Kite's ups and downs notation: Difference between revisions

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__FORCETOC__

~~=A 22edo example=~~

Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works with almost all EDOs. When extended with lifts and drops (/ and \), it works with all rank 2 ~~tunings~~ (see the [[pergen|pergens]] page). It only adds 3 symbols to standard notation~~, so it's very easy to learn~~. The ~~name comes from the~~ up symbol "^" and the down symbol "v". (The ~~third symbol is the~~ mid symbol, "~".)

== Definition ==

Ups and Downs (or ^v) is a notation system developed by [[KiteGiedraitis|Kite]] that works with almost all EDOs. When extended with lifts and drops (/ and \), it works with all rank-2 temperaments (see the [[pergen|pergens]] page). It only adds 3 symbols to standard notation. The up symbol "^" and the down symbol "v" indicate raising or lowering a note (or widening/narrowing an interval) by one edostep. The mid symbol, "~" is for intervals exactly midway between major and minor.

~~To understand the ups and downs notation~~, ~~let~~'~~s start with an EDO that doesn't need~~ it~~. 19-EDO is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDO-step. So C# is right next to C~~, ~~and~~ the ~~keyboard runs C C# Db D D# Eb E etc~~. ~~Conventional~~ notation ~~works perfectly with 19~~-~~EDO as long as you remember that C# and Db are different notes~~.

== Sorry everyone, this page is a freakin' mess. Until I fix it, see instead the [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Notation guide for edos 5-72] ==

In contrast, 22-~~EDO~~ is hard to notate because 7 fifths are three ~~EDO-steps~~, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the major 3rd becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really bad. A notation system should work in every key!

==Explanation -- a 22edo example==

To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-edo is easy to notate because 7 fifths reduced by 4 octaves adds up to one edostep. C# is right next to C, and the keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-edo as long as you remember that C# and Db are different notes.

In contrast, 22-edo is hard to notate because 7 fifths are three edosteps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the major 3rd becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really bad. A notation system should work in every key!

The solution is to use the sharp symbol to mean "raised by 7 fifths", and to use the up symbol "^" to mean "sharpened by one EDO-step". 22-EDO can be written C - Db - Db^ - Dv - D - Eb - Eb^ - Ev - E - F etc. The notes are pronounced "D-flat-up, D-down", etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp.

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The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.

Relative notation for 22-EDO is P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - d5 - ^d5 - vP5 - P5 etc. That's pronounced "upminor 2nd, downmajor 3rd", etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any key~~, with certain keys requiring double-sharps or even triple-sharps~~. The notes without ups or downs always form a chain of fifths.

Relative notation for 22-EDO is P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - d5 - ^d5 - vP5 - P5 etc. That's pronounced "upminor 2nd, downmajor 3rd", etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any 22-edo key. The notes without ups or downs always form a chain of fifths.

You can loosely relate the ups and downs to JI: major = ru or fifthward wa, downmajor = yo, upminor = gu, minor = zo or fourthwards wa. Or simply up = gu, down = yo, and ~~mid~~ = wa, zo or ru. (~~See [[Kite's_color_notation|Kite's color notation]] for an explanation of the colors~~.) These correlations are for 22-~~EDO~~ only, other EDOs have other correlations.

You can loosely relate the ups and downs to [[Kite's color notation|colors]]: major = ru or fifthward wa, downmajor = yo, upminor = gu, minor = zo or fourthwards wa. Or simply up = gu, down = yo, and plain = wa, zo or ru (plain means neither up nor down). These correlations are for 22-edo only, other EDOs have other correlations.

Conventionally, in C you use D# instead of Eb when you have a Gaug chord. You have the freedom to spell your notes how you like, to make your chords look right. Likewise, in 22-EDO, Db can be spelled C^ or B#v or even B^^ ("B double-up"). However avoid using both C# and Db, as the ascending Db-C# interval appears descending.

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'''Interval arithmetic'''

~~22-EDO interval~~ arithmetic ~~works out very neatly~~. Ups and downs are just added in:

Interval arithmetic is always preserved. Ups and downs are just added in:

C + M3 = E, C + ~~vM3~~ = Ev, C^ + M3 = E^

C + M3 = E, C^ + M3 = E^, C + ^M3 = E^

D-F# ~~is a~~ M3, Dv-F#v = M3

D-F# = M3, D^-F# = vM3, D-F#^ = ^M3

M2 + m2 = m3, M2 + ^m2 = ^m3, vM2 + m2 = vm3

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Gb^ - Db^ - Ab^ - Eb^ - Bb^ - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - Gv - Dv - Av - Ev - Bv

The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. By which is meant, 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary~~. Here's 22-EDO in pentatonic notation:~~

The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. By which is meant, 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary.

~~chain of "fifths": Gx Dx Ax F# C# G# D# A# F C G D A Fb Cb Gb Db Ab Fbb Cbb Gbb Dbb~~

~~scale in C: C C# Dbb Db D D# Dx Fbb Fb F F# Gbb Gb G G# Gx Ab A A# Ax Cbb Cb C~~

That's an awful lot of sharps and flats, but that does make a neat and tidy notation (except for the Dbb-Gx fifth). And it exists as an alternative, embedded within conventional notation, with a key signature with circled X's on the B and E spots.

So the chain of fifths has a few spots to watch out for. You have to remember that fifths sometimes appear as downminor 6ths, in the form of B-something to G-something. A little tricky, but manageable. Analogous to 12-ET, where G# to Eb is a fifth that looks like a sixth.

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etc.

=~~~~'''Other EDOs'''~~~~=

=='''Other EDOs'''==

The up symbol means "sharpened by one edo-step" in any edo that uses them. The size in cents of the up changes greatly depending on the edo, from 120¢ in 10-edo to ~17¢ in 72-edo. The sharp symbol's cents size also depends on the edo, ranging from 240¢ in 5-edo to ~26¢ in 47edo.

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EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:

'''supersharp''' = ~~EDOs~~ with fifths wider than 720¢

'''supersharp''' = edos with fifths wider than 720¢

'''pentatonic =''' ~~EDOs~~ with a fifth =720¢

'''pentatonic''' = edos with a fifth =720¢

'''~~regular~~''' = ~~EDOs~~ with a fifth that hits the "sweet spot" between 720¢ and 686¢

'''diatonic''' = edos with a fifth that hits the "sweet spot" between 720¢ and 686¢

'''perfect''' =~~`**`**EDOs~~ with a fifth four sevenths of an octave = 4\7 =`` 686¢

'''perfect''' = edos with a fifth four sevenths of an octave = 4\7 =`` 686¢

'''superflat''' = ~~EDOs~~ with a fifth less than 686¢

'''superflat''' = edos with a fifth less than 686¢

This is in addition to the '''trivial''' EDOs, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.

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|-

| style="text-align:center;" | 12edo

| style="text-align:center;" | ~~regular~~

| style="text-align:center;" |diatonic

| style="text-align:center;" |

| style="text-align:center;" | C

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|-

| style="text-align:center;" | 17edo

| style="text-align:center;" | ~~regular~~

| style="text-align:center;" |diatonic

| style="text-align:center;" |

| style="text-align:center;" | C

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|-

| style="text-align:center;" | 19edo

| style="text-align:center;" | ~~regular~~

| style="text-align:center;" | diatonic

| style="text-align:center;" |

| style="text-align:center;" | C

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|-

| style="text-align:center;" | 22edo

| style="text-align:center;" | ~~regular~~

| style="text-align:center;" | diatonic

| style="text-align:center;" |

| style="text-align:center;" | C

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|-

| style="text-align:center;" | 24edo

| style="text-align:center;" | ~~regular~~

| style="text-align:center;" | diatonic

| style="text-align:center;" |

| style="text-align:center;" | C

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|-

| style="text-align:center;" | 26edo

| style="text-align:center;" | ~~regular~~

| style="text-align:center;" | diatonic

| style="text-align:center;" |

| style="text-align:center;" | C

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|-

| style="text-align:center;" | 27edo

| style="text-align:center;" | ~~regular~~

| style="text-align:center;" | diatonic

| style="text-align:center;" |

| style="text-align:center;" | C

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|-

| style="text-align:center;" | 29edo

| style="text-align:center;" | ~~regular~~

| style="text-align:center;" | diatonic

| style="text-align:center;" |

| style="text-align:center;" | C

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|-

| style="text-align:center;" | 31edo

| style="text-align:center;" | ~~regular~~

| style="text-align:center;" | diatonic

| style="text-align:center;" |

| style="text-align:center;" | C

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|-

| style="text-align:center;" | 36edo

| style="text-align:center;" | ~~regular~~

| style="text-align:center;" | diatonic

| style="text-align:center;" |

| style="text-align:center;" | C

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='''Summary of EDO notation'''=

==~~Regular~~ EDOs==

==Diatonic EDOs==

(12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)

All ~~regular~~ EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.

All diatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.

Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.

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For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, i.e. not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't.

All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring ~~regular~~ frameworks with sharpness > 1 or < -1. If these are notated without ups and downs, the notes run out of order:

All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring diatonic frameworks with sharpness > 1 or < -1. If these are notated without ups and downs, the notes run out of order:

17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C

@@ Line 1: / Line 1: @@
 __FORCETOC__
-=<u>A 22edo example</u>=
-Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works with almost all EDOs. When extended with lifts and drops (/ and \), it works with all rank 2 tunings (see the [[pergen|pergens]] page). It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol "^" and the down symbol "v". (The third symbol is the mid symbol, "~".)
+== Definition ==
+Ups and Downs (or ^v) is a notation system developed by [[KiteGiedraitis|Kite]] that works with almost all EDOs. When extended with lifts and drops (/ and \), it works with all rank-2 temperaments (see the [[pergen|pergens]] page). It only adds 3 symbols to standard notation. The up symbol "^" and the down symbol "v" indicate raising or lowering a note (or widening/narrowing an interval) by one edostep. The mid symbol, "~" is for intervals exactly midway between major and minor.
-To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDO-step. So C# is right next to C, and the keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-EDO as long as you remember that C# and Db are different notes.
+== <u>Sorry everyone, this page is a freakin' mess. Until I fix it, see instead the [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Notation guide for edos 5-72]</u> ==
-In contrast, 22-EDO is hard to notate because 7 fifths are <u>three</u> EDO-steps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the major 3rd becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really bad. A notation system should work in every key!
+==Explanation -- a 22edo example==
+To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-edo is easy to notate because 7 fifths reduced by 4 octaves adds up to one edostep. C# is right next to C, and the keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-edo as long as you remember that C# and Db are different notes.
+In contrast, 22-edo is hard to notate because 7 fifths are <u>three</u> edosteps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the major 3rd becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really bad. A notation system should work in every key!
 The solution is to use the sharp symbol to mean "raised by 7 fifths", and to use the up symbol "^" to mean "sharpened by one EDO-step". 22-EDO can be written C - Db - Db^ - Dv - D - Eb - Eb^ - Ev - E - F etc. The notes are pronounced "D-flat-up, D-down", etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp.
@@ Line 14: / Line 18: @@
 The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.
-Relative notation for 22-EDO is P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - d5 - ^d5 - vP5 - P5 etc. That's pronounced "upminor 2nd, downmajor 3rd", etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The notes without ups or downs always form a chain of fifths.
+Relative notation for 22-EDO is P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - d5 - ^d5 - vP5 - P5 etc. That's pronounced "upminor 2nd, downmajor 3rd", etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any 22-edo key. The notes without ups or downs always form a chain of fifths.
-You can loosely relate the ups and downs to JI: major = ru or fifthward wa, downmajor = yo, upminor = gu, minor = zo or fourthwards wa. Or simply up = gu, down = yo, and mid = wa, zo or ru. (See [[Kite's_color_notation|Kite's color notation]] for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.
+You can loosely relate the ups and downs to [[Kite's color notation|colors]]: major = ru or fifthward wa, downmajor = yo, upminor = gu, minor = zo or fourthwards wa. Or simply up = gu, down = yo, and plain = wa, zo or ru (plain means neither up nor down). These correlations are for 22-edo only, other EDOs have other correlations.
 Conventionally, in C you use D# instead of Eb when you have a Gaug chord. You have the freedom to spell your notes how you like, to make your chords look right. Likewise, in 22-EDO, Db can be spelled C^ or B#v or even B^^ ("B double-up"). However avoid using both C# and Db, as the ascending Db-C# interval appears descending.
@@ Line 22: / Line 26: @@
 <u>'''Interval arithmetic'''</u>
--EDO interval arithmetic works out very neatly. Ups and downs are just added in:
+Interval arithmetic is <u>always</u> preserved. Ups and downs are just added in:
-C + M3 = E, C + vM3 = Ev, C^ + M3 = E^
+C + M3 = E, C^ + M3 = E^, C + ^M3 = E^
-D-F# is a M3, Dv-F#v = M3
+D-F# = M3, D^-F# = vM3, D-F#^ = ^M3
 M2 + m2 = m3, M2 + ^m2 = ^m3, vM2 + m2 = vm3
@@ Line 38: / Line 42: @@
 Gb^ - Db^ - Ab^ - Eb^ - Bb^ - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - Gv - Dv - Av - Ev - Bv
-The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. By which is meant, 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary. Here's 22-EDO in pentatonic notation:
+The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. By which is meant, 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary.
-chain of "fifths": Gx Dx Ax F# C# G# D# A# F C G D A Fb Cb Gb Db Ab Fbb Cbb Gbb Dbb
-scale in C: C C# Dbb Db D D# Dx Fbb Fb F F# Gbb Gb G G# Gx Ab A A# Ax Cbb Cb C
-That's an awful lot of sharps and flats, but that does make a neat and tidy notation (except for the Dbb-Gx fifth). And it exists as an alternative, embedded within conventional notation, with a key signature with circled X's on the B and E spots.
 So the chain of fifths has a few spots to watch out for. You have to remember that fifths sometimes appear as downminor 6ths, in the form of B-something to G-something. A little tricky, but manageable. Analogous to 12-ET, where G# to Eb is a fifth that looks like a sixth.
@@ Line 88: / Line 86: @@
 etc.
-=<u>'''Other EDOs'''</u>=
+=='''Other EDOs'''==
 The up symbol means "sharpened by one edo-step" in any edo that uses them. The size in cents of the up changes greatly depending on the edo, from 120¢ in 10-edo to ~17¢ in 72-edo. The sharp symbol's cents size also depends on the edo, ranging from 240¢ in 5-edo to ~26¢ in 47edo.
@@ Line 94: / Line 92: @@
 EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:
-'''supersharp''' = EDOs with fifths wider than 720¢
+'''supersharp''' = edos with fifths wider than 720¢
-'''pentatonic =''' EDOs with a fifth =720¢
+'''pentatonic''' = edos with a fifth =720¢
-'''regular''' = EDOs with a fifth that hits the "sweet spot" between 720¢ and 686¢
+'''diatonic''' = edos with a fifth that hits the "sweet spot" between 720¢ and 686¢
-'''perfect''' =`**`**EDOs with a fifth four sevenths of an octave = 4\7 =`` 686¢
+'''perfect''' = edos with a fifth four sevenths of an octave = 4\7 =`` 686¢
-'''superflat''' = EDOs with a fifth less than 686¢
+'''superflat''' = edos with a fifth less than 686¢
 This is in addition to the '''trivial''' EDOs, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.
@@ Line 1,201: / Line 1,199: @@
 |-
 | style="text-align:center;" | 12edo
-| style="text-align:center;" | regular
+| style="text-align:center;" |diatonic
 | style="text-align:center;" |
 | style="text-align:center;" | C
@@ Line 1,271: / Line 1,269: @@
 |-
 | style="text-align:center;" | 17edo
-| style="text-align:center;" | regular
+| style="text-align:center;" |diatonic
 | style="text-align:center;" |
 | style="text-align:center;" | C
@@ Line 1,299: / Line 1,297: @@
 |-
 | style="text-align:center;" | 19edo
-| style="text-align:center;" | regular
+| style="text-align:center;" | diatonic
 | style="text-align:center;" |
 | style="text-align:center;" | C
@@ Line 1,341: / Line 1,339: @@
 |-
 | style="text-align:center;" | 22edo
-| style="text-align:center;" | regular
+| style="text-align:center;" | diatonic
 | style="text-align:center;" |
 | style="text-align:center;" | C
@@ Line 1,369: / Line 1,367: @@
 |-
 | style="text-align:center;" | 24edo
-| style="text-align:center;" | regular
+| style="text-align:center;" | diatonic
 | style="text-align:center;" |
 | style="text-align:center;" | C
@@ Line 1,397: / Line 1,395: @@
 |-
 | style="text-align:center;" | 26edo
-| style="text-align:center;" | regular
+| style="text-align:center;" | diatonic
 | style="text-align:center;" |
 | style="text-align:center;" | C
@@ Line 1,411: / Line 1,409: @@
 |-
 | style="text-align:center;" | 27edo
-| style="text-align:center;" | regular
+| style="text-align:center;" | diatonic
 | style="text-align:center;" |
 | style="text-align:center;" | C
@@ Line 1,439: / Line 1,437: @@
 |-
 | style="text-align:center;" | 29edo
-| style="text-align:center;" | regular
+| style="text-align:center;" | diatonic
 | style="text-align:center;" |
 | style="text-align:center;" | C
@@ Line 1,467: / Line 1,465: @@
 |-
 | style="text-align:center;" | 31edo
-| style="text-align:center;" | regular
+| style="text-align:center;" | diatonic
 | style="text-align:center;" |
 | style="text-align:center;" | C
@@ Line 1,537: / Line 1,535: @@
 |-
 | style="text-align:center;" | 36edo
-| style="text-align:center;" | regular
+| style="text-align:center;" | diatonic
 | style="text-align:center;" |
 | style="text-align:center;" | C
@@ Line 1,791: / Line 1,789: @@
 =<u>'''Summary of EDO notation'''</u>=
-==<u>Regular EDOs</u>==
+==<u>Diatonic EDOs</u>==
 (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)
-All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.
+All diatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.
 Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.
@@ Line 2,402: / Line 2,400: @@
 For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, i.e. not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't.
-All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring regular frameworks with sharpness &gt; 1 or &lt; -1. If these are notated without ups and downs, the notes run out of order:
+All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring diatonic frameworks with sharpness &gt; 1 or &lt; -1. If these are notated without ups and downs, the notes run out of order:
 -tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C