viable 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;

In the &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials"&gt;Supplemental materials&lt;/a&gt; section, a program is linked to that performs these calculations and lists all pergens. It also lists suggested enharmonics. Experimenting with allowing y and i to range further does not produce any additional pergens.&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, ups and downs are used for the octave-splitting enharmonic, and highs/lows are used for the multigen-splitting enharmonic. But the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs could be 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. #1 is always (-11,7) = 2187/2048, by definition.&lt;br /&gt;

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, and the cents of any accidental pairs used, rounded off to the nearest integer. Ths gives the musician, who may not be well-versed in microtonal theory, the necessary information to play the score. Examples:&lt;br /&gt;

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19-edo: # = 63¢&lt;br /&gt;

21-edo: ^ = 57¢&lt;br /&gt;

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quarter-comma meantone: # = 76¢&lt;br /&gt;

fifth-comma meantone: # = 84¢&lt;br /&gt;

third-comma archy: # = 177¢&lt;br /&gt;

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Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.&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;

&lt;ul class="quotelist"&gt;&lt;li&gt;a = -11k + 19b&lt;/li&gt;&lt;li&gt;b = 7a - 12b&lt;/li&gt;&lt;/ul&gt;Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].&lt;br /&gt;

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For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The enharmonic can also be found using gedras: xE = M - n&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G = P5 - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m3 = [7,4] - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[3,2] = [7,4] - [6,4] = [1,0] = A1.&lt;br /&gt;

A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3).&lt;br /&gt;

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&lt;span style="display: block; text-align: center;"&gt;P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- E^=Fv\ -- Ab/=A\ -- C^/=Dbv -- F&lt;/span&gt;&lt;br /&gt;

One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling.&lt;br /&gt;

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&lt;span style="display: block; text-align: center;"&gt;P1 — \M3 — \\A5=/m6 — P8&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C — E\ — Ab/ — C&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;8=v/m9 — F&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C — E^\ — Ab^^/=Avv\ — Dbv/ — F&lt;/span&gt;&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;

&lt;span style="display: block; text-align: center;"&gt;P1 -- vM3 -- ^m6 -- P8&lt;br /&gt;

&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Ev -- Ab^ -- C&lt;br /&gt;

Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2. Thus the vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.&lt;br /&gt;

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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. Further study is needed.&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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Alt-pergenLister 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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Screenshot of the first 38 pergens:&lt;br /&gt;

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Given:&lt;br /&gt;