Kite's thoughts on pergens: Difference between revisions

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Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multigen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further Discussion-Supplemental materials-Notaion guide PDF|notation guide]]. It even allows every pergen to be numbered.

The enharmonic interval, or more briefly the '''enharmonic''', 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.

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.'')

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

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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:

<ul><li>For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 < |x| <= m/2 and 0 < |y| <= n/2</li><li>x is the count for E, with E occurring x times in one octave, and xE is the octave's '''multi-enharmonic''', or '''multi-E''' for short</li><li>y is the count for E', with E' occurring y times in one multigen, and yE' is the multigen's multi-E</li><li>For false doubles using single-pair notation, E = E', 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 E" and new counts, P8 = mP + x'E", and M' = n'G' + y'E"</li></ul>

<ul><li>For (P8/m, M/n), P8 = mP + xEI and M = nG + yEI', with 0 < |x| <= m/2 and 0 < |y| <= 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>

The '''keyspan''' of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The '''stepspan''' of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.

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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: xE = M - n⋅G = P5 - 2⋅m3 = [7,4] - 2⋅[3,2] = [7,4] - [6,4] = [1,0] = A1.

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⋅G = P5 - 2⋅m3 = [7,4] - 2⋅[3,2] = [7,4] - [6,4] = [1,0] = A1.

Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - mP = P8 - 5⋅M2 = [12,7] - 5⋅[2,1] = [2,2] = 2⋅[1,1] = 2⋅m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2⋅m2 = d3). The enharmonic's '''count''' is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v5m2. Since P8 = 5⋅P + 2⋅E, 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:

Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xEI = P8 - mP = P8 - 5⋅M2 = [12,7] - 5⋅[2,1] = [2,2] = 2⋅[1,1] = 2⋅m2. Because x = 2, EI will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2⋅m2 = d3). The enharmonic's '''count''' is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus EI = v5m2. Since P8 = 5⋅P + 2⋅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:

P1 -- ^^M2=v3m3 -- v4 -- ^5 -- ^3M6=vvm7 -- P8

C -- ^^D=v3Eb -- vF -- ^G -- ^3A=vvBb -- C

Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [1,1] = m2. The bare enharmonic is P4 - 5⋅m2 = [5,3] - 5⋅[1,1] = [5,3] - [5,5] = [0,-2] = -2⋅[0,1] = two descending d2's. The d2 must be upped, and E = ^5d2. Since P4 = 5⋅G - 2⋅E, G must be ^^m2. The genchain is:

Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [1,1] = m2. The bare enharmonic is P4 - 5⋅m2 = [5,3] - 5⋅[1,1] = [5,3] - [5,5] = [0,-2] = -2⋅[0,1] = two descending d2's. The d2 must be upped, and EI = ^5d2. Since P4 = 5⋅G - 2⋅EI, G must be ^^m2. The genchain is:

P1 -- ^^m2=v3A1 -- vM2 -- ^m3 -- ^3d4=vvM3 -- P4C -- ^^Db -- vD -- ^Eb -- vvE -- F

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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 bare alternate generator G' is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10⋅P1 = m2. It must be downed, thus E = v10m2. Since m2 = 10⋅G' + E, G' is ^1. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x⋅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⋅P + 2⋅E, and P = ^4M2. Next, find the original half-4th generator. P = P8/5 = ~240¢, and G = P4/2 = ~250¢. Because P < G, G' is not P - G but G - P, and G is not P - G' but P + G', which equals ^4M2 + ^1 = ^5M2. While the alternate multigen is more complex than the original multigen, the alternate generator is usually simpler than the original generator.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The bare alternate generator G' is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10⋅P1 = m2. It must be downed, thus EI = v10m2. Since m2 = 10⋅G' + EI, G' is ^1. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x⋅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⋅P + 2⋅EI, and P = ^4M2. Next, find the original half-4th generator. P = P8/5 = ~240¢, and G = P4/2 = ~250¢. Because P < G, G' is not P - G but G - P, and G is not P - G' but P + G', which equals ^4M2 + ^1 = ^5M2. While the alternate multigen is more complex than the original multigen, the alternate generator is usually simpler than the original generator.

P1- - ^4M2=v6m3 -- vv4 -- ^^5 -- ^6M6=v4m7 -- P8

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C -- ^5D=v5Eb -- F

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 E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2.

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 E = v4dd3, 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 bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v4\\d3 = 2·vv\m2, and E - E' = v4//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:

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 = v4dd3, 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 bare 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' = v4\\d3 = 2·vv\m2, and EI - EI' = v4//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:

P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11

C -- ^E=v\F -- /Ab=\A -- ^/C=vDb -- F

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 = v12m3. 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' = /3d2, 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. Enharmonic = v12m3 = /3d2 = v4/m2 = v4\\A1. Period = \M3 = v44 = //d4. Generator = ^\M3 = v34 = ^//d4.

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 = v12m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus EI' = /3d2, 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 = v12m3 = /3d2 = v4/m2 = v4\\A1. Period = \M3 = v44 = //d4. Generator = ^\M3 = v34 = ^//d4.

P1 — \M3 — \\A5=/m6 — P8C — \E — /Ab — CP1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^38=v/m9 — P11

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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 ^6d64 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 E = v3A1 and G'= vM2 = ^^m2. Half-4th has E' = \\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.

Sixth-4th with single-pair notation has an awkward ^6d64 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 = v3A1 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.

P1 — ^/1=^\m2 — ^^m2=vM2 — /M2=\m3 — ^m3=vvM3 — v/M3=v\4 — P4

C — ^/C=^\Db — ^^Db=vD — /D=\Eb — ^Eb=vvE — v/E=v\F — F

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, E must be an A1, with any number of downs except 3. v5A1 creates m ^m v~ ^~ vM M. 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.

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 any number of downs except 3. v5A1 creates m ^m v~ ^~ vM M. 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==

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C -- v3D -- v6E=^6Db -- ^3Eb -- F

Because G is a M2 and E is an A2, the equivalent generator G - E 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 v3D -- ^6Db is ascending. Double-pair notation may be preferable. This makes P = vM3, E = ^3d2, G = /m2, and E' = /4dd2.

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 v3D -- ^6Db is ascending. Double-pair notation may be preferable. This makes P = vM3, EI = ^3d2, G = /m2, and EI' = /4dd2.

P1 -- vM3 -- ^m6 -- P8

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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 E to a gedra, then multiply it by the count to get the multi-E (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 E 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-E. 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 E. 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-E. Add or subtract the new multi-E 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).

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 E = v3m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d32. Assuming a reasonably just 5th, E needs to be upped, so E = ^3d32. Add the multi-E ^3[-2,1] to the multigen P5 = [7,4] to get ^3[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^3[-2,1] = v3[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3d32 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d32 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- vD# -- ^Fb -- G. Because d32 = -200¢ - 26·c, ^ = (-d32) / 3 = 67¢ + 8.67·c, about a third-tone.

For example, (P8, P5/3) has n = 3, G = ^M2, and EI = v3m2 = [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] = d32. Assuming a reasonably just 5th, EI needs to be upped, so EI = ^3d32. Add the multi-EI ^3[-2,1] to the multigen P5 = [7,4] to get ^3[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^3[-2,1] = v3[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3d32 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d32 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- vD# -- ^Fb -- G. Because d32 = -200¢ - 26·c, ^ = (-d32) / 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 E, or the multi-E if the count is > 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 E = ^^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.

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 > 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.

Line 1,254:

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 v3M2, but if 11/8 is notated as a vA4, the enharmonic is ^3dd2.

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 E = 3·vd5 - P11 = v3dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.

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 = v3dd3. 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.

Line 1,264:

In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G ^A and C E G vBb are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.

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 E = ^^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.

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 E = 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.

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.

Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. This is in fact a requirement for MOS scales.

Line 1,272:

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 E = 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.

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.

Mizarian Porcupine Overture by Herman Miller (P8, P4/3) ^3C = C#

Line 1,280:

==Tipping points and sweet spots==

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 E = ^^d2. But Injera, also half-8ve, has a flat 5th, and thus E = 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 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 E of vvM2. The tipping point becomes 600¢, a very unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on.

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 very 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 bare enharmonic, which is implied by the vanishing comma. For example, Porcupine aka Triyo's 250/243 comma is an A1 = (-11,7), which implies a bare E 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.

The notation's tipping point is determined by the bare enharmonic, which is implied by the vanishing comma. For example, Porcupine aka Triyo's 250/243 comma is an A1 = (-11,7), which implies a bare 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 E could be either ^4dd2 or v4dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an E of ^4dd2 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.

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 ^4dd2 or v4dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an EI of ^4dd2 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.

Line 1,314:

! | 5/4 is

! | 4:5:6 chord

! | E

! | EI

! | ratio

! | cents

Line 1,421:

| | rank-2

| | 1

| | E = d2

| | EI = d2

|-

| | 19-edo

Line 1,429:

| | rank-2

| | 1

| | E = dd2

| | EI = dd2

|-

| | 15-edo

Line 1,437:

| | rank-3

| | 2

| | E = m2, E' = v3A1 = v3M2

| | EI = m2, EI' = v3A1 = v3M2

|-

| | 24-edo

Line 1,445:

| | rank-3

| | 2

| | E = d2, E' = vvA1 = vvm2

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

|-

| | 3-limit JI aka pythagorean

Line 1,469:

| | rank-3

| | 1

| | E = ^^d2

| | EI = ^^d2

|-

| | Semaphore aka Zozo

Line 1,477:

| | rank-3

| | 1

| | E = vvm2

| | EI = vvm2

|-

| | Decimal aka Yoyo & Zozo

Line 1,485:

| | rank-4

| | 2

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

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

|-

| | 5-limit JI

Line 1,509:

| | rank-4

| | 1

| | E = \\dd3

| | EI = \\dd3

|-

| | 7-limit JI

Line 1,603:

If using single-pair notation, Marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - vE - G - vvA#. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - vE - G - \Bb.

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 E = //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.

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)-2 · (64/63)2 · (-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 E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.

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)-2 · (64/63)2 · (-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 E. 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, E = ^^\\\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.

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 < 3, 5/4 is preferred.

Line 1,786:

| | (P8/5, ^1)

| | rank-2 5-edo

| | E = m2

| | EI = m2

| | D E=F G A B=C D

| | D vF#=vG vvB...

Line 1,795:

| | (P8/7, ^1)

| | rank-2 7-edo

| | E = A1

| | EI = A1

| | D E F G A B C D

| | D ^F ^^A...

Line 1,804:

| | (P8/10, /1)

| | rank-2 10-edo

| | E = m2, E' = vvA1 = vvM2

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

| | D ^D=vE E=F ^F=vG G...

| | D \F#=\G \\B...

Line 1,813:

| | (P8/12, ^1)

| | rank-2 12-edo

| | E = d2

| | EI = d2

| | D D#=Eb E F F#=Gb...

| | D ^G ^^C

Line 1,831:

| | (P8/17, /1)

| | rank-2 17-edo

| | E = dd3, E' = vm2 = vvA1

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

| | D ^D=Eb D#=vE E F...

| | D \F# \\A#=v\\B...

Line 1,862:

| | (P8/5, P5)

| | fifth-8ve

| | E = v5m2

| | EI = v5m2

| | D ^^E vG ^A vvC D

| | C G D A E...

Line 3,026:

Half-8ve in 22edo has P = vA4. But in 16edo, P = A4 + 2 edosteps, and ^1 = -2 edosteps. Negative edosteps means up is down, and should be avoided. The notation has tipped, and the period should be notated as ^A4, making ^1 = 2 edostep. Even better would be P = ^4, making ^1 = 1 edostep.

Half-5th has E = 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-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 E = 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.

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

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.

The relationship between a pergen's up and an EDO's up when the pergen uses double-pair notation is complex. Consider half-everything, notated with ups/downs in the perchain and lifts/drops in the genchain. 10edo has ^1 = 1 edostep and /1 = 0 edosteps, thus one simply ignores lifts and drops. But for 24edo, ^1 = 0 edosteps and /1 = 1 edostep. One must ignore ups and downs and convert lifts/drops to edosteps.

Line 4,317:

a'·b - a·b' works out to be k·s' - k'·s, and we have GCD ((k·s' - k'·s)·m/b, b'·n/b)

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

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

GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')

Line 4,390:

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

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 E of a 4th. But since sixth-4th is half-4th plus third-4th, and those two have a good E, sixth-4th can be notated with one pair from half-4th and another from third-4th.

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.

'''Expanding gedras'''

@@ Line 336: / Line 336: @@
 Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multigen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further Discussion-Supplemental materials-Notaion guide PDF|notation guide]]. It even allows every pergen to be numbered.
-The enharmonic interval, or more briefly the '''enharmonic''', 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.
+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.'')
 The '''genchain''' (chain of generators) in the table is only a short section of the full genchain.
@@ Line 1,171: / Line 1,171: @@
 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 + xE and M = nG + yE', with 0 &lt; |x| &lt;= m/2 and 0 &lt; |y| &lt;= n/2</li><li>x is the count for E, with E occurring x times in one octave, and xE is the octave's '''multi-enharmonic''', or '''multi-E''' for short</li><li>y is the count for E', with E' occurring y times in one multigen, and yE' is the multigen's multi-E</li><li>For false doubles using single-pair notation, E = E', 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 E" and new counts, P8 = mP + x'E", and M' = n'G' + y'E"</li></ul>
+<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>
 The '''keyspan''' of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The '''stepspan''' of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.
@@ Line 1,183: / Line 1,183: @@
 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: xE = 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.
+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. xE = 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, E 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 bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v<span style="vertical-align: super;">5</span>m2. Since P8 = 5<span style="">⋅</span>P + 2<span style="">⋅</span>E, 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:
+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 bare 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:
 <span style="display: block; text-align: center;">P1 -- ^^M2=v<span style="vertical-align: super;">3</span>m3 -- v4 -- ^5 -- ^<span style="vertical-align: super;">3</span>M6=vvm7 -- P8</span>
 <span style="display: block; text-align: center;">C -- ^^D=v<span style="vertical-align: super;">3</span>Eb -- vF -- ^G -- ^<span style="vertical-align: super;">3</span>A=vvBb -- C</span>
-Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [1,1] = m2. The bare 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 E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5<span style="">⋅</span>G - 2<span style="">⋅</span>E, G must be ^^m2. The genchain is:
+Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [1,1] = m2. The bare 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:
 <span style="display: block; text-align: center;">P1 -- ^^m2=v<span style="vertical-align: super;">3</span>A1 -- vM2 -- ^m3 -- ^<span style="vertical-align: super;">3</span>d4=vvM3 -- P4</span><span style="display: block; text-align: center;">C -- ^^Db -- vD -- ^Eb -- vvE -- F</span>
@@ Line 1,196: / Line 1,196: @@
 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 bare alternate generator G' is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10<span style="">⋅</span>P1 = m2. It must be downed, thus E = v<span style="vertical-align: super;">10</span>m2. Since m2 = 10<span style="">⋅</span>G' + E, 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>E, 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.
+For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The bare alternate generator G' is [1,1]/10 = [0,0] = P1. The bare 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.
 <span style="display: block; text-align: center;">P1- - ^<span style="vertical-align: super;">4</span>M2=v<span style="vertical-align: super;">6</span>m3 -- vv4 -- ^^5 -- ^<span style="vertical-align: super;">6</span>M6=v<span style="vertical-align: super;">4</span>m7 -- P8</span>
@@ Line 1,203: / Line 1,203: @@
 <span style="display: block; text-align: center;">C -- ^<span style="vertical-align: super;">5</span>D=v<span style="vertical-align: super;">5</span>Eb -- F</span>
-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 E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2.
+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 E = 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 bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = 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:
+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 bare 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:
 <span style="display: block; text-align: center;">P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11</span>
 <span style="display: block; text-align: center;">C -- ^E=v\F -- /Ab=\A -- ^/C=vDb -- F</span>
-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<span style="vertical-align: super;">12</span>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' = /<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 E + E' and E - 2·E'. 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.
+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 bare 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.
 <span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — \E — /Ab — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — P11
@@ Line 1,219: / Line 1,219: @@
 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 E = v<span style="vertical-align: super;">3</span>A1 and G'= vM2 = ^^m2. Half-4th has E' = \\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 v<sup>3</sup>\\M2 and ^<sup>3</sup>\\d2.
+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 v<sup>3</sup>\\M2 and ^<sup>3</sup>\\d2.
 <span style="display: block; text-align: center;">P1 — ^/1=^\m2 — ^^m2=vM2 — /M2=\m3 — ^m3=vvM3 — v/M3=v\4 — P4
 <span style="display: block; text-align: center;">C — ^/C=^\Db — ^^Db=vD — /D=\Eb — ^Eb=vvE — v/E=v\F — F
-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, E must be an A1, with any number of downs except 3. v<sup>5</sup>A1 creates m ^m v~ ^~ vM M. 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.
+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 any number of downs except 3. v<sup>5</sup>A1 creates m ^m v~ ^~ vM M. 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==
@@ Line 1,235: / Line 1,235: @@
 <span style="display: block; text-align: center;">C -- v<span style="vertical-align: super;">3</span>D -- v<span style="vertical-align: super;">6</span>E=^<span style="vertical-align: super;">6</span>Db -- ^<span style="vertical-align: super;">3</span>Eb -- F
-Because G is a M2 and E is an A2, the equivalent generator G - E 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, E = ^3d2, G = /m2, and E' = /4dd2.
+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.
 <span style="display: block; text-align: center;">P1 -- vM3 -- ^m6 -- P8
@@ Line 1,244: / Line 1,244: @@
 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 E to a gedra, then multiply it by the count to get the multi-E (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 E 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-E. 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 E. 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-E. Add or subtract the new multi-E 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).
+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 E = v<span style="vertical-align: super;">3</span>m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d<span style="vertical-align: super;">3</span>2. Assuming a reasonably just 5th, E needs to be upped, so E = ^<span style="vertical-align: super;">3</span>d<span style="vertical-align: super;">3</span>2. Add the multi-E ^<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·E 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.
+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·E 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 E, or the multi-E 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 E = ^^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.
+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.
@@ Line 1,254: / Line 1,254: @@
 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 E = 3·vd5 - P11 = v<span style="vertical-align: super;">3</span>dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.
+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.
@@ Line 1,264: / Line 1,264: @@
 In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G ^A and C E G vBb are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.
-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 E = ^^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.
+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 E = 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.
+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.
 Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. This is in fact a requirement for MOS scales.
@@ Line 1,272: / Line 1,272: @@
 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 E = v<sup>3</sup>A1. 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.
+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 = v<sup>3</sup>A1. 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.
 <u><span style="font-size: 110%;">Mizarian Porcupine Overture by Herman Miller (P8, P4/3) ^<sup>3</sup>C = C#</span></u>
@@ Line 1,280: / Line 1,280: @@
 ==Tipping points and sweet spots==
-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 E = ^^d2. But Injera, also half-8ve, has a flat 5th, and thus E = 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 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 E of vvM2. The tipping point becomes 600¢, a <u>very</u> unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on.
+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 bare enharmonic, which is implied by the vanishing comma. For example, Porcupine aka Triyo's 250/243 comma is an A1 = (-11,7), which implies a bare E 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.
+The notation's tipping point is determined by the bare enharmonic, which is implied by the vanishing comma. For example, Porcupine aka Triyo's 250/243 comma is an A1 = (-11,7), which implies a bare 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 E 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 E 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.
+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.
@@ Line 1,314: / Line 1,314: @@
 ! | 5/4 is
 ! | 4:5:6 chord
-! | E
+! | EI
 ! | ratio
 ! | cents
@@ Line 1,421: / Line 1,421: @@
 | | rank-2
 | | 1
-| | E = d2
+| | EI = d2
 |-
 | | 19-edo
@@ Line 1,429: / Line 1,429: @@
 | | rank-2
 | | 1
-| | E = dd2
+| | EI = dd2
 |-
 | | 15-edo
@@ Line 1,437: / Line 1,437: @@
 | | rank-3
 | | 2
-| | E = m2, E' = v<span style="vertical-align: super;">3</span>A1 = v<span style="vertical-align: super;">3</span>M2
+| | EI = m2, EI' = v<span style="vertical-align: super;">3</span>A1 = v<span style="vertical-align: super;">3</span>M2
 |-
 | | 24-edo
@@ Line 1,445: / Line 1,445: @@
 | | rank-3
 | | 2
-| | E = d2, E' = vvA1 = vvm2
+| | EI = d2, EI' = vvA1 = vvm2
 |-
 | | 3-limit JI aka pythagorean
@@ Line 1,469: / Line 1,469: @@
 | | rank-3
 | | 1
-| | E = ^^d2
+| | EI = ^^d2
 |-
 | | Semaphore aka Zozo
@@ Line 1,477: / Line 1,477: @@
 | | rank-3
 | | 1
-| | E = vvm2
+| | EI = vvm2
 |-
 | | Decimal aka Yoyo & Zozo
@@ Line 1,485: / Line 1,485: @@
 | | rank-4
 | | 2
-| | E = vvd2, E' = \\m2 = ^^\\A1
+| | EI = vvd2, EI' = \\m2 = ^^\\A1
 |-
 | | 5-limit JI
@@ Line 1,509: / Line 1,509: @@
 | | rank-4
 | | 1
-| | E = \\dd3
+| | EI = \\dd3
 |-
 | | 7-limit JI
@@ Line 1,603: / Line 1,603: @@
 If using single-pair notation, Marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - vE - G - vvA#. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - vE - G - \Bb.
-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 E = //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.
+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 E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.
+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 E. 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, E = ^^\\\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.
+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.
@@ Line 1,786: / Line 1,786: @@
 | | (P8/5, ^1)
 | | rank-2 5-edo
-| | E = m2
+| | EI = m2
 | | D E=F G A B=C D
 | | D vF#=vG vvB...
@@ Line 1,795: / Line 1,795: @@
 | | (P8/7, ^1)
 | | rank-2 7-edo
-| | E = A1
+| | EI = A1
 | | D E F G A B C D
 | | D ^F ^^A...
@@ Line 1,804: / Line 1,804: @@
 | | (P8/10, /1)
 | | rank-2 10-edo
-| | E = m2, E' = vvA1 = vvM2
+| | EI = m2, EI' = vvA1 = vvM2
 | | D ^D=vE E=F ^F=vG G...
 | | D \F#=\G \\B...
@@ Line 1,813: / Line 1,813: @@
 | | (P8/12, ^1)
 | | rank-2 12-edo
-| | E = d2
+| | EI = d2
 | | D D#=Eb E F F#=Gb...
 | | D ^G ^^C
@@ Line 1,831: / Line 1,831: @@
 | | (P8/17, /1)
 | | rank-2 17-edo
-| | E = dd3, E' = vm2 = vvA1
+| | EI = dd3, EI' = vm2 = vvA1
 | | D ^D=Eb D#=vE E F...
 | | D \F# \\A#=v\\B...
@@ Line 1,862: / Line 1,862: @@
 | | (P8/5, P5)
 | | fifth-8ve
-| | E = v<span style="vertical-align: super;">5</span>m2
+| | EI = v<span style="vertical-align: super;">5</span>m2
 | | D ^^E vG ^A vvC D
 | | C G D A E...
@@ Line 3,026: / Line 3,026: @@
 Half-8ve in 22edo has P = vA4. But in 16edo, P = A4 + 2 edosteps, and ^1 = -2 edosteps. Negative edosteps means up is down, and should be avoided. The notation has tipped, and the period should be notated as ^A4, making ^1 = 2 edostep. Even better would be P = ^4, making ^1 = 1 edostep.
-Half-5th has E = 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-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 E = 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.
+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 E = v<sup>3</sup>A1. Doubled EDOs are the same ones as half-5th's tripled EDOs. Third-5th has E = v<sup>3</sup>m2. Doubled EDOs are the same as half-4th's tripled EDOs.
+Third-4th has EI = v<sup>3</sup>A1. Doubled EDOs are the same ones as half-5th's tripled EDOs. Third-5th has EI = v<sup>3</sup>m2. Doubled EDOs are the same as half-4th's tripled EDOs.
 The relationship between a pergen's up and an EDO's up when the pergen uses double-pair notation is complex. Consider half-everything, notated with ups/downs in the perchain and lifts/drops in the genchain. 10edo has ^1 = 1 edostep and /1 = 0 edosteps, thus one simply ignores lifts and drops. But for 24edo, ^1 = 0 edosteps and /1 = 1 edostep. One must ignore ups and downs and convert lifts/drops to edosteps.
@@ Line 4,317: / Line 4,317: @@
 a'·b - a·b' works out to be k·s' - k'·s, and we have GCD ((k·s' - k'·s)·m/b, b'·n/b)
-If s is a multiple of n (happens when E is an A1) and s' is a multiple of n, let s = x·n and s' = y·n
+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
 GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')
@@ Line 4,390: / Line 4,390: @@
 However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.
-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 E of a 4th. But since sixth-4th is half-4th plus third-4th, and those two have a good E, sixth-4th can be notated with one pair from half-4th and another from third-4th.
+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>'''Expanding gedras'''</u>