Kite's ups and downs notation

Definition

Ups and Downs (or ^v) is a notation system developed by Kite that can notate almost every EDO. The up symbol "^" and the down symbol "v" indicate raising or lowering a note (or widening/narrowing an interval) by one edostep. The mid symbol, "~" is for intervals exactly midway between major and minor, e.g. 3\24 is a mid 2nd. The mid 4th (~4) is midway between perfect and augmented, i.e. half-augmented, and the mid 5th (~5) is a half-diminished 5th.

Ups and downs can also notate any rank-2 temperament, although some temperaments require an additional pair of accidentals, lifts and drops (/ and \). In this context, an up or a lift represents sharpening by a comma that has been tempered, but not tempered out. For example, in Triyo aka Porcupine, an up/down represents raising/lowering by a tempered 81/80, and lifts/drops aren't used. In practice, the two uses of the notation often coincide perfectly. Triyo is supported by both 15edo and 22edo, and both edos map 81/80 to one edostep. Thus if Triyo is tuned to 15edo, an up simultaneously means both a tempered 81/80 and 1\15. Likewise, if tuned to 22-edo, the up means both 81/80 and 1\22. If not tuned to an edo at all, then the up only means 81/80. Thus a piece written in Triyo can be converted to a piece written in 22-edo by simply writing "22-edo" on the top of the page.

Ups and downs notation as it applies to edos is fully documented in the Notation guide for edos 5-72, which also covers chord names, staff notation, key signatures, and scale trees. Ups and downs notation as it applies to rank-2 and rank-3 temperaments is fully documented in the pergens article.

Explanation -- a 22-edo example

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

In contrast, 22-edo is hard to notate because 7 fifths are three edosteps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the 5/4, usually a 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 edostep". 22-edo can be written C - Db - ^Db - vD - D - Eb - ^Eb - vE - E - F etc. The notes are pronounced up-D-flat, down-D, 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 up or down comes before the note name to make naming chords easy.

The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So the B scale is B - C - ^C - vC# - C# - D - ^D - vD# - 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.

Relative notation for 22-edo is P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - d5 - ^d5 - v5 - P5 etc. That's pronounced upminor 2nd, downmajor 3rd, etc. You can apply this pattern to any 22-edo key. The notes without ups or downs always form a chain of fifths.

Enharmonic equivalents

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 vB# or even ^^B (double-up B).

Interval arithmetic

A core principle of ups and downs notation is that interval arithmetic is always preserved. Ups and downs are just added in:

	conventional	with ups and downs
interval between two notes	C to E = M3	^C to E = vM3	C to ^E = ^M3
note plus an interval	C + M3 = E	^C + M3 = ^E	C + ^M3 = ^E
sum of two intervals	M2 + m2 = m3	M2 + ^m2 = ^m3	vM2 + m2 = vm3

EDOs 12-24, 31 and 41

12-edo: D * E F * G * A * B C * D, 1 key per sharp/flat, ups and downs not needed

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

13b-edo: D E * * * F G A B * * * C D, 2 keys per sharp/flat

with sharp lowering the pitch, and major/aug narrower than minor/dim:

D E ^E/F# vEb/^F# Eb/vF F G A B ^B/C# vBb/^C# Bb/vC C D

P1 M2 ^M2/M3 vm2/^M3 m2/vm3 m3 P4 P5 M6 ^M6/M7 vm6/^M7 m6/vm7 m7 P8

13b-edo with sharp raising the pitch, and major/aug wider than minor/dim:

D E ^E/Fb vE#/^Fb E#/vF F G A B ^B/Cb vB#/^Cb B#/vC C D

P1 m2 ^m2/m3 vM2/^m3 M2/vM3 M3 P4 P5 m6 ^m6/m7 vM6/^m7 M6/vM7 M7 P8

14-edo: D * E * F * G * A * B * C * D, zero keys per sharp/flat. Because every interval is perfect, the quality can be omitted

D ^D/vE E ^E/vF F ^F/vG G ^G/vA A ^A/vB B ^B/vC C ^C/vD D

1 ^1/v2 2 ^2/v3 3 ^3/v4 4 ^4/v5 5 ^5/v6 6 ^6/v7 7 ^7/v8 8

15-edo: D * * E/F * * G * * A * * B/C * * D, 3 keys per sharp/flat

D ^D vE E/F ^F vG G ^G vA A ^A vB B/C ^C vD D

P1 ^m2 vM2 M2/m3 ^m3 vM3 M3/P4 ^4 v5 P5 ^m6 vM6 M6/m7 ^m7 vM7 P8

16-edo: D * E * * F * G * A * B * * C * D, 1 key per sharp/flat, ups and downs not needed

with sharp lowering the pitch, and major/aug narrower than minor/dim:

D Db/E# E Eb F# F Fb/G# G Gb/A# A Ab/B# B Bb C# C Cb/D# D

P1 A2 M2 m2/A3 M3 m3 d3/A4 P4 d4/A5 P5 d5/A6 M6 m6/A7 M7 m7 d7 P8

16-edo with sharp raising the pitch, and major/aug wider than minor/dim:

D D#/Eb E E# Fb F F#/Gb G G#/Ab A A#/Bb B B# Cb C C#/Db D

P1 d2 m2 M2 m3 M3 A3 P4 A4/d5 P5 d6 m6 M6/d7 m7 M7 A7 P8

17edo: D * * E F * * G * * A * * B C * * D, 2 keys per sharp/flat

D ^D/Eb D#/vE E F ^F/Gb F#/vG G ^G/Ab G#/vA A ^A/Bb A#/vB B C ^C/Db C#/vD D

P1 ^1/m2 A1/~2 M2 m3 ~3 M3 P4 ^4/d5 A4/v5 P5 m6 ~6 M6 m7 ~7 M7 P8

18b-edo: D * E * * * F * G * A * B * * * C * D, 2 keys per sharp/flat

with sharp lowering the pitch, and major/aug narrower than minor/dim:

D ^D/vE E ^E Eb/F# vF F ^F/vG G ^G/vA A ^A/vB B ^B Bb/C# vC C ^C/vD D

P1 ^P1/vM2 M2 ~2 m2/M3 ~3 m3 ^m3/v4 P4 ^4/v5 P5 ^5/vM6 M6 ~6 m6/M7 ~7 m7 ^m2/d8 P8

18b-edo with sharp raising the pitch, and major/aug wider than minor/dim:

D ^D/vE E ^E E#/Fb vF F ^F/vG G ^G/vA A ^A/vB B ^B B#/Cb vC C ^C/vD D

P1 ^P1/vm2 m2 ~2 mM2/m3 ~3 M3 ^M3/v4 P4 ^4/v5 P5 ^5/vm6 m6 ~6 M6/m7 ~7 M7 ^M7/d8 P8

19-edo: D * * E * F * * G * * A * * B * C * * D, 1 key per sharp/flat, ups and downs not needed

D D# Eb E E#/Fb F F# Gb G G# Ab A A# Bb B B#/Cb C C# Db D

P1 d2 m2 M2 d3 m3 M3 A3 P4 A4 d5 P5 A5 m6 M6 d7 m7 M7 A7 P8

20-edo: D * * * E/F * * * G * * * A * * * B/C * * * D, 4 keys per sharp/flat

D ^D ^^D/vvE vE E/F ^F ^^F/vvG vG G ^G ^^G/vvA vA A ^A ^^A/vvB vB B/C ^C ^^C/vvD vD D

P1/m2 ^m2 ~2 vM2 M2/m3 ^m3 ~3 vM3 M3/P4 ^4 ^^4/vv5 v5 P5/m6 ^m6 ~6 vM6 M6/m7 ^m7 ~7 vM7 P8

21-edo: D * * E * * F * * G * * A * * B * * C * * D, zero keys per sharp/flat. Because every interval is perfect, the quality can be omitted

D ^D vE E ^E vF F ^F vG G ^G vA A ^A vB B ^B vC C ^C vD D

1 ^1 v2 2 ^2 v3 3 ^3 v4 4 ^4 v5 5 ^5 v6 6 ^6 v7 7 ^7 v8 8

22-edo: D * * * E F * * * G * * * A * * * B C * * * D, 3 keys per sharp/flat

D ^D/Eb vD#/^Eb D#/vE E F ^F/Gb vF#/^Gb F#/vG G ^G/Ab vG#/^Ab G#/vA A etc.

P1 ^1/m2 A1/^m2 vM2 M2 m3 ^m3 vM3 M3 P4 ^4/d5 vA4/^d5 A4/v5 P5 etc.

23-edo: D * * E * * * F * * G * * A * * B * * * C * * D, 1 key per sharp/flat, ups or downs not needed

with sharp lowering the pitch, and major/aug narrower than minor/dim:

D Db E# E Eb Ebb/Fx F# F Fb G# G Gb A# A Ab B# B Bb Bbb/Cx C# C Cb D# D

P1 d1 A2 M2 m2 d2/A3 M3 m3 d3 A4 P4 d4 A5 P5 d5 A6 M6 m6 d6/A7 M7 m7 d7 A8 P8

23-edo with sharp raising the pitch, and major/aug wider than minor/dim:

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

24-edo: C * * * D * * * E * F * * * G * * * A * * * B * C, 2 keys per sharp/flat

C ^C/vDb C#/Db ^C#/vD D ^D/vEb D#/Eb ^D#/vE E ^E/vF F ^F F#/Gb vG G etc.

P1 ^1/vm2 A1/m2 ~2 M2 ^M2/vm3 m3 ~3 M3 ^M3/v4 P4 ^4 A4/d5 v5 P5 etc.

31-edo: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C, 2 keys per sharp/flat

C ^C C#/vDb ^C#/Db vD D ^D D#/vEb ^D#/Eb vE E ^E vF F ^F F#/vGb ^F#/Gb vG G etc.

P1 ^1/d2 A1/vm2 m2 ~2 M2 ^M2 vm3 m3 ~3 M3 ^M3 v4 P4 ^4 A4 d5 v5 P5 etc.

41-edo: C * * * * * * D * * * * * * E * * F * * * * * * G * * * * * * A * * * * * * B * * C, 4 keys per sharp/flat

P1 ^1 vm2 m2 ^m2 ~2 vM2 M2 ^M2 vm3 m3 ^m3 ~3 vM3 M3 ^M3 v4 P4 ^4 ~4 vA4/d5 A4/^d5 ~5 v5 P5 etc.

C ^C vDb Db ^Db vvD vD D ^D vEb Eb ^Eb vvE vE E ^E vF F ^F ^^F vF#/Gb F#/^Gb vvG vG G etc.

Chords and Chord Progressions

Chord names are based on jazz chord names. See Jim Aiken's book A Player's Guide to Chords & Harmony . Alterations are enclosed in parentheses, additions never are.

In perfect EDOs (7, 14, 21, 28 and 35), every interval is perfect, and there is no major or minor. In the following list of chord names, omit major, minor, dim and aug. Substitute up for upmajor and upminor, and down for downmajor and downminor. The C-E-G chord is called "C perfect" or simply "C".

An up or down between the chord root and the chord type (e.g. C^m7) is called a "global" up or down. Adding one to a chord raises or lowers the 3rd, and also the 6th, 7th or 11th, if present. Thus C down-nine is the usual C9 chord with the 3rd and 7th downed: Cv9 = C vE G vBb D. A "global mid" chord has a mid 3rd, 6th, 7th, and/or 11th. Global affects every other note of a stacked-3rds chord with a 6th below the root: 6 - 1 - 3 - 5 - 7 - 9 - 11 - 13.

The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd. Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the 5th. By the same logic, if the 7th is upped or downed, the 11th would be too.

There's one exception: it's pointless to apply a global down to a C5 chord, because there is no 3rd, 6th or 7th to alter. Thus Cv5 is invalid, and "C down-5" means C(v5) = C E vG.

Chord progressions use ups/downs notation to name the roots, e.g. Cv - Gv - vA^m - F or Iv - Vv - vVI^m - IVv. In relative notation, never use lower case roman numerals for minor chords. Write either vIIm or VIIm, never vii.

The major chord and various alterations of it:

C E G = C = "C" or "C major" (in perfect EDOs, "C" or "C perfect")

C ^E G = C^ = "C up" or "C upmajor"

C vE G = Cv = "C down" or "C downmajor" (in EDOs 10, 17, 24, 31, etc., C~ = "C mid")

C vvE G = Cvv = "C double-down" (in EDOs 20, 27, 34, 41, etc., C~ = "C mid", in EDOs 25, 32, 39, 46, etc. C^~ = "C upmid")

C E vG = C(v5) = "C down-five" (in EDOs 10, 17, 24, 31, etc., same or C(~5) = "C mid-five")

C vE vG = Cv(v5) = "C down down-five" (in EDOs 10, 17, 24, 31, etc., C~(~5) = "C mid mid-five")

All triads follow a similar pattern.

	maj	min	sus4	sus2	dim	aug
what's downed	C E G	C Eb G	C F G	C D G	C Eb Gb	C E G#
nothing	C	Cm	C4	C2	Co	Caug
3rd	Cv	Cvm	Cv4	Cv2	Cvo	Cvaug
5th	C(v5)	Cm(v5)	C4(v5)	C2(v5)	Co(v5)	Caug(v5)
3rd, 5th	Cv(v5)	Cvm(v5)	Cv4(v5)	Cv2(v5)	Cvo(v5)	Cvaug(v5)

Many EDOs have notes between the major 3rd and the perfect 4th, creating triads impossible in 12-edo, such as:

C Fb G = C(d4) or C(b4) = "C dim-four" or "C flat-four"

C E# G = C(A3) or C(#3) = "C aug-three" or "C sharp-three"

C Ebb G = C(d3) or C(bb3) = "C dim-three" or "C double-flat-three"

C D# G = C(A2) or C(#2) = "C aug-two" or "C sharp-two"

Care must be taken in pronouncing the latter of the two names, that C(#2) doesn't sound like C#2 = C# D# G#. More non-12-edo triads:

C D# Gb = C(A2,d5) or C(#2,b5) = "C aug-two dim-five" or "C sharp-two, flat-five"

C Ebb Gb = Cdim(d3) or Cdim(bb3) = "C dim dim-three" or "C dim double-flat-three"

C Eb G# = Cmin(#5) = "C minor sharp-five"

C E# G# = Caug(#3) = "C aug sharp-three"

C Fb G# = C(b4,#5) = "C flat-four sharp-five"

Sixth and seventh chords:

If the 7th is not a perfect 5th or a dim 5th above the 3rd, the chord is named as a triad with an added 7th.

C E G Bb = C7 = "C seven"

C vE G Bb = Cv,7 = "C down add-seven"

C E G vBb = C,v7 = "C add down-seven"

C vE G vBb = Cv7 = "C down seven"

All 7th chords follow this same pattern. Likewise, if a 6th is not a P4 or A4 above the 3rd, it's an "add-6" chord. Permitting add-7 chords has the added benefit that the wordy "minor-7 flat-5" can be replaced with the more concise "dim add-7", written Co,7. "Half-dim" can still be used, e.g. C uphalf-dim is C ^Eb Gb ^Bb. But half-dim is problematic because the slash-circle abbreviation for half-dim is untypable.

In the table below, if a chord is bolded, the comma must be spoken as "add".

Question: should a dim 7th be written as d7 or o7?

Question: for all tetrads and pentads, should a comma that needn't be spoken as "add" be omitted from the written name?

	maj7	dom7	min7	half-dim	dim7	min-maj	maj6	min6
what's downed	C E G B	C E G Bb	C Eb G Bb	C Eb Gb Bb	C Eb Gb Bbb	C Eb G B	C E G A	C Eb G A
nothing	CM7	C7	Cm7	Co,7	Co7	Cm,M7	C6	Cm6
3rd	Cv,M7	Cv,7	Cvm,7	Cvo,7	Cvo,d7	Cvm,M7	Cv,6	Cvm,6
6th/7th	C,vM7	C,v7	Cm,v7	Co,v7	Co,vd7	Cm,vM7	C,v6	Cm,v6
3rd, 6th/7th	CvM7	Cv7	Cvm7	Cvo,v7	Cvo7	Cvm,vM7	Cv6	Cvm6
5th	CM7(v5)	C7(v5)	Cm7(v5)	Co(v5)7	Co7(v5)	Cm(v5)M7	C6(v5)	Cm6(v5)
3rd, 5th	Cv(v5)M7	Cv(v5)7	Cvm(v5)7	Cvo(v5)7	Cvo(v5)d7	Cvm(v5)M7	Cv(v5)6	Cvm(v5)6
5th, 6th/7th	C(v5)vM7	C(v5)v7	Cm(v5)v7	Co(v5)v7	Co(v5)vd7	Cm(v5)vM7	C(v5)v6	Cm(v5)v6
3rd, 5th, 6th/7th	CvM7(v5)	Cv7(v5)	Cvm7(v5)	Cvo(v5)v7	Cvo7(v5)	Cvm(v5)vM7	Cv6(v5)	Cvm6(v5)

Some non-12-edo sixth and seventh chords:

C E G Ab = C,b6 or C,m6 = "C add flat-six" or "C add minor-six"

C E G A# = C,#6 or C,A6 = "C add sharp-six" or "C add aug-six"

C vE G ^Bb = Cv,^7 = "C down up-seven" (in certain EDOs, C~7 = "C mid-seven")

C E G Bbb = C,d7 or C,bb7 = "C add dim-seven" or "C add double-flat-seven" (19-edo's 4:5:6:7 chord)

C E G B# is C,#7 or C,A7 = "C add sharp-seven" or "C add aug-seven"

C E G Cb = C,b8 or C,d8 = "C add flat-eight" or "C add dim-eight"

Ninth chords:

In bolded chords, the comma is spoken as "add". Double additions need only a single comma, e.g. C E G vBb vD is named C,v7v9. Double alterations need only a single pair of parentheses, e.g. C vE vG B D is named CM9(v3v5).

Question: Should double alterations like (v3v5) be spoken "down three and five"? Should they be written (v3,5) or (v3&5)?

Question: What to do when the 3rd is downed but the 7th isn't? Is it a down chord with an added 7th, or a 7th chord with a downed 3rd?

	add9	maj9	dom9	min9	dom7b9	maj6/9	min6/9
what's downed	C E G D	C E G B D	C E G Bb D	C Eb G Bb D	C E G Bb Db	C E G A D	C Eb G A D
nothing	C,9	CM9	C9	Cm9	C7,b9	C6,9?	Cm6,9?
3rd	Cv,9	CM9(v3)	C9(v3)	Cm9(v3)	Cv,7b9 or C7(v3)b9 or C7,b9(v3)	Cv,6,9?	Cvm,6,9?
6th/7th	----	CM9(v7)	C9(v7)	Cm9(v7)	C,v7b9	C,v6,9?	Cm,v6
3rd, 6th/7th	---	CvM9	Cv9	Cvm9	Cv7,b9	Cv6,9?	Cvm6,9
9th	C,v9	CM7,v9	C7,v9	Cm7,v9	C7,vb9	C6,v9	Cm6,v9
3rd, 9th	Cv,v9	Cv,M7v9 or CM7(v3)v9	Cv,7v9 or C7(v3)v9	Cvm,7v9 or Cm7(v3)v9	Cv,7vb9 or C7(v3)vb9	Cv,6v9 or C6(v3)v9	Cvm,6v9 or Cm6(v3)v9
6th/7th, 9th	---	C,vM7v9	C,v7v9	Cm,v7v9	C,v7vb9	C,v6v9	Cm,v6v9
3rd, 6th/7th, 9th	---	CvM7,v9	Cv7,v9	Cvm7,v9	Cv7,vb9	Cv6,v9	Cvm6,v9
5th	C,9(v5)	CM9(v5)	C9(v5)	Cm9(v5)	C7(v5)b9	C6(v5)9	Cm6(v5)9
3rd, 5th	Cv(v5)9	CM9(v3v5)	C9(v3v5)	Cm9(v3v5)	Cv(v5)7b9 or C7(v3v5)b9	Cv(v5)6,9	Cvm(v5)6,9
5th, 6th/7th	---	CM9(v5v7)	C9(v5v7)	Cm9(v5v7)	C(v5)v7b9	C(v5)v6,9	Cm(v5)v6,9
3rd, 5th, 6th/7th	---	CvM9(v5)	Cv9(v5)	Cvm9(v5)	Cv7(v5)b9	Cv6(v5)9	Cvm6(v5)9
5th, 9th	C(v5)v9	CM7(v5)v9	C7(v5)v9	Cm7(v5)v9	C7(v5)vb9	C6(v5)v9	Cm6(v5)v9
3rd, 5th, 9th	Cv(v5)v9	Cv(v5)M7v9 or CM7(v3v5)v9	Cv(v5)7v9 or C7(v3v5)v9	Cvm(v5)7v9 or Cm7(v3v5)v9	Cv(v5)7vb9 or C7(v3v5)b9	Cv(v5)6v9 or C6(v3v5)v9	Cvm(v5)6v9 or Cm6(v3v5)v9
5th, 6th/7th, 9th	---	C(v5)vM7v9	C(v5)v7v9	Cm(v5)v7v9	C(v5)v7vb9	C(v5)v6v9	Cm(v5)v6v9
3rd, 5th, 6th/7th, 9th	---	CvM7(v5)v9	Cv7(v5)v9	Cvm7(v5)v9	Cv7(v5)vb9	Cv6(v5)v9	Cvm6(v5)v9

Staff Notation

For staff notation, put an up or down to the left of the note and any sharp or flat it might have. Like sharps and flats, an up or down applies to any similar note that follows in the measure. If F is upped, any other F in the same octave inherits the up, but an F# doesn't.

Cross-EDO considerations

In 22-edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19-edo, 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. 22-edo major chords sound ru (7-under) and 19-edo major chords sound yo (5-over).

A chord quality like "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 only major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22-edo, the chord that you need sounds like a ru chord.

In other words, I - VIm - IIm - V - I in just intonation implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in those EDOs in which major sounds yo. Because 22-edo's yo chord 0-7-13 = 0¢-382¢-709¢ is downmajor, 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 22-edo, that 7th sounds zo (7-over). In 19-edo, it sounds gu (5-under). If you want to play this progression in 19-edo, using the downminor (zo) 7th will cause shifts or drifts in the progression.

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, and 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 optionally separated from the first by an "h", a "w", or a "y"

Thus vC# is Deo, pronounced as 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, Dae = ^C, Dai = ^^C, Dao = vC, Dau = vvC, Daa = ^^^C

De = C#, Dee = ^C#, Dei = ^^C#, Deo = vC#, Deu = vvC#, Dea = vvvC#

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

Dae = ^1, Dai = ^^1, Dao = v1, Dau = vv1, Daa = ^^^1

etc.

Imperfect intervals (2nd, 3rd, 6th and 7th): a = mid, e = major, i = aug, o = minor, u = dim

Ra = ~2, Re = M2, Ri = A2, Ro = m2, Ru = d2

Ree = ^M2, Rei = ^^M2, Reo = vM2, Reu = vvM2, Rea = ^^^M2

etc.