Kite's ups and downs notation

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author TallKite and made on 2015-09-03 17:42:02 UTC.

The original revision id was 558211375.

The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

="Ups and Downs" Notation= 

Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works very well with almost all EDOs and rank 2 tunings. 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". There's also the mid symbol "~" which undoes ups and downs.

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 adds up to one EDO-step. So C# is right next to C, and your 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__ 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!

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.

The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So in B, we get B-C-C^-C#v-C#-D-D^-D#v-D#-E etc.

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.

The basic pattern 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 mid notes always form a chain of fifths.

You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See [[Kite's color notation]] for an explanation of the colors.) 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# looks descending.

__**Interval arithmetic**__
In ups and downs notation, as in conventional notation, the chain of fifths runs:
Ebb-Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#-Fx-Cx etc.
This chain can be expressed in relative notation:
d2-d6-d3-d7-d4-d1-d5-m2-m6-m3-m7-P4-P1-P5-M2-M6-M3-M7-A4-A1-A5-A2-A6-A3-A7 etc.
To name the interval between any two notes, superimpose one chain onto the other, with P1 lining up with the lower note. For example C-E = M3 because M3 means "raised by 4 fifths" and E is 4 fifths away from C. Likewise, C + M3 = E.
C - G - D - A - E
P1-P5-M2-M6-M3

To add any two intervals, superimpose two copies of the relative chain. m3 + M2 = P4:
m3-m7-P4-P1
P1-P5-M2
Line up the lower P1 with m3 and look for what lies above M2.

22-EDO interval arithmetic works out very neatly. Ups and downs are just added in:
C + M3 = E, C + vM3 = Ev, C^ + M3 = E^
D-F# is a M3, Dv-F#v = M3
M2 + m2 = m3, M2 + ^m2 = ^m3, vM2 + m2 = vm3

There are some exceptions. Take this scale:
C Db Db^ Dv D Eb Eb^ Ev E F Gb Gb^ Gv G Ab Ab^ Av A Bb Bb^ Bv B C
Here's our fifths: C-G, Db-Ab, Db^-Ab^, Dv-Av, D-A, etc. Most fifths *look* like fifths and are easy to find. So do the 4ths. Our 4\22 maj 2nds are C-D, Db-Eb, Db^-Eb^, Dv-Ev, D-E, Eb-F, good until we reach Eb^-Gb, which looks like a min 3rd. Here's this scale's chain of 5ths:

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:

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

Now that's an awful lot of sharps and flats, but that does make a neat and tidy notation (except for the Gbb-Gx fifth). And it exists as an alternative, embedded within our standard 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 B-something to G-something is sometimes a fifth, sometimes a sixth. A little tricky, but manageable. Analogous to 12-ET, where G# to Eb is a fifth that looks like a sixth.

__**Staff Notation**__
For staff notation, just put an up or down to the left of the note and any standard accidental it might have. To write Db^ followed by Db in the same measure, use the mid sign: Db^ Db~. All 22 possible keys can be written out. The tonic is always a mid note, i.e. not up or down. Just as conventionally each black key produces both a sharp key and a flat key (Db major and C# minor), each of the 15 black keys of 22-EDO produces both, and there are 37 possible keys. The 2 most remote are Bbbb and F###, and triple-sharps and triple-flat keys seem rather extreme. Avoiding those, we have 35 possible tonics that run from Fbb to Bx. Some of the key signatures will have double-sharps or double-flats in them, or even triple-sharps.

C: no sharps
G: 1 sharp on F
D: 2 sharps on F and C
...
C#: 7 sharps
G#: 6 sharps, 1 double-sharp on F
D#: 5 sharps, 2 double-sharps on F and C
...
B#: 2 sharps, 5 double-sharps on F, C, G, D and A
...
Bx: 2 double-sharps on E and B, 5 triple-sharps on F, C, G, D and A

__**Other EDOs**__
This notation works for almost every EDO. 9, 11, 16, and 23 have weird interval arithmetic because of the narrow fifth, but they can be notated. 13 and 18 are best notated using the narrower of the 2 possible fifths, which makes them like 9, 11, 16 and 23. 8-EDO is hard. It works with pentatonic notation, if you don't mind learning pentatonic interval arithmetic. (Big if!)

EDOs come in 5 categories, based on the size of the fifth:
supersharp EDOs, with fifths wider than 720¢
pentatonic EDOs, with a fifth = 720¢
"sweet" EDOs, so-called because the fifth hits the "sweet spot" between 720¢ and 686¢
"perfect" EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢
superflat EDOs or Mavila EDOs, with a fifth less than 686¢

This is in addition to the trivial EDOs, 1, 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. This section will cover sweet EDOs and the other categories will be covered in other sections.

As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.

**__17-EDO__:** (2 keys per sharp/flat)
Black and white keys: C * * D * * E F * * G * * A * * B C
Relative notation: P1 m2 vM2 M2 m3 vM3 M3 P4 d5 vP5 P5 m6 vM6 M6 m7 vM7 M7 P8
or with upminors instead of downmajors: P1 m2 ^m2 M2 m3 ^m3 M3 P4 d5 ^d5 P5 m6 ^m6 M6 m7 ^m7 M7 P8
The d5 could instead be an A4: P4 ^P4 A4 P5 or P4 vA4 A4 P5
Many other variations are possible, much freedom of spelling.
In C, with downmajors: C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C
In B, with upminors: B C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B
One can't associate ups and downs with yellow and green because of the poor approximation of the 5-limit. However major = red or fifthward white, minor = blue or fourthward white, and downmajor = upminor = jade or amber.

**__24-EDO__:** (2 keys per sharp/flat)
black and white keys: C * * * D * * * E * F * * * G * * * A * * * B * C
Relative notation: P1 vm2 m2 vM2 M2 vm3 m3 vM3 M3 vP4 P4 ^P4 d5 vP5 P5 etc.
Many alternate spellings available, for example vm3 = ^M2, vM3 = ^m3, ^P4 = vd5, etc.
In C: C Dbv Db Dv D Ebv Eb Ev E Fv F F^ Gb Gv G etc.
JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber.

24-EDO is an example of a closed EDO. An EDO is closed if the keyspan of the fifth isn't coprime with the keyspan of the octave, and open if it is. 24-EDO has a fifth of 14 steps, and 14 isn't coprime with 24, because they have a common divisor of 2. 24-EDO is said to close at 12 (1/2 of 24), because the circle of fifths has only 12 notes. There are actually 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one:
Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#
Eb^-Bb^-F^-C^-G^-D^-A^-E^-B^-F#^-C#^-G#^
Just as G# could be written as Ab, all the up notes could be written as down notes.

In open EDOs, we can require that the tonic be a mid note. For example in 22-EDO, rather than using C#v as a tonic, we use B#. But closed EDOs force the use of tonics that are not a mid note. For example, the key of C^ runs:
C^ Db Db^ D D^ Eb Eb^ E E^ F F^ F^^ Gb^ G G^ etc.

**__31-EDO__:** (2 keys per sharp/flat)
Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C
relative notation: P1 ^P1 vm2 m2 ^m2 M2 ^M2 vm3 m3 ^m3 M3 ^M3 vP4 P4 ^P4 A4 d5 ^d5 P5 etc.
alternate spellings: A1=vm2, ^m2=vM2, ^M3=vP4, ^P4=vA4, etc.
In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.
JI associations: Perfect = white, major = yellow or fifthward white, minor = green or fourthward white, downminor = blue, upmajor = red, downmajor = upminor = jade or amber (same as 24-EDO).

**__41-EDO__:** (4 keys per sharp/flat)
Black and white keys: C * * * * * * D * * * * * * E * * F * * * * * * G * * * * * * A * * * * * * B * * C
P1 ^P1 vm2 m2 ^m2 ^^m2 vM2 M2 ^M2 vm3 m3 ^m3 ^^m3 vM3 M3 ^M3 vP4 P4 ^P4 ^^P4 d5 ^d5 vvP5 vP5 P5 etc.
alternate spellings: A1=^m2, ^^m2=vvM2, ^M3=vP4, vA4=d5, A4=^d5, etc.
In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.
JI associations: Perfect = white, major = fifthward white, minor = fourthward white, downmajor = yellow, upminor = green, downminor = blue, upmajor = red, double-downmajor = double-upminor = jade or amber.

**__53-EDO__:** (5 keys per sharp/flat)
Black and white keys: C * * * * * * * * D * * * * * * * * E * * * F * * * * * * * * G * * * * * * * * A * * * * * * * * B * * * C

=__Naming Chords__= 

Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths (or more generally, the chain of generators). Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.

There are 3 special cases to be addressed. The first is when the edo's 5th is narrower than 4\7, as in 16edo. Major is defined as wider than minor, so major is not fifthwards but fourthwards:

The fourthwards chain of fifths in superflat aka Mavila EDOs (3/2 maps to less than 4\7):
M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 - A1 etc.
F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.
16edo: P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8
16edo: C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C

In other words, sharp/flat, major/minor, and aug/dim all retain their melodic meaning but the chain-of-fifths meaning is reversed. Perfect and natural are unaffected. Interval arithmetic in fourthwards edos is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.
M2 + M2 --> m2 + m2 = dim3 --> aug3
D to F# --> D to Fb = dim3 --> aug3
Eb + m3 --> E# + M3 = G## --> Gbb

The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. That's because the scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect.

The chain of fifths in "perfect" EDOs (3/2 maps to 4\7):
P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.
21edo: P1 - ^P1 - vP2 - P2 - ^P2 - vP3 - P3 - ^P3 - vP4 - P4 - ^P4 - vP5 - P5 - ^P5 - vP6 - P6 - ^P6 - vP7 - P7 - ^P7 - vP8 - P8
Because everything is perfect, the quality can be omitted:
21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8
21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C
Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But they are two different concepts that must be kept distinct. In this redefined notation, B - F# isn't a perfect fifth because it's really B - F^.

The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. Such EDOs are dealt with below.

Chord names are based entirely on the ups/downs interval names, not on JI ratios. This avoids identifying one EDOstep with multiple ratios, as happens in 22edo when 0-7-18 implies 4:5:7 but 0-9-18 implies 9:12:16. 18\22 is neither 7/4 nor 16/9, it's 18\22!

==__22edo chord names__== 

Let's review the 22edo interval names:
0\22 = P1
1\22 = m2
2\22 = ^m2
3\22 = vM2
4\22 = M2
5\22 = m3
6\22 = ^m3
7\22 = vM3
8\22 = M3
9\22 = P4
10\22 = ^P4, d5
11\22 = vA4, ^d5
12\22 = A4, vP5
13\22 = P5
14\22 = m6
15\22 = ^m6
16\22 = vM6
17\22 = M6
18\22 = m7
19\22 = ^m7
20\22 = vM7
21\22 = M7
22\22 = P8

These are pronounced "downmajor second", "upminor third", etc. For 4ths and 5ths, "perfect" is implied and can be omitted: ^P4 = "up-four" and vP5 = "down-five". In larger edos there may be "down-octave", "up-unison", etc.

0-7-13-18 in C is "C,vM,m7", pronounced "C downmajor, minor seventh". The comma between the C and the down symbol is needed because Cv is a note, and "Cv,M,m7" is a different chord. That chord is pronounced "C down, major, minor 7th", so one has to "speak the comma". The extra comma isn't needed when there's no ups or downs immediately after the note name, e.g. Cm.

The conventional chord naming system uses a lot of "shorthand" like dom7 for M3,m7 and min6 for m3,M6. This causes problems in 22edo where there are so many choices for the 3rd, the 6th, the 7th and the 9th. For example, min6 could mean m3,vM6 = approximate 6:7:9:10 chord, or it could mean ^m3,M6 = approximate 1/1-6/5-3/2-12/7 chord. Larger edos would present even greater problems. Furthermore there's some ambiguity in the shorthand, e.g. in 12edo, both 0-3-6 and 0-3-6-9 are called dim chords.

Thus the shorthand should be largely abandoned and all the components of the chord should be explicitly spelled out, with a few exceptions: 1) The root, obviously. 2) The perfect 5th is assumed present unless otherwise specified. Thus 0-7-18 is C,vM,m7,-5 and 0-6-11 is C,^m,^d5. 3) The 3rd is also assumed to be present, and is implied by a quality with no degree. Thus 0-7-13 is C,vM. 4) The 3rd isn't spelled out if the 6th or 7th has the same quality as the 3rd. Thus 0-7-13-16 is C,vM6, but 0-7-13-17 is C,vM,M6. Thirdless chords: 0-13-18 is either Cm7,-3 or C5,m7.

The 6th, the 7th, the 9th, the 11th, etc. are explicitly written out, including their qualities. Thus the 9th isn't assumed to be major, and the presence of a 9th doesn't imply the presence of a 7th.

Sus chords: as usual, "sus" means the 3rd is replaced by the named note, a 2nd or 4th. "Sus4" implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-perfect 4th, etc. Some larger edos would have susv4, susvv4, etc. "Sus2" implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See 16edo below for an exception.

0-5-13 = m
0-6-13 = ^m
0-7-13 = vM
0-8-13 = M
0-9-13 = sus4
0-10-13 = sus^4
0-4-13 = sus2
0-3-13 = susvM2

0-5-11 = m,^d5
0-5-12 = m,vP5 (or possibly m,A4)

0-5-11-14 = m6,^d5
0-6-11-15 = ^m6,^d5
0-7-13-16 = vM6
0-8-13-17 = M6

0-5-13-18 = m7
0-6-13-19 = ^m7
0-7-13-20 = vM7
0-8-13-21 = M7

0-5-13-16 = m,vM6
0-8-13-19 = M,^m7
0-7-13-18-26 = vM,m7,M9
0-7-13-18-26-32 = vM,m7,M9,^P11

You can write out chord progressions using the ups/downs notation for note names. Here's the first 4 chords of Paul Erlich's 22edo composition Tibia:
G,vM7,-5 = "G downmajor seven, no five""
Eb^,vM,M9 = "E flat up, downmajor, major nine"
Gm7,-5 (no comma needed) = "G minor seven, no five"
A,vM,m7 = "A downmajor, minor seven"

To use relative notation, first write out all possible 22edo chord roots relatively. This is equivalent to the interval notation with Roman numerals substituted for Arabic, # for aug, and b for minor. Dim from perfect is b, but dim from minor is bb. Enharmonic equivalents like ^I = bII are used in certain chord progressions like Im - ^IIIM - ^VIIM - ^IVm - ^Im.
I ^I/bII v#I/^bII #I/vII II ^II/bIII v#II/^bIII #II/vIII III IV ^IV/bV v#IV/^bV #IV/vV V ^V/bVI v#V/^bVI #V/vVI VI ^VI/bVII v#VI/^bVII #VI/vVII VII/vI
These are pronounced "down-two", "up-flat-three", "down-sharp-four", etc.

Here's the Tibia chords. No comma is needed after the root because ups and downs are always leading, never trailing.
IvM7,-5 = "one downmajor seven, no five"
^bVIvM,M9 = "up-flat six downmajor, major nine"
Im7,-5 = "one minor seven, no five"
IIvM,m7 = "two downmajor, minor seven"

==__Chord names in other EDOs__== 

15edo: 3 keys per #/b, so ups and downs are needed.
keyboard/fretboard: D * * E/F * * G * * A * * B/C * * D
(the chain of fifths is always centered on D)
chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 M3/P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7 P8
chord roots: I ^bII vII II/bIII ^bIII vIII III/IV ^IV vV V ^bVI vVI VI/bVII ^bVII vVII
0-3-9 = m or sus2
0-4-9 = ^m
0-5-9 = vM
0-6-9 = M or sus4
0-5-9-12 = vM,m7

16edo: D * E * * F * G * A * B * * C * D, 1 key per #/b, ups and downs not needed. # is fourthward.
chord components: P1 d2 m2 M2 m3 M3 A3/d4 P4 A4/d5 P5 d6 m6 M6/d7 m7 M7 A7 P8
chord roots: I #I/bbII bII II bIII III #III/vIV IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI
0-3-9 = sus2
0-4-9 = m
0-5-9 = M
0-5-10 = M,A5 (the conventional aug chord)
0-6-9 = A (aug 3rd, perfect 5th)
0-7-9 = sus4
0-4-8-12 = m,d5,d7 (the conventional dim tetrad)

17edo: D * * E F * * G * * A * * B C * * D, 2 keys per #/b.
chord components: P1 m2 ^m2/vM2 M2 m3 ^m3/vM3 M3 P4 ^P4/d5 A4/vP5 P5 m6 ^m6/vM6 M6 m7 ^m7/vM7 M7 P8
chord roots: I v#I/bII #I/vII II bIII vIII III IV ^IV/bV #IV/vV V #V/bVI vVI VI bVII vVII VII
0-4-10 = m
0-5-10 = ^m or vM (probably choose vM over ^m whenever possible)
0-6-10 = M
0-7-10 = sus4
0-4-10-14 = m7
0-5-10-15 = vM7
0-6-10-16 = M7
Alternatively, one could replace downmajor with n = neutral or somesuch.

19edo: D * * E * F * * G * * A * * B * C * * D, ups and downs not needed.
chord components: P1 d2 m2 M2 d3 m3 M3 A3 P4 A4 d5 P5 d6 m6 M6 d7 m7 M7 A7 P8
chord roots: I v#I/bII #I/vII II bIII vIII III IV ^IV/bV #IV/vV V #V/bVI vVI VI bVII vVII VII
The possibility of a dim 3rd or an aug 3rd changes the meaning of "dim chord" and "aug chord".
0-4-11 = d (dim 3rd, perfect 5th, not a conventional dim chord)
0-4-10 = d,d5
0-5-11 = m
0-5-10 = m,d5 (conventional dim chord)
0-6-11 = M
0-7-11 = A (aug 3rd, perfect 5th, not a conventional aug chord)
0-6-12 = M,A5 (conventional aug chord)
0-7-12 = A,A5
0-8-11 = sus4

21edo: D * * E * * F * * G * * A * * B * * C * * D, zero keys per #/b.
chord components: P1 ^P1/vvP2 vP2 P2 ^P2 vP3 P3 ^P3 vP4 P4 ^P4 vP5 P5 ^P5 vP6 P6 ^P6 vP7 P7 ^P7 ^^P7/vP8
Because everything is perfect, the quality can be omitted.
chord components: 1 ^1/vv2 v2 2 ^2 v3 3 ^3 v4 4 ^4 v5 5 ^5 v6 6 ^6 v7 7 ^7 ^^7/v8
chord roots: I ^I vII II ^II vIII III vIII vIV IV ^IV vV V ^V vVI VI ^VI vVII VII ^VII vI
Quality can also be omitted in the chord names if we use the mid symbol "~":
0-3-12 = sus2
0-4-12 = vv or sus^2
0-5-12 = v (a down chord, e.g. C,v = "C down")
0-6-12 = ~ (e.g. D,~ = "D mid")
0-7-12 = ^ (e.g. Eb,^ = "E flat up")
0-8-12 = ^^ or susv4
0-9-12 = sus4
0-6-11 = ~,v5

0-7-12-19 = ^7
0-7-12-18 = ^,~7
0-7-12-17 = ^,v7
0-7-12-16 = ^6
0-7-12-15 = ^,~6
0-7-12-14 = ^,v6

24edo: D * * * E * F * * * G * * * A * * * B * C * * * D, 2 keys per #/b.
chord components: P1 vm2 m2 vM2 M2 vm3 m3 vM3 M3 ^M3/vP4 P4 ^P4 A4/d5 vP5 P5 vm6 m6 vM6 M6 ^M6/vm7 m7 vM7 M7 ^M7
chord roots: I v#I/vbII #I/bII vII II vbIII bIII vIII III ^III/vIV IV ^IV #IV/bV vV V ^#V/vbVI bVI vVI VI ^VI/vbVII bVII vVII VII ^VII/vI
0-5-14 = vm
0-6-14 = m
0-7-14 = vM
0-8-14 = M
0-9-14 = ^M
0-10-14 = sus4

31edo: D * * * * E * * F * * * * G * * * * A etc. 2 keys per #/b.
P1 ^P1 vm2 m2 vM2 M2 ^M2 vm3 m3 vM3 M3 ^M3 vP4 P4 ^P4 A4 d5 vP5 P5 etc.
I ^I vbII bII vII II ^II vbIII bIII vIII III ^III vIV IV ^IV #IV bV vV V etc.
0-7-18 = vm
0-8-18 = m
0-9-18 = vM
0-10-18 = M
0-11-18 = ^M
0-12-18 = sus-v4

==**__Cross-EDO considerations__**== 

In 22edo, the major chord is 0¢-436¢-709¢. In 19edo, it's 0¢-379¢-695¢. The two chords sound quite different, because "major 3rd" is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.

The name "major" refers not to the sound but to the function of the chord. If you want to pump an 81/80 with a I - VIm - IIm - V - I progression, you can do that in any edo, as long as you use major and minor chords.


==__EDOs with an inaccurate 3/2__== 

Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in this diagram, they are the ones to the left of the central line in the light blue region, plus the ones to the right of the central line in the orange region. The ones on the left edge of the blue region are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.

[[image:The 5th of EDOs 5-53.png width="640" height="802"]]


There are two strategies for notating these "oddball" EDOs, besides heptatonic fifth-based notation with ups and downs. One is to switch from heptatonic notation to some other type. The orange region contains edos for which pentatonic notation is a natural fit, in the sense that no ups or downs are needed. This includes 8edo, 13edo and 18edo.

The other approach is to use some interval other than the fifth to generate the notation. Above I said 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C.


__**Theoretical alternatives for 8edo, 11edo, 13edo and 18edo**__


8edo octatonic (every note is a generator)
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9

11edo heptatonic narrow-fifth-based, fourthwards with ^/v, 2 keys per #/b (3/2 maps to 6\11 5th):
P1 - m2 - vM2/m3 - M2/^m3 - M3 - P4 - P5 - m6 - vM6/m7 - M6/^m7 - M7 - P8
problematic because m3 = 2\11 is narrower than M2 = 3\11

11edo nonotonic narrow-fifth-based, fourthwards with no ups and downs (3/2 maps to 6\11 6th):
nonotonic fourthwards chain of sixths:
M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9 - d5 etc.
P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8
requires learning nonotonic interval arithmetic and devising a nonotonic staff notation

11edo pentatonic wide-fifth-based, fifthwards using ^/v, 2 keys per #/b (3/2 maps to 7\11 6th):
P1 - ms3 - ^ms3/vMs3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ^ms7/vMs7 - Ms7 - P8d
pentatonic plus ups and downs is doubly confusing!

11edo octatonic wide-fifth-based, fifthwards, no ^/v (3/2 maps to 7\11 = perfect 6th):
octatonic chain of 6ths: m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7
P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9
requires learning octatonic interval arithmetic and notation

13edo heptatonic narrow-fifth-based, fourthwards, 3 keys per #/b, (3/2 maps to 7\13 = perfect 5th):
P1 - m2 - m3 - vM2/^m3 - M2 - M3 - P4 - P5 - m6 - m7 - vM6/^m7 - M6 - M7 - P8
problematic because m3 = 2\13 is narrower than M2 = 4\13

(13edo undecatonic narrow-fifth-based, 3/2 maps to 7\13 = perfect 7th)

(13edo octatonic wide-fifth-based, fourthwards)

18edo heptatonic narrow-fifth-based, fourthwards, 2 keys per #/b (3/2 maps to 10\18 = perfect 5th)
P1 - vm2 - m2 - vM2 - M2/m3 - vM3 - M3 - ^M3 - P4 - ^P4/vP5 - P5 - vm6 - m6 - vM6 - M6/m7 - vM7 - M7 - ^M7 - P8
fourthwards plus ups and downs is doubly confusing!

18edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th)
P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10
requires learning nonotonic interval arithmetic and devising a nonotonic staff notation



=__**Summary of EDO notation**__= 

There are five EDO categories, based on the size of the fifth:
"Fifth-less" EDOs (8, 11, 13 and 18)
Fourthward EDOs (9, 16 and 23)
"Perfect" EDOs (7, 14, 21, 28 and 35)
Pentatonic EDOs (5, 10, 15, 20, 25 and 30)
"Sweet" EDOs (all others)
The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The last may or may not. 

To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. The C-C# interval is the augmented unison, and if they are not adjacent, ups and downs are required.The C-Db interval is the minor 2nd
The C-D interval is the major 2nd (P2 for perfect EDOs, m2 for fourthward EDOs).



===__**"Fifth-less" EDOs (8, 11, 13 and 18)**__=== 

**__8edo__:** (generator = 1\8 = perfect 2nd = 150¢)
D E F G * A B C D
D - E - F - G - G#/Ab - A -B - C - D
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8
seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc.

__**11edo**__: (generator = 3\11 = perfect 3rd)
D * E F * G A * B C * D
D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb

__**13edo**__**:** (generator = 2\13 = perfect 2nd)
D * E * F * G A * B * C * D
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8
secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#

**__18edo__:** (generator = 5\18 = perfect 3rd)
D * * E * F * * G * A * * B * C * * D
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb

===__**Alternate pentatonic notation for EDOs 8, 13 and 18**__=== 

All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.
Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# etc.

__**8edo**__**:** (generator = 5\8 = perfect 5thoid) C C#/Db D
D * E G * A C * D
D - D#/Eb - E - G - G#/Ab - A - C - C#/Db - D
P1 - ms3 - Ms3 - P4d - A4d/d5d - P5d - ms7 - Ms7 - P8d

__**13edo**__**:** (generator = 8\13 = perfect 5thoid) C C# Db D
D * * E * G * * A * C * * D
D - D# - Eb - E - E#/Gb - G - G# - Ab - A - A#/Cb - C - C# - Db - D
P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d

__**18edo**__**:** (generator = 11\18 = perfect 5thoid) C C# * Db D
D * * * E * * G * * * A * * C * * * D
D - D# - Dx/Ebb - Eb - E - E# - Gb - G - G# - Gx/Abb - Ab - A - A# - Cb - C - C# - Cx/Dbb - Db - D
P1 - A1 - ds3 - ms3 - Ms3 - As3 - d4d - P4d - A4d - AA4d/dd5d - d5d - P5d - A5d - ds7 - ms7 - Ms7 - As7 - d8d - P8d


===__Fourthward EDOs (9, 16 and 23)__=== 

All fourthwards EDOs use the same chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 etc.
F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.

**__9edo__:** C/Db C#/D 
D E * F G A B * C D
D - E - E#/Fb - F - G - A - B - B#/Cb - C - D
P1 - m2 - M2/m3 - M3 - P4 - P5 - m6 - M6/m7 - M7 - P8

**__16edo__:** C C#/Db D 
D * E * * F * G * A * B * * C * D
D - D#/Eb - E - E# - Fb - F F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C - C#/Db - D
P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8

**__23edo__:** C C# * Db D
D * * E * * * F * * G * * A * * B * * * C * * D
D - D# - Eb - E - E# - Ex/Fbb - Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B# - Bx/Cbb - Cb - C - C# - Db - D
P1 - A1 - d2 - m2 - M2 - A2/d3 - m3 - M3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - m6 - M6 - A6/d7 - m7 - M7 - A7 - d8 - P8


===__"Perfect" EDOs (7, 14, 21, 28 and 35)__=== 

All perfect EDOs use the same chain of fifths: P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.

**__7edo__:** C/Db C#/D
D E F G A B C D
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8
Because everything is perfect, the quality can be omitted: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8

**__14edo__:** 2 keys per sharp/flat: C/C# * Db/D 
D * E * F * G * A * B * C * D
D - D^/Ev - E - E/ Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^/Cv - C - C^/Dv - D
1 - ^1/v2 - 2 - ^2/v3 - 3 - ^3/v4 - 4 - ^4/v5 - 5 - ^5/v6 - 6 - ^6/v7 - 7 - ^7/v8 - 8

**__21edo__:** 3 keys per sharp/flat: C/C# * * Db/D
D * * E * * F * * G * * A * * B * * C * * D
D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C - C^ - Dv - D
1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8

**__28edo__:** 4 keys per sharp/flat: C/C# * * * Db/D
D * * * E * * * F * * * G * * * A * * * B * * * C * * * D
D - D^ - D^^/Evv - Ev - E - E^ - E^^/Fvv - Fv - F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A etc.
1 - ^1 - ^^1/vv2 - v2 - 2 - ^2 - ^^2/vv3 - v3 - 3 - ^3 - ^^3/vv4 - v4 - 4 - ^4 - ^^4/vv5 - v5 - 5 etc.

**__35edo__:** 5 keys per sharp/flat: C/C# * * * * Db/D
D * * * * E * * * * F * * * * G * * * * A * * * * B * * * * C * * * * D
D - D^ - D^^ - Evv - Ev - E - E^ - E^^ - Fvv - Fv - F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A etc.
1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc.


===__Pentatonic EDOs (5, 10, 15, 20, 25 and 30)__=== 

All pentatonic 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.
In all pentatonic EDOs, the minor 2nd = unison and the major 3rd = perfect fourth.

**__5edo__:** C/Db C#/D
D E/F G A B/C D
P1 - M2/m3 - P4 - P5 - M6/m7 - P8

**__10edo__:** 2 keys per sharp/flat: C/Db * C#/D
D * E/F * G * A * B/C * D
D - D^/Ev - E/F - F^/Gv - G - G^/Av - A - A^/Bv - B/C - C^/Dv - D
P1/m2 - ^m2/vM2 - M2/m3 - ^m3/vM3 - M3/P4 - ^P4/vP5 - P5/m6 - ^m6/vM6 - M6/m7 - ^m7/vM7 - P8

**__15edo__:** 3 keys per sharp/flat: C/Db [*] [*] C#/D
D * * E/F * * G * * A * * B/C * * D
D - D^ - Ev - E/F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B/C - C^ - Dv - D
P1/m2 - ^m2 - vM2 - M2/m3 - ^m3 - vM3 - M3/P4 - ^P4 - vP5 - P5/m6 - ^m6 - vM6 - M6/m7 - ^m7 - vM7 - P8

**__20edo__:** 4 keys per sharp/flat: C/Db * * * C#/D
D * * * E/F * * * G * * * A * * * B/C * * * D
D - D^ - D^^/Evv - Ev - E/F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A - A^ - A^^/Bvv - Bv - B/C - C^ - C^^/Dvv - Dv - D
P1/m2 - ^m2 - ^^m2/vvM2 - vM2 - M2/m3 - ^m3 - ^^m3/vvM3 - vM3 - M3/P4 - ^P4 - ^^P4/vvP5 - vP5 - P5/m6 - ^m6 - ^^m6/vvM6 - vM6 - M6/m7 - ^m7 - ^^m7/vvM7 - vM7 - P8

**__25edo__:** 5 keys per sharp/flat: C/Db * * * * C#/D
D * * * * E/F * * * * G * * * * A * * * * B/C * * * * D
D - D^ - D^^ - Evv - Ev - E/F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A - A^ - A^^ - Bvv - Bv - B/C - C^ - C^^ - Dvv - Dv - D
P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8

**__30edo__:** 6 keys per sharp/flat: C/Db * * * * * C#/D
D * * * * * E/F * * * * * G * * * * * A * * * * * B/C * * * * * D
D - D^ - D^^ - Evv - Ev - E/F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A - A^ - A^^ - Bvv - Bv - B/C - C^ - C^^ - Dvv - Dv - D
P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8


===__"Sweet" EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31, etc.)__=== 

All sweet 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.

**__12edo__:** sharp/flat = 1 key, no ups and downs: C C#/Db D
D * E F * G * A * B C * D
D - D#/Eb - E - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - C - C#/Db - D
P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8

**__17edo__:** sharp = 2 keys: C Db C# C
D * * E F * * G * * A * * B C * * D
D - D^/Eb - D#/Ev - Eb - E - F - F^/Gb - F#/Gv - G - G^/Ab - G#/Av - A - A^/Bb - A#/Bv - B - C - C^/Db - C#/Dv - D
P1 - m2 - ^m2/vM2 - M2 - m3 - ^m3/vM3 - M3 - P4 - ^P4/d5 - A4/vP5 - P5 - m6 - ^m6/vM6 - M6 - m7 ^m7/vM7 - M7 - P8

**__19edo__:** no ups and downs C C# Db D
D * * E * F * * G * * A * * B * C * * D
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D
P1 - A1/d2 - m2 - M2 - A2/d3 - m3 - M3 - A3/d4 - P4 - A4 - d5 - P5 - A5/d6 - m6 - M6 - A6/d7 - m7 - M7 - A7/d8 - P8

**__22edo__:** sharp = 3 keys: C Db * C# C
D * * * E F * * * G * * * A * * * B C * * * D
D - D^/Eb - D#v/Eb^ - D#/Ev - E - F - F^/Gb - F#v/Gb^ - F#/Gv - G - G^/Ab - G#v/Ab^ - G#/Av - A etc.
P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - ^P4/d5 - vA4/^d5 - A4/vP5 - P5 etc.

**__24edo__:** sharp = 2 keys: C * C#/Db * C
D * * * E * F * * * G * * * A * * * B * C * * * D
D - D^/Ebv - D#/Eb - D#^/Ev - E - E^/Fv - F - F^/Gbv - F#/Gb - F#^/Gv - G - G^/Abv - G#/Ab - G#^/Av - A etc.
P1 - ^P1/vm2 - m2 - ^m2/vM2 - M2 - ^M2/vm3 - m3 - ^m3/vM3 - M3 - ^M3/vP4 - P4 - ^P4/vd5 - A4/d5 - ^A4/vP5 - P5 etc.

Original HTML content:

<html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->&quot;Ups and Downs&quot; Notation</h1>
 <br />
Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works very well with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol &quot;^&quot; and the down symbol &quot;v&quot;. There's also the mid symbol &quot;~&quot; which undoes ups and downs.<br />
<br />
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 adds up to one EDO-step. So C# is right next to C, and your 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.<br />
<br />
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!<br />
<br />
The solution is to use the sharp symbol to mean &quot;raised by 7 fifths&quot;, and to use the up symbol &quot;^&quot; to mean &quot;sharpened by one EDO-step&quot;. 22-EDO can be written C-Db-Db^-Dv-D-Eb-Eb^-Ev-E-F etc. The notes are pronounced &quot;D-flat-up, D-down&quot;, 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.<br />
<br />
The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So in B, we get B-C-C^-C#v-C#-D-D^-D#v-D#-E etc.<br />
<br />
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.<br />
<br />
The basic pattern for 22-EDO is P1-m2-^m2-vM2-M2-m3-^m3-vM3-M3-P4-d5-^d5-vP5-P5 etc. That's pronounced &quot;upminor 2nd, downmajor 3rd&quot;, 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 mid notes always form a chain of fifths.<br />
<br />
You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See <a class="wiki_link" href="/Kite%27s%20color%20notation">Kite's color notation</a> for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.<br />
<br />
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^^ (&quot;B double-up&quot;). However avoid using both C# and Db, as the ascending Db-C# looks descending.<br />
<br />
<u><strong>Interval arithmetic</strong></u><br />
In ups and downs notation, as in conventional notation, the chain of fifths runs:<br />
Ebb-Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#-Fx-Cx etc.<br />
This chain can be expressed in relative notation:<br />
d2-d6-d3-d7-d4-d1-d5-m2-m6-m3-m7-P4-P1-P5-M2-M6-M3-M7-A4-A1-A5-A2-A6-A3-A7 etc.<br />
To name the interval between any two notes, superimpose one chain onto the other, with P1 lining up with the lower note. For example C-E = M3 because M3 means &quot;raised by 4 fifths&quot; and E is 4 fifths away from C. Likewise, C + M3 = E.<br />
C - G - D - A - E<br />
P1-P5-M2-M6-M3<br />
<br />
To add any two intervals, superimpose two copies of the relative chain. m3 + M2 = P4:<br />
m3-m7-P4-P1<br />
P1-P5-M2<br />
Line up the lower P1 with m3 and look for what lies above M2.<br />
<br />
22-EDO interval arithmetic works out very neatly. Ups and downs are just added in:<br />
C + M3 = E, C + vM3 = Ev, C^ + M3 = E^<br />
D-F# is a M3, Dv-F#v = M3<br />
M2 + m2 = m3, M2 + ^m2 = ^m3, vM2 + m2 = vm3<br />
<br />
There are some exceptions. Take this scale:<br />
C Db Db^ Dv D Eb Eb^ Ev E F Gb Gb^ Gv G Ab Ab^ Av A Bb Bb^ Bv B C<br />
Here's our fifths: C-G, Db-Ab, Db^-Ab^, Dv-Av, D-A, etc. Most fifths *look* like fifths and are easy to find. So do the 4ths. Our 4\22 maj 2nds are C-D, Db-Eb, Db^-Eb^, Dv-Ev, D-E, Eb-F, good until we reach Eb^-Gb, which looks like a min 3rd. Here's this scale's chain of 5ths:<br />
<br />
Gb^ Db^ Ab^ Eb^ Bb^ Gb Db Ab Eb Bb F C G D A E B Gv Dv Av Ev Bv<br />
<br />
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:<br />
<br />
Gx Dx Ax F# C# G# D# A# F C G D A Fb Cb Gb Db Ab Fbb Cbb Gbb Dbb<br />
C C# Dbb Db D D# Dx Fbb Fb F F# Gbb Gb G G# Gx Ab A A# Ax Cbb Cb C<br />
<br />
Now that's an awful lot of sharps and flats, but that does make a neat and tidy notation (except for the Gbb-Gx fifth). And it exists as an alternative, embedded within our standard notation, with a key signature with circled X's on the B and E spots.<br />
<br />
So the chain of fifths has a few spots to watch out for. You have to remember that B-something to G-something is sometimes a fifth, sometimes a sixth. A little tricky, but manageable. Analogous to 12-ET, where G# to Eb is a fifth that looks like a sixth.<br />
<br />
<u><strong>Staff Notation</strong></u><br />
For staff notation, just put an up or down to the left of the note and any standard accidental it might have. To write Db^ followed by Db in the same measure, use the mid sign: Db^ Db~. All 22 possible keys can be written out. The tonic is always a mid note, i.e. not up or down. Just as conventionally each black key produces both a sharp key and a flat key (Db major and C# minor), each of the 15 black keys of 22-EDO produces both, and there are 37 possible keys. The 2 most remote are Bbbb and F###, and triple-sharps and triple-flat keys seem rather extreme. Avoiding those, we have 35 possible tonics that run from Fbb to Bx. Some of the key signatures will have double-sharps or double-flats in them, or even triple-sharps.<br />
<br />
C: no sharps<br />
G: 1 sharp on F<br />
D: 2 sharps on F and C<br />
...<br />
C#: 7 sharps<br />
G#: 6 sharps, 1 double-sharp on F<br />
D#: 5 sharps, 2 double-sharps on F and C<br />
...<br />
B#: 2 sharps, 5 double-sharps on F, C, G, D and A<br />
...<br />
Bx: 2 double-sharps on E and B, 5 triple-sharps on F, C, G, D and A<br />
<br />
<u><strong>Other EDOs</strong></u><br />
This notation works for almost every EDO. 9, 11, 16, and 23 have weird interval arithmetic because of the narrow fifth, but they can be notated. 13 and 18 are best notated using the narrower of the 2 possible fifths, which makes them like 9, 11, 16 and 23. 8-EDO is hard. It works with pentatonic notation, if you don't mind learning pentatonic interval arithmetic. (Big if!)<br />
<br />
EDOs come in 5 categories, based on the size of the fifth:<br />
supersharp EDOs, with fifths wider than 720¢<br />
pentatonic EDOs, with a fifth = 720¢<br />
&quot;sweet&quot; EDOs, so-called because the fifth hits the &quot;sweet spot&quot; between 720¢ and 686¢<br />
&quot;perfect&quot; EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢<br />
superflat EDOs or Mavila EDOs, with a fifth less than 686¢<br />
<br />
This is in addition to the trivial EDOs, 1, 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. This section will cover sweet EDOs and the other categories will be covered in other sections.<br />
<br />
As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.<br />
<br />
<strong><u>17-EDO</u>:</strong> (2 keys per sharp/flat)<br />
Black and white keys: C * * D * * E F * * G * * A * * B C<br />
Relative notation: P1 m2 vM2 M2 m3 vM3 M3 P4 d5 vP5 P5 m6 vM6 M6 m7 vM7 M7 P8<br />
or with upminors instead of downmajors: P1 m2 ^m2 M2 m3 ^m3 M3 P4 d5 ^d5 P5 m6 ^m6 M6 m7 ^m7 M7 P8<br />
The d5 could instead be an A4: P4 ^P4 A4 P5 or P4 vA4 A4 P5<br />
Many other variations are possible, much freedom of spelling.<br />
In C, with downmajors: C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C<br />
In B, with upminors: B C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B<br />
One can't associate ups and downs with yellow and green because of the poor approximation of the 5-limit. However major = red or fifthward white, minor = blue or fourthward white, and downmajor = upminor = jade or amber.<br />
<br />
<strong><u>24-EDO</u>:</strong> (2 keys per sharp/flat)<br />
black and white keys: C * * * D * * * E * F * * * G * * * A * * * B * C<br />
Relative notation: P1 vm2 m2 vM2 M2 vm3 m3 vM3 M3 vP4 P4 ^P4 d5 vP5 P5 etc.<br />
Many alternate spellings available, for example vm3 = ^M2, vM3 = ^m3, ^P4 = vd5, etc.<br />
In C: C Dbv Db Dv D Ebv Eb Ev E Fv F F^ Gb Gv G etc.<br />
JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber.<br />
<br />
24-EDO is an example of a closed EDO. An EDO is closed if the keyspan of the fifth isn't coprime with the keyspan of the octave, and open if it is. 24-EDO has a fifth of 14 steps, and 14 isn't coprime with 24, because they have a common divisor of 2. 24-EDO is said to close at 12 (1/2 of 24), because the circle of fifths has only 12 notes. There are actually 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one:<br />
Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#<br />
Eb^-Bb^-F^-C^-G^-D^-A^-E^-B^-F#^-C#^-G#^<br />
Just as G# could be written as Ab, all the up notes could be written as down notes.<br />
<br />
In open EDOs, we can require that the tonic be a mid note. For example in 22-EDO, rather than using C#v as a tonic, we use B#. But closed EDOs force the use of tonics that are not a mid note. For example, the key of C^ runs:<br />
C^ Db Db^ D D^ Eb Eb^ E E^ F F^ F^^ Gb^ G G^ etc.<br />
<br />
<strong><u>31-EDO</u>:</strong> (2 keys per sharp/flat)<br />
Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C<br />
relative notation: P1 ^P1 vm2 m2 ^m2 M2 ^M2 vm3 m3 ^m3 M3 ^M3 vP4 P4 ^P4 A4 d5 ^d5 P5 etc.<br />
alternate spellings: A1=vm2, ^m2=vM2, ^M3=vP4, ^P4=vA4, etc.<br />
In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.<br />
JI associations: Perfect = white, major = yellow or fifthward white, minor = green or fourthward white, downminor = blue, upmajor = red, downmajor = upminor = jade or amber (same as 24-EDO).<br />
<br />
<strong><u>41-EDO</u>:</strong> (4 keys per sharp/flat)<br />
Black and white keys: C * * * * * * D * * * * * * E * * F * * * * * * G * * * * * * A * * * * * * B * * C<br />
P1 ^P1 vm2 m2 ^m2 ^^m2 vM2 M2 ^M2 vm3 m3 ^m3 ^^m3 vM3 M3 ^M3 vP4 P4 ^P4 ^^P4 d5 ^d5 vvP5 vP5 P5 etc.<br />
alternate spellings: A1=^m2, ^^m2=vvM2, ^M3=vP4, vA4=d5, A4=^d5, etc.<br />
In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.<br />
JI associations: Perfect = white, major = fifthward white, minor = fourthward white, downmajor = yellow, upminor = green, downminor = blue, upmajor = red, double-downmajor = double-upminor = jade or amber.<br />
<br />
<strong><u>53-EDO</u>:</strong> (5 keys per sharp/flat)<br />
Black and white keys: C * * * * * * * * D * * * * * * * * E * * * F * * * * * * * * G * * * * * * * * A * * * * * * * * B * * * C<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Naming Chords"></a><!-- ws:end:WikiTextHeadingRule:2 --><u>Naming Chords</u></h1>
 <br />
Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths (or more generally, the chain of generators). Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.<br />
<br />
There are 3 special cases to be addressed. The first is when the edo's 5th is narrower than 4\7, as in 16edo. Major is defined as wider than minor, so major is not fifthwards but fourthwards:<br />
<br />
The fourthwards chain of fifths in superflat aka Mavila EDOs (3/2 maps to less than 4\7):<br />
M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 - A1 etc.<br />
F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.<br />
16edo: P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8<br />
16edo: C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C<br />
<br />
In other words, sharp/flat, major/minor, and aug/dim all retain their melodic meaning but the chain-of-fifths meaning is reversed. Perfect and natural are unaffected. Interval arithmetic in fourthwards edos is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.<br />
M2 + M2 --&gt; m2 + m2 = dim3 --&gt; aug3<br />
D to F# --&gt; D to Fb = dim3 --&gt; aug3<br />
Eb + m3 --&gt; E# + M3 = G## --&gt; Gbb<br />
<br />
The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. That's because the scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect.<br />
<br />
The chain of fifths in &quot;perfect&quot; EDOs (3/2 maps to 4\7):<br />
P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.<br />
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.<br />
21edo: P1 - ^P1 - vP2 - P2 - ^P2 - vP3 - P3 - ^P3 - vP4 - P4 - ^P4 - vP5 - P5 - ^P5 - vP6 - P6 - ^P6 - vP7 - P7 - ^P7 - vP8 - P8<br />
Because everything is perfect, the quality can be omitted:<br />
21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8<br />
21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C<br />
Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But they are two different concepts that must be kept distinct. In this redefined notation, B - F# isn't a perfect fifth because it's really B - F^.<br />
<br />
The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. Such EDOs are dealt with below.<br />
<br />
Chord names are based entirely on the ups/downs interval names, not on JI ratios. This avoids identifying one EDOstep with multiple ratios, as happens in 22edo when 0-7-18 implies 4:5:7 but 0-9-18 implies 9:12:16. 18\22 is neither 7/4 nor 16/9, it's 18\22!<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Naming Chords-22edo chord names"></a><!-- ws:end:WikiTextHeadingRule:4 --><u>22edo chord names</u></h2>
 <br />
Let's review the 22edo interval names:<br />
0\22 = P1<br />
1\22 = m2<br />
2\22 = ^m2<br />
3\22 = vM2<br />
4\22 = M2<br />
5\22 = m3<br />
6\22 = ^m3<br />
7\22 = vM3<br />
8\22 = M3<br />
9\22 = P4<br />
10\22 = ^P4, d5<br />
11\22 = vA4, ^d5<br />
12\22 = A4, vP5<br />
13\22 = P5<br />
14\22 = m6<br />
15\22 = ^m6<br />
16\22 = vM6<br />
17\22 = M6<br />
18\22 = m7<br />
19\22 = ^m7<br />
20\22 = vM7<br />
21\22 = M7<br />
22\22 = P8<br />
<br />
These are pronounced &quot;downmajor second&quot;, &quot;upminor third&quot;, etc. For 4ths and 5ths, &quot;perfect&quot; is implied and can be omitted: ^P4 = &quot;up-four&quot; and vP5 = &quot;down-five&quot;. In larger edos there may be &quot;down-octave&quot;, &quot;up-unison&quot;, etc.<br />
<br />
0-7-13-18 in C is &quot;C,vM,m7&quot;, pronounced &quot;C downmajor, minor seventh&quot;. The comma between the C and the down symbol is needed because Cv is a note, and &quot;Cv,M,m7&quot; is a different chord. That chord is pronounced &quot;C down, major, minor 7th&quot;, so one has to &quot;speak the comma&quot;. The extra comma isn't needed when there's no ups or downs immediately after the note name, e.g. Cm.<br />
<br />
The conventional chord naming system uses a lot of &quot;shorthand&quot; like dom7 for M3,m7 and min6 for m3,M6. This causes problems in 22edo where there are so many choices for the 3rd, the 6th, the 7th and the 9th. For example, min6 could mean m3,vM6 = approximate 6:7:9:10 chord, or it could mean ^m3,M6 = approximate 1/1-6/5-3/2-12/7 chord. Larger edos would present even greater problems. Furthermore there's some ambiguity in the shorthand, e.g. in 12edo, both 0-3-6 and 0-3-6-9 are called dim chords.<br />
<br />
Thus the shorthand should be largely abandoned and all the components of the chord should be explicitly spelled out, with a few exceptions: 1) The root, obviously. 2) The perfect 5th is assumed present unless otherwise specified. Thus 0-7-18 is C,vM,m7,-5 and 0-6-11 is C,^m,^d5. 3) The 3rd is also assumed to be present, and is implied by a quality with no degree. Thus 0-7-13 is C,vM. 4) The 3rd isn't spelled out if the 6th or 7th has the same quality as the 3rd. Thus 0-7-13-16 is C,vM6, but 0-7-13-17 is C,vM,M6. Thirdless chords: 0-13-18 is either Cm7,-3 or C5,m7.<br />
<br />
The 6th, the 7th, the 9th, the 11th, etc. are explicitly written out, including their qualities. Thus the 9th isn't assumed to be major, and the presence of a 9th doesn't imply the presence of a 7th.<br />
<br />
Sus chords: as usual, &quot;sus&quot; means the 3rd is replaced by the named note, a 2nd or 4th. &quot;Sus4&quot; implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-perfect 4th, etc. Some larger edos would have susv4, susvv4, etc. &quot;Sus2&quot; implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See 16edo below for an exception.<br />
<br />
0-5-13 = m<br />
0-6-13 = ^m<br />
0-7-13 = vM<br />
0-8-13 = M<br />
0-9-13 = sus4<br />
0-10-13 = sus^4<br />
0-4-13 = sus2<br />
0-3-13 = susvM2<br />
<br />
0-5-11 = m,^d5<br />
0-5-12 = m,vP5 (or possibly m,A4)<br />
<br />
0-5-11-14 = m6,^d5<br />
0-6-11-15 = ^m6,^d5<br />
0-7-13-16 = vM6<br />
0-8-13-17 = M6<br />
<br />
0-5-13-18 = m7<br />
0-6-13-19 = ^m7<br />
0-7-13-20 = vM7<br />
0-8-13-21 = M7<br />
<br />
0-5-13-16 = m,vM6<br />
0-8-13-19 = M,^m7<br />
0-7-13-18-26 = vM,m7,M9<br />
0-7-13-18-26-32 = vM,m7,M9,^P11<br />
<br />
You can write out chord progressions using the ups/downs notation for note names. Here's the first 4 chords of Paul Erlich's 22edo composition Tibia:<br />
G,vM7,-5 = &quot;G downmajor seven, no five&quot;&quot;<br />
Eb^,vM,M9 = &quot;E flat up, downmajor, major nine&quot;<br />
Gm7,-5 (no comma needed) = &quot;G minor seven, no five&quot;<br />
A,vM,m7 = &quot;A downmajor, minor seven&quot;<br />
<br />
To use relative notation, first write out all possible 22edo chord roots relatively. This is equivalent to the interval notation with Roman numerals substituted for Arabic, # for aug, and b for minor. Dim from perfect is b, but dim from minor is bb. Enharmonic equivalents like ^I = bII are used in certain chord progressions like Im - ^IIIM - ^VIIM - ^IVm - ^Im.<br />
I ^I/bII v#I/^bII #I/vII II ^II/bIII v#II/^bIII #II/vIII III IV ^IV/bV v#IV/^bV #IV/vV V ^V/bVI v#V/^bVI #V/vVI VI ^VI/bVII v#VI/^bVII #VI/vVII VII/vI<br />
These are pronounced &quot;down-two&quot;, &quot;up-flat-three&quot;, &quot;down-sharp-four&quot;, etc.<br />
<br />
Here's the Tibia chords. No comma is needed after the root because ups and downs are always leading, never trailing.<br />
IvM7,-5 = &quot;one downmajor seven, no five&quot;<br />
^bVIvM,M9 = &quot;up-flat six downmajor, major nine&quot;<br />
Im7,-5 = &quot;one minor seven, no five&quot;<br />
IIvM,m7 = &quot;two downmajor, minor seven&quot;<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Naming Chords-Chord names in other EDOs"></a><!-- ws:end:WikiTextHeadingRule:6 --><u>Chord names in other EDOs</u></h2>
 <br />
15edo: 3 keys per #/b, so ups and downs are needed.<br />
keyboard/fretboard: D * * E/F * * G * * A * * B/C * * D<br />
(the chain of fifths is always centered on D)<br />
chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 M3/P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7 P8<br />
chord roots: I ^bII vII II/bIII ^bIII vIII III/IV ^IV vV V ^bVI vVI VI/bVII ^bVII vVII<br />
0-3-9 = m or sus2<br />
0-4-9 = ^m<br />
0-5-9 = vM<br />
0-6-9 = M or sus4<br />
0-5-9-12 = vM,m7<br />
<br />
16edo: D * E * * F * G * A * B * * C * D, 1 key per #/b, ups and downs not needed. # is fourthward.<br />
chord components: P1 d2 m2 M2 m3 M3 A3/d4 P4 A4/d5 P5 d6 m6 M6/d7 m7 M7 A7 P8<br />
chord roots: I #I/bbII bII II bIII III #III/vIV IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI<br />
0-3-9 = sus2<br />
0-4-9 = m<br />
0-5-9 = M<br />
0-5-10 = M,A5 (the conventional aug chord)<br />
0-6-9 = A (aug 3rd, perfect 5th)<br />
0-7-9 = sus4<br />
0-4-8-12 = m,d5,d7 (the conventional dim tetrad)<br />
<br />
17edo: D * * E F * * G * * A * * B C * * D, 2 keys per #/b.<br />
chord components: P1 m2 ^m2/vM2 M2 m3 ^m3/vM3 M3 P4 ^P4/d5 A4/vP5 P5 m6 ^m6/vM6 M6 m7 ^m7/vM7 M7 P8<br />
chord roots: I v#I/bII #I/vII II bIII vIII III IV ^IV/bV #IV/vV V #V/bVI vVI VI bVII vVII VII<br />
0-4-10 = m<br />
0-5-10 = ^m or vM (probably choose vM over ^m whenever possible)<br />
0-6-10 = M<br />
0-7-10 = sus4<br />
0-4-10-14 = m7<br />
0-5-10-15 = vM7<br />
0-6-10-16 = M7<br />
Alternatively, one could replace downmajor with n = neutral or somesuch.<br />
<br />
19edo: D * * E * F * * G * * A * * B * C * * D, ups and downs not needed.<br />
chord components: P1 d2 m2 M2 d3 m3 M3 A3 P4 A4 d5 P5 d6 m6 M6 d7 m7 M7 A7 P8<br />
chord roots: I v#I/bII #I/vII II bIII vIII III IV ^IV/bV #IV/vV V #V/bVI vVI VI bVII vVII VII<br />
The possibility of a dim 3rd or an aug 3rd changes the meaning of &quot;dim chord&quot; and &quot;aug chord&quot;.<br />
0-4-11 = d (dim 3rd, perfect 5th, not a conventional dim chord)<br />
0-4-10 = d,d5<br />
0-5-11 = m<br />
0-5-10 = m,d5 (conventional dim chord)<br />
0-6-11 = M<br />
0-7-11 = A (aug 3rd, perfect 5th, not a conventional aug chord)<br />
0-6-12 = M,A5 (conventional aug chord)<br />
0-7-12 = A,A5<br />
0-8-11 = sus4<br />
<br />
21edo: D * * E * * F * * G * * A * * B * * C * * D, zero keys per #/b.<br />
chord components: P1 ^P1/vvP2 vP2 P2 ^P2 vP3 P3 ^P3 vP4 P4 ^P4 vP5 P5 ^P5 vP6 P6 ^P6 vP7 P7 ^P7 ^^P7/vP8<br />
Because everything is perfect, the quality can be omitted.<br />
chord components: 1 ^1/vv2 v2 2 ^2 v3 3 ^3 v4 4 ^4 v5 5 ^5 v6 6 ^6 v7 7 ^7 ^^7/v8<br />
chord roots: I ^I vII II ^II vIII III vIII vIV IV ^IV vV V ^V vVI VI ^VI vVII VII ^VII vI<br />
Quality can also be omitted in the chord names if we use the mid symbol &quot;~&quot;:<br />
0-3-12 = sus2<br />
0-4-12 = vv or sus^2<br />
0-5-12 = v (a down chord, e.g. C,v = &quot;C down&quot;)<br />
0-6-12 = ~ (e.g. D,~ = &quot;D mid&quot;)<br />
0-7-12 = ^ (e.g. Eb,^ = &quot;E flat up&quot;)<br />
0-8-12 = ^^ or susv4<br />
0-9-12 = sus4<br />
0-6-11 = ~,v5<br />
<br />
0-7-12-19 = ^7<br />
0-7-12-18 = ^,~7<br />
0-7-12-17 = ^,v7<br />
0-7-12-16 = ^6<br />
0-7-12-15 = ^,~6<br />
0-7-12-14 = ^,v6<br />
<br />
24edo: D * * * E * F * * * G * * * A * * * B * C * * * D, 2 keys per #/b.<br />
chord components: P1 vm2 m2 vM2 M2 vm3 m3 vM3 M3 ^M3/vP4 P4 ^P4 A4/d5 vP5 P5 vm6 m6 vM6 M6 ^M6/vm7 m7 vM7 M7 ^M7<br />
chord roots: I v#I/vbII #I/bII vII II vbIII bIII vIII III ^III/vIV IV ^IV #IV/bV vV V ^#V/vbVI bVI vVI VI ^VI/vbVII bVII vVII VII ^VII/vI<br />
0-5-14 = vm<br />
0-6-14 = m<br />
0-7-14 = vM<br />
0-8-14 = M<br />
0-9-14 = ^M<br />
0-10-14 = sus4<br />
<br />
31edo: D * * * * E * * F * * * * G * * * * A etc. 2 keys per #/b.<br />
P1 ^P1 vm2 m2 vM2 M2 ^M2 vm3 m3 vM3 M3 ^M3 vP4 P4 ^P4 A4 d5 vP5 P5 etc.<br />
I ^I vbII bII vII II ^II vbIII bIII vIII III ^III vIV IV ^IV #IV bV vV V etc.<br />
0-7-18 = vm<br />
0-8-18 = m<br />
0-9-18 = vM<br />
0-10-18 = M<br />
0-11-18 = ^M<br />
0-12-18 = sus-v4<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Naming Chords-Cross-EDO considerations"></a><!-- ws:end:WikiTextHeadingRule:8 --><strong><u>Cross-EDO considerations</u></strong></h2>
 <br />
In 22edo, the major chord is 0¢-436¢-709¢. In 19edo, it's 0¢-379¢-695¢. The two chords sound quite different, because &quot;major 3rd&quot; is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.<br />
<br />
The name &quot;major&quot; refers not to the sound but to the function of the chord. If you want to pump an 81/80 with a I - VIm - IIm - V - I progression, you can do that in any edo, as long as you use major and minor chords.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Naming Chords-EDOs with an inaccurate 3/2"></a><!-- ws:end:WikiTextHeadingRule:10 --><u>EDOs with an inaccurate 3/2</u></h2>
 <br />
Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in this diagram, they are the ones to the left of the central line in the light blue region, plus the ones to the right of the central line in the orange region. The ones on the left edge of the blue region are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:26:&lt;img src=&quot;/file/view/The%205th%20of%20EDOs%205-53.png/558356515/640x802/The%205th%20of%20EDOs%205-53.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 802px; width: 640px;&quot; /&gt; --><img src="/file/view/The%205th%20of%20EDOs%205-53.png/558356515/640x802/The%205th%20of%20EDOs%205-53.png" alt="The 5th of EDOs 5-53.png" title="The 5th of EDOs 5-53.png" style="height: 802px; width: 640px;" /><!-- ws:end:WikiTextLocalImageRule:26 --><br />
<br />
<br />
There are two strategies for notating these &quot;oddball&quot; EDOs, besides heptatonic fifth-based notation with ups and downs. One is to switch from heptatonic notation to some other type. The orange region contains edos for which pentatonic notation is a natural fit, in the sense that no ups or downs are needed. This includes 8edo, 13edo and 18edo.<br />
<br />
The other approach is to use some interval other than the fifth to generate the notation. Above I said 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C.<br />
<br />
<br />
<u><strong>Theoretical alternatives for 8edo, 11edo, 13edo and 18edo</strong></u><br />
<br />
<br />
8edo octatonic (every note is a generator)<br />
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9<br />
<br />
11edo heptatonic narrow-fifth-based, fourthwards with ^/v, 2 keys per #/b (3/2 maps to 6\11 5th):<br />
P1 - m2 - vM2/m3 - M2/^m3 - M3 - P4 - P5 - m6 - vM6/m7 - M6/^m7 - M7 - P8<br />
problematic because m3 = 2\11 is narrower than M2 = 3\11<br />
<br />
11edo nonotonic narrow-fifth-based, fourthwards with no ups and downs (3/2 maps to 6\11 6th):<br />
nonotonic fourthwards chain of sixths:<br />
M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9 - d5 etc.<br />
P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8<br />
requires learning nonotonic interval arithmetic and devising a nonotonic staff notation<br />
<br />
11edo pentatonic wide-fifth-based, fifthwards using ^/v, 2 keys per #/b (3/2 maps to 7\11 6th):<br />
P1 - ms3 - ^ms3/vMs3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ^ms7/vMs7 - Ms7 - P8d<br />
pentatonic plus ups and downs is doubly confusing!<br />
<br />
11edo octatonic wide-fifth-based, fifthwards, no ^/v (3/2 maps to 7\11 = perfect 6th):<br />
octatonic chain of 6ths: m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7<br />
P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9<br />
requires learning octatonic interval arithmetic and notation<br />
<br />
13edo heptatonic narrow-fifth-based, fourthwards, 3 keys per #/b, (3/2 maps to 7\13 = perfect 5th):<br />
P1 - m2 - m3 - vM2/^m3 - M2 - M3 - P4 - P5 - m6 - m7 - vM6/^m7 - M6 - M7 - P8<br />
problematic because m3 = 2\13 is narrower than M2 = 4\13<br />
<br />
(13edo undecatonic narrow-fifth-based, 3/2 maps to 7\13 = perfect 7th)<br />
<br />
(13edo octatonic wide-fifth-based, fourthwards)<br />
<br />
18edo heptatonic narrow-fifth-based, fourthwards, 2 keys per #/b (3/2 maps to 10\18 = perfect 5th)<br />
P1 - vm2 - m2 - vM2 - M2/m3 - vM3 - M3 - ^M3 - P4 - ^P4/vP5 - P5 - vm6 - m6 - vM6 - M6/m7 - vM7 - M7 - ^M7 - P8<br />
fourthwards plus ups and downs is doubly confusing!<br />
<br />
18edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th)<br />
P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10<br />
requires learning nonotonic interval arithmetic and devising a nonotonic staff notation<br />
<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Summary of EDO notation"></a><!-- ws:end:WikiTextHeadingRule:12 --><u><strong>Summary of EDO notation</strong></u></h1>
 <br />
There are five EDO categories, based on the size of the fifth:<br />
&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)<br />
Fourthward EDOs (9, 16 and 23)<br />
&quot;Perfect&quot; EDOs (7, 14, 21, 28 and 35)<br />
Pentatonic EDOs (5, 10, 15, 20, 25 and 30)<br />
&quot;Sweet&quot; EDOs (all others)<br />
The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The last may or may not. <br />
<br />
To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. The C-C# interval is the augmented unison, and if they are not adjacent, ups and downs are required.The C-Db interval is the minor 2nd<br />
The C-D interval is the major 2nd (P2 for perfect EDOs, m2 for fourthward EDOs).<br />
<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Summary of EDO notation--&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)"></a><!-- ws:end:WikiTextHeadingRule:14 --><u><strong>&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)</strong></u></h3>
 <br />
<strong><u>8edo</u>:</strong> (generator = 1\8 = perfect 2nd = 150¢)<br />
D E F G * A B C D<br />
D - E - F - G - G#/Ab - A -B - C - D<br />
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8<br />
seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.<br />
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc.<br />
<br />
<u><strong>11edo</strong></u>: (generator = 3\11 = perfect 3rd)<br />
D * E F * G A * B C * D<br />
D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D<br />
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8<br />
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.<br />
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br />
<br />
<u><strong>13edo</strong></u><strong>:</strong> (generator = 2\13 = perfect 2nd)<br />
D * E * F * G A * B * C * D<br />
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D<br />
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8<br />
secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.<br />
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#<br />
<br />
<strong><u>18edo</u>:</strong> (generator = 5\18 = perfect 3rd)<br />
D * * E * F * * G * A * * B * C * * D<br />
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D<br />
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8<br />
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.<br />
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Summary of EDO notation--Alternate pentatonic notation for EDOs 8, 13 and 18"></a><!-- ws:end:WikiTextHeadingRule:16 --><u><strong>Alternate pentatonic notation for EDOs 8, 13 and 18</strong></u></h3>
 <br />
All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.<br />
Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# etc.<br />
<br />
<u><strong>8edo</strong></u><strong>:</strong> (generator = 5\8 = perfect 5thoid) C C#/Db D<br />
D * E G * A C * D<br />
D - D#/Eb - E - G - G#/Ab - A - C - C#/Db - D<br />
P1 - ms3 - Ms3 - P4d - A4d/d5d - P5d - ms7 - Ms7 - P8d<br />
<br />
<u><strong>13edo</strong></u><strong>:</strong> (generator = 8\13 = perfect 5thoid) C C# Db D<br />
D * * E * G * * A * C * * D<br />
D - D# - Eb - E - E#/Gb - G - G# - Ab - A - A#/Cb - C - C# - Db - D<br />
P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d<br />
<br />
<u><strong>18edo</strong></u><strong>:</strong> (generator = 11\18 = perfect 5thoid) C C# * Db D<br />
D * * * E * * G * * * A * * C * * * D<br />
D - D# - Dx/Ebb - Eb - E - E# - Gb - G - G# - Gx/Abb - Ab - A - A# - Cb - C - C# - Cx/Dbb - Db - D<br />
P1 - A1 - ds3 - ms3 - Ms3 - As3 - d4d - P4d - A4d - AA4d/dd5d - d5d - P5d - A5d - ds7 - ms7 - Ms7 - As7 - d8d - P8d<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"></a><!-- ws:end:WikiTextHeadingRule:18 --><u>Fourthward EDOs (9, 16 and 23)</u></h3>
 <br />
All fourthwards EDOs use the same chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 etc.<br />
F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.<br />
<br />
<strong><u>9edo</u>:</strong> C/Db C#/D <br />
D E * F G A B * C D<br />
D - E - E#/Fb - F - G - A - B - B#/Cb - C - D<br />
P1 - m2 - M2/m3 - M3 - P4 - P5 - m6 - M6/m7 - M7 - P8<br />
<br />
<strong><u>16edo</u>:</strong> C C#/Db D <br />
D * E * * F * G * A * B * * C * D<br />
D - D#/Eb - E - E# - Fb - F F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C - C#/Db - D<br />
P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8<br />
<br />
<strong><u>23edo</u>:</strong> C C# * Db D<br />
D * * E * * * F * * G * * A * * B * * * C * * D<br />
D - D# - Eb - E - E# - Ex/Fbb - Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B# - Bx/Cbb - Cb - C - C# - Db - D<br />
P1 - A1 - d2 - m2 - M2 - A2/d3 - m3 - M3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - m6 - M6 - A6/d7 - m7 - M7 - A7 - d8 - P8<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Summary of EDO notation--&quot;Perfect&quot; EDOs (7, 14, 21, 28 and 35)"></a><!-- ws:end:WikiTextHeadingRule:20 --><u>&quot;Perfect&quot; EDOs (7, 14, 21, 28 and 35)</u></h3>
 <br />
All perfect EDOs use the same chain of fifths: P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.<br />
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.<br />
<br />
<strong><u>7edo</u>:</strong> C/Db C#/D<br />
D E F G A B C D<br />
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8<br />
Because everything is perfect, the quality can be omitted: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8<br />
<br />
<strong><u>14edo</u>:</strong> 2 keys per sharp/flat: C/C# * Db/D <br />
D * E * F * G * A * B * C * D<br />
D - D^/Ev - E - E/ Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^/Cv - C - C^/Dv - D<br />
1 - ^1/v2 - 2 - ^2/v3 - 3 - ^3/v4 - 4 - ^4/v5 - 5 - ^5/v6 - 6 - ^6/v7 - 7 - ^7/v8 - 8<br />
<br />
<strong><u>21edo</u>:</strong> 3 keys per sharp/flat: C/C# * * Db/D<br />
D * * E * * F * * G * * A * * B * * C * * D<br />
D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C - C^ - Dv - D<br />
1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8<br />
<br />
<strong><u>28edo</u>:</strong> 4 keys per sharp/flat: C/C# * * * Db/D<br />
D * * * E * * * F * * * G * * * A * * * B * * * C * * * D<br />
D - D^ - D^^/Evv - Ev - E - E^ - E^^/Fvv - Fv - F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A etc.<br />
1 - ^1 - ^^1/vv2 - v2 - 2 - ^2 - ^^2/vv3 - v3 - 3 - ^3 - ^^3/vv4 - v4 - 4 - ^4 - ^^4/vv5 - v5 - 5 etc.<br />
<br />
<strong><u>35edo</u>:</strong> 5 keys per sharp/flat: C/C# * * * * Db/D<br />
D * * * * E * * * * F * * * * G * * * * A * * * * B * * * * C * * * * D<br />
D - D^ - D^^ - Evv - Ev - E - E^ - E^^ - Fvv - Fv - F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A etc.<br />
1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Summary of EDO notation--Pentatonic EDOs (5, 10, 15, 20, 25 and 30)"></a><!-- ws:end:WikiTextHeadingRule:22 --><u>Pentatonic EDOs (5, 10, 15, 20, 25 and 30)</u></h3>
 <br />
All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br />
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.<br />
In all pentatonic EDOs, the minor 2nd = unison and the major 3rd = perfect fourth.<br />
<br />
<strong><u>5edo</u>:</strong> C/Db C#/D<br />
D E/F G A B/C D<br />
P1 - M2/m3 - P4 - P5 - M6/m7 - P8<br />
<br />
<strong><u>10edo</u>:</strong> 2 keys per sharp/flat: C/Db * C#/D<br />
D * E/F * G * A * B/C * D<br />
D - D^/Ev - E/F - F^/Gv - G - G^/Av - A - A^/Bv - B/C - C^/Dv - D<br />
P1/m2 - ^m2/vM2 - M2/m3 - ^m3/vM3 - M3/P4 - ^P4/vP5 - P5/m6 - ^m6/vM6 - M6/m7 - ^m7/vM7 - P8<br />
<br />
<strong><u>15edo</u>:</strong> 3 keys per sharp/flat: C/Db [*] [*] C#/D<br />
D * * E/F * * G * * A * * B/C * * D<br />
D - D^ - Ev - E/F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B/C - C^ - Dv - D<br />
P1/m2 - ^m2 - vM2 - M2/m3 - ^m3 - vM3 - M3/P4 - ^P4 - vP5 - P5/m6 - ^m6 - vM6 - M6/m7 - ^m7 - vM7 - P8<br />
<br />
<strong><u>20edo</u>:</strong> 4 keys per sharp/flat: C/Db * * * C#/D<br />
D * * * E/F * * * G * * * A * * * B/C * * * D<br />
D - D^ - D^^/Evv - Ev - E/F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A - A^ - A^^/Bvv - Bv - B/C - C^ - C^^/Dvv - Dv - D<br />
P1/m2 - ^m2 - ^^m2/vvM2 - vM2 - M2/m3 - ^m3 - ^^m3/vvM3 - vM3 - M3/P4 - ^P4 - ^^P4/vvP5 - vP5 - P5/m6 - ^m6 - ^^m6/vvM6 - vM6 - M6/m7 - ^m7 - ^^m7/vvM7 - vM7 - P8<br />
<br />
<strong><u>25edo</u>:</strong> 5 keys per sharp/flat: C/Db * * * * C#/D<br />
D * * * * E/F * * * * G * * * * A * * * * B/C * * * * D<br />
D - D^ - D^^ - Evv - Ev - E/F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A - A^ - A^^ - Bvv - Bv - B/C - C^ - C^^ - Dvv - Dv - D<br />
P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8<br />
<br />
<strong><u>30edo</u>:</strong> 6 keys per sharp/flat: C/Db * * * * * C#/D<br />
D * * * * * E/F * * * * * G * * * * * A * * * * * B/C * * * * * D<br />
D - D^ - D^^ - Evv - Ev - E/F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A - A^ - A^^ - Bvv - Bv - B/C - C^ - C^^ - Dvv - Dv - D<br />
P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Summary of EDO notation--&quot;Sweet&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31, etc.)"></a><!-- ws:end:WikiTextHeadingRule:24 --><u>&quot;Sweet&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31, etc.)</u></h3>
 <br />
All sweet EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br />
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.<br />
<br />
<strong><u>12edo</u>:</strong> sharp/flat = 1 key, no ups and downs: C C#/Db D<br />
D * E F * G * A * B C * D<br />
D - D#/Eb - E - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - C - C#/Db - D<br />
P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8<br />
<br />
<strong><u>17edo</u>:</strong> sharp = 2 keys: C Db C# C<br />
D * * E F * * G * * A * * B C * * D<br />
D - D^/Eb - D#/Ev - Eb - E - F - F^/Gb - F#/Gv - G - G^/Ab - G#/Av - A - A^/Bb - A#/Bv - B - C - C^/Db - C#/Dv - D<br />
P1 - m2 - ^m2/vM2 - M2 - m3 - ^m3/vM3 - M3 - P4 - ^P4/d5 - A4/vP5 - P5 - m6 - ^m6/vM6 - M6 - m7 ^m7/vM7 - M7 - P8<br />
<br />
<strong><u>19edo</u>:</strong> no ups and downs C C# Db D<br />
D * * E * F * * G * * A * * B * C * * D<br />
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D<br />
P1 - A1/d2 - m2 - M2 - A2/d3 - m3 - M3 - A3/d4 - P4 - A4 - d5 - P5 - A5/d6 - m6 - M6 - A6/d7 - m7 - M7 - A7/d8 - P8<br />
<br />
<strong><u>22edo</u>:</strong> sharp = 3 keys: C Db * C# C<br />
D * * * E F * * * G * * * A * * * B C * * * D<br />
D - D^/Eb - D#v/Eb^ - D#/Ev - E - F - F^/Gb - F#v/Gb^ - F#/Gv - G - G^/Ab - G#v/Ab^ - G#/Av - A etc.<br />
P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - ^P4/d5 - vA4/^d5 - A4/vP5 - P5 etc.<br />
<br />
<strong><u>24edo</u>:</strong> sharp = 2 keys: C * C#/Db * C<br />
D * * * E * F * * * G * * * A * * * B * C * * * D<br />
D - D^/Ebv - D#/Eb - D#^/Ev - E - E^/Fv - F - F^/Gbv - F#/Gb - F#^/Gv - G - G^/Abv - G#/Ab - G#^/Av - A etc.<br />
P1 - ^P1/vm2 - m2 - ^m2/vM2 - M2 - ^M2/vm3 - m3 - ^m3/vM3 - M3 - ^M3/vP4 - P4 - ^P4/vd5 - A4/d5 - ^A4/vP5 - P5 etc.</body></html>