Kite's thoughts on pergens: Difference between revisions

[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf '''TallKite.com/misc_files/notation guide for rank-2 pergens.pdf'''] for practical notation examples.  

''See also: [[Rank-2 temperaments by mapping of 3]]''

A '''pergen''' (pronounced "peer-gen") is a way of classifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.

For example, the Srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The Dicot aka Yoyo temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine aka Triyo is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore aka Zozo, a pun on "semi-fourth", is of course half-fourth.

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both Srutal aka Sagugu and Injera aka Gu & Biruyo sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using '''ups and downs''' (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and Downs Notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.

| | Gu

| | Ru

| | Layo

| | Sagugu

| | Gu & Biruyo

| | Yoyo

| | Lulu

| | Zozo

| | Yoyo & Zozo

| | Triyo

| | Satrilu

| | Quadgu

| | Laquadlo

| | Laquinyo

For example, Marvel aka Ruyoyo (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.

More examples: Trizogu (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Biruyo (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, Biruyo Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.

For example, consider Sawa & Ruyoyo (2.3.5.7 with commas 256/243 and 225/224). The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below).

For example, Porcupine aka Triyo (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 <= i <= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).

Rank-3 pergens are trickier to find. For example, Breedsmic aka Bizozogu 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:

Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is Triyo, and the second one is Triyo & Ru. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups_and_Downs_Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Lulu and Dicot aka Yoyo are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and 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 aka Layo is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C vE G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.

The enharmonic interval, or more briefly the '''enharmonic''' or the '''EI''', can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. The pergen and the enharmonic together define the notation. (''Edited to add: not quite accurate, see the Addenda.'')

interval(s)

| | Srutal aka Sagugu

| | Injera aka Gu & Biruyo

| | Thotho, if 13/8 = M6

| | Semaphore aka Zozo

| | Lala-yoyo

| | Mohajira aka Lulu

| | Zozo & Lulu

| | Sagugu & Zozo

| | Sagugu & Lulu

| | Augmented aka Trigu

| | Porcupine aka Triyo

| | Slendric aka Latrizo

| | Satrilu, if 11/8 = A4

| | Satrilu, if 11/8 = P4

| | Tribilo, if 11/8 = P4

| | Triforce aka Trigu & Zozo

| | Satribizo

| | Latribiru

| | Lartribiyo

| | Lemba aka Latrizo & Biruyo

| | Latribilo, if 11/8 = P4

| | Triyo & Triru

| | Trigu & Latrizo

| | Triyo & Latrizo

| | Diminished aka Quadgu

| | Negri aka Laquadyo

| | Tetracot aka Saquadyo

| | Squares aka Laquadru

| | Vulture aka Sasa-quadyo

Removing the ups and downs from an enharmonic interval makes an '''uninflected''' enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the uninflected enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore <u>'''up may need to be swapped with down, depending on the size of the 5th'''</u> in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every uninflected enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the uninflected enharmonic.

! colspan="2" | uninflected EI

Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka Triyo) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:

More remote intervals include A1, d4, d7 and d10. These unfortunately require a very long genchain. The most interesting melodically is A1: C - ^C - vC# - C#. From C to C# is seven 5ths, which equals 21 generators, so the genchain would have to contain 22 notes if it had no gaps. Note that A1 is the uninflected enharmonic of third-4th. The uninflected enharmonic is always a secondary split.

Given a pergen (P8/m, (a,b)/n), an interval (a',b') splits into GCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof [[pergen#Further Discussion-Various proofs|below]]). For an unsplit pergen, we have the naturally occurring split of GCD (a', b'). If only the 8ve is split, we have GCD (a'·m, b'). If m = n (an nth-everything pergen), we have n·GCD (a',b'). If the enharmonic is an A1, every interval with a degree of n+1 will be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. "Every" means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.

| | (P83, P4/3)

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a '''single-split''' pergen. If it has two fractions, it's a '''double-split''' 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 '''single-pair''' notation because it adds only a single pair of accidentals to conventional notation. '''Double-pair''' notation uses both ups/downs and lifts/drops. 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, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.

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 color depth of 1 (i.e. exponent of ±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<span style="">⋅</span>P and P8. If P is 6/5, the comma is 4<span style="">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4<span style="">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>, the Quadru temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be x. Then n<span style="">⋅x</span> gens = n<span style="">⋅</span>I = x<span style="">⋅</span>M, where M is the multigen and M/n is the generator. The comma can be found from this equation, if n and x are coprime. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5<span style="">⋅</span>(7/6) = 2<span style="">⋅P5. Thus </span>2<span style="">⋅P</span>5 - 5<span style="">⋅</span>(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7<span style="">⋅</span>(11/9) = 2<span style="">⋅</span>P8, and the comma is (-2, -14, 0, 0, 7), Saseplo.

Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an '''alternate''' generator. A generator or period plus or minus any number of enharmonics makes an '''equivalent''' generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. For example: (P8, P5/2) has generator ^m3 and equivalent generator vM3. Another example, half-8ve (P8/2, P5) has period vA4 and equivalent period ^d5. It has generator P5 and alternate generators P4 and vA1. vA1 is equivalent to ^m2.

Of the two equivalent generators, the preferred generator is always the smaller one (smaller degree, or if the degrees are the same, more diminished). This is because the equivalent generator can be more easily found by adding the enharmonic, rather than subtracting it. For example, ^m3 + vvA1 = vM3 is an easier calculation than vM3 - vvA1 = ^m3. This is particularly true with complex enharmonics like ^<span style="vertical-align: super;">6</span>dd2.

There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2^x · 3^y · P^±1, where P is a higher prime. They are called mapping commas because they equate or map P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for prime 5 include 81/80, 135/128, and the schisma = Ly-2 = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 (besides 1/1 of course) are the currently employed commas and combinations of them.

If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & Biruyo, where ^1 equals 64/63 minus 81/80.

Cents can be assigned to each accidental symbol, even if no specific commas are specified. 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. The sharp always equals A1 = 100¢ + 7c. Since the enharmonic = 0¢, we can derive the cents of the up symbol. 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.

In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the enharmonic is an A1.

* third-comma Archy aka Ru: # = 177¢

* eighth-comma Porcupine aka Triyo: # = 157¢, ^ = 52¢ (^ = third-sharp)

* seventh-comma Srutal aka Sagugu & Zoquadyo: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #, seventh-comma means 1/7 of 2048/2025)

* third-comma Injera aka Gu & Biruyo: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)

* eighth-comma Hedgehog aka Triyo & Biruyo: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)

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:

<ul><li>For (P8/m, M/n), P8 = mP + xEI and M = nG + yEI', with 0 &lt; |x| &lt;= m/2 and 0 &lt; |y| &lt;= n/2</li><li>x is the count for EI, with EI occurring x times in one octave, and xEI is the octave's '''multi-enharmonic''', or '''multi-EI''' for short</li><li>y is the count for EI', with EI' occurring y times in one multigen, and yEI' is the multigen's multi-EI</li><li>For false doubles using single-pair notation, EI = EI', but x and y are usually different, making different multi-enharmonics</li><li>The unreduced pergen is (P8/m, M'/n'), with a new enharmonic EI" and new counts, P8 = mP + x'EI", and M' = n'G' + y'EI"</li></ul>

Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up.

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: xEI = M - n<span style="">⋅</span>G = P5 - 2<span style="">⋅</span>m3 = [7,4] - 2<span style="">⋅</span>[3,2] = [7,4] - [6,4] = [1,0] = A1.

Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xEI = P8 - mP = P8 - 5<span style="">⋅</span>M2 = [12,7] - 5<span style="">⋅</span>[2,1] = [2,2] = 2<span style="">⋅</span>[1,1] = 2<span style="">⋅</span>m2. Because x = 2, EI will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2<span style="">⋅</span>m2 = d3). The enharmonic's '''count''' is 2. The uninflected enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus EI = v<span style="vertical-align: super;">5</span>m2. Since P8 = 5<span style="">⋅</span>P + 2<span style="">⋅</span>EI, the period must be ^^M2, to make the ups and downs come out even. The number of the period's (or generator's) ups or downs always equals the count. Equipped with the period and the enharmonic, the perchain is easily found:

Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has an uninflected generator [5,3]/5 = [1,1] = m2. The uninflected enharmonic is P4 - 5<span style="">⋅</span>m2 = [5,3] - 5<span style="">⋅</span>[1,1] = [5,3] - [5,5] = [0,-2] = -2<span style="">⋅</span>[0,1] = two descending d2's. The d2 must be upped, and EI = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5<span style="">⋅</span>G - 2<span style="">⋅</span>EI, G must be ^^m2. The genchain is:

To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional multigen as before. Then deduce the period from the enharmonic. If the multigen was changed by unreducing, find the original generator from the period and the alternate generator.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The uninflected alternate generator G' is [1,1]/10 = [0,0] = P1. The uninflected enharmonic is m2 - 10<span style="">⋅</span>P1 = m2. It must be downed, thus EI = v<span style="vertical-align: super;">10</span>m2. Since m2 = 10<span style="">⋅</span>G' + EI, G' is ^1. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x<span style="">⋅</span>m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5<span style="">⋅</span>P + 2<span style="">⋅</span>EI, and P = ^<span style="vertical-align: super;">4</span>M2. Next, find the original half-4th generator. P = P8/5 = ~240¢, and G = P4/2 = ~250¢. Because P &lt; G, G' is not P - G but G - P, and G is not P - G' but P + G', which equals ^<span style="vertical-align: super;">4</span>M2 + ^1 = ^<span style="vertical-align: super;">5</span>M2. While the alternate multigen is more complex than the original multigen, the alternate generator is usually simpler than the original generator.

To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic. For (P8/2, P4/2), the split octave implies P = vA4 and EI = ^^d2, and the split 4th implies G = /M2 and EI' = \\m2.

A false-double pergen can optionally 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).

Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So EI = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The uninflected enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: EI' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: EI + EI' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and EI - EI' = v<span style="vertical-align: super;">4</span>//ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:

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 EI = v<span style="vertical-align: super;">12</span>m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the uninflected enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus EI' = /<span style="vertical-align: super;">3</span>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 EI + EI' and EI - 2·EI'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling. Enharmonic = v<span style="vertical-align: super;">12</span>m3 = /<span style="vertical-align: super;">3</span>d2 = v<span style="vertical-align: super;">4</span>/m2 = v<span style="vertical-align: super;">4</span>\\A1. Period = \M3 = v<span style="vertical-align: super;">4</span>4 = //d4. Generator = ^\M3 = v<span style="vertical-align: super;">3</span>4 = ^//d4.

It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan.

Sixth-4th with single-pair notation has an awkward ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">6</span>4 enharmonic. This pergen might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore  commas, aka Triyo & Zozo), and its double-pair notation can also combine both. Third-4th has EI = v<span style="vertical-align: super;">3</span>A1 and G'= vM2 = ^^m2. Half-4th has EI' = \\m2 and G' = /M2 = \m3. G' - G = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G' - G = /M2 - vM2 = ^/1. Equivalent enharmonics are v³\\M2 and ^³\\d2.  

When ups and downs are used to notate edos, a third symbol is used, a '''mid''' , written ~. The mid is only for relative notation (intervals), never for absolute notation (notes). For imperfect intervals, the mid interval is the interval exactly midway between major and minor. In addition, the mid-4th is midway between perfect and augmented, i.e. half-augmented, and the mid-5th is half-diminished. For example, 17edo's 2nds and 3rds run minor-mid-major. This avoids having to constantly choose between the equivalent terms upminor and downmajor. 72edo's 2nds and 3rds run minor, upminor, downmid, mid, upmid, downmajor, major. This makes the terms more concise. Mids can be used in pergen notation too. For single-pair, EI must be an A1, with an even number of downs. Using mids with double-pair notation is trickier. Mids never appear in the perchain. If one accidental pair appears only in the perchain and the other pair only in the genchain, then the mids appear only in the genchain, if at all.

==Alternate enharmonics==

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, ccM6/12), a false double. The uninflected alternate generator is ccM6/12 = [33,19]/12 = [3,2] = m3. The uninflected 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<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.

Because G is a M2 and EI is an A2, the equivalent generator G - EI is a descending A1. Ascending intervals that look descending can be confusing, one has to take into account the eighth-tone ups to see that v<span style="vertical-align: super;">3</span>D -- ^<span style="vertical-align: super;">6</span>Db is ascending. Double-pair notation may be preferable. This makes P = vM3, EI = ^3d2, G = /m2, and EI' = /4dd2.

Because the enharmonic found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic.

To search for alternate enharmonics, convert EI to a gedra, then multiply it by the count to get the multi-EI (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and EI is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-EI. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new EI. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see [[pergen#Applications-Tipping points|tipping points]] above. Add n'''·'''count ups or downs to the new multi-EI. Add or subtract the new multi-EI from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period).

For example, (P8, P5/3) has n = 3, G = ^M2, and EI = v<span style="vertical-align: super;">3</span>m2 = [1,1]. G is upped only once, so the count is 1, and the multi-EI is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-EI [-2,1]. This can't be simplified, so the new EI is also [-2,1] = d<span style="vertical-align: super;">3</span>2. Assuming a reasonably just 5th, EI needs to be upped, so EI = ^<span style="vertical-align: super;">3</span>d<span style="vertical-align: super;">3</span>2. Add the multi-EI ^<span style="vertical-align: super;">3</span>[-2,1] to the multigen P5 = [7,4] to get ^<span style="vertical-align: super;">3</span>[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^<span style="vertical-align: super;">3</span>[-2,1] = v<span style="vertical-align: super;">3</span>[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·EI to check: 3·vA2 + 1·^3d<span style="vertical-align: super;">3</span>2 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d<span style="vertical-align: super;">3</span>2 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- vD# -- ^Fb -- G. Because d<span style="vertical-align: super;">3</span>2 = -200¢ - 26·c, ^ = (-d<span style="vertical-align: super;">3</span>2) / 3 = 67¢ + 8.67·c, about a third-tone.

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 EI, or the multi-EI if the count is &gt; 1. For example, consider Semaphore aka Zozo (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- ^D=vEb -- F. But consider Lala-yoyo, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus EI = ^^dd2, G = vA2, and the genchain is C -- vD#=^Ebb -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.

Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.

For example, Satrilu tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the enharmonic is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as a vA4, the enharmonic is ^<span style="vertical-align: super;">3</span>dd2.

Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, Liese aka Gu & Trizogu (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and EI = 3·vd5 - P11 = v<span style="vertical-align: super;">3</span>dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.

This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.

Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups_and_Downs_Notation|ups and downs]] 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.

Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka Sagugu & Ru (2.3.5.7 with 2048/2025 and 64/63) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and EI = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C vE G Bb = Cv,7.

A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, Injera aka Gu & Biruyo (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and EI = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G vBb = C,v7.

Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. Iv -- vIIIv -- vVI^m -- Iv. A Porcupine aka Triyo (third-4th) comma pump can be written out like so: Cv -- vA^m -- vDv -- [vvB=^Bb]^m -- ^Ebv -- G^m -- Gv -- Cv. Brackets are used to show that vvB and ^Bb are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. vvB is written first to show that this root is a vM6 above the previous root, vD. ^Bb is second to show the P4 relationship to the next root, ^Eb. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either vvB^m or ^Bb^m, or possibly ^BbvvM = ^Bb vD ^F.

Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with EI = v³A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps.

The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. As noted above, the 5th of Pajara (half-8ve) tends to be sharp, thus it has EI = ^^d2. But Injera, also half-8ve, has a flat 5th, and thus EI = vvd2. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But if a single temperament "tips over", either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.

The tipping point depends on the choice of enharmonic. It's not the temperament that tips, it's the notation. Half-8ve could be notated with an EI of vvM2. The tipping point becomes 600¢, a <u>very</u> unlikely 5th, and tipping is impossible. For single-comma temperaments, EI usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of EI is a given. However, the mapping of primes 11 and 13 is not agreed on.

The notation's tipping point is determined by the uninflected enharmonic, which is implied by the vanishing comma. For example, Porcupine aka Triyo's 250/243 comma is an A1 = (-11,7), which implies an uninflected EI of A1, which implies 7-edo, and a 685.7¢ tipping point. Dicot aka Yoyo's 25/24 comma is also an A1, and has the same tipping point. Semaphore aka Zozo's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. See [[pergen#Applications-Tipping points|tipping points]] above for a more complete list.

Double-pair notation has two enharmonics, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two enharmonics can be either A1, m2 or d2. One can choose to use whichever two enharmonics best avoid tipping.

An example of a temperament that tips easily is Negri aka Laquadyo, 2.3.5 and (-14,3,4). Because Negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with Negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and EI could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an EI of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma Negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.

Another "tippy" temperament is found by adding the mapping comma 81/80 to the Negri comma and getting the Latriyo comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.

! | EI

| | Meantone aka Gu

| | Mavila aka Layobi

| | Lagu

| | Schismic aka Layo

| | Lalagu

| | Father aka Gubi

| | Superpyth aka Sasayo

A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Ru aka Archy (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C vE G vBb.

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonics are like commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:

! | notation's rank<br>if no enharmonics

! | # of enharmonics<br>needed

! | enharmonics

| | EI = d2

| | EI = dd2

| | EI = m2, EI' = v<span style="vertical-align: super;">3</span>A1 = v<span style="vertical-align: super;">3</span>M2

| | EI = d2, EI' = vvA1 = vvm2

| | Meantone aka Gu

| | Diaschismic aka Sagugu

| | EI = ^^d2

| | Semaphore aka Zozo

| | EI = vvm2

| | Decimal aka Yoyo & Zozo

| | EI = vvd2, EI' = \\m2 = ^^\\A1

| | Marvel aka Ruyoyo

| | Breedsmic aka Bizozogu

| | EI = \\dd3

When there is more than one enharmonic, the first one can be added to or subtracted from the 2nd one, to make an equivalent 2nd enharmonic.

Even the unsplit rank-3 pergen requires single-pair notation, for the 2nd generator. Single-splits and false double-splits require double-pair, true doubles and false triples require triple-pair, and true triples require quadruple-pair. Some false triples and some true doubles may use quadruple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third or fourth pair of symbols, and a third or fourth pair of adjectives to describe them, one might simply use colors.

! | enharmonic

| | Marvel aka Ruyoyo

| | Biruyo

| | Trizogu

| | Breedsmic aka Bizozogu

| | Demeter aka Trizo-agugu

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, Biruyo is half-8ve, with a d2 comma, like Srutal. Using the same notation as Srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and EI = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C vE G v/Bb. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C vE G \Bb.

With this standardization, the enharmonic can be derived directly from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)<span style="vertical-align: super;">-2</span> · (64/63)<span style="vertical-align: super;">2</span> · (-19,12). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore EI = ^^\\d2. The period is found by adding/subtracting EI from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both EI and P become more complex, but the ratio for /1 becomes simpler.

This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For Biruyo, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyo doesn't tip. Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.

Unlike the previous examples, Demeter aka Trizo-agugu's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no EI. But the 4:5:6:7 chord would be spelled C -- ^^^Fbbb -- G -- ^^Bbb, very awkward! Standard double-pair notation is better. Gen2 = v/A1, EI = ^^\\\dd3, and ^^\\\C = A##. Genchain2 is C -- v/C# -- \Eb -- vE -- \\Gb -- v\G -- vvG#=\\\Bbb -- v\\Bb... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals.

There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For Demeter, any combination of \m3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because there will always be a 3-limit comma small enough to be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the '''DOL''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred.

If ^1 = 81/80, possible half-split gen2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.

| | half-8ve half-upminor 3rd

| | half-8ve half-downmajor 3rd

| | etc.

| | (P8/2, P5, ^m6/2)

| | half-8ve half-upminor 6th

| | (P8/2, P5, ^M6/2)

| |  

| |  

|-

| |  

| | 18

| | (P8, P5/2, ^m3/2)

| |  

| |  

| | 19

| |  

|-

| |  

| | 22

| |  

A multi-EDO pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood aka Sawa plus Ya is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:

! | enharmonics

| | Blackwood aka Sawa+ya

| | EI = m2

| | Whitewood aka Lawa+ya

| | EI = A1

| | EI = m2, EI' = vvA1 = vvM2

| | EI = d2

| | EI = dd3, EI' = vm2 = vvA1

! | enharmonic

| | Laquinzo

| | EI = v<span style="vertical-align: super;">5</span>m2

| | Saquinru

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.

But in non-8ve and no-5ths pergens, not every name has a note. For example, Biruyo Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - vF# - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.

Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.

| | (P8/2, P5) [10]

| | half-8ve decatonic

The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the Sensei aka Sepgu & Ruyoyo generator would be (P8, ccP5/7) [5]. The rationale would be that two Sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be Roulette [7] aka Zozoquingu Nowa [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4.

Half-5th has EI = vvA1, hence any EDO in which A1 = 4 edosteps will have ^1 = 2 edosteps. These "doubled EDOs" are 20, 27, 34, 41, 48, 55, etc. The "tripled EDOs" with A1 = 6 edosteps and ^1 = 3 edosteps are every 7th EDO from 30 to 72.

Half-4th has EI = vvm2. Doubled EDOs have m2 = 4 edosteps. These are 23, 28, 33, 38, 43, 48, 53, etc. Tripled EDOs are every 5th one from 47 to 72.

Third-4th has EI = v³A1. Doubled EDOs are the same ones as half-5th's tripled EDOs. Third-5th has EI = v³m2. Doubled EDOs are the same as half-4th's tripled EDOs.

This PDF is a rank-2 notation guide that shows the full lattice for the first 32 pergens, up through the quarter-splits block. It also includes the single-split pergens from the fifth-split, sixth-split and seventh-split blocks. It includes alternate enharmonics for many pergens.

PergenLister lists out thousands of rank-2 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 or edo pair.

The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.

Screenshots of the first 69 pergens:

[[File:alt-pergenLister_1.png|alt=alt-pergenLister 1.png|800x427px|alt-pergenLister 1.png]]

[[File:alt-pergenLister_2.png|alt=alt-pergenLister 2.png|800x455px|alt-pergenLister 2.png]]

[[File:alt-pergenLister_12edo.png|alt=alt-pergenLister 12edo.png|800x449px|alt-pergenLister 12edo.png]]

Some of the pergens supported by 15edo. A red asterisk means partial support.

[[File:alt-pergenLister_15edo.png|alt=alt-pergenLister 15edo.png|800x493px|alt-pergenLister 15edo.png]]

[[File:alt-pergenLister_19edo.png|alt=alt-pergenLister 19edo.png|800x459px|alt-pergenLister 19edo.png]]

If s is a multiple of n (happens when EI is an A1) and s' is a multiple of n, let s = x·n and s' = y·n

enharmonic interval, EI<br />

<u>'''Combining pergens'''</u>

 

If a false double pergen can be broken down into two simpler ones, that may help with finding a double-pair notation with an enharmonic of a 2nd or less. For example, sixth-4th's single pair notation has an EI of a 4th. But since sixth-4th is half-4th plus third-4th, and those two have a good EI, sixth-4th can be notated with one pair from half-4th and another from third-4th.

 

<u>'''Height of a pergen'''</u>

The LCM of the pergen's two splitting fractions could be 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 <u>minimum</u> 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.

<u>'''Credits'''</u>

 

Pergens were discovered by [[KiteGiedraitis|Kite Giedraitis]] in 2017, and developed with the help of [[PraveenVenkataramana|Praveen Venkataramana]].

All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyo has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Sagugu has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.

===The EI(s) define the pergen===

The pergen can be derived directly from the EI(s). Thus the EI(s) define both the pergen and the notation. An EI can be thought of as a comma in the 2.3.^ subgroup. One derives a mapping from this comma as one would for any 3-prime comma, and one derives the pergen by inverting the first 2 columns of the mapping.  

For example, if the EI is vvd2, the 2.3.^ monzo is [19 -12 -2]. There are many possible mappings, but only one that gives a canonical pergen: [(2 2 7) (0 1 -6)]. Discarding the last column and inverting gives us [(1/2 0) (-1 1)] = (P8/2, P5). Another example: vvA1 = [-11 7 -2]. The mapping [(1 1 -2) (0 2 7)] reduces to [(1 1) (0 2)], which inverts to [(1 0) (-1/2 1/2)] = (P8, P5/2).

One can explore the universe of possible EI's, and thus possible pergen notations, more easily if using the gedra format, expressing the EI as a combination of A1's, d2's and arrows. Thus vvA1 = [1 0 -2], v³m2 = [1 1 -3], etc. Unlike a conventional 2.3.^ monzo, the first two numbers in a A1.d2.^1 monzo are fairly small, and the second number is never negative (since it's an EI). And we can require that the first two numbers be coprime (see the next section). All this facilitates one's search.

===Simplifying a "squared" EI===

Consider an uninflected EI of AA1. AA1 is "squared" in the sense that AA1 = A1 + A1, thus both numbers in its ratio are square numbers. If it had an even upspan (the number of ups), it would obviously be an invalid EI, for if v⁴AA1 = 0¢, then so does vvA1, and v⁴AA1 could be replaced with vvA1. So the upspan must be odd.

Consider an EI of v³AA1. The pergen is (P8, P4/3). Here are the [[twin squares]].

The two red numbers in the lower left are both even, because AA1 is a squared ratio. As a result, the two red numbers in the upper right must both be even to ensure their rows' dot products with the EI are zero, which is an even number. If we halve all the red numbers and double all the green numbers, all the various row dot products will be unchanged, and the twin squares will remain valid:

But while the EI has become simpler, the generator has become more complex in that it is dup not up. This is remedied by adding the EI to it. Changes are in red:

Following this procedure, it's always possible to simplify a squared (or cubed, etc.) EI.

Consider Triyo/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EI is v³A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyo/Porcupine with an EI of v³A1, the vAC is v(81/80) or [-4 4 -1 -1].

Thus when we consider a single-comma temperament along with its notation, there are <u>three</u> commas of interest, the VC, the vAC and the EI. In a 5-limit rank-2 temperament, they can all be expressed as 2.3.5.^ monzos.

Any two of these 3 commas determines the third comma. Thus we can approach temperaments and notations from 6 different angles. We can start with any one of these three commas and give it a specific monzo. Then we can choose one of the other two commas, experiment with a variety of specific monzos and examine the results. Let's start with specifying the VC, experimenting with the vAC and seeing what the EI turns out to be.

The EI always equals the VC (possibly inverted) plus or minus some number of vAC's. That number is whatever is needed to eliminate the higher prime. The VC must be inverted if the resulting EI would otherwise have a negative stepspan, or is a diminished unison.  

In our Triyo example, 250/243 plus 3 downed syntonic commas = v³A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EI always has a zero prime-5-count.

Visualizing the three commas in the lattice, one starts at the VC and heads towards the row of 3-limit intervals via the vAC. Wherever one lands is the uninflected EI. One can try other AC's besides 81/80. The AC's prime-5-count must be ±1, so [[135/128|Layobi]] (135/128) or [[Schisma|Layo]] (the schisma) are possibilities. But either of these would lead to a very remote EI (v³AA1 and v³d⁴4 respectively), making a very awkward notation.  

Next let's specify the AC, experiment with the VC and see what the EI turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EI (and thus the notation) from the VC. For example, Sagugu/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EI for (P8/2, P5).

More examples: Laquinyo/Magic is (P8, P12/5) and has VC = [-10 -1 5 0]. Adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^⁵dd2. Gugu/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2. Again, this is the recommended EI.  

Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozo/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2.{{todo|complete section|comment=apply to multi-comma temperaments|inline=1}}