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 2016-06-04 01:45:10 UTC.
- The original revision id was 584785481.
- 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 optional mid symbol "~" which undoes ups and downs (see the Cancelling section).
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 is a major 2nd that is spelled as a downminor 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:
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
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 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.
__**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~. Or just write 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. Double-sharps and double-flats are avoided, as are Fb, Cb, B# and E#. Also Fb^, Cb^, B#v and E#v. This is so that the three black keys between, say, C and D are only notated as some version of C or D, never as some version of B or E. To achieve this, ups and downs are allowed in tonic names and key signatures.
major keys: C, Db, Db^, Dv, D, Eb, Eb^, Ev, E, F, F^, Gb^, Gv, G, Ab, Ab^, Av, A, Bb, Bb^, Bv, B
minor keys: C, C^, C#v, C#, D, D^, Eb^, Ev, E, F, F^, F#v, F#, G, G^, G#v, G#, A, Bb, Bb^, Bv, B
Major keys are almost entirely natural, down, flat or upflat. The one exception is F^ major, needed because Gb major would use Cb. Likewise, minor keys are mostly natural, up, sharp or downsharp. Exceptions: Ev minor for D# minor, and Bv minor for A# minor, to avoid E#. In addition, three minor keys are named to match their relative major. This isn't as strict a rule, and the other names may be used as alternatives. Thus Bb minor and Bb^ minor are preferred over A^ minor and A#v minor, to match their relative majors Db major and Db^ major. Also Eb^ minor is preferred over D#v minor, to match its relative major Gb^ major. These two keys<span style="line-height: 1.5;"> break the rule for naming black keys because they have a Cb^.There is unfortunately no way to notate these keys and follow the rule!</span>
<span style="line-height: 1.5;">Key signatures: </span>
<span style="line-height: 1.5;">C major: all natural</span>
<span style="line-height: 1.5;">Db major: B, E, A, D and G are flat</span>
<span style="line-height: 1.5;">Db^ major: </span>B, E, A, D and G are upflat, C and F are up
Dv major: F and C are downsharp, G, D, A, E and B are down
D major: F and C are sharp
Eb major: B, E, and A are flat
etc.
C minor: B, E and A are flat
C^ minor: B, E and A are upflat, D, G, C and F are up
C#v minor: F, C, G and D are downsharp, A, E and B are down
C# minor: F, C, G and D are sharp
D minor: B is flat
D^ minor: B is upflat, E, A, D, G, C and F are up
etc.
__**Other EDOs**__
EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:
"fifth-less" 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¢
fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢
[[image:The fifth of EDOs 5-53.png width="800" height="1035"]]
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.
The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same "generation" occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call **kites**. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.
Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs.
The diagram only shows part of the full Stern-Brocot tree, which extends sideways from 0¢ (0\1) to 1200¢ (1\1). The full tree contains four pentatonic kites and six heptatonic kites. The blue kite is the 4\7 kite; the others are the 1\7, 2\7 3\7, 5\7 and 6\7 kites. The 3\7 kite is the mirror image of the 4\7 kite, 5\7 mirrors 2\7, and 6\7 mirrors 1\7. The 4\7 kite contains EDOs best notated by heptatonic notation generated by the fifth (i.e., to sharpen or augment means to add seven fifths, octave-reduced). The octave inverse of the generator is also a generator, thus fourth-generated is equivalent to fifth-generated, and the 3\7 kite contains the exact same EDOs as the 4\7 kite. The 2\7 kite is for notation generated by thirds, and the 1\7 kite is for notation generated by seconds.
Every EDO larger than 7edo will appear on only one of these three mirror-pairs of kites. The only exception is perfect EDOs, which appear on the spine of every heptatonic kite. This means that every non-perfect EDO above 7edo has a "natural" (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.
This section will cover sweet EDOs and the other categories will be covered in later 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 (generator) 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 alternatively be written as Ab, all the up notes could alternatively 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. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. 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 this would be confusing because 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-fourth" and vP5 = "down-fifth". In larger edos there may be "down-octave", "up-unison", etc.
There are some alternate names. The dim7 of a dim7 chord would be three EDOsteps below a min7 = 15\22 = ^m6. 14\22 could be written as m6 or as vd7. However double-ups and double-downs are to be avoided in 22edo. In larger edos, they would be necessary. Thus 7\22 would never be written ^^m3.
0-8-13 in C has C E & G, and is written "C" and pronounced "C" or "C major".
0-7-13 = C Ev G is written "C.v", spoken as "C downmajor" or possibly "C dot down".
The period is needed because "Cv", spoken as "C down", is a note, not a chord.
0-6-13 = C Eb^ G is "C.^m", "C upminor"
0-5-13 = C Eb G is "Cm", "C minor"
The period isn't needed here because there's no ups or downs immediately after the note name.
0-8-13-18 = C E G Bb is "C7", "C seven", a standard C7 chord with a M3 and a m7.
0-7-13-18 = C Ev G Bb is "C7(v3)", "C seven, down third". The altered note or notes are in parentheses.
0-8-13-21 = C E G B is "CM7", "C major seven".
0-7-13-20 = C Ev G Bv is "C.vM7", "C downmajor seven". The down symbol affects both the 3rd and the 7th.
Often the root of a chord will not be a mid note. The root in the next two examples is Cv.
0-8-13-21 = Cv Ev Gv Bv is "Cv.M7", "C down, major seven"
To distinguish between C.vM7 and Cv.M7, one has to pronounce the period with a small pause.
0-7-13-20 = Cv Evv Gv Bvv is "Cv.vM7", "C down, downmajor seven".
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-fourth, 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 the fourthwards EDOs below for an exception.
"Aug" and "dim" chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. An A3,P5 chord is A3 = "aug three chord", not "aug chord", to distinguish it from the conventional aug chord M3,A5. That chord is still called an aug chord. Likewise d3,P5 is a "dim three chord", and m3,d5 is a "dim" chord.
0-3-13 = C Dv G = Csusv2
0-4-13 = C D G = Csus2
0-5-13 = C Eb G = Cm
0-6-13 = C Eb^ G = C.^m
0-7-13 = C Ev G = C.vM
0-8-13 = C E G = C
0-9-13 = C F G = Csus4
0-10-13 = C F^ G = Csus^4
0-5-10 = C Eb Gb = Cdim
0-5-11 = C Eb Gb^ = Cdim(^5)
0-5-12 = C Eb Gv = Cm(v5)
0-5-10-15 = C Eb Gb Bbb = Cdim7
0-5-11-14 = C Eb Gb^ Bbbv = Cdim7(^5,v7)
0-6-11-15 = C Eb^ Gb^ Bbb = Cdim7(^3,^5)
0-6-11-16 = C Eb^ Gb^ Bbb^ = C.^dim7 (the up symbol applies to m3, d5 and d7)
0-5-13-17 = C Eb G A = Cm6
Sometimes doubled ups/downs are unavoidable:
0-6-12-15 = C Eb^ Gv Avv = Cm6(^3,v5,vv6), or C Eb^ Gb^^ Bbb = Cdim7(^3,^^5)
0-7-13-16 = C Ev G Av = C.vM6 (the down symbol applies to both the 3rd and the 6th)
0-8-13-17 = C E G A = C6
0-7-13-16 = C Ev G Av = C.v6
0-5-13-18 = C Eb G Bb = Cm7
0-6-13-19 = C Eb^ G Bb^ = C.^m7
0-7-13-20 = C Ev G Bv = C.vM7
0-8-13-21 = C E G B = CM7
0-5-13-16 = C Eb G Av = Cm6(v6)
0-8-13-19 = C E G Bb^ = C7(^7)
0-7-13-18-26 = C Ev G Bb D = C9(v3)
0-7-13-18-26-32 = C Ev G Bb D F^ = C11(v3,^11)
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(no5) = "G downmajor seven, no five"
Eb^.v(add9) = "E flat up, downmajor, add nine"
C7sus4 = "C seven, sus four"
A7(v3) = "A seven, down three"
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 might be used in certain chord progressions like Im - ^IIIM - ^VIIM - ^IVm - ^Im.
I
^I or bII
v#I or ^bII
#I or vII
II
^II or bIII
v#II or ^bIII
#II or vIII
III
IV
^IV or bV
v#IV or ^bV
#IV or vV
V
^V or bVI
v#V or ^bVI
#V or vVI
VI
^VI or bVII
v#VI or ^bVII
#VI or vVII
VII or vI
These are pronounced "down-two", "up-flat-three", "down-sharp-four", etc.
Here's the Tibia chords. Periods are never needed after the root in relative notation because ups and downs are always leading, never trailing.
IvM7(no5) = "one downmajor seven, no five"
^bVIv(add9) = "up-flat six downmajor, add nine"
IV7sus4 = "four seven, sus four"
II7(v3) = "two seven, down three"
[[image:Tibia in G with ^v, rygb 1.jpg width="800" height="1035"]]
==[[image:Tibia in G with ^v, rygb 2.jpg width="800" height="957"]]==
== ==
== ==
==__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
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 possibly sus2)
0-4-9 = ^m
0-5-9 = vM
0-6-9 = M (or possibly sus4)
0-5-9-12 = 7(v3)
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 P4 A4/d5 P5 d6 m6 M6/d7 m7 M7 A7
chord roots: I #I/bbII bII II bIII III #III/bIV IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI
0-3-9 = sus2
0-4-9 = m
0-5-9 = M (in practice, no symbol, as in "C" for the C chord)
0-5-10 = aug (the conventional aug chord)
0-6-9 = (A3) (aug 3rd, perfect 5th)
0-7-9 = sus4
0-4-8-12 = dim7 (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
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
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
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
0-4-11 = (d3) (dim 3rd, perfect 5th)
0-4-10 = dim(d3)
0-5-11 = m
0-5-10 = dim (conventional dim chord)
0-6-11 = M
0-7-11 = (A3) (aug 3rd, perfect 5th)
0-6-12 = aug (conventional aug chord)
0-7-12 = aug(A3)
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 dot down")
0-6-12 = ~ (a mid chord, e.g. D.~ = "D dot mid")
0-7-12 = ^ (an up chord, e.g. E.^ = "E dot up")
0-8-12 = ^^ or susv4
0-9-12 = sus4
0-6-11 = ~(v5)
0-6-12-18 = 7
0-7-12-18 = 7(^3)
0-7-12-19 = ^7
0-7-12-17 = ^(v7)
0-6-12-15 = 6
0-7-12-15 = 6(^3)
0-7-12-16 = ^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 = ^m or vM or ~
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 = ^m or vM or ~
0-10-18 = M
0-11-18 = ^M
0-12-18 = susv4
==**__Cross-EDO considerations__**==
In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 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 play a I - VIm - IIm - V - I progression without pitch shifts or tonic drift, you can do that in any edo, as long as you use major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22edo, the chord that you need sounds like a red chord.
In other words, I - VIm - IIm - V - I in JI implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in certain EDOs. The notation tells you which ones. If 22edo's downmajor chord 0-7-13 = 0¢-382¢-709¢ were called "major", you wouldn't know that it doesn't work in that progression.
Another example: I7 - bVII7 - IV7 - I7. To make this work, the 7th in the I7 chord must be a minor 7th. in 22edo, that 7th sounds blue. In 19edo, it sounds green. If you want a blue 7th in 19edo, you have to use the downminor 7th, which will cause shifts or drifts in the progression.
==__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 the above 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.
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 notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But these EDOs don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like.
__**Theoretical alternatives for 8edo, 11edo, 13edo and 18edo**__
8edo octatonic (every note is a generator)
D E F G H A B C D
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9
requires learning octatonic interval arithmetic and staff notation
11edo heptatonic narrow-fifth-based, fourthwards, # is ^^ (3/2 maps to 6\11 perfect 5th):
D E * * F G A B * * C D
fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7
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, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):
nonotonic fifthwards chain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9
P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8
requires learning nonotonic interval arithmetic and staff notation
11edo pentatonic wide-fifth-based, fifthwards, # is ^^ (3/2 maps to 7\11 6th):
D * * E G * * A C * * D
pentatonic fifthwards chain of fifthoids: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7
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, sharp = ^^^ (3/2 maps to 7\13 perfect 5th):
D E * * * F G A B * * * C D
fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7
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, fourthwards, 3/2 maps to 7\13 = perfect 7th
undecatonic sixthwards chain of sevenths:
M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11
P1 - m2 - M2/m3 - M3/m4 - M4/m5 - M5 - P6 - P7 - m8 - M8/m9 - M9/m10 - M10/m11 - M11 - P12
requires learning undecatonic interval arithmetic and notation
13edo octatonic wide-fifth-based, fourthwards, 3/2 maps to 8\13 = perfect 6th
octotonic chain of sixths: 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
18edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^ (3/2 maps to 10\18 perfect 5th)
D * E * * * F * G * A * B * * * C * D
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 plus closed is triply 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 staff notation
__**Alternate notation for other edos:**__
23edo pentatonic wide-fifth-based, fifthwards, 3/2 maps to 14\23 = perfect fifthoid:
D * * * * E * * * G * * * * A * * * C * * * * D
35edo heptatonic wide-fifth-based, sharp = seven ups, 3/2 maps to 21\35 = perfect fifth:
D * * * * * * E/F * * * * * * G * * * * * * A * * * * * * B/C * * * * * * D
42edo heptatonic narrow-fifth-based, sharp = six ups, 3/2 maps to 24\42 = perfect fifth:
D * * * * * E * * * * * F * * * * * G * * * * * A * * * * * B * * * * * C * * * * * D
=__**Summary of EDO notation**__=
Besides the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO, 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 sweet EDOs may or may not.
To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. Examples:
C C#/Db D (12edo)
C Db C# D (17edo)
C C# Db D (19edo)
C C# _ Db D (26edo)
C _ C# Db _ D (31edo)
The scale fragment concisely conveys the "flavor" of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For fourthward EDOs, C-Db = d2 and C-D = m2. D# is included for these EDOs because C-D# is a M2 just like E-F. For fifthless EDOs, the scale fragment isn't as helpful because you can't deduce the entire keyboard layout from it.
Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: "wide-fifth" 35edo (which uses 21\35 as a fifth) has a different scale fragment than "narrow-fifth" 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute M2 for m2.)
||= 5edo ||= pentatonic ||= ||= C/Db ||= C#/D ||= ||= ||= ||= ||= ||= ||= ||= ||
||= 7edo ||= perfect ||= ||= C/C# ||= Db/D ||= ||= ||= ||= ||= ||= ||= ||= ||
||= 8edo ||= fifthless ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= 9edo ||= fourthward ||= ||= C/Db ||= C#/D ||= D# ||= ||= ||= ||= ||= ||= ||= ||
||= 10edo ||= pentatonic ||= ||= C/Db ||= * ||= C#/D ||= ||= ||= ||= ||= ||= ||= ||
||= 11edo ||= fifthless ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= 12edo ||= sweet ||= ||= C ||= C#/Db ||= D ||= ||= ||= ||= ||= ||= ||= ||
||= 13edo ||= fifthless ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= 14edo ||= perfect ||= ||= C/C# ||= * ||= Db/D ||= ||= ||= ||= ||= ||= ||= ||
||= 15edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= ||= ||
||= 16edo ||= fourthward ||= ||= C ||= C#/Db ||= D ||= D# ||= ||= ||= ||= ||= ||= ||
||= 17edo ||= sweet ||= ||= C ||= Db ||= C# ||= D ||= ||= ||= ||= ||= ||= ||
||= 18edo ||= fifthless ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= 19edo ||= sweet ||= ||= C ||= C# ||= Db ||= D ||= ||= ||= ||= ||= ||= ||
||= 20edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= ||
||= 21edo ||= perfect ||= ||= C/C# ||= * ||= * ||= Db/D ||= ||= ||= ||= ||= ||= ||
||= 22edo ||= sweet ||= ||= C ||= Db ||= * ||= C# ||= D ||= ||= ||= ||= ||= ||
||= 23edo ||= fourthward ||= ||= C ||= C# ||= Db ||= D ||= D# ||= ||= ||= ||= ||= ||
||= 24edo ||= sweet ||= ||= C ||= * ||= C#/Db ||= * ||= D ||= ||= ||= ||= ||= ||
||= 25edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||= ||
||= 26edo ||= sweet ||= ||= C ||= C# ||= * ||= Db ||= D ||= ||= ||= ||= ||= ||
||= 27edo ||= sweet ||= ||= C ||= Db ||= * ||= * ||= C# ||= D ||= ||= ||= ||= ||
||= 28edo ||= perfect ||= ||= C/C# ||= * ||= * ||= * ||= Db/D ||= ||= ||= ||= ||= ||
||= 29edo ||= sweet ||= ||= C ||= * ||= Db ||= C# ||= * ||= D ||= ||= ||= ||= ||
||= 30edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||
||= 31edo ||= sweet ||= ||= C ||= * ||= C# ||= Db ||= * ||= D ||= ||= ||= ||= ||
||= 32edo ||= sweet ||= ||= C ||= Db ||= * ||= * ||= * ||= C# ||= D ||= ||= ||= ||
||= 33edo ||= sweet ||= ||= C ||= C# ||= * ||= * ||= Db ||= D ||= ||= ||= ||= ||
||= 34edo ||= sweet ||= ||= C ||= * ||= Db ||= * ||= C# ||= * ||= D ||= ||= ||= ||
||= 35edo ||= perfect ||= ||= C/C# ||= * ||= * ||= * ||= * ||= Db/D ||= ||= ||= ||= ||
||= 36edo ||= sweet ||= ||= C ||= * ||= * ||= C#/Db ||= * ||= * ||= D ||= ||= ||= ||
||= 37edo ||= sweet ||= ||= C ||= Db ||= * ||= * ||= * ||= * ||= C# ||= D ||= ||= ||
||= 38edo ||= sweet ||= ||= C ||= * ||= C# ||= * ||= Db ||= * ||= D ||= ||= ||= ||
||= 39edo ||= sweet ||= ||= C ||= * ||= Db ||= * ||= * ||= C# ||= * ||= D ||= ||= ||
||= 40edo ||= sweet ||= ||= C ||= C# ||= * ||= * ||= * ||= Db ||= D ||= ||= ||= ||
||= 41edo ||= sweet ||= ||= C ||= * ||= * ||= Db ||= C# ||= * ||= * ||= D ||= ||= ||
||= 42edo ||= sweet ||= ||= C ||= Db ||= * ||= * ||= * ||= * ||= * ||= C# ||= D ||= ||
||= 43edo ||= sweet ||= ||= C ||= * ||= * ||= C# ||= Db ||= * ||= * ||= D ||= ||= ||
||= 44ddo ||= sweet ||= ||= C ||= * ||= Db ||= * ||= * ||= * ||= C# ||= * ||= D ||= ||
||= 45edo ||= sweet ||= ||= C ||= * ||= Db ||= * ||= * ||= C# ||= * ||= D ||= ||= ||
||= 46edo ||= sweet ||= ||= C ||= * ||= * ||= Db ||= * ||= C# ||= * ||= * ||= D ||= ||
||= 47edo ||= sweet ||= ||= C ||= C# ||= * ||= * ||= * ||= * ||= Db ||= D ||= ||= ||
||= 48edo ||= sweet ||= ||= C ||= * ||= * ||= * ||= C#/Db ||= * ||= * ||= * ||= D ||= ||
||= 49edo ||= sweet ||= ||= C ||= * ||= Db ||= * ||= * ||= * ||= * ||= C# ||= * ||= D ||
||= 50edo ||= sweet ||= ||= C ||= * ||= * ||= C# ||= * ||= Db ||= * ||= * ||= D ||= ||
||= 51edo ||= sweet ||= ||= C ||= * ||= * ||= Db ||= * ||= * ||= C# ||= * ||= * ||= D ||
||= 52edo ||= sweet ||= ||= C ||= * ||= C# ||= * ||= * ||= * ||= Db ||= * ||= D ||= ||
||= 53edo ||= sweet ||= ||= C ||= * ||= * ||= * ||= Db ||= C# ||= * ||= * ||= * ||= D ||
===__**"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 = the unison and the major 3rd = the 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
===__Alternative pentatonic notation for pentatonic EDOs:__===
Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.
C# - G# - D# - A# - E# - C - G - D - A - E - Cb - Gb - Db - Ab - Eb etc.
All intervals are perfect, so quality can be omitted.
__**5edo**__**:** zero keys per sharp/flat: C/C# Db/D
D E G A C D
1 - s3 - 4d - 5d - s7 - 8d
__**10edo**__**:** zero keys per sharp/flat: C/C# * Db/D
D * E * G * A * C * D
D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D
1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d
__**15edo**__**:** zero keys per sharp/flat: C/C# * * Db/D
D * * E * * G * * A * * C * * D
D - D^ - Ev - E - E^ - Gv - G - G^ - Av - A - A^ - Cv - C - C^ - Dv - D
1 - ^1 - vs3 - s3 - ^s3 - v4d - 4d - ^4d - v5d - 5d - ^5d - vs7 - s7 - ^s7 - v8d - 8d
etc.
===__"Sweet" EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)__===
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
perfect = white, major = red, yellow and fifthward white, minor = green, blue and fourthwards white
**__17edo__:** sharp = 2 keys: C Db C# D
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
perfect = white, major = yellow and fifthward white, minor = green and fourthward white, aug/dim = red/blue.
**__22edo__:** sharp = 3 keys: C Db * C# D
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 * D
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.
==__Ups and downs solfege__==
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:
The initial consonant remains as before: D, R, M, F, S, L and T
The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb
The vowels are pronounced as in Spanish or Italian
The pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u
The optional 2nd vowel indicates up/down: a = ^^^, e = ^, i = ^^, o = v, u = vv
The 2nd vowel is separated from the first by either a glottal stop, an "h", a "w", or a "y"
Thus C#v is Deo, pronounced as De'o or Deho or Dewo or Deyo.
This suffices for many but not all edos, as some require triple sharps or quadruple ups.
Fixed-do solfege:
Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb
Da = C, Da'e = C^, Da'i = C^^, Da'o = Cv, Da'u = Cvv, Da'a = C^^^
De = C#, De'e = C#^, De'i = C#^^, De'o = C#v, De'u = C#vv, De'a = C#^^^
etc.
Moveable-do solfege:
The 2nd vowel is as before. The 1st vowel's meaning depends on the interval.
Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim
Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1
Da'e = ^P1, Da'i = ^^P1, Da'o = vP1, Da'u = vvP1, Da'a = ^^^P1
etc.
Imperfect intervals (2nd, 3rd, 6th and 7th): a = major, e = aug, i = double-aug, o = minor, u = dim
Ra = M2, Re = A2, Ri = AA2, Ro = m2, Ru = d2
Ra'e = ^M2, Ra'i = ^^M2, Ra'o = vM2, Ra'u = vvM2, Ra'a = ^^^M2
etc.
==__Rank-2 Notation__==
Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)
Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:
...Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# ...
Fifth-generated rank-2 tunings can be written without ups and downs in any EDO on either side of the 4\7 kite:
12-tone genchain Eb to G#: C C# D Eb E F F# G G# A Bb B C
12-tone genchain Ab to C#: C C# D Eb E F F# G Ab A Bb B C
12-tone genchain C to E#: C C# D D# E E# F# G G# A A# B C
In __12edo__, C# and Db are identical, but in __12-tone__, they may not be, and usually aren't.
19-tone genchain Gb to B#: C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B B# C
19-tone genchain Fb to A#: C C# Db D D# Eb E Fb F F# Gb G G# Ab A A# Bb B Cb C
19-tone genchain F to Ax: C C# Cx D D# Dx E E# F F# Fx G G# Gx A A# Ax B B# C
Fourthward frameworks can be notated with the # sign meaning harmonically sharp but melodically flat:
16-tone genchain Db to A#: C D# D Db E Eb F# F G# G A# A Ab B Bb C# C
16-tone genchain Fb to C#: C Cb D Db E Eb F# F Fb G Gb A Ab B Bb C# C
For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. Each node in the Stern-Brocot EDO chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the fifth's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).
All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone. These two are easily notated without ups and downs:
5-tone genchain C to E: C D E G A C
5-tone genchain F to A: C D F G A C
7-tone genchain C to F#: C D E F# G A B C
7-tone genchain Bb to E: C D E F G A Bb C
All fifthless frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd is a descending interval.
If the sharp's keyspan is 1 or -1, as with 12-tone, 19-tone, and all fourthward frameworks, ups and downs aren't needed to notate rank-2. They also aren't needed for 5-tone and 7-tone. Since perfect, pentatonic and fifthless frameworks are incompatible, we need only consider sweet frameworks, excluding those that lie on the side of the heptatonic kite and those that lie on the spine of any kite.
To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a **keyspan** (always +1) but also a **genspan**, which indicates how many steps forward or backwards along the generator chain, or **genchain**, one must travel to find the interval.
For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is C^ + m2 = (C + m2)^, exactly equivalent to Db^. However, C^^ is not equivalent to Dvv, even though they occuy the same key on the keyboard, just as C# may not equal Db in 12-tone.
The usual genchain note names will run out of order when mapped to the 22-tone framework. For example, we might have C Db B# C# D. So ups and downs are used to provide alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D.
Here's part of the 22-tone genchain. There are more than 22 notes because the genchain is theoretically infinite:
||= ||= ||= Fb ||= Eb^ ||= D^^ ||
||= ||= ||= Cb ||= Bb^ ||= A^^ ||
||= ||= ||= Gb ||= F^ ||= E^^ ||
||= ||= ||= Db ||= C^ ||= B^^ ||
||= ||= ||= Ab ||= G^ ||= F#^^ ||
||= ||= Fbv ||= Eb ||= D^ ||= C#^^ ||
||= ||= Cbv ||= Bb ||= A^ ||= G#^^ ||
||= ||= Gbv ||= F ||= E^ ||= A#^^ ||
||= ||= Dbv ||= C ||= B^ ||= E#^^ ||
||= ||= Abv ||= G ||= F#^ ||= B#^^ ||
||= Fbvv ||= Ebv ||= D ||= C#^ ||= ||
||= Cbvv ||= Bbv ||= A ||= G#^ ||= ||
||= Gbvv ||= Fv ||= E ||= D#^ ||= ||
||= Dbvv ||= Cv ||= B ||= A#^ ||= ||
||= Abvv ||= Gv ||= F# ||= E#^ ||= ||
||= Ebvv ||= Dv ||= C# ||= B#^ ||= ||
||= Bbvv ||= Av ||= G# ||= ||= ||
||= Fvv ||= Ev ||= D# ||= ||= ||
||= Cvv ||= Bv ||= A# ||= ||= ||
||= Gvv ||= F#v ||= E# ||= ||= ||
||= Dvv ||= C#v ||= B# ||= ||= ||
||= Avv ||= G#v ||= Fx ||= ||= ||
||= Evv ||= D#v ||= Cx ||= ||= ||
||= Bvv ||= A#v ||= Gx ||= ||= ||
||= F#vv ||= E#v ||= Dx ||= ||= ||
||= C#vv ||= B#v ||= Ax ||= ||= ||
|| || || etc. || || ||
Here's each note, with alternate tunings for the black keys, with the keyspan and genspan both measured from C:
||= keyspan ||= genspan ||= note ||= genspan ||= note ||
||= 0 ||= 0 ||= C ||= ||= ||
||= 1 ||= -5 ||= Db = C^ ||= +17 ||= C#vv = Dv3 ||
||= 2 ||= -10 ||= Db^ = C^^ ||= +12 ||= C#v = Dvv ||
||= 3 ||= -15 ||= Db^^ = C^3 ||= +7 ||= C# = Dv ||
||= 4 ||= +2 ||= D ||= ||= ||
||= 5 ||= -3 ||= Eb = D^ ||= +19 ||= D#vv = Ev3 ||
||= 6 ||= -8 ||= Eb^ = D^^ ||= +14 ||= D#v = Evv ||
||= 7 ||= -13 ||= Eb^^ = D^3 ||= +9 ||= D# = Ev ||
||= 8 ||= +4 ||= E ||= ||= ||
||= 9 ||= -1 ||= F ||= ||= ||
||= 10 ||= -6 ||= Gb = F^ ||= +16 ||= F#vv = Gv3 ||
||= 11 ||= -11 ||= Gb^ = F^^ ||= +11 ||= F#v = Gvv ||
||= 12 ||= -16 ||= Gb^^ = F^3 ||= +6 ||= F# = Gv ||
||= 13 ||= +1 ||= G ||= ||= ||
||= 14 ||= -4 ||= Ab = G^ ||= +18 ||= G#vv = Av3 ||
||= 15 ||= -9 ||= Ab^ = G^^ ||= +13 ||= G#v = Avv ||
||= 16 ||= -14 ||= Ab^^ = G^3 ||= +8 ||= G# = Av ||
||= 17 ||= +3 ||= A ||= ||= ||
||= 18 ||= -2 ||= Bb = A^ ||= +20 ||= A#vv = Bv3 ||
||= 19 ||= -7 ||= Bb^ = A^^ ||= +15 ||= A#v = Bvv ||
||= 20 ||= -12 ||= Bb^^ = A^3 ||= +10 ||= A# = Bv ||
||= 21 ||= +5 ||= B ||= ||= ||
||= 22 ||= 0 ||= C ||= ||= ||
"^3" means three ups. Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.
The genspan for the up symbol in 22-tone is calculated from the keyspans:
K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)
K(#) = X, K(b) = -X (X = keyspan of the sharp symbol, i.e., how many keys wide it is. For 22-tone, X = 3)
K(#vX) = K(#) + X * K(v) = 0 (going up X keys using a sharp, then going down X keys using X downs, must cancel out)
"#vX" means one sharp plus X downs. Zero keyspans in the genchain only occur on every Nth step for a N-tone framework. E.g., 12-tone keyspans:
|| genchain of fifths || C || G || D || A || E || B || F# || C# || G# || D# || A# || E# || B# ||
|| genspan from C || 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 ||
|| 12-tone keyspan from C || 0 || 7 || 2 || 9 || 4 || 11 || 6 || 1 || 8 || 3 || 10 || 5 || 0 ||
B#, genspan 12, has a zero keyspan, as does Dbb, genspan -12, and A###, genspan 24. Thus the final equation means that the genspan resulting from going up a sharp and down X downs must be zero, N, -N, 2N, -2N, etc. Thus this genspan mod N must be zero.
G(#) = 7 (by definition, the sharp's genspan = 7, since we're assuming heptatonic notation)
G(#vX) = G(#) + X * G(v) = G(#) - X * G(^) = 7 - X * G(^)
G(#vX) mod N = 0, thus G(#vX) = i * N for some integer i
7 - X * G(^) = i * N
G(^) = - (i * N - 7) / X
For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. Thus i = 1, G(^) = -5, and ^ = min 2nd.
For 17-tone, X = 2, i = 1, G(^) = -5, and ^ = min 2nd
For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd.
||= 5edo ||= pentatonic ||= K(#) || || ||
||= 17edo ||= sweet ||= || -5 || min 2nd ||
||= 19edo ||= sweet ||= || || ||
||= 22edo ||= sweet ||= || -5 || min 2nd ||
||= 26edo ||= sweet ||= || || ||
||= 27edo ||= sweet ||= || || ||
||= 29edo ||= sweet ||= || || ||
||= 31edo ||= sweet ||= || -12 || dim 2nd ||
||= 32edo ||= sweet ||= || || ||
||= 33edo ||= sweet ||= || || ||
||= 34edo ||= sweet ||= || || ||
||= 37edo ||= sweet ||= || || ||
||= 38edo ||= sweet ||= || || ||
||= 39edo ||= sweet ||= || || ||
||= 40edo ||= sweet ||= || || ||
||= 41edo ||= sweet ||= || || ||
||= 42edo ||= sweet ||= || || ||
||= 43edo ||= sweet ||= || || ||
||= 44ddo ||= sweet ||= || || ||
||= 45edo ||= sweet ||= || || ||
||= 46edo ||= sweet ||= || || ||
||= 47edo ||= sweet ||= || || ||
||= 49edo ||= sweet ||= || || ||
||= 50edo ||= sweet ||= || || ||
||= 51edo ||= sweet ||= || || ||
||= 52edo ||= sweet ||= || || ||
||= 53edo ||= sweet ||= || || ||Original HTML content:
<html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x"Ups and Downs" Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->"Ups and Downs" 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 "^" and the down symbol "v". There's also the optional mid symbol "~" which undoes ups and downs (see the Cancelling section).<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 "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.<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 "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.<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^^ ("B double-up"). 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 "raised by 4 fifths" 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 is a major 2nd that is spelled as a downminor 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 />
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<br />
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<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 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.<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~. Or just write 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. Double-sharps and double-flats are avoided, as are Fb, Cb, B# and E#. Also Fb^, Cb^, B#v and E#v. This is so that the three black keys between, say, C and D are only notated as some version of C or D, never as some version of B or E. To achieve this, ups and downs are allowed in tonic names and key signatures.<br />
<br />
major keys: C, Db, Db^, Dv, D, Eb, Eb^, Ev, E, F, F^, Gb^, Gv, G, Ab, Ab^, Av, A, Bb, Bb^, Bv, B<br />
minor keys: C, C^, C#v, C#, D, D^, Eb^, Ev, E, F, F^, F#v, F#, G, G^, G#v, G#, A, Bb, Bb^, Bv, B<br />
<br />
Major keys are almost entirely natural, down, flat or upflat. The one exception is F^ major, needed because Gb major would use Cb. Likewise, minor keys are mostly natural, up, sharp or downsharp. Exceptions: Ev minor for D# minor, and Bv minor for A# minor, to avoid E#. In addition, three minor keys are named to match their relative major. This isn't as strict a rule, and the other names may be used as alternatives. Thus Bb minor and Bb^ minor are preferred over A^ minor and A#v minor, to match their relative majors Db major and Db^ major. Also Eb^ minor is preferred over D#v minor, to match its relative major Gb^ major. These two keys<span style="line-height: 1.5;"> break the rule for naming black keys because they have a Cb^.There is unfortunately no way to notate these keys and follow the rule!</span><br />
<br />
<br />
<span style="line-height: 1.5;">Key signatures: </span><br />
<span style="line-height: 1.5;">C major: all natural</span><br />
<span style="line-height: 1.5;">Db major: B, E, A, D and G are flat</span><br />
<span style="line-height: 1.5;">Db^ major: </span>B, E, A, D and G are upflat, C and F are up<br />
Dv major: F and C are downsharp, G, D, A, E and B are down<br />
D major: F and C are sharp<br />
Eb major: B, E, and A are flat<br />
etc.<br />
<br />
C minor: B, E and A are flat<br />
C^ minor: B, E and A are upflat, D, G, C and F are up<br />
C#v minor: F, C, G and D are downsharp, A, E and B are down<br />
C# minor: F, C, G and D are sharp<br />
D minor: B is flat<br />
D^ minor: B is upflat, E, A, D, G, C and F are up<br />
etc.<br />
<br />
<br />
<u><strong>Other EDOs</strong></u><br />
<br />
EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:<br />
"fifth-less" EDOs, with fifths wider than 720¢<br />
pentatonic EDOs, with a fifth = 720¢<br />
"sweet" EDOs, so-called because the fifth hits the "sweet spot" between 720¢ and 686¢<br />
"perfect" EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢<br />
fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:2418:<img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png" alt="" title="" style="height: 1035px; width: 800px;" /> --><img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png" alt="The fifth of EDOs 5-53.png" title="The fifth of EDOs 5-53.png" style="height: 1035px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:2418 --><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.<br />
<br />
The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same "generation" occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call <strong>kites</strong>. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.<br />
<br />
Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs.<br />
<br />
The diagram only shows part of the full Stern-Brocot tree, which extends sideways from 0¢ (0\1) to 1200¢ (1\1). The full tree contains four pentatonic kites and six heptatonic kites. The blue kite is the 4\7 kite; the others are the 1\7, 2\7 3\7, 5\7 and 6\7 kites. The 3\7 kite is the mirror image of the 4\7 kite, 5\7 mirrors 2\7, and 6\7 mirrors 1\7. The 4\7 kite contains EDOs best notated by heptatonic notation generated by the fifth (i.e., to sharpen or augment means to add seven fifths, octave-reduced). The octave inverse of the generator is also a generator, thus fourth-generated is equivalent to fifth-generated, and the 3\7 kite contains the exact same EDOs as the 4\7 kite. The 2\7 kite is for notation generated by thirds, and the 1\7 kite is for notation generated by seconds.<br />
<br />
Every EDO larger than 7edo will appear on only one of these three mirror-pairs of kites. The only exception is perfect EDOs, which appear on the spine of every heptatonic kite. This means that every non-perfect EDO above 7edo has a "natural" (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.<br />
<br />
This section will cover sweet EDOs and the other categories will be covered in later 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 (generator) 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 alternatively be written as Ab, all the up notes could alternatively 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:<h1> --><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 --> m2 + m2 = dim3 --> aug3<br />
D to F# --> D to Fb = dim3 --> aug3<br />
Eb + m3 --> E# + M3 = G## --> 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 "perfect" 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. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. 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 this would be confusing because 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:<h2> --><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 "downmajor second", "upminor third", etc. For 4ths and 5ths, "perfect" is implied and can be omitted: ^P4 = "up-fourth" and vP5 = "down-fifth". In larger edos there may be "down-octave", "up-unison", etc.<br />
<br />
There are some alternate names. The dim7 of a dim7 chord would be three EDOsteps below a min7 = 15\22 = ^m6. 14\22 could be written as m6 or as vd7. However double-ups and double-downs are to be avoided in 22edo. In larger edos, they would be necessary. Thus 7\22 would never be written ^^m3.<br />
<br />
0-8-13 in C has C E & G, and is written "C" and pronounced "C" or "C major".<br />
0-7-13 = C Ev G is written "C.v", spoken as "C downmajor" or possibly "C dot down".<br />
The period is needed because "Cv", spoken as "C down", is a note, not a chord.<br />
0-6-13 = C Eb^ G is "C.^m", "C upminor"<br />
0-5-13 = C Eb G is "Cm", "C minor"<br />
The period isn't needed here because there's no ups or downs immediately after the note name.<br />
<br />
0-8-13-18 = C E G Bb is "C7", "C seven", a standard C7 chord with a M3 and a m7.<br />
0-7-13-18 = C Ev G Bb is "C7(v3)", "C seven, down third". The altered note or notes are in parentheses.<br />
<br />
0-8-13-21 = C E G B is "CM7", "C major seven".<br />
0-7-13-20 = C Ev G Bv is "C.vM7", "C downmajor seven". The down symbol affects both the 3rd and the 7th.<br />
Often the root of a chord will not be a mid note. The root in the next two examples is Cv.<br />
0-8-13-21 = Cv Ev Gv Bv is "Cv.M7", "C down, major seven"<br />
To distinguish between C.vM7 and Cv.M7, one has to pronounce the period with a small pause.<br />
0-7-13-20 = Cv Evv Gv Bvv is "Cv.vM7", "C down, downmajor seven".<br />
<br />
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-fourth, 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 the fourthwards EDOs below for an exception.<br />
<br />
"Aug" and "dim" chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. An A3,P5 chord is A3 = "aug three chord", not "aug chord", to distinguish it from the conventional aug chord M3,A5. That chord is still called an aug chord. Likewise d3,P5 is a "dim three chord", and m3,d5 is a "dim" chord.<br />
<br />
0-3-13 = C Dv G = Csusv2<br />
0-4-13 = C D G = Csus2<br />
0-5-13 = C Eb G = Cm<br />
0-6-13 = C Eb^ G = C.^m<br />
0-7-13 = C Ev G = C.vM<br />
0-8-13 = C E G = C<br />
0-9-13 = C F G = Csus4<br />
0-10-13 = C F^ G = Csus^4<br />
<br />
0-5-10 = C Eb Gb = Cdim<br />
0-5-11 = C Eb Gb^ = Cdim(^5)<br />
0-5-12 = C Eb Gv = Cm(v5)<br />
<br />
0-5-10-15 = C Eb Gb Bbb = Cdim7<br />
0-5-11-14 = C Eb Gb^ Bbbv = Cdim7(^5,v7)<br />
0-6-11-15 = C Eb^ Gb^ Bbb = Cdim7(^3,^5)<br />
0-6-11-16 = C Eb^ Gb^ Bbb^ = C.^dim7 (the up symbol applies to m3, d5 and d7)<br />
0-5-13-17 = C Eb G A = Cm6<br />
<br />
Sometimes doubled ups/downs are unavoidable:<br />
0-6-12-15 = C Eb^ Gv Avv = Cm6(^3,v5,vv6), or C Eb^ Gb^^ Bbb = Cdim7(^3,^^5)<br />
0-7-13-16 = C Ev G Av = C.vM6 (the down symbol applies to both the 3rd and the 6th)<br />
0-8-13-17 = C E G A = C6<br />
0-7-13-16 = C Ev G Av = C.v6<br />
<br />
0-5-13-18 = C Eb G Bb = Cm7<br />
0-6-13-19 = C Eb^ G Bb^ = C.^m7<br />
0-7-13-20 = C Ev G Bv = C.vM7<br />
0-8-13-21 = C E G B = CM7<br />
<br />
0-5-13-16 = C Eb G Av = Cm6(v6)<br />
0-8-13-19 = C E G Bb^ = C7(^7)<br />
0-7-13-18-26 = C Ev G Bb D = C9(v3)<br />
0-7-13-18-26-32 = C Ev G Bb D F^ = C11(v3,^11)<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(no5) = "G downmajor seven, no five"<br />
Eb^.v(add9) = "E flat up, downmajor, add nine"<br />
C7sus4 = "C seven, sus four"<br />
A7(v3) = "A seven, down three"<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 might be used in certain chord progressions like Im - ^IIIM - ^VIIM - ^IVm - ^Im.<br />
<br />
I<br />
^I or bII<br />
v#I or ^bII<br />
#I or vII<br />
II<br />
^II or bIII<br />
v#II or ^bIII<br />
#II or vIII<br />
III<br />
IV<br />
^IV or bV<br />
v#IV or ^bV<br />
#IV or vV<br />
V<br />
^V or bVI<br />
v#V or ^bVI<br />
#V or vVI<br />
VI<br />
^VI or bVII<br />
v#VI or ^bVII<br />
#VI or vVII<br />
VII or vI<br />
These are pronounced "down-two", "up-flat-three", "down-sharp-four", etc.<br />
<br />
Here's the Tibia chords. Periods are never needed after the root in relative notation because ups and downs are always leading, never trailing.<br />
IvM7(no5) = "one downmajor seven, no five"<br />
^bVIv(add9) = "up-flat six downmajor, add nine"<br />
IV7sus4 = "four seven, sus four"<br />
II7(v3) = "two seven, down three"<br />
<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:2419:<img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="" title="" style="height: 1035px; width: 800px;" /> --><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:2419 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextLocalImageRule:2420:<img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="" title="" style="height: 957px; width: 800px;" /> --><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:2420 --></h2>
<!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><!-- ws:end:WikiTextHeadingRule:8 --> </h2>
<!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><!-- ws:end:WikiTextHeadingRule:10 --> </h2>
<!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Naming Chords-Chord names in other EDOs"></a><!-- ws:end:WikiTextHeadingRule:12 --><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<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 possibly sus2)<br />
0-4-9 = ^m<br />
0-5-9 = vM<br />
0-6-9 = M (or possibly sus4)<br />
0-5-9-12 = 7(v3)<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 P4 A4/d5 P5 d6 m6 M6/d7 m7 M7 A7<br />
chord roots: I #I/bbII bII II bIII III #III/bIV 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 (in practice, no symbol, as in "C" for the C chord)<br />
0-5-10 = aug (the conventional aug chord)<br />
0-6-9 = (A3) (aug 3rd, perfect 5th)<br />
0-7-9 = sus4<br />
0-4-8-12 = dim7 (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<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 />
<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<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 />
0-4-11 = (d3) (dim 3rd, perfect 5th)<br />
0-4-10 = dim(d3)<br />
0-5-11 = m<br />
0-5-10 = dim (conventional dim chord)<br />
0-6-11 = M<br />
0-7-11 = (A3) (aug 3rd, perfect 5th)<br />
0-6-12 = aug (conventional aug chord)<br />
0-7-12 = aug(A3)<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 "~":<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 = "C dot down")<br />
0-6-12 = ~ (a mid chord, e.g. D.~ = "D dot mid")<br />
0-7-12 = ^ (an up chord, e.g. E.^ = "E dot up")<br />
0-8-12 = ^^ or susv4<br />
0-9-12 = sus4<br />
0-6-11 = ~(v5)<br />
<br />
0-6-12-18 = 7<br />
0-7-12-18 = 7(^3)<br />
0-7-12-19 = ^7<br />
0-7-12-17 = ^(v7)<br />
0-6-12-15 = 6<br />
0-7-12-15 = 6(^3)<br />
0-7-12-16 = ^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 = ^m or vM or ~<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 = ^m or vM or ~<br />
0-10-18 = M<br />
0-11-18 = ^M<br />
0-12-18 = susv4<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Naming Chords-Cross-EDO considerations"></a><!-- ws:end:WikiTextHeadingRule:14 --><strong><u>Cross-EDO considerations</u></strong></h2>
<br />
In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 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.<br />
<br />
The name "major" refers not to the sound but to the function of the chord. If you want to play a I - VIm - IIm - V - I progression without pitch shifts or tonic drift, you can do that in any edo, as long as you use major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22edo, the chord that you need sounds like a red chord.<br />
<br />
In other words, I - VIm - IIm - V - I in JI implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in certain EDOs. The notation tells you which ones. If 22edo's downmajor chord 0-7-13 = 0¢-382¢-709¢ were called "major", you wouldn't know that it doesn't work in that progression.<br />
<br />
Another example: I7 - bVII7 - IV7 - I7. To make this work, the 7th in the I7 chord must be a minor 7th. in 22edo, that 7th sounds blue. In 19edo, it sounds green. If you want a blue 7th in 19edo, you have to use the downminor 7th, which will cause shifts or drifts in the progression.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="Naming Chords-EDOs with an inaccurate 3/2"></a><!-- ws:end:WikiTextHeadingRule:16 --><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 the above 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 />
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.<br />
<br />
The other approach is to use some interval other than the fifth to generate the notation. Above I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But these EDOs don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like.<br />
<br />
<br />
<u><strong>Theoretical alternatives for 8edo, 11edo, 13edo and 18edo</strong></u><br />
<br />
8edo octatonic (every note is a generator)<br />
D E F G H A B C D<br />
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9<br />
requires learning octatonic interval arithmetic and staff notation<br />
<br />
11edo heptatonic narrow-fifth-based, fourthwards, # is ^^ (3/2 maps to 6\11 perfect 5th):<br />
D E * * F G A B * * C D<br />
fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7<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, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):<br />
nonotonic fifthwards chain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9<br />
P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8<br />
requires learning nonotonic interval arithmetic and staff notation<br />
<br />
11edo pentatonic wide-fifth-based, fifthwards, # is ^^ (3/2 maps to 7\11 6th):<br />
D * * E G * * A C * * D<br />
pentatonic fifthwards chain of fifthoids: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7<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, sharp = ^^^ (3/2 maps to 7\13 perfect 5th):<br />
D E * * * F G A B * * * C D<br />
fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7<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, fourthwards, 3/2 maps to 7\13 = perfect 7th<br />
undecatonic sixthwards chain of sevenths:<br />
M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11<br />
P1 - m2 - M2/m3 - M3/m4 - M4/m5 - M5 - P6 - P7 - m8 - M8/m9 - M9/m10 - M10/m11 - M11 - P12<br />
requires learning undecatonic interval arithmetic and notation<br />
<br />
13edo octatonic wide-fifth-based, fourthwards, 3/2 maps to 8\13 = perfect 6th<br />
octotonic chain of sixths: 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 />
18edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^ (3/2 maps to 10\18 perfect 5th)<br />
D * E * * * F * G * A * B * * * C * D<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 plus closed is triply 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 staff notation<br />
<br />
<br />
<u><strong>Alternate notation for other edos:</strong></u><br />
23edo pentatonic wide-fifth-based, fifthwards, 3/2 maps to 14\23 = perfect fifthoid:<br />
D * * * * E * * * G * * * * A * * * C * * * * D<br />
35edo heptatonic wide-fifth-based, sharp = seven ups, 3/2 maps to 21\35 = perfect fifth:<br />
D * * * * * * E/F * * * * * * G * * * * * * A * * * * * * B/C * * * * * * D<br />
42edo heptatonic narrow-fifth-based, sharp = six ups, 3/2 maps to 24\42 = perfect fifth:<br />
D * * * * * E * * * * * F * * * * * G * * * * * A * * * * * B * * * * * C * * * * * D<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:<h1> --><h1 id="toc9"><a name="Summary of EDO notation"></a><!-- ws:end:WikiTextHeadingRule:18 --><u><strong>Summary of EDO notation</strong></u></h1>
<br />
Besides the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO, there are five EDO categories, based on the size of the fifth:<br />
"Fifth-less" EDOs (8, 11, 13 and 18)<br />
"Fourthward" EDOs (9, 16 and 23)<br />
"Perfect" EDOs (7, 14, 21, 28 and 35)<br />
"Pentatonic" EDOs (5, 10, 15, 20, 25 and 30)<br />
"Sweet" EDOs (all others)<br />
The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The sweet EDOs may or may not.<br />
<br />
To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. Examples:<br />
C C#/Db D (12edo)<br />
C Db C# D (17edo)<br />
C C# Db D (19edo)<br />
C C# _ Db D (26edo)<br />
C _ C# Db _ D (31edo)<br />
<br />
The scale fragment concisely conveys the "flavor" of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For fourthward EDOs, C-Db = d2 and C-D = m2. D# is included for these EDOs because C-D# is a M2 just like E-F. For fifthless EDOs, the scale fragment isn't as helpful because you can't deduce the entire keyboard layout from it.<br />
<br />
Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: "wide-fifth" 35edo (which uses 21\35 as a fifth) has a different scale fragment than "narrow-fifth" 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute M2 for m2.)<br />
<br />
<table class="wiki_table">
<tr>
<td style="text-align: center;">5edo<br />
</td>
<td style="text-align: center;">pentatonic<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C/Db<br />
</td>
<td style="text-align: center;">C#/D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">7edo<br />
</td>
<td style="text-align: center;">perfect<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C/C#<br />
</td>
<td style="text-align: center;">Db/D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">8edo<br />
</td>
<td style="text-align: center;">fifthless<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">9edo<br />
</td>
<td style="text-align: center;">fourthward<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C/Db<br />
</td>
<td style="text-align: center;">C#/D<br />
</td>
<td style="text-align: center;">D#<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">10edo<br />
</td>
<td style="text-align: center;">pentatonic<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C/Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#/D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">11edo<br />
</td>
<td style="text-align: center;">fifthless<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">12edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">C#/Db<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">13edo<br />
</td>
<td style="text-align: center;">fifthless<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">14edo<br />
</td>
<td style="text-align: center;">perfect<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C/C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db/D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">15edo<br />
</td>
<td style="text-align: center;">pentatonic<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C/Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#/D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">16edo<br />
</td>
<td style="text-align: center;">fourthward<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">C#/Db<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;">D#<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">17edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">18edo<br />
</td>
<td style="text-align: center;">fifthless<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">19edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">20edo<br />
</td>
<td style="text-align: center;">pentatonic<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C/Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#/D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">21edo<br />
</td>
<td style="text-align: center;">perfect<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C/C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db/D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">22edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">23edo<br />
</td>
<td style="text-align: center;">fourthward<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;">D#<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">24edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#/Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">25edo<br />
</td>
<td style="text-align: center;">pentatonic<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C/Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#/D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">26edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">27edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">28edo<br />
</td>
<td style="text-align: center;">perfect<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C/C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db/D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">29edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">30edo<br />
</td>
<td style="text-align: center;">pentatonic<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C/Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#/D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">31edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">32edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">33edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">34edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">35edo<br />
</td>
<td style="text-align: center;">perfect<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C/C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db/D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">36edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#/Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">37edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">38edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">39edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">40edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">41edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">42edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">43edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">44ddo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">45edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">46edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">47edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">48edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#/Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">49edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
</tr>
<tr>
<td style="text-align: center;">50edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">51edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
</tr>
<tr>
<td style="text-align: center;">52edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">53edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">*<br />
</td>
<td style="text-align: center;">D<br />
</td>
</tr>
</table>
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:<h3> --><h3 id="toc10"><a name="Summary of EDO notation--"Fifth-less" EDOs (8, 11, 13 and 18)"></a><!-- ws:end:WikiTextHeadingRule:20 --><u><strong>"Fifth-less" 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:22:<h3> --><h3 id="toc11"><a name="Summary of EDO notation--Alternate pentatonic notation for EDOs 8, 13 and 18"></a><!-- ws:end:WikiTextHeadingRule:22 --><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:24:<h3> --><h3 id="toc12"><a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"></a><!-- ws:end:WikiTextHeadingRule:24 --><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:26:<h3> --><h3 id="toc13"><a name="Summary of EDO notation--"Perfect" EDOs (7, 14, 21, 28 and 35)"></a><!-- ws:end:WikiTextHeadingRule:26 --><u>"Perfect" 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:28:<h3> --><h3 id="toc14"><a name="Summary of EDO notation--Pentatonic EDOs (5, 10, 15, 20, 25 and 30)"></a><!-- ws:end:WikiTextHeadingRule:28 --><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 = the unison and the major 3rd = the 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 />
<!-- ws:start:WikiTextHeadingRule:30:<h3> --><h3 id="toc15"><a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"></a><!-- ws:end:WikiTextHeadingRule:30 --><u>Alternative pentatonic notation for pentatonic EDOs:</u></h3>
<br />
Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.<br />
C# - G# - D# - A# - E# - C - G - D - A - E - Cb - Gb - Db - Ab - Eb etc.<br />
All intervals are perfect, so quality can be omitted.<br />
<br />
<u><strong>5edo</strong></u><strong>:</strong> zero keys per sharp/flat: C/C# Db/D<br />
D E G A C D<br />
1 - s3 - 4d - 5d - s7 - 8d<br />
<br />
<u><strong>10edo</strong></u><strong>:</strong> zero keys per sharp/flat: C/C# * Db/D<br />
D * E * G * A * C * D<br />
D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D<br />
1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d<br />
<br />
<u><strong>15edo</strong></u><strong>:</strong> zero keys per sharp/flat: C/C# * * Db/D<br />
D * * E * * G * * A * * C * * D<br />
D - D^ - Ev - E - E^ - Gv - G - G^ - Av - A - A^ - Cv - C - C^ - Dv - D<br />
1 - ^1 - vs3 - s3 - ^s3 - v4d - 4d - ^4d - v5d - 5d - ^5d - vs7 - s7 - ^s7 - v8d - 8d<br />
<br />
etc.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:32:<h3> --><h3 id="toc16"><a name="Summary of EDO notation--"Sweet" EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)"></a><!-- ws:end:WikiTextHeadingRule:32 --><u>"Sweet" EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)</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 />
perfect = white, major = red, yellow and fifthward white, minor = green, blue and fourthwards white<br />
<br />
<strong><u>17edo</u>:</strong> sharp = 2 keys: C Db C# D<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 />
perfect = white, major = yellow and fifthward white, minor = green and fourthward white, aug/dim = red/blue.<br />
<br />
<strong><u>22edo</u>:</strong> sharp = 3 keys: C Db * C# D<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 * D<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.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:<h2> --><h2 id="toc17"><a name="Summary of EDO notation-Ups and downs solfege"></a><!-- ws:end:WikiTextHeadingRule:34 --><u>Ups and downs solfege</u></h2>
<br />
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:<br />
The initial consonant remains as before: D, R, M, F, S, L and T<br />
The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb<br />
The vowels are pronounced as in Spanish or Italian<br />
The pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u<br />
The optional 2nd vowel indicates up/down: a = ^^^, e = ^, i = ^^, o = v, u = vv<br />
The 2nd vowel is separated from the first by either a glottal stop, an "h", a "w", or a "y"<br />
Thus C#v is Deo, pronounced as De'o or Deho or Dewo or Deyo.<br />
This suffices for many but not all edos, as some require triple sharps or quadruple ups.<br />
<br />
Fixed-do solfege:<br />
Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb<br />
Da = C, Da'e = C^, Da'i = C^^, Da'o = Cv, Da'u = Cvv, Da'a = C^^^<br />
De = C#, De'e = C#^, De'i = C#^^, De'o = C#v, De'u = C#vv, De'a = C#^^^<br />
etc.<br />
<br />
Moveable-do solfege:<br />
The 2nd vowel is as before. The 1st vowel's meaning depends on the interval.<br />
Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim<br />
Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1<br />
Da'e = ^P1, Da'i = ^^P1, Da'o = vP1, Da'u = vvP1, Da'a = ^^^P1<br />
etc.<br />
<br />
Imperfect intervals (2nd, 3rd, 6th and 7th): a = major, e = aug, i = double-aug, o = minor, u = dim<br />
Ra = M2, Re = A2, Ri = AA2, Ro = m2, Ru = d2<br />
Ra'e = ^M2, Ra'i = ^^M2, Ra'o = vM2, Ra'u = vvM2, Ra'a = ^^^M2<br />
etc.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:36:<h2> --><h2 id="toc18"><a name="Summary of EDO notation-Rank-2 Notation"></a><!-- ws:end:WikiTextHeadingRule:36 --><u>Rank-2 Notation</u></h2>
<br />
Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)<br />
<br />
Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:<br />
...Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# ...<br />
<br />
Fifth-generated rank-2 tunings can be written without ups and downs in any EDO on either side of the 4\7 kite:<br />
<br />
12-tone genchain Eb to G#: C C# D Eb E F F# G G# A Bb B C<br />
12-tone genchain Ab to C#: C C# D Eb E F F# G Ab A Bb B C<br />
12-tone genchain C to E#: C C# D D# E E# F# G G# A A# B C<br />
<br />
In <u>12edo</u>, C# and Db are identical, but in <u>12-tone</u>, they may not be, and usually aren't.<br />
<br />
19-tone genchain Gb to B#: C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B B# C<br />
19-tone genchain Fb to A#: C C# Db D D# Eb E Fb F F# Gb G G# Ab A A# Bb B Cb C<br />
19-tone genchain F to Ax: C C# Cx D D# Dx E E# F F# Fx G G# Gx A A# Ax B B# C<br />
<br />
Fourthward frameworks can be notated with the # sign meaning harmonically sharp but melodically flat:<br />
<br />
16-tone genchain Db to A#: C D# D Db E Eb F# F G# G A# A Ab B Bb C# C<br />
16-tone genchain Fb to C#: C Cb D Db E Eb F# F Fb G Gb A Ab B Bb C# C<br />
<br />
For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. Each node in the Stern-Brocot EDO chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the fifth's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).<br />
<br />
All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone. These two are easily notated without ups and downs:<br />
<br />
5-tone genchain C to E: C D E G A C<br />
5-tone genchain F to A: C D F G A C<br />
<br />
7-tone genchain C to F#: C D E F# G A B C<br />
7-tone genchain Bb to E: C D E F G A Bb C<br />
<br />
All fifthless frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd is a descending interval.<br />
<br />
If the sharp's keyspan is 1 or -1, as with 12-tone, 19-tone, and all fourthward frameworks, ups and downs aren't needed to notate rank-2. They also aren't needed for 5-tone and 7-tone. Since perfect, pentatonic and fifthless frameworks are incompatible, we need only consider sweet frameworks, excluding those that lie on the side of the heptatonic kite and those that lie on the spine of any kite.<br />
<br />
To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a <strong>keyspan</strong> (always +1) but also a <strong>genspan</strong>, which indicates how many steps forward or backwards along the generator chain, or <strong>genchain</strong>, one must travel to find the interval.<br />
<br />
For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is C^ + m2 = (C + m2)^, exactly equivalent to Db^. However, C^^ is not equivalent to Dvv, even though they occuy the same key on the keyboard, just as C# may not equal Db in 12-tone.<br />
<br />
The usual genchain note names will run out of order when mapped to the 22-tone framework. For example, we might have C Db B# C# D. So ups and downs are used to provide alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D.<br />
<br />
Here's part of the 22-tone genchain. There are more than 22 notes because the genchain is theoretically infinite:<br />
<table class="wiki_table">
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">Fb<br />
</td>
<td style="text-align: center;">Eb^<br />
</td>
<td style="text-align: center;">D^^<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">Cb<br />
</td>
<td style="text-align: center;">Bb^<br />
</td>
<td style="text-align: center;">A^^<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">Gb<br />
</td>
<td style="text-align: center;">F^<br />
</td>
<td style="text-align: center;">E^^<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td style="text-align: center;">C^<br />
</td>
<td style="text-align: center;">B^^<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">Ab<br />
</td>
<td style="text-align: center;">G^<br />
</td>
<td style="text-align: center;">F#^^<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">Fbv<br />
</td>
<td style="text-align: center;">Eb<br />
</td>
<td style="text-align: center;">D^<br />
</td>
<td style="text-align: center;">C#^^<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">Cbv<br />
</td>
<td style="text-align: center;">Bb<br />
</td>
<td style="text-align: center;">A^<br />
</td>
<td style="text-align: center;">G#^^<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">Gbv<br />
</td>
<td style="text-align: center;">F<br />
</td>
<td style="text-align: center;">E^<br />
</td>
<td style="text-align: center;">A#^^<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">Dbv<br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">B^<br />
</td>
<td style="text-align: center;">E#^^<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">Abv<br />
</td>
<td style="text-align: center;">G<br />
</td>
<td style="text-align: center;">F#^<br />
</td>
<td style="text-align: center;">B#^^<br />
</td>
</tr>
<tr>
<td style="text-align: center;">Fbvv<br />
</td>
<td style="text-align: center;">Ebv<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;">C#^<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Cbvv<br />
</td>
<td style="text-align: center;">Bbv<br />
</td>
<td style="text-align: center;">A<br />
</td>
<td style="text-align: center;">G#^<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Gbvv<br />
</td>
<td style="text-align: center;">Fv<br />
</td>
<td style="text-align: center;">E<br />
</td>
<td style="text-align: center;">D#^<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Dbvv<br />
</td>
<td style="text-align: center;">Cv<br />
</td>
<td style="text-align: center;">B<br />
</td>
<td style="text-align: center;">A#^<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Abvv<br />
</td>
<td style="text-align: center;">Gv<br />
</td>
<td style="text-align: center;">F#<br />
</td>
<td style="text-align: center;">E#^<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Ebvv<br />
</td>
<td style="text-align: center;">Dv<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td style="text-align: center;">B#^<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Bbvv<br />
</td>
<td style="text-align: center;">Av<br />
</td>
<td style="text-align: center;">G#<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Fvv<br />
</td>
<td style="text-align: center;">Ev<br />
</td>
<td style="text-align: center;">D#<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Cvv<br />
</td>
<td style="text-align: center;">Bv<br />
</td>
<td style="text-align: center;">A#<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Gvv<br />
</td>
<td style="text-align: center;">F#v<br />
</td>
<td style="text-align: center;">E#<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Dvv<br />
</td>
<td style="text-align: center;">C#v<br />
</td>
<td style="text-align: center;">B#<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Avv<br />
</td>
<td style="text-align: center;">G#v<br />
</td>
<td style="text-align: center;">Fx<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Evv<br />
</td>
<td style="text-align: center;">D#v<br />
</td>
<td style="text-align: center;">Cx<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">Bvv<br />
</td>
<td style="text-align: center;">A#v<br />
</td>
<td style="text-align: center;">Gx<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">F#vv<br />
</td>
<td style="text-align: center;">E#v<br />
</td>
<td style="text-align: center;">Dx<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">C#vv<br />
</td>
<td style="text-align: center;">B#v<br />
</td>
<td style="text-align: center;">Ax<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td>etc.<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
Here's each note, with alternate tunings for the black keys, with the keyspan and genspan both measured from C:<br />
<table class="wiki_table">
<tr>
<td style="text-align: center;">keyspan<br />
</td>
<td style="text-align: center;">genspan<br />
</td>
<td style="text-align: center;">note<br />
</td>
<td style="text-align: center;">genspan<br />
</td>
<td style="text-align: center;">note<br />
</td>
</tr>
<tr>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">-5<br />
</td>
<td style="text-align: center;">Db = C^<br />
</td>
<td style="text-align: center;">+17<br />
</td>
<td style="text-align: center;">C#vv = Dv3<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">-10<br />
</td>
<td style="text-align: center;">Db^ = C^^<br />
</td>
<td style="text-align: center;">+12<br />
</td>
<td style="text-align: center;">C#v = Dvv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">3<br />
</td>
<td style="text-align: center;">-15<br />
</td>
<td style="text-align: center;">Db^^ = C^3<br />
</td>
<td style="text-align: center;">+7<br />
</td>
<td style="text-align: center;">C# = Dv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4<br />
</td>
<td style="text-align: center;">+2<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">5<br />
</td>
<td style="text-align: center;">-3<br />
</td>
<td style="text-align: center;">Eb = D^<br />
</td>
<td style="text-align: center;">+19<br />
</td>
<td style="text-align: center;">D#vv = Ev3<br />
</td>
</tr>
<tr>
<td style="text-align: center;">6<br />
</td>
<td style="text-align: center;">-8<br />
</td>
<td style="text-align: center;">Eb^ = D^^<br />
</td>
<td style="text-align: center;">+14<br />
</td>
<td style="text-align: center;">D#v = Evv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">7<br />
</td>
<td style="text-align: center;">-13<br />
</td>
<td style="text-align: center;">Eb^^ = D^3<br />
</td>
<td style="text-align: center;">+9<br />
</td>
<td style="text-align: center;">D# = Ev<br />
</td>
</tr>
<tr>
<td style="text-align: center;">8<br />
</td>
<td style="text-align: center;">+4<br />
</td>
<td style="text-align: center;">E<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">9<br />
</td>
<td style="text-align: center;">-1<br />
</td>
<td style="text-align: center;">F<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">10<br />
</td>
<td style="text-align: center;">-6<br />
</td>
<td style="text-align: center;">Gb = F^<br />
</td>
<td style="text-align: center;">+16<br />
</td>
<td style="text-align: center;">F#vv = Gv3<br />
</td>
</tr>
<tr>
<td style="text-align: center;">11<br />
</td>
<td style="text-align: center;">-11<br />
</td>
<td style="text-align: center;">Gb^ = F^^<br />
</td>
<td style="text-align: center;">+11<br />
</td>
<td style="text-align: center;">F#v = Gvv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">12<br />
</td>
<td style="text-align: center;">-16<br />
</td>
<td style="text-align: center;">Gb^^ = F^3<br />
</td>
<td style="text-align: center;">+6<br />
</td>
<td style="text-align: center;">F# = Gv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">13<br />
</td>
<td style="text-align: center;">+1<br />
</td>
<td style="text-align: center;">G<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">14<br />
</td>
<td style="text-align: center;">-4<br />
</td>
<td style="text-align: center;">Ab = G^<br />
</td>
<td style="text-align: center;">+18<br />
</td>
<td style="text-align: center;">G#vv = Av3<br />
</td>
</tr>
<tr>
<td style="text-align: center;">15<br />
</td>
<td style="text-align: center;">-9<br />
</td>
<td style="text-align: center;">Ab^ = G^^<br />
</td>
<td style="text-align: center;">+13<br />
</td>
<td style="text-align: center;">G#v = Avv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">16<br />
</td>
<td style="text-align: center;">-14<br />
</td>
<td style="text-align: center;">Ab^^ = G^3<br />
</td>
<td style="text-align: center;">+8<br />
</td>
<td style="text-align: center;">G# = Av<br />
</td>
</tr>
<tr>
<td style="text-align: center;">17<br />
</td>
<td style="text-align: center;">+3<br />
</td>
<td style="text-align: center;">A<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">18<br />
</td>
<td style="text-align: center;">-2<br />
</td>
<td style="text-align: center;">Bb = A^<br />
</td>
<td style="text-align: center;">+20<br />
</td>
<td style="text-align: center;">A#vv = Bv3<br />
</td>
</tr>
<tr>
<td style="text-align: center;">19<br />
</td>
<td style="text-align: center;">-7<br />
</td>
<td style="text-align: center;">Bb^ = A^^<br />
</td>
<td style="text-align: center;">+15<br />
</td>
<td style="text-align: center;">A#v = Bvv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">20<br />
</td>
<td style="text-align: center;">-12<br />
</td>
<td style="text-align: center;">Bb^^ = A^3<br />
</td>
<td style="text-align: center;">+10<br />
</td>
<td style="text-align: center;">A# = Bv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">21<br />
</td>
<td style="text-align: center;">+5<br />
</td>
<td style="text-align: center;">B<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">22<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
"^3" means three ups. Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.<br />
<br />
The genspan for the up symbol in 22-tone is calculated from the keyspans:<br />
<br />
K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)<br />
K(#) = X, K(b) = -X (X = keyspan of the sharp symbol, i.e., how many keys wide it is. For 22-tone, X = 3)<br />
K(#vX) = K(#) + X * K(v) = 0 (going up X keys using a sharp, then going down X keys using X downs, must cancel out)<br />
<br />
"#vX" means one sharp plus X downs. Zero keyspans in the genchain only occur on every Nth step for a N-tone framework. E.g., 12-tone keyspans:<br />
<table class="wiki_table">
<tr>
<td>genchain of fifths<br />
</td>
<td>C<br />
</td>
<td>G<br />
</td>
<td>D<br />
</td>
<td>A<br />
</td>
<td>E<br />
</td>
<td>B<br />
</td>
<td>F#<br />
</td>
<td>C#<br />
</td>
<td>G#<br />
</td>
<td>D#<br />
</td>
<td>A#<br />
</td>
<td>E#<br />
</td>
<td>B#<br />
</td>
</tr>
<tr>
<td>genspan from C<br />
</td>
<td>0<br />
</td>
<td>1<br />
</td>
<td>2<br />
</td>
<td>3<br />
</td>
<td>4<br />
</td>
<td>5<br />
</td>
<td>6<br />
</td>
<td>7<br />
</td>
<td>8<br />
</td>
<td>9<br />
</td>
<td>10<br />
</td>
<td>11<br />
</td>
<td>12<br />
</td>
</tr>
<tr>
<td>12-tone keyspan from C<br />
</td>
<td>0<br />
</td>
<td>7<br />
</td>
<td>2<br />
</td>
<td>9<br />
</td>
<td>4<br />
</td>
<td>11<br />
</td>
<td>6<br />
</td>
<td>1<br />
</td>
<td>8<br />
</td>
<td>3<br />
</td>
<td>10<br />
</td>
<td>5<br />
</td>
<td>0<br />
</td>
</tr>
</table>
B#, genspan 12, has a zero keyspan, as does Dbb, genspan -12, and A###, genspan 24. Thus the final equation means that the genspan resulting from going up a sharp and down X downs must be zero, N, -N, 2N, -2N, etc. Thus this genspan mod N must be zero.<br />
<br />
G(#) = 7 (by definition, the sharp's genspan = 7, since we're assuming heptatonic notation)<br />
G(#vX) = G(#) + X * G(v) = G(#) - X * G(^) = 7 - X * G(^)<br />
G(#vX) mod N = 0, thus G(#vX) = i * N for some integer i<br />
7 - X * G(^) = i * N<br />
G(^) = - (i * N - 7) / X<br />
<br />
For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. Thus i = 1, G(^) = -5, and ^ = min 2nd.<br />
<br />
For 17-tone, X = 2, i = 1, G(^) = -5, and ^ = min 2nd<br />
<br />
For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd.<br />
<br />
<table class="wiki_table">
<tr>
<td style="text-align: center;">5edo<br />
</td>
<td style="text-align: center;">pentatonic<br />
</td>
<td style="text-align: center;">K(#)<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">17edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td>-5<br />
</td>
<td>min 2nd<br />
</td>
</tr>
<tr>
<td style="text-align: center;">19edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">22edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td>-5<br />
</td>
<td>min 2nd<br />
</td>
</tr>
<tr>
<td style="text-align: center;">26edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">27edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">29edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">31edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td>-12<br />
</td>
<td>dim 2nd<br />
</td>
</tr>
<tr>
<td style="text-align: center;">32edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">33edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">34edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">37edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">38edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">39edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">40edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">41edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">42edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">43edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">44ddo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">45edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">46edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">47edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">49edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">50edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">51edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">52edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: center;">53edo<br />
</td>
<td style="text-align: center;">sweet<br />
</td>
<td style="text-align: center;"><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
</table>
</body></html>