Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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==Naming very large intervals==

So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by "W". Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For ~~(P8~~, ~~M/n), valid multigens are any~~ voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M can be P4, P5, P11, P12, WWP4 or WWP5.

So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by "W". Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M is less than 3 octaves, and can be P4, P5, P11, P12, WWP4 or WWP5.

==Secondary splits==

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22-edo: # ``=`` 164¢, ^ = 55¢ (^ = 1/3 #)

quarter-comma meantone: # = 76¢

quarter-comma meantone: # ``=`` 76¢

fifth-comma meantone: # = 84¢

fifth-comma meantone: # ``=`` 84¢

third-comma archy: # = 177¢

third-comma archy: # ``=`` 177¢

eighth-comma porcupine: # ``=`` 157¢, ^ = 52¢ (^ = 1/3 #)

sixth-comma srutal: # ``=`` 139¢, ^ = 33¢ (no fixed relationship between ^ and #)

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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 -- D#v -- 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 > 1. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.

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 (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^=Ebv -- F. But consider double large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.

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.

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Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese (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 -- Gbv -- 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 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.

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.

==Chord names and scale names==

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In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G A^ and C E G Bbv 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~~'s pergen~~ 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)] = [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = WWm7, 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 Ev G Bb = C7(v3).

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 (2.3.5.7 with 50/49 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 = WWm7, 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 Ev G Bb = C7(v3).

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 (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 Bbv = C,v7.

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==Notating unsplit pergens==

An unsplit pergen doesn't __require__ ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a comma ~~that maps to P1~~, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (the monzo has a final exponent of ±1).

An unsplit pergen doesn't __require__ ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a mapping comma, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (i.e. the monzo has a final exponent of ±1).

The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate.

The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate. The up symbol's ratio is always the mapping comma, or its inverse.

||~ __5-limit temperament__ ||~ __comma__ ||~ __sweet spot__ ||||~ __no ups or downs__ ||||||~ __with ups and downs__ ||||~ __up symbol__ ||

||~ (pergen is unsplit) ||~ ||~ (5th = 700¢ + c) ||~ 5/4 is ||~ 4:5:6 chord ||~ 5/4 is ||~ 4:5:6 chord ||~ E ||~ ratio ||~ cents ||

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The schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad.

For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, ~~simply by adding~~ or ~~subtracting~~ the vanishing comma ~~from 81/80 or 80/81~~. This 3-limit comma is tempered, as is every ratio. It can be found directly from the 3-limit mapping of the vanishing comma. ~~Because~~ the schismic comma is a descending d2, and d2 = (19,-12), the 3-comma is the pythagorean comma (-19,12).

For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-comma is the pythagorean comma (-19,12).

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

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

==Notating rank-3 pergens==

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||= breedsmic ||= (P8, P5/2, ^1) ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 ||

||= 7-limit JI ||= (P8, P5, ^1, /1) ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= --- ||

When there is more than one enharmonic, the first one can be ~~combined with~~ the 2nd to make an equivalent 2nd enharmonic.

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

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas ~~and two new accidental pairs~~. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.)

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas. There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.

~~All~~ true triples require triple-pair ~~notation~~, ~~but some~~ false triples and some ~~double-splits~~ may use ~~triple~~-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors.

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

~~Quadruple-pair notation is needed for some true triples like (P8/2, P5/2, vM3/2).~~ A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.

A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.

Some examples of 7-limit rank-3 temperaments:

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

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish 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 (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. 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 Ev G Bb\.

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish 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 Ev G Bbv/. 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 Ev G Bb\.

With this standardization, the enharmonic can be derived 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)<span style="vertical-align: super;">-1</span>. This can be rewritten as vv1 + ``//``1 - d2 = vv``//``-d2 = -^^\\d2. The comma is negative, but the enharmonic never is, therefore E = ^^\\d2. 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 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)<span style="vertical-align: super;">-1</span>. 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.

~~The~~ 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 deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish 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.

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 deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish 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.

Demeter ~~is unusual in that its~~ 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, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (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^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... 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/highs/lows only for the other kind of accidentals.

Unlike the previous examples, Demeter'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, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (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 -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... 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/highs/lows 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 ~~the 2nd generator~~ to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For demeter, any combination of vm3, 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 any 3-limit comma can 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** (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred.

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 vm3, 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 any 3-limit comma can 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** (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred.

If ^1 = 81/80, possible half-split ~~gens2~~'s are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.

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

All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens:

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==Notating Blackwood-like pergens*==

A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.

A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.

A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma ~~divides~~ the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).

A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).

Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:

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||= ||= ||= ||= ||= ||= ||= ||= ||

If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or 13/12.

If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15, or 12/11, or 13/12.

The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and ^1 = 81/80 or equivalently, 16/15.

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

But in "no-5ths" or "minus white" pergens, not every name has a note. For example, deep reddish minus white (2.5.7 ~~with~~ 50/49 ~~tempered out~~) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.

But in "no-5ths" or "minus white" pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.

Note-less names can be avoided in any pergen with a 5th. The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.

Note-less names can be avoided in any pergen with a 5th (or 4th or 11th, etc.). The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.

||~ tuning ||~ pergen ||~ spoken name ||~ enharmonic ||~ perchain ||~ genchain ||~ notes ||

||~ tuning ||~ comma ||~ pergen ||~ spoken name ||~ enharmonic ||~ perchain ||~ genchain ||~ notes ||

||= large quintuple blue,

||= large quintuple blue

small quintuple red ||= (P8/5, P5) ||= fifth-8ve ||= E = v<span style="vertical-align: super;">5</span>m2 ||= C D^^ Fv G^ Bbvv C ||= C G D A E... ||= a valid notation ||

or small quintuple red ||= (-14,0,0,5)

||= Blackwood ||= (P8/5, ^1) ||= rank-2 5-edo ||= E = m2 ||= C D=Eb E=F G A=Bb B=C ||= C Ev G#vv... ||= a valid notation ||

or (22,-5,0,-5) ||= (P8/5, P5) ||= fifth-8ve ||= E = v<span style="vertical-align: super;">5</span>m2 ||= C D^^ Fv G^ Bbvv C ||= C G D A E... ||= a valid notation ||

||= " ||= (P8/5, M3) ||= fifth-8ve major 3rd ||= E = v<span style="vertical-align: super;">5</span>m2 ||= C D^^ Fv G^ Bbvv C ||= C E G#... ||= invalid, no G, D or A notes ||

||= Blackwood ||= 256/243 ||= (P8/5, ^1) ||= rank-2 5-edo ||= E = m2 ||= C D=Eb E=F G A=Bb B=C ||= C Ev G#vv... ||= a valid notation ||

||= " ||= " ||= (P8/5, M3) ||= fifth-8ve major 3rd ||= E = v<span style="vertical-align: super;">5</span>m2 ||= C D^^ Fv G^ Bbvv C ||= C E G#... ||= invalid, no G,

D or A notes ||

== ==

edo-subset notations

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Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of note-less names. The composer may well want to use a notation that isn't backwards compatible.

Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all.

Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all...

A similar chart could be made of all rank-3 pergen cubes.

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The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.

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

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4.

==Pergens and EDOs*==

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For each edo, find the nearest edomapping (also known as the patent val) for the 2.3 subgroup. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If |d| = m, the generator is the 5th, and the pergen is simply (P8/m, P5).

For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, ~~and~~ the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.

For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.

Sometimes the second-nearest edomapping is preferred, more on this later.

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If |d| ≠ m, we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree, find the two ancestors and the two descendants of this ratio. Choose one of the four (more on this below), and let this new ratio be g/g'. The generator G maps to both g\N and g'\N'. The two edos come closest to coinciding here.

For example, for 7-edo and 17-edo, m = 1 but d = 2. The ancestors and ~~descendents~~ of 7/17 are 2/5, 5/12, 9/22 and 12/29. Choose the smallest for now, 2/5. The generator maps to both 2\7 and 5\17. 2\7 is 343¢ and 5\17 is 353¢, ~~both neutral 3rds. Their~~ difference is ~10¢ = 1\119. (119 = LCM (7, 17)). This is the ~~smallest~~ difference ~~possible~~ between 7edo~~'s notes~~ and 17edo~~'s notes~~, except of course ~~for~~ 0\7 and 0\17~~. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths~~.

For example, for 7-edo and 17-edo, m = 1 but d = 2. The ancestors and descendants of 7/17 are 2/5, 5/12, 9/22 and 12/29. Choose the smallest for now, 2/5. The generator maps to both 2\7 and 5\17, which are both neutral 3rds. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths. 2\7 = 343¢ and 5\17 = 353¢, and their difference is only ~10¢ = 1\119. (119 = LCM (7, 17)). This 10¢ is the least difference between any 7edo note and any 17edo note, except that the two neutral 6ths differ by the same 10¢, and of course excluding note pairs that coincide exactly such as 0\7 and 0\17.

Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, we extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found earlier, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this vector with either edomapping is zero. Treating this new vector as a monzo, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for.

Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, we extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found earlier, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for.

For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two ~~neutral 3rds~~ with a 5th. The pergen is obviously (P8, P5/2).

For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2).

To verify the validity of this approach, one can find a specific ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 = 71¢. A smaller comma can be found with a ratio more in the center of the range, such as 11/9, which yeilds 242/243. Because the direction of the cross product vector depends on the order of the two vectors multiplied, the sign of the comma is arbitrary, and the comma may be a descending ~~interval~~. In that ~~case~~, invert the comma to make 243/242. Using unreduced edomappings gives the same result. If the basis is (2/1, 3/1, 5/1), we have (7, 11, 16) x (17, 27, 39) = (-3, -1, 2) = 25/24.

To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 = 71¢. A smaller comma can be found with a ratio more in the center of the range, such as 11/9, which yeilds 242/243. Because the direction of the cross product vector depends on the order of the two vectors multiplied, the sign of the comma is arbitrary, and the comma may be a descending ratio. If that happens, invert the comma to make 243/242.

Using unreduced edomappings gives the same result. If the basis is (2/1, 3/1, 5/1), we have (7, 11, 16) x (17, 27, 39) = (-3, -1, 2) = 25/24. If the basis is (2/1, 3/1, 11/9)...

If the octave is split,

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check references to mapping commas

finish proofs

make a glossary of all bolded terms?

link from: ups and downs page, Kite Giedraitis page, MOS scale names page,

http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments

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http://www.tallkite.com/misc_files/alt-pergenLister.zip

Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. Screenshots of the first 38 pergens:

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

Screenshots of the first 38 pergens:

[[image:alt-pergenLister.png width="704" height="460"]]

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Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><h1 id="toc0"> </h1>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><h1 id="toc0"> </h1>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#toc0"> </a></div>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#toc0"> </a></div>

<div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

<div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

<div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>

<div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>

<div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>

<div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>

<div style="margin-left: 2em;"><a href="#Applications-Tipping points">Tipping points</a></div>

<div style="margin-left: 2em;"><a href="#Applications-Tipping points">Tipping points</a></div>

<div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>

<div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Naming very large intervals">Naming very large intervals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Naming very large intervals">Naming very large intervals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Secondary splits">Secondary splits</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Secondary splits">Secondary splits</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating unsplit pergens">Notating unsplit pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating unsplit pergens">Notating unsplit pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating rank-3 pergens">Notating rank-3 pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating rank-3 pergens">Notating rank-3 pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating Blackwood-like pergens*">Notating Blackwood-like pergens*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating Blackwood-like pergens*">Notating Blackwood-like pergens*</a></div>

<div style="margin-left: 2em;"><a href="#toc18"> </a></div>

<div style="margin-left: 2em;"><a href="#toc18"> </a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating non-8ve and no-5ths pergens*">Notating non-8ve and no-5ths pergens*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating non-8ve and no-5ths pergens*">Notating non-8ve and no-5ths pergens*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating tunings with an arbitrary generator">Notating tunings with an arbitrary generator</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating tunings with an arbitrary generator">Notating tunings with an arbitrary generator</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs*">Pergens and EDOs*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs*">Pergens and EDOs*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials*">Supplemental materials*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials*">Supplemental materials*</a></div>

<div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-Notaion guide PDF">Notaion guide PDF</a></div>

<div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-Notaion guide PDF">Notaion guide PDF</a></div>

<div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-pergenLister app">pergenLister app</a></div>

<div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-pergenLister app">pergenLister app</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Miscellaneous Notes">Miscellaneous Notes</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Miscellaneous Notes">Miscellaneous Notes</a></div>

</div>

</div>

<h1 id="toc1"><a name="Definition"></a><u><strong>Definition</strong></u></h1>

<h1 id="toc1"><a name="Definition"></a><u><strong>Definition</strong></u></h1>

<br />

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<br />

<h1 id="toc2"><a name="Derivation"></a><u>Derivation</u></h1>

<h1 id="toc2"><a name="Derivation"></a><u>Derivation</u></h1>

<br />

For any comma, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents, where GCD (0,x) = x. The comma will split the octave into m parts, and if n &gt; m, it will split some 3-limit interval into n parts.<br />

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<br />

<h1 id="toc3"><a name="Applications"></a><u>Applications</u></h1>

<h1 id="toc3"><a name="Applications"></a><u>Applications</u></h1>

<br />

Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.<br />

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Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).<br />

<br />

<h2 id="toc4"><a name="Applications-Tipping points"></a>Tipping points</h2>

<h2 id="toc4"><a name="Applications-Tipping points"></a>Tipping points</h2>

<br />

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

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<br />

<h1 id="toc5"><a name="Further Discussion"></a><u>Further Discussion</u></h1>

<h1 id="toc5"><a name="Further Discussion"></a><u>Further Discussion</u></h1>

<br />

<h2 id="toc6"><a name="Further Discussion-Naming very large intervals"></a>Naming very large intervals</h2>

<h2 id="toc6"><a name="Further Discussion-Naming very large intervals"></a>Naming very large intervals</h2>

<br />

So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by &quot;W&quot;. Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For ~~(P8~~, ~~M/n), valid multigens are any~~ voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M can be P4, P5, P11, P12, WWP4 or WWP5.<br />

So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by &quot;W&quot;. Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M is less than 3 octaves, and can be P4, P5, P11, P12, WWP4 or WWP5.<br />

<br />

<h2 id="toc7"><a name="Further Discussion-Secondary splits"></a>Secondary splits</h2>

<h2 id="toc7"><a name="Further Discussion-Secondary splits"></a>Secondary splits</h2>

<br />

Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with:<br />

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<br />

<h2 id="toc8"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

<h2 id="toc8"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

<br />

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

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

<br />

<h2 id="toc9"><a name="Further Discussion-Finding an example temperament"></a>Finding an example temperament</h2>

<h2 id="toc9"><a name="Further Discussion-Finding an example temperament"></a>Finding an example temperament</h2>

<br />

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

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

<br />

<h2 id="toc10"><a name="Further Discussion-Ratio and cents of the accidentals"></a>Ratio and cents of the accidentals</h2>

<h2 id="toc10"><a name="Further Discussion-Ratio and cents of the accidentals"></a>Ratio and cents of the accidentals</h2>

<br />

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, the up symbol often equals only a few ratios. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 11-limit temperaments use either 33/32 or 729/704. These <strong>mapping commas</strong> are used to map higher primes to 3-limit intervals, and are essential for notation. They also determine where a ratio &quot;lands&quot; on a keyboard. By definition they are a P1, and the only intervals that map to P1 are these commas and combinations of them.<br />

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22-edo: # = 164¢, ^ = 55¢ (^ = 1/3 #)<br />

<br />

quarter-comma meantone: # = 76¢<br />

quarter-comma meantone: # = 76¢<br />

fifth-comma meantone: # = 84¢<br />

fifth-comma meantone: # = 84¢<br />

third-comma archy: # = 177¢<br />

third-comma archy: # = 177¢<br />

eighth-comma porcupine: # = 157¢, ^ = 52¢ (^ = 1/3 #)<br />

eighth-comma porcupine: # = 157¢, ^ = 52¢ (^ = 1/3 #)<br />

sixth-comma srutal: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #)<br />

sixth-comma srutal: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #)<br />

third-comma injera: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)<br />

third-comma injera: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)<br />

eighth-comma hedgehog: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)<br />

eighth-comma hedgehog: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)<br />

<br />

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

<br />

<h2 id="toc11"><a name="Further Discussion-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

<h2 id="toc11"><a name="Further Discussion-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

<br />

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:<br />

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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).<br />

<br />

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:<br />

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:<br />

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.<br />

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.<br />

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.<br />

<br />

This is a lot of math, but it only needs to be done once for each pergen!<br />

<br />

<h2 id="toc12"><a name="Further Discussion-Alternate enharmonics"></a>Alternate enharmonics</h2>

<h2 id="toc12"><a name="Further Discussion-Alternate enharmonics"></a>Alternate enharmonics</h2>

<br />

Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.<br />

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</span><span style="display: block; text-align: center;">C -- Ev -- Ab^ -- C<br />

</span><span style="display: block; text-align: center;">P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4<br />

</span><span style="display: block; text-align: center;">P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4<br />

</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb//=D#\\ -- E\ -- F</span><br />

</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb//=D#\\ -- E\ -- F</span><br />

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.<br />

<br />

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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 -- D#v -- 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.<br />

<br />

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 (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^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.<br />

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 (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^=Ebv -- F. But consider double large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.<br />

<br />

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.<br />

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Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese (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 -- Gbv -- B^ -- F.<br />

<br />

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 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.<br />

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.<br />

<br />

<h2 id="toc13"><a name="Further Discussion-Chord names and scale names"></a>Chord names and scale names</h2>

<h2 id="toc13"><a name="Further Discussion-Chord names and scale names"></a>Chord names and scale names</h2>

<br />

Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.<br />

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

<br />

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~~'s pergen~~ 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)] = [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = WWm7, 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 Ev G Bb = C7(v3).<br />

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 (2.3.5.7 with 50/49 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 = WWm7, 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 Ev G Bb = C7(v3).<br />

<br />

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 (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 Bbv = C,v7.<br />

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Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. 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 Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.<br />

<br />

<h2 id="toc14"><a name="Further Discussion-Tipping points and sweet spots"></a>Tipping points and sweet spots</h2>

<h2 id="toc14"><a name="Further Discussion-Tipping points and sweet spots"></a>Tipping points and sweet spots</h2>

<br />

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 &quot;tips over&quot;, 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 &quot;sweet spot&quot;, where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.<br />

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Another &quot;tippy&quot; temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow 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.<br />

<br />

<h2 id="toc15"><a name="Further Discussion-Notating unsplit pergens"></a>Notating unsplit pergens</h2>

<h2 id="toc15"><a name="Further Discussion-Notating unsplit pergens"></a>Notating unsplit pergens</h2>

<br />

An unsplit pergen doesn't <u>require</u> ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a comma ~~that maps to P1~~, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (the monzo has a final exponent of ±1).<br />

An unsplit pergen doesn't <u>require</u> ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a mapping comma, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (i.e. the monzo has a final exponent of ±1).<br />

<br />

The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate.<br />

The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate. The up symbol's ratio is always the mapping comma, or its inverse.<br />

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The schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad.<br />

<br />

For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, ~~simply by adding~~ or ~~subtracting~~ the vanishing comma ~~from 81/80 or 80/81~~. This 3-limit comma is tempered, as is every ratio. It can be found directly from the 3-limit mapping of the vanishing comma. ~~Because~~ the schismic comma is a descending d2, and d2 = (19,-12), the 3-comma is the pythagorean comma (-19,12).<br />

For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-comma is the pythagorean comma (-19,12).<br />

<br />

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

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

<br />

<h2 id="toc16"><a name="Further Discussion-Notating rank-3 pergens"></a>Notating rank-3 pergens</h2>

<h2 id="toc16"><a name="Further Discussion-Notating rank-3 pergens"></a>Notating rank-3 pergens</h2>

<br />

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:<br />

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<th>notation's rank<br />

</th>

<th># of enharmonics needed<br />

<th># of enharmonics needed<br />

</th>

<th>enharmonics<br />

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</table>

When there is more than one enharmonic, the first one can be ~~combined with~~ the 2nd to make an equivalent 2nd enharmonic.<br />

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

<br />

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas ~~and two new accidental pairs~~. (There are even &quot;superfalse&quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.)<br />

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas. There are even &quot;superfalse&quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.<br />

<br />

~~All~~ true triples require triple-pair ~~notation~~, ~~but some~~ false triples and some ~~double-splits~~ may use ~~triple~~-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors.<br />

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

<br />

~~Quadruple-pair notation is needed for some true triples like (P8/2, P5/2, vM3/2).~~ A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.<br />

A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.<br />

<br />

Some examples of 7-limit rank-3 temperaments:<br />

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</td>

</td>

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If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - Ev - G - Bb\.<br />

<br />

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish 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 (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. 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 Ev G Bb\.<br />

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish 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 Ev G Bbv/. 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 Ev G Bb\.<br />

<br />

With this standardization, the enharmonic can be derived 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)<span style="vertical-align: super;">-1</span>. This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative, but the enharmonic never is, therefore E = ^^\\d2. P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.<br />

With this standardization, the enharmonic can be derived 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)<span style="vertical-align: super;">-1</span>. 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.<br />

<br />

~~The~~ 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 deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish 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.<br />

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 deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish 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.<br />

<br />

Demeter ~~is unusual in that its~~ 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, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (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^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... 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/highs/lows only for the other kind of accidentals.<br />

Unlike the previous examples, Demeter'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, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (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 -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... 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/highs/lows only for the other kind of accidentals.<br />

<br />

There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from ~~the 2nd generator~~ to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For demeter, any combination of vm3, 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 any 3-limit comma can 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 <strong>DOL</strong> (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred.<br />

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 vm3, 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 any 3-limit comma can 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 <strong>DOL</strong> (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred.<br />

<br />

If ^1 = 81/80, possible half-split ~~gens2~~'s are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.<br />

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

<br />

All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens:<br />

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There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.<br />

<br />

<h2 id="toc17"><a name="Further Discussion-Notating Blackwood-like pergens*"></a>Notating Blackwood-like pergens*</h2>

<h2 id="toc17"><a name="Further Discussion-Notating Blackwood-like pergens*"></a>Notating Blackwood-like pergens*</h2>

<br />

A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.<br />

A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.<br />

<br />

A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma ~~divides~~ the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).<br />

A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).<br />

<br />

Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:<br />

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</table>

If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or 13/12.<br />

If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15, or 12/11, or 13/12.<br />

<br />

The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and ^1 = 81/80 or equivalently, 16/15.<br />

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

<br />

But in &quot;no-5ths&quot; or &quot;minus white&quot; pergens, not every name has a note. For example, deep reddish minus white (2.5.7 ~~with~~ 50/49 ~~tempered out~~) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.<br />

But in &quot;no-5ths&quot; or &quot;minus white&quot; pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.<br />

<br />

Note-less names can be avoided in any pergen with a 5th. The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.<br />

Note-less names can be avoided in any pergen with a 5th (or 4th or 11th, etc.). The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.<br />

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<tr>

<th>tuning<br />

</th>

<th>comma<br />

</th>

<th>pergen<br />

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</tr>

<tr>

<td style="text-align: center;">large quintuple blue,<br />

<td style="text-align: center;">large quintuple blue<br />

small quintuple red<br />

or small quintuple red<br />

</td>

or (22,-5,0,-5)<br />

</td>

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<tr>

<td style="text-align: center;">Blackwood<br />

</td>

</td>

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</tr>

<tr>

</td>

</td>

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</td>

<td style="text-align: center;">invalid, no G, D or A notes<br />

<td style="text-align: center;">invalid, no G, <br />

D or A notes<br />

</td>

</tr>

</table>

<h2 id="toc18"> </h2>

<h2 id="toc18"> </h2>

edo-subset notations<br />

P8/6, P8/8<br />

<br />

<h2 id="toc19"><a name="Further Discussion-Notating non-8ve and no-5ths pergens*"></a>Notating non-8ve and no-5ths pergens*</h2>

<h2 id="toc19"><a name="Further Discussion-Notating non-8ve and no-5ths pergens*"></a>Notating non-8ve and no-5ths pergens*</h2>

<br />

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. The pergens are grouped into blocks and sections as before:<br />

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Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a <u>huge</u> number of note-less names. The composer may well want to use a notation that isn't backwards compatible.<br />

<br />

Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all.<br />

Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all...<br />

<br />

A similar chart could be made of all rank-3 pergen cubes.<br />

<br />

<h2 id="toc20"><a name="Further Discussion-Notating tunings with an arbitrary generator"></a>Notating tunings with an arbitrary generator</h2>

<h2 id="toc20"><a name="Further Discussion-Notating tunings with an arbitrary generator"></a>Notating tunings with an arbitrary generator</h2>

<br />

Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.<br />

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See also the <a class="wiki_link" href="/Map%20of%20rank-2%20temperaments">map of rank-2 temperaments</a>.<br />

<br />

<h2 id="toc21"><a name="Further Discussion-Pergens and MOS scales"></a>Pergens and MOS scales</h2>

<h2 id="toc21"><a name="Further Discussion-Pergens and MOS scales"></a>Pergens and MOS scales</h2>

<br />

Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.<br />

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The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.<br />

<br />

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, ~~(5/4)~~/2), where 5·G = 7/4.<br />

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4.<br />

<br />

<h2 id="toc22"><a name="Further Discussion-Pergens and EDOs*"></a>Pergens and EDOs*</h2>

<h2 id="toc22"><a name="Further Discussion-Pergens and EDOs*"></a>Pergens and EDOs*</h2>

<br />

Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.<br />

Line 6,559:

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For each edo, find the nearest edomapping (also known as the patent val) for the 2.3 subgroup. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If |d| = m, the generator is the 5th, and the pergen is simply (P8/m, P5).<br />

<br />

For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, ~~and~~ the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.<br />

For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.<br />

<br />

Sometimes the second-nearest edomapping is preferred, more on this later.<br />

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If |d| ≠ m, we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree, find the two ancestors and the two descendants of this ratio. Choose one of the four (more on this below), and let this new ratio be g/g'. The generator G maps to both g\N and g'\N'. The two edos come closest to coinciding here.<br />

<br />

For example, for 7-edo and 17-edo, m = 1 but d = 2. The ancestors and ~~descendents~~ of 7/17 are 2/5, 5/12, 9/22 and 12/29. Choose the smallest for now, 2/5. The generator maps to both 2\7 and 5\17. 2\7 is 343¢ and 5\17 is 353¢, ~~both neutral 3rds. Their~~ difference is ~10¢ = 1\119. (119 = LCM (7, 17)). This is the ~~smallest~~ difference ~~possible~~ between 7edo~~'s notes~~ and 17edo~~'s notes~~, except of course ~~for~~ 0\7 and 0\17~~. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths~~.<br />

For example, for 7-edo and 17-edo, m = 1 but d = 2. The ancestors and descendants of 7/17 are 2/5, 5/12, 9/22 and 12/29. Choose the smallest for now, 2/5. The generator maps to both 2\7 and 5\17, which are both neutral 3rds. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths. 2\7 = 343¢ and 5\17 = 353¢, and their difference is only ~10¢ = 1\119. (119 = LCM (7, 17)). This 10¢ is the least difference between any 7edo note and any 17edo note, except that the two neutral 6ths differ by the same 10¢, and of course excluding note pairs that coincide exactly such as 0\7 and 0\17.<br />

<br />

Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, we extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found earlier, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this vector with either edomapping is zero. Treating this new vector as a monzo, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for.<br />

Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, we extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found earlier, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for.<br />

<br />

For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two ~~neutral 3rds~~ with a 5th. The pergen is obviously (P8, P5/2).<br />

For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2).<br />

<br />

To verify the validity of this approach, one can find a specific ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 = 71¢. A smaller comma can be found with a ratio more in the center of the range, such as 11/9, which yeilds 242/243. Because the direction of the cross product vector depends on the order of the two vectors multiplied, the sign of the comma is arbitrary, and the comma may be a descending ~~interval~~. In that ~~case~~, invert the comma to make 243/242. Using unreduced edomappings gives the same result. If the basis is (2/1, 3/1, 5/1), we have (7, 11, 16) x (17, 27, 39) = (-3, -1, 2) = 25/24.<br />

To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 = 71¢. A smaller comma can be found with a ratio more in the center of the range, such as 11/9, which yeilds 242/243. Because the direction of the cross product vector depends on the order of the two vectors multiplied, the sign of the comma is arbitrary, and the comma may be a descending ratio. If that happens, invert the comma to make 243/242. <br />

<br />

Using unreduced edomappings gives the same result. If the basis is (2/1, 3/1, 5/1), we have (7, 11, 16) x (17, 27, 39) = (-3, -1, 2) = 25/24. If the basis is (2/1, 3/1, 11/9)...<br />

<br />

If the octave is split,<br />

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<br />

<h2 id="toc23"><a name="Further Discussion-Supplemental materials*"></a>Supplemental materials*</h2>

<h2 id="toc23"><a name="Further Discussion-Supplemental materials*"></a>Supplemental materials*</h2>

<br />

needs more screenshots, including 12-edo's pergens and a page of the pdf<br />

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check references to mapping commas<br />

finish proofs<br />

make a glossary of all bolded terms?<br />

link from: ups and downs page, Kite Giedraitis page, MOS scale names page,<br />

<a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a><br />

<a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a><br />

<a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a><br />

<a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a><br />

<br />

<h3 id="toc24"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a>Notaion guide PDF</h3>

<h3 id="toc24"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a>Notaion guide PDF</h3>

<br />

This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.<br />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><br />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><br />

(screenshot)<br />

<br />

<h3 id="toc25"><a name="Further Discussion-Supplemental materials*-pergenLister app"></a>pergenLister app</h3>

<h3 id="toc25"><a name="Further Discussion-Supplemental materials*-pergenLister app"></a>pergenLister app</h3>

<br />

Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a><br />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a><br />

<br />

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

<br />

~~Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.~~ Screenshots of the first 38 pergens:<br />

Screenshots of the first 38 pergens:<br />

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><br />

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><br />

<br />

Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br />

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The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.<br />

<br />

<h2 id="toc26"><a name="Further Discussion-Various proofs (unfinished)"></a>Various proofs (unfinished)</h2>

<h2 id="toc26"><a name="Further Discussion-Various proofs (unfinished)"></a>Various proofs (unfinished)</h2>

<br />

The interval P8/2 has a &quot;ratio&quot; of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the <strong>pergen matrix</strong> [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.<br />

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Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.<br />

<br />

<h2 id="toc27"><a name="Further Discussion-Miscellaneous Notes"></a>Miscellaneous Notes</h2>

<h2 id="toc27"><a name="Further Discussion-Miscellaneous Notes"></a>Miscellaneous Notes</h2>

<br />

<u><strong>Staff notation</strong></u><br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-22 16:44:33 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-22 19:29:07 UTC</tt>.<br>
-: The original revision id was <tt>627918511</tt>.<br>
+: The original revision id was <tt>627923165</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 328: / Line 328: @@
 ==Naming very large intervals==
-So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by "W". Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M can be P4, P5, P11, P12, WWP4 or WWP5.
+So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by "W". Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M is less than 3 octaves, and can be P4, P5, P11, P12, WWP4 or WWP5.
 ==Secondary splits==
@@ Line 431: / Line 431: @@
 -edo: # ``=`` 164¢, ^ = 55¢ (^ = 1/3 #)
-quarter-comma meantone: # = 76¢
+quarter-comma meantone: # ``=`` 76¢
-fifth-comma meantone: # = 84¢
+fifth-comma meantone: # ``=`` 84¢
-third-comma archy: # = 177¢
+third-comma archy: # ``=`` 177¢
 eighth-comma porcupine: # ``=`` 157¢, ^ = 52¢ (^ = 1/3 #)
 sixth-comma srutal: # ``=`` 139¢, ^ = 33¢ (no fixed relationship between ^ and #)
@@ Line 503: / Line 503: @@
 For example, (P8, P5/3) has n = 3, G = ^M2, and E = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;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&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. Assuming a reasonably just 5th, E needs to be upped, so E = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. Add the multi-E ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[-2,1] to the multigen P5 = [7,4] to get ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[-2,1] = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- D#v -- Fb^ -- G. Because d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 = -200¢ - 26·c, ^ = (-d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;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 (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^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.
+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 (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^=Ebv -- F. But consider double large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.
 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 511: / Line 511: @@
 Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese (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&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- Gbv -- 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 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.
+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.
 ==Chord names and scale names==
@@ Line 519: / Line 519: @@
 In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G A^ and C E G Bbv 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's pergen 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)] = [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = WWm7, 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 Ev G Bb = C7(v3).
+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 (2.3.5.7 with 50/49 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 = WWm7, 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 Ev G Bb = C7(v3).
 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 (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 Bbv = C,v7.
@@ Line 543: / Line 543: @@
 ==Notating unsplit pergens==
-An unsplit pergen doesn't __require__ ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a comma that maps to P1, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (the monzo has a final exponent of ±1).
+An unsplit pergen doesn't __require__ ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a mapping comma, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (i.e. the monzo has a final exponent of ±1).
-The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate.
+The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate. The up symbol's ratio is always the mapping comma, or its inverse.
 ||~ __5-limit temperament__ ||~ __comma__ ||~ __sweet spot__ ||||~ __no ups or downs__ ||||||~ __with ups and downs__ ||||~ __up symbol__ ||
 ||~ (pergen is unsplit) ||~   ||~ (5th = 700¢ + c) ||~ 5/4 is ||~ 4:5:6 chord ||~ 5/4 is ||~ 4:5:6 chord ||~ E ||~ ratio ||~ cents ||
@@ Line 557: / Line 557: @@
 The schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad.
-For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, simply by adding or subtracting the vanishing comma from 81/80 or 80/81. This 3-limit comma is tempered, as is every ratio. It can be found directly from the 3-limit mapping of the vanishing comma. Because the schismic comma is a descending d2, and d2 = (19,-12), the 3-comma is the pythagorean comma (-19,12).
+For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-comma is the pythagorean comma (-19,12).
-A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv.
+A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for archy (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv.
 ==Notating rank-3 pergens==
@@ Line 578: / Line 578: @@
 ||= breedsmic ||= (P8, P5/2, ^1) ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 ||
 ||= 7-limit JI ||= (P8, P5, ^1, /1) ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= --- ||
-When there is more than one enharmonic, the first one can be combined with the 2nd to make an equivalent 2nd enharmonic.
+When there is more than one enharmonic, the first one can be added to or subtracted from the 2nd one, to make an equivalent 2nd enharmonic.
-A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.)
+A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas. There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.
-All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors.
+Even the unsplit rank-3 pergen requires single-pair notation, for the 2nd generator. Single-splits and false double-splits require double-pair, true doubles and false triples require triple-pair, and true triples require quadruple-pair. Some false triples and some true doubles may use quadruple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third or fourth pair of symbols, and a third or fourth pair of adjectives to describe them, one might simply use colors.
-Quadruple-pair notation is needed for some true triples like (P8/2, P5/2, vM3/2). A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.
+A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.
 Some examples of 7-limit rank-3 temperaments:
@@ Line 596: / Line 596: @@
 If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - Ev - G - Bb\.
-There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish 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 (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. 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 Ev G Bb\.
+There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish 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 Ev G Bbv/. 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 Ev G Bb\.
-With this standardization, the enharmonic can be derived 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)&lt;span style="vertical-align: super;"&gt;-2&lt;/span&gt; · (64/63)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; · (19,-12)&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;. This can be rewritten as vv1 + ``//``1 - d2 = vv``//``-d2 = -^^\\d2. The comma is negative, but the enharmonic never is, therefore E = ^^\\d2. 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 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)&lt;span style="vertical-align: super;"&gt;-2&lt;/span&gt; · (64/63)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; · (19,-12)&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;. 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.
-The 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 deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish 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.
+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 deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish 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.
-Demeter is unusual in that its 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, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (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^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... 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/highs/lows only for the other kind of accidentals.
+Unlike the previous examples, Demeter'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, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (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 -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... 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/highs/lows 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 the 2nd generator to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For demeter, any combination of vm3, 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 any 3-limit comma can 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** (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred.
+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 vm3, 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 any 3-limit comma can 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** (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred.
-If ^1 = 81/80, possible half-split gens2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.
+If ^1 = 81/80, possible half-split gen2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.
 All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens:
@@ Line 637: / Line 637: @@
 ==Notating Blackwood-like pergens*==
-A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.
+A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.
-A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).
+A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).
 Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:
@@ Line 655: / Line 655: @@
 ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
 ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
-If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or 13/12.
+If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15, or 12/11, or 13/12.
 The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and ^1 = 81/80 or equivalently, 16/15.
@@ Line 667: / Line 667: @@
 In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.
-But in "no-5ths" or "minus white" pergens, not every name has a note. For example, deep reddish minus white (2.5.7 with 50/49 tempered out) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.
+But in "no-5ths" or "minus white" pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.
-Note-less names can be avoided in any pergen with a 5th. The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.
+Note-less names can be avoided in any pergen with a 5th (or 4th or 11th, etc.). The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.
-||~ tuning ||~ pergen ||~ spoken name ||~ enharmonic ||~ perchain ||~ genchain ||~ notes ||
+||~ tuning ||~ comma ||~ pergen ||~ spoken name ||~ enharmonic ||~ perchain ||~ genchain ||~ notes ||
-||= large quintuple blue,
+||= large quintuple blue
-small quintuple red ||= (P8/5, P5) ||= fifth-8ve ||= E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2 ||= C D^^ Fv G^ Bbvv C ||= C G D A E... ||= a valid notation ||
+or small quintuple red ||= (-14,0,0,5)
-||= Blackwood ||= (P8/5, ^1) ||= rank-2 5-edo ||= E = m2 ||= C D=Eb E=F G A=Bb B=C ||= C Ev G#vv... ||= a valid notation ||
+or (22,-5,0,-5) ||= (P8/5, P5) ||= fifth-8ve ||= E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2 ||= C D^^ Fv G^ Bbvv C ||= C G D A E... ||= a valid notation ||
-||= " ||= (P8/5, M3) ||= fifth-8ve major 3rd ||= E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2 ||= C D^^ Fv G^ Bbvv C ||= C E G#... ||= invalid, no G, D or A notes ||
+||= Blackwood ||= 256/243 ||= (P8/5, ^1) ||= rank-2 5-edo ||= E = m2 ||= C D=Eb E=F G A=Bb B=C ||= C Ev G#vv... ||= a valid notation ||
+||= " ||= " ||= (P8/5, M3) ||= fifth-8ve major 3rd ||= E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2 ||= C D^^ Fv G^ Bbvv C ||= C E G#... ||= invalid, no G,
+D or A notes ||
 == ==
 edo-subset notations
@@ Line 725: / Line 727: @@
 Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of note-less names. The composer may well want to use a notation that isn't backwards compatible.
-Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all.
+Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all...
 A similar chart could be made of all rank-3 pergen cubes.
@@ Line 856: / Line 858: @@
 The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.
-Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.
+Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4.
 ==Pergens and EDOs*==
@@ Line 918: / Line 920: @@
 For each edo, find the nearest edomapping (also known as the patent val) for the 2.3 subgroup. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If |d| = m, the generator is the 5th, and the pergen is simply (P8/m, P5).
-For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, and the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.
+For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.
 Sometimes the second-nearest edomapping is preferred, more on this later.
@@ Line 924: / Line 926: @@
 If |d| ≠ m, we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree, find the two ancestors and the two descendants of this ratio. Choose one of the four (more on this below), and let this new ratio be g/g'. The generator G maps to both g\N and g'\N'. The two edos come closest to coinciding here.
-For example, for 7-edo and 17-edo, m = 1 but d = 2. The ancestors and descendents of 7/17 are 2/5, 5/12, 9/22 and 12/29. Choose the smallest for now, 2/5. The generator maps to both 2\7 and 5\17. 2\7 is 343¢ and 5\17 is 353¢, both neutral 3rds. Their difference is ~10¢ = 1\119. (119 = LCM (7, 17)). This is the smallest difference possible between 7edo's notes and 17edo's notes, except of course for 0\7 and 0\17. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths.
+For example, for 7-edo and 17-edo, m = 1 but d = 2. The ancestors and descendants of 7/17 are 2/5, 5/12, 9/22 and 12/29. Choose the smallest for now, 2/5. The generator maps to both 2\7 and 5\17, which are both neutral 3rds. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths. 2\7 = 343¢ and 5\17 = 353¢, and their difference is only ~10¢ = 1\119. (119 = LCM (7, 17)). This 10¢ is the least difference between any 7edo note and any 17edo note, except that the two neutral 6ths differ by the same 10¢, and of course excluding note pairs that coincide exactly such as 0\7 and 0\17.
-Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, we extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found earlier, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this vector with either edomapping is zero. Treating this new vector as a monzo, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for.
+Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, we extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found earlier, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for.
-For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two neutral 3rds with a 5th. The pergen is obviously (P8, P5/2).
+For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2).
-To verify the validity of this approach, one can find a specific ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 = 71¢. A smaller comma can be found with a ratio more in the center of the range, such as 11/9, which yeilds 242/243. Because the direction of the cross product vector depends on the order of the two vectors multiplied, the sign of the comma is arbitrary, and the comma may be a descending interval. In that case, invert the comma to make 243/242. Using unreduced edomappings gives the same result. If the basis is (2/1, 3/1, 5/1), we have (7, 11, 16) x (17, 27, 39) = (-3, -1, 2) = 25/24.
+To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 = 71¢. A smaller comma can be found with a ratio more in the center of the range, such as 11/9, which yeilds 242/243. Because the direction of the cross product vector depends on the order of the two vectors multiplied, the sign of the comma is arbitrary, and the comma may be a descending ratio. If that happens, invert the comma to make 243/242.
+Using unreduced edomappings gives the same result. If the basis is (2/1, 3/1, 5/1), we have (7, 11, 16) x (17, 27, 39) = (-3, -1, 2) = 25/24. If the basis is (2/1, 3/1, 11/9)...
 If the octave is split,
@@ Line 967: / Line 971: @@
 check references to mapping commas
 finish proofs
+make a glossary of all bolded terms?
 link from: ups and downs page, Kite Giedraitis page, MOS scale names page,
 http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments
@@ Line 982: / Line 987: @@
 http://www.tallkite.com/misc_files/alt-pergenLister.zip
-Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. Screenshots of the first 38 pergens:
+The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.
+Screenshots of the first 38 pergens:
 [[image:alt-pergenLister.png width="704" height="460"]]
@@ Line 1,186: / Line 1,193: @@
 Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.</pre></div>
 <h4>Original HTML content:</h4>
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+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;pergen&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt; &lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextTocRule:113:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:113 --&gt;&lt;!-- ws:start:WikiTextTocRule:114: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#toc0"&gt; &lt;/a&gt;&lt;/div&gt;
+  &lt;!-- ws:start:WikiTextTocRule:116:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:116 --&gt;&lt;!-- ws:start:WikiTextTocRule:117: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#toc0"&gt; &lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:117 --&gt;&lt;!-- ws:start:WikiTextTocRule:118: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:115 --&gt;&lt;!-- ws:start:WikiTextTocRule:116: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:118 --&gt;&lt;!-- ws:start:WikiTextTocRule:119: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:116 --&gt;&lt;!-- ws:start:WikiTextTocRule:117: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:119 --&gt;&lt;!-- ws:start:WikiTextTocRule:120: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:117 --&gt;&lt;!-- ws:start:WikiTextTocRule:118: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Applications-Tipping points"&gt;Tipping points&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:120 --&gt;&lt;!-- ws:start:WikiTextTocRule:121: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Applications-Tipping points"&gt;Tipping points&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:118 --&gt;&lt;!-- ws:start:WikiTextTocRule:119: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Further Discussion"&gt;Further Discussion&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:121 --&gt;&lt;!-- ws:start:WikiTextTocRule:122: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Further Discussion"&gt;Further Discussion&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:119 --&gt;&lt;!-- ws:start:WikiTextTocRule:120: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Naming very large intervals"&gt;Naming very large intervals&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:122 --&gt;&lt;!-- ws:start:WikiTextTocRule:123: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Naming very large intervals"&gt;Naming very large intervals&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:120 --&gt;&lt;!-- ws:start:WikiTextTocRule:121: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Secondary splits"&gt;Secondary splits&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:123 --&gt;&lt;!-- ws:start:WikiTextTocRule:124: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Secondary splits"&gt;Secondary splits&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:121 --&gt;&lt;!-- ws:start:WikiTextTocRule:122: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Singles and doubles"&gt;Singles and doubles&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:124 --&gt;&lt;!-- ws:start:WikiTextTocRule:125: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Singles and doubles"&gt;Singles and doubles&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:122 --&gt;&lt;!-- ws:start:WikiTextTocRule:123: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding an example temperament"&gt;Finding an example temperament&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:125 --&gt;&lt;!-- ws:start:WikiTextTocRule:126: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding an example temperament"&gt;Finding an example temperament&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:123 --&gt;&lt;!-- ws:start:WikiTextTocRule:124: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Ratio and cents of the accidentals"&gt;Ratio and cents of the accidentals&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:126 --&gt;&lt;!-- ws:start:WikiTextTocRule:127: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Ratio and cents of the accidentals"&gt;Ratio and cents of the accidentals&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:124 --&gt;&lt;!-- ws:start:WikiTextTocRule:125: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding a notation for a pergen"&gt;Finding a notation for a pergen&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:127 --&gt;&lt;!-- ws:start:WikiTextTocRule:128: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding a notation for a pergen"&gt;Finding a notation for a pergen&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:125 --&gt;&lt;!-- ws:start:WikiTextTocRule:126: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate enharmonics"&gt;Alternate enharmonics&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:128 --&gt;&lt;!-- ws:start:WikiTextTocRule:129: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate enharmonics"&gt;Alternate enharmonics&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:126 --&gt;&lt;!-- ws:start:WikiTextTocRule:127: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Chord names and scale names"&gt;Chord names and scale names&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:129 --&gt;&lt;!-- ws:start:WikiTextTocRule:130: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Chord names and scale names"&gt;Chord names and scale names&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:127 --&gt;&lt;!-- ws:start:WikiTextTocRule:128: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Tipping points and sweet spots"&gt;Tipping points and sweet spots&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:130 --&gt;&lt;!-- ws:start:WikiTextTocRule:131: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Tipping points and sweet spots"&gt;Tipping points and sweet spots&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:128 --&gt;&lt;!-- ws:start:WikiTextTocRule:129: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating unsplit pergens"&gt;Notating unsplit pergens&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:131 --&gt;&lt;!-- ws:start:WikiTextTocRule:132: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating unsplit pergens"&gt;Notating unsplit pergens&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:129 --&gt;&lt;!-- ws:start:WikiTextTocRule:130: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating rank-3 pergens"&gt;Notating rank-3 pergens&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:132 --&gt;&lt;!-- ws:start:WikiTextTocRule:133: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating rank-3 pergens"&gt;Notating rank-3 pergens&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:130 --&gt;&lt;!-- ws:start:WikiTextTocRule:131: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating Blackwood-like pergens*"&gt;Notating Blackwood-like pergens*&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:133 --&gt;&lt;!-- ws:start:WikiTextTocRule:134: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating Blackwood-like pergens*"&gt;Notating Blackwood-like pergens*&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:131 --&gt;&lt;!-- ws:start:WikiTextTocRule:132: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#toc18"&gt; &lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:134 --&gt;&lt;!-- ws:start:WikiTextTocRule:135: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#toc18"&gt; &lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:132 --&gt;&lt;!-- ws:start:WikiTextTocRule:133: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating non-8ve and no-5ths pergens*"&gt;Notating non-8ve and no-5ths pergens*&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:135 --&gt;&lt;!-- ws:start:WikiTextTocRule:136: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating non-8ve and no-5ths pergens*"&gt;Notating non-8ve and no-5ths pergens*&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:133 --&gt;&lt;!-- ws:start:WikiTextTocRule:134: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating tunings with an arbitrary generator"&gt;Notating tunings with an arbitrary generator&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:136 --&gt;&lt;!-- ws:start:WikiTextTocRule:137: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating tunings with an arbitrary generator"&gt;Notating tunings with an arbitrary generator&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:134 --&gt;&lt;!-- ws:start:WikiTextTocRule:135: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and MOS scales"&gt;Pergens and MOS scales&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:137 --&gt;&lt;!-- ws:start:WikiTextTocRule:138: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and MOS scales"&gt;Pergens and MOS scales&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:135 --&gt;&lt;!-- ws:start:WikiTextTocRule:136: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and EDOs*"&gt;Pergens and EDOs*&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:138 --&gt;&lt;!-- ws:start:WikiTextTocRule:139: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and EDOs*"&gt;Pergens and EDOs*&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:136 --&gt;&lt;!-- ws:start:WikiTextTocRule:137: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*"&gt;Supplemental materials*&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:139 --&gt;&lt;!-- ws:start:WikiTextTocRule:140: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*"&gt;Supplemental materials*&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:137 --&gt;&lt;!-- ws:start:WikiTextTocRule:138: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;Notaion guide PDF&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:140 --&gt;&lt;!-- ws:start:WikiTextTocRule:141: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;Notaion guide PDF&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:138 --&gt;&lt;!-- ws:start:WikiTextTocRule:139: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-pergenLister app"&gt;pergenLister app&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:141 --&gt;&lt;!-- ws:start:WikiTextTocRule:142: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-pergenLister app"&gt;pergenLister app&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:139 --&gt;&lt;!-- ws:start:WikiTextTocRule:140: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Various proofs (unfinished)"&gt;Various proofs (unfinished)&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:142 --&gt;&lt;!-- ws:start:WikiTextTocRule:143: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Various proofs (unfinished)"&gt;Various proofs (unfinished)&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:140 --&gt;&lt;!-- ws:start:WikiTextTocRule:141: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Miscellaneous Notes"&gt;Miscellaneous Notes&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:143 --&gt;&lt;!-- ws:start:WikiTextTocRule:144: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Miscellaneous Notes"&gt;Miscellaneous Notes&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:141 --&gt;&lt;!-- ws:start:WikiTextTocRule:142: --&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:145 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 For any comma, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents, where GCD (0,x) = x. The comma will split the octave into m parts, and if n &amp;gt; m, it will split some 3-limit interval into n parts.&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:63:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:63 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.&lt;br /&gt;
@@ Line 2,472: / Line 2,479: @@
 Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:65:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Applications-Tipping points"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:65 --&gt;Tipping points&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Applications-Tipping points"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;Tipping points&lt;/h2&gt;
   &lt;br /&gt;
 Removing the ups and downs from an enharmonic interval makes a &amp;quot;bare&amp;quot; enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &amp;quot;sweet spot&amp;quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the &amp;quot;tipping point&amp;quot;: if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore &lt;u&gt;&lt;strong&gt;ups and downs may need to be swapped, depending on the size of the 5th&lt;/strong&gt;&lt;/u&gt; in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Further Discussion"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;&lt;u&gt;Further Discussion&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:69:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Further Discussion-Naming very large intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:69 --&gt;Naming very large intervals&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Further Discussion-Naming very large intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;Naming very large intervals&lt;/h2&gt;
   &lt;br /&gt;
-So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by &amp;quot;W&amp;quot;. Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M can be P4, P5, P11, P12, WWP4 or WWP5.&lt;br /&gt;
+So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by &amp;quot;W&amp;quot;. Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M is less than 3 octaves, and can be P4, P5, P11, P12, WWP4 or WWP5.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:71:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Further Discussion-Secondary splits"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:71 --&gt;Secondary splits&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:74:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Further Discussion-Secondary splits"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:74 --&gt;Secondary splits&lt;/h2&gt;
   &lt;br /&gt;
 Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with:&lt;br /&gt;
@@ Line 2,891: / Line 2,898: @@
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:73:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Further Discussion-Singles and doubles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:73 --&gt;Singles and doubles&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:76:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Further Discussion-Singles and doubles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:76 --&gt;Singles and doubles&lt;/h2&gt;
   &lt;br /&gt;
 If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a &lt;strong&gt;single-split&lt;/strong&gt; pergen. If it has two fractions, it's a &lt;strong&gt;double-split&lt;/strong&gt; pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called &lt;strong&gt;single-pair&lt;/strong&gt; notation because it adds only a single pair of accidentals to conventional notation. &lt;strong&gt;Double-pair&lt;/strong&gt; notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.&lt;br /&gt;
@@ Line 2,901: / Line 2,908: @@
 A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:75:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Further Discussion-Finding an example temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:75 --&gt;Finding an example temperament&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:78:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Further Discussion-Finding an example temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:78 --&gt;Finding an example temperament&lt;/h2&gt;
   &lt;br /&gt;
 To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P and P8. If P is 6/5, the comma is 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P - P8 = (6/5)&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P = (2/1) · (7/6)&lt;span style="vertical-align: super;"&gt;-4&lt;/span&gt;, the quadruple red temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.&lt;br /&gt;
@@ Line 2,923: / Line 2,930: @@
 There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:77:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Further Discussion-Ratio and cents of the accidentals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:77 --&gt;Ratio and cents of the accidentals&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:80:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Further Discussion-Ratio and cents of the accidentals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:80 --&gt;Ratio and cents of the accidentals&lt;/h2&gt;
   &lt;br /&gt;
 The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, the up symbol often equals only a few ratios. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 11-limit temperaments use either 33/32 or 729/704. These &lt;strong&gt;mapping commas&lt;/strong&gt; are used to map higher primes to 3-limit intervals, and are essential for notation. They also determine where a ratio &amp;quot;lands&amp;quot; on a keyboard. By definition they are a P1, and the only intervals that map to P1 are these commas and combinations of them.&lt;br /&gt;
@@ Line 2,945: / Line 2,952: @@
 -edo: # &lt;!-- ws:start:WikiTextRawRule:042:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:042 --&gt; 164¢, ^ = 55¢ (^ = 1/3 #)&lt;br /&gt;
 &lt;br /&gt;
-quarter-comma meantone: # = 76¢&lt;br /&gt;
+quarter-comma meantone: # &lt;!-- ws:start:WikiTextRawRule:043:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:043 --&gt; 76¢&lt;br /&gt;
-fifth-comma meantone: # = 84¢&lt;br /&gt;
+fifth-comma meantone: # &lt;!-- ws:start:WikiTextRawRule:044:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:044 --&gt; 84¢&lt;br /&gt;
-third-comma archy: # = 177¢&lt;br /&gt;
+third-comma archy: # &lt;!-- ws:start:WikiTextRawRule:045:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:045 --&gt; 177¢&lt;br /&gt;
-eighth-comma porcupine: # &lt;!-- ws:start:WikiTextRawRule:043:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:043 --&gt; 157¢, ^ = 52¢ (^ = 1/3 #)&lt;br /&gt;
+eighth-comma porcupine: # &lt;!-- ws:start:WikiTextRawRule:046:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:046 --&gt; 157¢, ^ = 52¢ (^ = 1/3 #)&lt;br /&gt;
-sixth-comma srutal: # &lt;!-- ws:start:WikiTextRawRule:044:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:044 --&gt; 139¢, ^ = 33¢ (no fixed relationship between ^ and #)&lt;br /&gt;
+sixth-comma srutal: # &lt;!-- ws:start:WikiTextRawRule:047:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:047 --&gt; 139¢, ^ = 33¢ (no fixed relationship between ^ and #)&lt;br /&gt;
-third-comma injera: # &lt;!-- ws:start:WikiTextRawRule:045:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:045 --&gt; 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)&lt;br /&gt;
+third-comma injera: # &lt;!-- ws:start:WikiTextRawRule:048:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:048 --&gt; 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)&lt;br /&gt;
-eighth-comma hedgehog: # &lt;!-- ws:start:WikiTextRawRule:046:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:046 --&gt; 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)&lt;br /&gt;
+eighth-comma hedgehog: # &lt;!-- ws:start:WikiTextRawRule:049:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:049 --&gt; 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)&lt;br /&gt;
 &lt;br /&gt;
 Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:79:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Further Discussion-Finding a notation for a pergen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:79 --&gt;Finding a notation for a pergen&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:82:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Further Discussion-Finding a notation for a pergen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:82 --&gt;Finding a notation for a pergen&lt;/h2&gt;
   &lt;br /&gt;
 There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:&lt;br /&gt;
@@ Line 2,981: / Line 2,988: @@
 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).&lt;br /&gt;
 &lt;br /&gt;
-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&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;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&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\d3 = 2·vv\m2, and E - E' = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;&lt;!-- ws:start:WikiTextRawRule:047:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:047 --&gt;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:&lt;br /&gt;
+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&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;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&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\d3 = 2·vv\m2, and E - E' = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;&lt;!-- ws:start:WikiTextRawRule:050:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:050 --&gt;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:&lt;br /&gt;
 &lt;span style="display: block; text-align: center;"&gt;P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- E^=Fv\ -- Ab/=A\ -- C^/=Dbv -- F&lt;/span&gt;&lt;br /&gt;
 One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling.&lt;br /&gt;
-Enharmonic = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3 = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;/m2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\A1. Period = \M3 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;4 = &lt;!-- ws:start:WikiTextRawRule:048:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:048 --&gt;d4. Generator = ^\M3 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4 = ^&lt;!-- ws:start:WikiTextRawRule:049:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:049 --&gt;d4.&lt;br /&gt;
+Enharmonic = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3 = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;/m2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\A1. Period = \M3 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;4 = &lt;!-- ws:start:WikiTextRawRule:051:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:051 --&gt;d4. Generator = ^\M3 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4 = ^&lt;!-- ws:start:WikiTextRawRule:052:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:052 --&gt;d4.&lt;br /&gt;
 &lt;br /&gt;
 &lt;span style="display: block; text-align: center;"&gt;P1 — \M3 — \\A5=/m6 — P8&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C — E\ — Ab/ — C&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;8=v/m9 — F&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C — E^\ — Ab^^/=Avv\ — Dbv/ — F&lt;/span&gt;&lt;br /&gt;
 This is a lot of math, but it only needs to be done once for each pergen!&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:81:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Further Discussion-Alternate enharmonics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:81 --&gt;Alternate enharmonics&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:84:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Further Discussion-Alternate enharmonics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:84 --&gt;Alternate enharmonics&lt;/h2&gt;
   &lt;br /&gt;
 Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;A2, which is an improvement but still awkward. The period is ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m3 and the generator is v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.&lt;br /&gt;
@@ Line 3,000: / Line 3,007: @@
 &lt;span style="display: block; text-align: center;"&gt;P1 -- vM3 -- ^m6 -- P8&lt;br /&gt;
 &lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Ev -- Ab^ -- C&lt;br /&gt;
-&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 -- /m2 -- &lt;!-- ws:start:WikiTextRawRule:050:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:050 --&gt;d3=\\A2 -- \M3 -- P4&lt;br /&gt;
+&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 -- /m2 -- &lt;!-- ws:start:WikiTextRawRule:053:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:053 --&gt;d3=\\A2 -- \M3 -- P4&lt;br /&gt;
-&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Db/ -- Ebb&lt;!-- ws:start:WikiTextRawRule:051:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:051 --&gt;=D#\\ -- E\ -- F&lt;/span&gt;&lt;br /&gt;
+&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Db/ -- Ebb&lt;!-- ws:start:WikiTextRawRule:054:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:054 --&gt;=D#\\ -- E\ -- F&lt;/span&gt;&lt;br /&gt;
 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.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 3,008: / Line 3,015: @@
 For example, (P8, P5/3) has n = 3, G = ^M2, and E = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;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&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. Assuming a reasonably just 5th, E needs to be upped, so E = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. Add the multi-E ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[-2,1] to the multigen P5 = [7,4] to get ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[-2,1] = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- D#v -- Fb^ -- G. Because d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 = -200¢ - 26·c, ^ = (-d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2) / 3 = 67¢ + 8.67·c, about a third-tone.&lt;br /&gt;
 &lt;br /&gt;
-Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to E, or the multi-E if the count is &amp;gt; 1. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.&lt;br /&gt;
+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 &amp;gt; 1. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.&lt;br /&gt;
 &lt;br /&gt;
 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.&lt;br /&gt;
@@ Line 3,016: / Line 3,023: @@
 Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese (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&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- Gbv -- B^ -- F.&lt;br /&gt;
 &lt;br /&gt;
-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 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.&lt;br /&gt;
+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.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:83:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Further Discussion-Chord names and scale names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:83 --&gt;Chord names and scale names&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:86:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Further Discussion-Chord names and scale names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:86 --&gt;Chord names and scale names&lt;/h2&gt;
   &lt;br /&gt;
 Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt; page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.&lt;br /&gt;
@@ Line 3,024: / Line 3,031: @@
 In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.&lt;br /&gt;
 &lt;br /&gt;
-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's pergen 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)] = [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = WWm7, 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 Ev G Bb = C7(v3).&lt;br /&gt;
+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 (2.3.5.7 with 50/49 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 = WWm7, 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 Ev G Bb = C7(v3).&lt;br /&gt;
 &lt;br /&gt;
 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 (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 Bbv = C,v7.&lt;br /&gt;
@@ Line 3,032: / Line 3,039: @@
 Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. 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 Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:85:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Tipping points and sweet spots"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:85 --&gt;Tipping points and sweet spots&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:88:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Tipping points and sweet spots"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:88 --&gt;Tipping points and sweet spots&lt;/h2&gt;
   &lt;br /&gt;
 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 &amp;quot;tips over&amp;quot;, 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 &amp;quot;sweet spot&amp;quot;, where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.&lt;br /&gt;
@@ Line 3,046: / Line 3,053: @@
 Another &amp;quot;tippy&amp;quot; temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow 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.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:87:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Further Discussion-Notating unsplit pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:87 --&gt;Notating unsplit pergens&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:90:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Further Discussion-Notating unsplit pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:90 --&gt;Notating unsplit pergens&lt;/h2&gt;
   &lt;br /&gt;
-An unsplit pergen doesn't &lt;u&gt;require&lt;/u&gt; ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a comma that maps to P1, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (the monzo has a final exponent of ±1).&lt;br /&gt;
+An unsplit pergen doesn't &lt;u&gt;require&lt;/u&gt; ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a mapping comma, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (i.e. the monzo has a final exponent of ±1).&lt;br /&gt;
 &lt;br /&gt;
-The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate.&lt;br /&gt;
+The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate. The up symbol's ratio is always the mapping comma, or its inverse.&lt;br /&gt;
@@ Line 3,248: / Line 3,255: @@
 The schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad.&lt;br /&gt;
 &lt;br /&gt;
-For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, simply by adding or subtracting the vanishing comma from 81/80 or 80/81. This 3-limit comma is tempered, as is every ratio. It can be found directly from the 3-limit mapping of the vanishing comma. Because the schismic comma is a descending d2, and d2 = (19,-12), the 3-comma is the pythagorean comma (-19,12).&lt;br /&gt;
+For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-comma is the pythagorean comma (-19,12).&lt;br /&gt;
 &lt;br /&gt;
-A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv.&lt;br /&gt;
+A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for archy (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:89:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Further Discussion-Notating rank-3 pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:89 --&gt;Notating rank-3 pergens&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:92:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Further Discussion-Notating rank-3 pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:92 --&gt;Notating rank-3 pergens&lt;/h2&gt;
   &lt;br /&gt;
 Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:&lt;br /&gt;
@@ Line 3,269: / Line 3,276: @@
          &lt;th&gt;notation's rank&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;&lt;!-- ws:start:WikiTextRawRule:052:``#`` --&gt;#&lt;!-- ws:end:WikiTextRawRule:052 --&gt; of enharmonics needed&lt;br /&gt;
+         &lt;th&gt;&lt;!-- ws:start:WikiTextRawRule:055:``#`` --&gt;#&lt;!-- ws:end:WikiTextRawRule:055 --&gt; of enharmonics needed&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;enharmonics&lt;br /&gt;
@@ Line 3,484: / Line 3,491: @@
 &lt;/table&gt;
-When there is more than one enharmonic, the first one can be combined with the 2nd to make an equivalent 2nd enharmonic.&lt;br /&gt;
+When there is more than one enharmonic, the first one can be added to or subtracted from the 2nd one, to make an equivalent 2nd enharmonic.&lt;br /&gt;
 &lt;br /&gt;
-A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &amp;quot;superfalse&amp;quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.)&lt;br /&gt;
+A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas. There are even &amp;quot;superfalse&amp;quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.&lt;br /&gt;
 &lt;br /&gt;
-All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors.&lt;br /&gt;
+Even the unsplit rank-3 pergen requires single-pair notation, for the 2nd generator. Single-splits and false double-splits require double-pair, true doubles and false triples require triple-pair, and true triples require quadruple-pair. Some false triples and some true doubles may use quadruple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third or fourth pair of symbols, and a third or fourth pair of adjectives to describe them, one might simply use colors.&lt;br /&gt;
 &lt;br /&gt;
-Quadruple-pair notation is needed for some true triples like (P8/2, P5/2, vM3/2). A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.&lt;br /&gt;
+A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.&lt;br /&gt;
 &lt;br /&gt;
 Some examples of 7-limit rank-3 temperaments:&lt;br /&gt;
@@ Line 3,609: / Line 3,616: @@
          &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;v&lt;!-- ws:start:WikiTextRawRule:053:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:053 --&gt;A2 = 60/49&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;v&lt;!-- ws:start:WikiTextRawRule:056:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:056 --&gt;A2 = 60/49&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;/1 = 64/63&lt;br /&gt;
@@ Line 3,640: / Line 3,647: @@
 If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - Ev - G - Bb\.&lt;br /&gt;
 &lt;br /&gt;
-There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish 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 = &lt;!-- ws:start:WikiTextRawRule:054:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:054 --&gt;d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. 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 Ev G Bb\.&lt;br /&gt;
+There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish 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 = &lt;!-- ws:start:WikiTextRawRule:057:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:057 --&gt;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 Ev G Bbv/. 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 Ev G Bb\.&lt;br /&gt;
 &lt;br /&gt;
-With this standardization, the enharmonic can be derived 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)&lt;span style="vertical-align: super;"&gt;-2&lt;/span&gt; · (64/63)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; · (19,-12)&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;. This can be rewritten as vv1 + &lt;!-- ws:start:WikiTextRawRule:055:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:055 --&gt;1 - d2 = vv&lt;!-- ws:start:WikiTextRawRule:056:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:056 --&gt;-d2 = -^^\\d2. The comma is negative, but the enharmonic never is, therefore E = ^^\\d2. P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.&lt;br /&gt;
+With this standardization, the enharmonic can be derived 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)&lt;span style="vertical-align: super;"&gt;-2&lt;/span&gt; · (64/63)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; · (19,-12)&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;. This can be rewritten as vv1 + &lt;!-- ws:start:WikiTextRawRule:058:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:058 --&gt;1 - d2 = vv&lt;!-- ws:start:WikiTextRawRule:059:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:059 --&gt;-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.&lt;br /&gt;
 &lt;br /&gt;
-The 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 deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish 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.&lt;br /&gt;
+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 deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish 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.&lt;br /&gt;
 &lt;br /&gt;
-Demeter is unusual in that its 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, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (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^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... 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/highs/lows only for the other kind of accidentals.&lt;br /&gt;
+Unlike the previous examples, Demeter'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, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (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 -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... 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/highs/lows only for the other kind of accidentals.&lt;br /&gt;
 &lt;br /&gt;
-There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from the 2nd generator to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For demeter, any combination of vm3, 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 any 3-limit comma can 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 &lt;strong&gt;DOL&lt;/strong&gt; (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &amp;lt; 3, 5/4 is preferred.&lt;br /&gt;
+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 vm3, 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 any 3-limit comma can 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 &lt;strong&gt;DOL&lt;/strong&gt; (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &amp;lt; 3, 5/4 is preferred.&lt;br /&gt;
 &lt;br /&gt;
-If ^1 = 81/80, possible half-split gens2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.&lt;br /&gt;
+If ^1 = 81/80, possible half-split gen2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.&lt;br /&gt;
 &lt;br /&gt;
 All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens:&lt;br /&gt;
@@ Line 3,948: / Line 3,955: @@
 There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:91:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Further Discussion-Notating Blackwood-like pergens*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:91 --&gt;Notating Blackwood-like pergens*&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:94:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Further Discussion-Notating Blackwood-like pergens*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:94 --&gt;Notating Blackwood-like pergens*&lt;/h2&gt;
   &lt;br /&gt;
-A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.&lt;br /&gt;
+A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.&lt;br /&gt;
 &lt;br /&gt;
-A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).&lt;br /&gt;
+A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).&lt;br /&gt;
 &lt;br /&gt;
 Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:&lt;br /&gt;
@@ Line 4,194: / Line 4,201: @@
 &lt;/table&gt;
-If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or 13/12.&lt;br /&gt;
+If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15, or 12/11, or 13/12.&lt;br /&gt;
 &lt;br /&gt;
 The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and ^1 = 81/80 or equivalently, 16/15.&lt;br /&gt;
@@ Line 4,206: / Line 4,213: @@
 In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.&lt;br /&gt;
 &lt;br /&gt;
-But in &amp;quot;no-5ths&amp;quot; or &amp;quot;minus white&amp;quot; pergens, not every name has a note. For example, deep reddish minus white (2.5.7 with 50/49 tempered out) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.&lt;br /&gt;
+But in &amp;quot;no-5ths&amp;quot; or &amp;quot;minus white&amp;quot; pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.&lt;br /&gt;
 &lt;br /&gt;
-Note-less names can be avoided in any pergen with a 5th. The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.&lt;br /&gt;
+Note-less names can be avoided in any pergen with a 5th (or 4th or 11th, etc.). The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.&lt;br /&gt;
@@ Line 4,214: / Line 4,221: @@
      &lt;tr&gt;
          &lt;th&gt;tuning&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;comma&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;pergen&lt;br /&gt;
@@ Line 4,229: / Line 4,238: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;large quintuple blue,&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;large quintuple blue&lt;br /&gt;
-small quintuple red&lt;br /&gt;
+or small quintuple red&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(-14,0,0,5)&lt;br /&gt;
+or (22,-5,0,-5)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/5, P5)&lt;br /&gt;
@@ Line 4,247: / Line 4,259: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;Blackwood&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;256/243&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/5, ^1)&lt;br /&gt;
@@ Line 4,262: / Line 4,276: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 4,274: / Line 4,290: @@
          &lt;td style="text-align: center;"&gt;C E G#...&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;invalid, no G, D or A notes&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;invalid, no G, &lt;br /&gt;
+D or A notes&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
 &lt;/table&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:93:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;!-- ws:end:WikiTextHeadingRule:93 --&gt; &lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:96:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;!-- ws:end:WikiTextHeadingRule:96 --&gt; &lt;/h2&gt;
   edo-subset notations&lt;br /&gt;
 P8/6, P8/8&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:95:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Further Discussion-Notating non-8ve and no-5ths pergens*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:95 --&gt;Notating non-8ve and no-5ths pergens*&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:98:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Further Discussion-Notating non-8ve and no-5ths pergens*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:98 --&gt;Notating non-8ve and no-5ths pergens*&lt;/h2&gt;
   &lt;br /&gt;
 Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. The pergens are grouped into blocks and sections as before:&lt;br /&gt;
@@ Line 4,790: / Line 4,807: @@
 Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a &lt;u&gt;huge&lt;/u&gt; number of note-less names. The composer may well want to use a notation that isn't backwards compatible.&lt;br /&gt;
 &lt;br /&gt;
-Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all.&lt;br /&gt;
+Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all...&lt;br /&gt;
 &lt;br /&gt;
 A similar chart could be made of all rank-3 pergen cubes.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:97:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Further Discussion-Notating tunings with an arbitrary generator"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:97 --&gt;Notating tunings with an arbitrary generator&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:100:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Further Discussion-Notating tunings with an arbitrary generator"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:100 --&gt;Notating tunings with an arbitrary generator&lt;/h2&gt;
   &lt;br /&gt;
 Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.&lt;br /&gt;
@@ Line 5,139: / Line 5,156: @@
 See also the &lt;a class="wiki_link" href="/Map%20of%20rank-2%20temperaments"&gt;map of rank-2 temperaments&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:99:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:99 --&gt;Pergens and MOS scales&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:102:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:102 --&gt;Pergens and MOS scales&lt;/h2&gt;
   &lt;br /&gt;
 Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.&lt;br /&gt;
@@ Line 6,242: / Line 6,259: @@
 The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.&lt;br /&gt;
 &lt;br /&gt;
-Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.&lt;br /&gt;
+Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:101:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Further Discussion-Pergens and EDOs*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:101 --&gt;Pergens and EDOs*&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:104:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Further Discussion-Pergens and EDOs*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:104 --&gt;Pergens and EDOs*&lt;/h2&gt;
   &lt;br /&gt;
 Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.&lt;br /&gt;
@@ Line 6,559: / Line 6,576: @@
 For each edo, find the nearest edomapping (also known as the patent val) for the 2.3 subgroup. Form a 2x2 matrix from these edomappings. Let d be the determinant of this matrix. If |d| = m, the generator is the 5th, and the pergen is simply (P8/m, P5).&lt;br /&gt;
 &lt;br /&gt;
-For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, and the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.&lt;br /&gt;
+For example, 12edo's 3-limit edomapping is (12, 19), and 16edo's is (16, 25). The determinant of [(12 19) (16 25)] is -4, thus the pergen for 12edo and 16edo is (P8/4, P5). To make the calculations easier, octave-reduced edomappings can be used, which indicate the number of edosteps that 3/2 maps to, not 3/1. For 12 and 16, we have [(12 7) (16 9)], and d is again -4.&lt;br /&gt;
 &lt;br /&gt;
 Sometimes the second-nearest edomapping is preferred, more on this later.&lt;br /&gt;
@@ Line 6,565: / Line 6,582: @@
 If |d| ≠ m, we must find the multigen. Take the ratio of the two edos N/N' and reduce it by m. In the scale tree, find the two ancestors and the two descendants of this ratio. Choose one of the four (more on this below), and let this new ratio be g/g'. The generator G maps to both g\N and g'\N'. The two edos come closest to coinciding here.&lt;br /&gt;
 &lt;br /&gt;
-For example, for 7-edo and 17-edo, m = 1 but d = 2. The ancestors and descendents of 7/17 are 2/5, 5/12, 9/22 and 12/29. Choose the smallest for now, 2/5. The generator maps to both 2\7 and 5\17. 2\7 is 343¢ and 5\17 is 353¢, both neutral 3rds. Their difference is ~10¢ = 1\119. (119 = LCM (7, 17)). This is the smallest difference possible between 7edo's notes and 17edo's notes, except of course for 0\7 and 0\17. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths.&lt;br /&gt;
+For example, for 7-edo and 17-edo, m = 1 but d = 2. The ancestors and descendants of 7/17 are 2/5, 5/12, 9/22 and 12/29. Choose the smallest for now, 2/5. The generator maps to both 2\7 and 5\17, which are both neutral 3rds. Using the other ancestor always gives the octave inverses, in this case 5/12 giving 5\7 and 12\17, which are neutral 6ths. 2\7 = 343¢ and 5\17 = 353¢, and their difference is only ~10¢ = 1\119. (119 = LCM (7, 17)). This 10¢ is the least difference between any 7edo note and any 17edo note, except that the two neutral 6ths differ by the same 10¢, and of course excluding note pairs that coincide exactly such as 0\7 and 0\17.&lt;br /&gt;
 &lt;br /&gt;
-Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, we extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found earlier, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this vector with either edomapping is zero. Treating this new vector as a monzo, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for.&lt;br /&gt;
+Next we must find a comma that both edos temper out, and find the pergen from that comma. To do this, we extend the edomappings to three entries. Rather than finding the mapping of 5/4 or 7/4, we use the generator we found earlier, without concerning ourselves about which ratio it corresponds to, in keeping with the higher-prime-agnostic nature of pergens. Treating the two edomappings as 3-D vectors, take the cross product of them, whch makes a new vector which is perpendicular to both. Thus the dot product of this vector with either edomapping is zero. Treating this new vector as a monzo, since the dot product is zero, both edos map this monzo to zero edosteps. In other words, they temper out this monzo, and this monzo is the comma we're looking for.&lt;br /&gt;
 &lt;br /&gt;
-For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two neutral 3rds with a 5th. The pergen is obviously (P8, P5/2).&lt;br /&gt;
+For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2).&lt;br /&gt;
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-To verify the validity of this approach, one can find a specific ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 = 71¢. A smaller comma can be found with a ratio more in the center of the range, such as 11/9, which yeilds 242/243. Because the direction of the cross product vector depends on the order of the two vectors multiplied, the sign of the comma is arbitrary, and the comma may be a descending interval. In that case, invert the comma to make 243/242. Using unreduced edomappings gives the same result. If the basis is (2/1, 3/1, 5/1), we have (7, 11, 16) x (17, 27, 39) = (-3, -1, 2) = 25/24.&lt;br /&gt;
+To verify the validity of this approach, one can find a specific JI ratio that maps to both 2\7 and 5\17 and use it to construct a comma. The ratio must contain only one higher prime, and must have color depth 1. If desired, a ratio of the form p/t can always be found, where p is a higher prime and t is a power of two. Any ratio between 3\14 and 5\14 maps to 2\7, and any ratio between 9\34 and 11\34 maps to 5\17. (The formula is (2n±1)/2d.) This gives us the ranges 257-429¢ and 318-388¢. Thus 5/4 barely works. The comma is (0, -1, 2) dot (2/1, 3/2, 5/4) = 25/24 = 71¢. A smaller comma can be found with a ratio more in the center of the range, such as 11/9, which yeilds 242/243. Because the direction of the cross product vector depends on the order of the two vectors multiplied, the sign of the comma is arbitrary, and the comma may be a descending ratio. If that happens, invert the comma to make 243/242. &lt;br /&gt;
+&lt;br /&gt;
+Using unreduced edomappings gives the same result. If the basis is (2/1, 3/1, 5/1), we have (7, 11, 16) x (17, 27, 39) = (-3, -1, 2) = 25/24. If the basis is (2/1, 3/1, 11/9)...&lt;br /&gt;
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 If the octave is split,&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:106:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="Further Discussion-Supplemental materials*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:106 --&gt;Supplemental materials*&lt;/h2&gt;
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 needs more screenshots, including 12-edo's pergens and a page of the pdf&lt;br /&gt;
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 check references to mapping commas&lt;br /&gt;
 finish proofs&lt;br /&gt;
+make a glossary of all bolded terms?&lt;br /&gt;
 link from: ups and downs page, Kite Giedraitis page, MOS scale names page,&lt;br /&gt;
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 This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.&lt;br /&gt;
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 Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
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+&lt;br /&gt;
+The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. &lt;br /&gt;
 &lt;br /&gt;
-Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. Screenshots of the first 38 pergens:&lt;br /&gt;
+Screenshots of the first 38 pergens:&lt;br /&gt;
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 Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:&lt;br /&gt;
@@ Line 6,823: / Line 6,845: @@
 The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.&lt;br /&gt;
 &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:112:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc26"&gt;&lt;a name="Further Discussion-Various proofs (unfinished)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:112 --&gt;Various proofs (unfinished)&lt;/h2&gt;
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 The interval P8/2 has a &amp;quot;ratio&amp;quot; of the square root of 2, which equals 2&lt;span style="vertical-align: super;"&gt;1/2&lt;/span&gt;, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the &lt;strong&gt;pergen matrix&lt;/strong&gt; [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.&lt;br /&gt;
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 Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:114:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc27"&gt;&lt;a name="Further Discussion-Miscellaneous Notes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:114 --&gt;Miscellaneous Notes&lt;/h2&gt;
   &lt;br /&gt;
 &lt;u&gt;&lt;strong&gt;Staff notation&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;