View source for Kite's ups and downs notation

== Definition ==
Ups and Downs (or ^v) is a notation system that can notate almost every [[EDO|edo]]. The up symbol "^" and the down symbol "v" indicate raising/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. halfway-augmented, and the mid 5th (~5) is a halfway-diminished 5th. 

Ups and downs can also notate any [[Tour of Regular Temperaments|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 [[Porcupine|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 22edo, 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 22edo by simply writing "22edo" on the top of the page. 

Ups and downs can also be used to notate rank-3 just intonation subgroups such as 2.3.5 or 2.3.7 or 2.3.11. See [[Ups and downs notation for Rank-3 JI]]. 

'''<u>This page only discusses notation of edos.</u>''' However, the notation of chords and chord progressions applies to all situations. For notation of rank-2 and rank-3 temperaments, see the [[pergen|pergens]] article.  

For more on edo notation, see the [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf '''Notation guide for edos 5-72'''], which also covers chord names, slash chords, staff notation, key signatures, and scale trees. 

== Explanation (a 22edo example) ==
To understand the ups and downs notation, let's start with an edo that doesn't need it. 19edo 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 {{nowrap|C, C♯, D♭, D, D♯, E♭, E}} etc. Conventional notation works perfectly with 19edo as long as you remember that C♯ and D♭ are different notes.

In contrast, 22edo is hard to notate because 7 fifths reduces to ''three'' EDOsteps, and the usual chain of fifths {{dash|E♭, B♭, F, C, G, D, A, E, B, F♯, C♯}} etc. creates the scale {{dash|C, D♭, B♯, C♯, D, E♭, F♭, D♯, E, F}}. That's very confusing because B♯–D♭ looks ascending on the page but sounds descending, and a 4:5:6 major chord is written {{dash|C, D♯, G}}, and the 5/4, usually a major third, becomes an augmented second. Some people forgo the chain of fifths for a maximally even scale like {{dash|C, D, E, F, G, A, B, C}}. But that's confusing because G–D and A–E are diminished 5ths. And if your piece is in G or A, that's really confusing. 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-arrow symbol to mean "sharpened by one EDOstep". 22edo can be written {{dash|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 <u>before</u> 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 {{dash|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 and interval arithmetic ===
Ups and downs can be used not only for absolute notation (note names) but also for relative notation (intervals, chords and scales). Relative notation for 22edo intervals: {{dash|P1, m2, ^m2, vM2, M2, m3, ^m3, vM3, M3, P4, ^4/d5, vA4/^d5, A4/v5, P5}} etc. That's pronounced upminor 2nd, downmajor 3rd, etc. You can apply this pattern to any 22edo key. The '''plain''' notes (those without ups or downs) always form a chain of fifths.

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

{| class="wikitable" style="text-align: center;" 
|-
! 
! Interval between<br />two notes
! Note plus<br />an interval
! Sum of two<br />intervals
|-
! conventional
| C to E = M3
| C + M3 = E
| M2 + M2 = M3
|-
! rowspan="2" | with ups<br />and downs
| ^C to E = vM3
| ^C + M3 = ^E
| ^M2 + M2 = ^M3
|-
| C to ^E = ^M3
| C + ^M3 = ^E
| M2 + vM2 = vM3
|-
! (cancelling)
| ^C to ^E = M3
| ^C + vM3 = E
| ^M2 + vM2 = M3
|-
! (combining)
| ^C to vE = vvM3
| ^C + ^M3 = ^^E
| vM2 + vM2 = vvM3
|}

The same logic holds for a note minus an interval (C - vm3 = ^A) or one interval minus another interval (M3 - vM2 = ^M2).

=== "Arrow" as a term for EDOstep ===
Up and down are short for up-arrow and down-arrow, and arrow refers to both. Sometimes the name of a notation symbol comes to mean that which the symbol indicates. Just as "bar" (the vertical line that separates measures) has come to mean "measure", "[[arrow]]" has also come to mean "EDOstep".

=== Enharmonic unisons ===
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 22edo, D♭ can be spelled ^C or vB♯ or even ^^B (double-up B, or '''dup''' B for short, rhymes with "cup"). Respelling is done by adding or subtracting an [[Enharmonic unisons in ups and downs notation|enharmonic unison]], '''EU''' for short. 

From the [[Pergen|pergens]] article: "Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonic unisons are like vanishing commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of EUs needed always equals the difference between the notation's rank and the tuning's rank." 

Since 22edo is rank-1, and conventional notation plus ups and downs is rank-3, two EUs are needed to define the notation: v<sup>3</sup>A1 and vm2. Either EU can be added to or subtracted from any note to respell the note. For example, ^C + vm2 = Db and ^^Eb + v<sup>3</sup>A1 = vE. Any combination of these two EUs is also an EU, for example their sum v<sup>4</sup>M2. Thus ^^F = ^^F + v<sup>4</sup>M2 =  vvG (double-down G, or '''dud''' G for short, rhymes with "cud"). 

=== Larger EDOs ===
In larger edos, triple-arrows, quadruple-arrows, etc. can occur. Up, dup, trup and quup all rhyme, as do dud, trud and quud.
 
{| class="wikitable" style="text-align:center;" 
|+ style="font-size: 105%;" | Symbols and words for multiple arrows
|-
! Written
! Spoken
! Etymology
! 
! Written
! Spoken
! Etymology
|-
| ^
| up
| 
! 1 arrow
| v
| down
| 
|-
| ^^
| dup
| '''<u>d</u>'''ouble-'''<u>up</u>'''
! 2 arrows
| vv
| dud
| rowspan="4" | "-d" for down <br />replaces<br />"-p" for up
|-
| ^^^
| trup
| '''<u>tr</u>'''iple-'''<u>up</u>'''
! 3 arrows
| vvv
| trud
|-
| v>
| quup<br>"kwup"
| '''<u>qu</u>'''adruple-'''<u>up</u>'''
! 4 arrows
| ^<
| quud<br>"kwud"
|-
| >
| quip
| '''<u>qui</u>'''ntuple-u'''<u>p</u>'''
! 5 arrows
| <
| quid
|}

(In addition to dup, trup, etc. there is dub, trip, quad and quin, used for multiple sharps/flats and multiple lifts/drops, e.g. dubsharp or triplift.)

Very large edos can go well beyond 5 arrows. The sequence of names resembles tally counting I, II, III, IIII, <s>||||</s>. But the sequence of ''symbols'' resembles roman numerals I, II, III, IV, V. Thus 4 ups is spoken quup but written v>.
{| class="wikitable" style="text-align:center;"
|-
| up<br />^
| dup<br />^^
| trup<br />^^^
| quup<br />v>
| quip<br />>
| upquip<br />^>
| dupquip<br />^^>
| trupquip<br />^^^>
| quupquip<br />v>>
| quipquip<br />>>
| upquipquip<br />^>>
| dupquipquip<br />^^>>
| trupquipquip<br />^^^>>
| quupquipquip<br />v>>>
| triplequip<br />>>>
|-
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
|-
| down<br />v
| dud<br />vv
| trud<br />vvv
| quud<br />^<
| quid<br /><
| downquid<br />v<
| dudquid<br />vv<
| trudquid<br />vvv<
| quudquid<br />^<<
| quidquid<br /><<
| downquidquid<br />v<<
| dudquidquid<br />vv<<
| trudquidquid<br />vvv<<
| quudquidquid<br />^<<<
| triplequid<br /><<<
|}

Lifts and drops (/ and \) can be used for microinflections of less than an edostep, since they look like part of an arrow.
{| class="wikitable"
|+
|^
|up
| rowspan="2" |arrow
| rowspan="4" |inflection
| rowspan="6" |alteration
|-
|v
|down
|-
|/
|lift
| rowspan="2" |slash
|-
|\
|drop
|-
|#
|sharp
| colspan="2" rowspan="2" |accidental
|-
|b
|flat
|}
For very large edos in which commas like 81/80 and 64/63 are many edosteps, the color notation accidental pairs yo/gu and zo/ru can be "edoized" to stand for a certain number of edosteps. For example, in [[311edo]], 81/80 is 6 edosteps. Thus g means ^> and y means v<. The colors can be combined with arrows as in upyo or dudgu (^y or vvg). Likewise, 64/63 is 7 edosteps, thus r means ^^> and z means vv<.

===Staff Notation===
For staff notation, put an arrow to the left of the note and any sharp or flat it might have. Like sharps and flats, an arrow applies to any similar note that follows in the measure. If C is upped, any other C in the same octave inherits the up. If an up-C is followed by a down-C, the down-arrow replaces the up-arrow.  

But what happens when accidentals are mixed with arrows? What if the key signature makes that upped C be sharp? Or what if there is a C with a sharp just before the upped C? Does the up-arrow override or "cancel" the sharp? And what if an upped C is followed by a sharpened C?

There are several possible ways to handle this issue. The default is the simplest way, to explicitly specify both arrows and accidentals every time. Thus any accidental or arrow cancels any previous ones. An arrow by itself implies a natural sign.

{| class="wikitable" style="text-align:center;"
|-
! rowspan="2" | Start with this
! colspan="6" | Turn it into this
|-
! C
! ^C
! ^^C
! C#
! ^C#
! ^^C#
|-
! C
| &nbsp; &nbsp; &nbsp; &nbsp;
| ^
| ^^
| #
| ^#
| ^^#
|-
! ^C
| <big>♮</big>
| &nbsp; &nbsp; &nbsp; &nbsp;
| ^^
| #
| ^#
| ^^#
|-
! ^^C
| <big>♮</big>
| ^
| &nbsp; &nbsp; &nbsp; &nbsp;
| #
| ^#
| ^^#
|-
! C#
| <big>♮</big>
| ^
| ^^
| &nbsp; &nbsp; &nbsp; &nbsp;
| ^#
| ^^#
|-
! ^C#
| <big>♮</big>
| ^
| ^^
| #
| &nbsp; &nbsp; &nbsp; &nbsp;
| ^^#
|-
! ^^C#
| <big>♮</big>
| ^
| ^^
| #
| ^#
| &nbsp; &nbsp; &nbsp; &nbsp;
|}

See [[Kite Guitar originals#Cancelling rules]] for another way.

For more on staff notation, see the [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Notation Guide for EDOs 5-72]. 

=== Key signatures ===
Key signatures follow the conventional practice, expanded to allow for double-sharps and double flats in some edos. For example, 19edo has the key of Bbb with a key signature of B𝄫 E𝄫 A♭ D♭ G♭ C♭ F♭. Some edos have upped/downed tonics, e.g. 24edo has the key of vD with a key signature of F♯ C♯ (v). The (v) is a "global down" that downs all 7 notes of the vD scale. See also [[Kite Guitar originals#Scales and key signatures]] for the use of '''arrow stacks'''.

=== Placement of the arrow ===
It might seem more natural to place the arrow after the note, for example B^ or Bb^. But the arrow must come first, to make chord names unambiguous. Otherwise B^m could mean either a minor chord rooted on B^ or an upminor chord rooted on B. (Chord names are explained fully below.)

The issue arises because while English normally places the adjective before the noun, it doesn't do so with sharps and flats. A flattened B should logically be called "flat B" not "B flat", and be written bB not Bb. If it were, then it would seem very natural to have the up come first, as in ^bB. This would be the typical English adjective-adjective-noun construction. Instead we must use ^Bb, an unnatural adjective-noun-adjective construction. This issue fortunately arises only for note names. On the staff, the flat comes before the note, so naturally the up comes before the flat. In relative notation, the quality comes before the interval, as in minor 3rd and augmented 4th, or in jazz terms flat 3rd and sharp 4th. So terms like upminor 3rd and downsharp 4th have a natural adjective-adjective-noun construction.

=== Further notes ===
Edo intervals are often written as 7\22. This can also be written as vM3\22. This is useful when comparing edos, e.g. vM3\22 vs. vM3\15.

== Examples: edos 12-24 ==
Sharp-1, flat-2, etc. refer to the [[sharpness]], the number of arrows made by seven 5ths minus four 8ves. All sharp-1 and flat-1 edos can be notated without ups and downs, because the up is exactly equivalent to a sharp or flat. 

A ring is a circle of 5ths. In multi-ring (aka ringy) edos like 14, 15 and 24, a single ring doesn't contain all the edo's notes. In contrast, edos like 12, 19 and 22 are single-ring. It's possible to notate any single-ring edo with conventional notation if notes are permitted to be out of order (e.g. 22edo could have C Db B# C# D). But multi-ring edos absolutely require ups and downs. 

13edo and 18edo aren't compatible with heptatonic notation, because the minor 2nd is descending. Thus the minor 3rd is flatter than the major 2nd, the 4th is flatter than the major 3rd, etc. These edos are best notated using the 2nd best fifth, as 13b and 18b. 

There are four flat-N edos on this list. 16edo and 23edo are flat-1, 18b is flat-2 and 13b is flat-3. There are two ways to notate such edos: with sharp lowering the pitch, and major/aug narrower than minor/dim, or with sharp raising the pitch, and major/aug wider than minor/dim. Both notations are shown. In the 2nd notation, note that a fifth above B is Fb, not F#.  

12edo is sharp-1, thus doesn't need ups and downs. Enharmonic unison: d2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[12-edo|12edo]]<br />{{normal|sharp-1}}
| '''D'''
| D#/Eb
| '''E'''
| '''F'''
| F#/Gb
| '''G'''
| G#/Ab
| '''A'''
| A#/Bb
| '''B'''
| '''C'''
| C#/Db
| '''D'''
|-
| P1
| A1/m2
| M2
| m3
| M3
| P4
| A4/d5
| P5
| m6
| M6
| m7
| M7
| P8
|}

There are two ways to notate 13b-edo. The enharmonic unisons for the 1st notation are ^<sup>3</sup>A1 and vM2. For the 2nd they are v<sup>3</sup>A1 and vm2.

{| class="wikitable" style="text-align:center;"
|-
! rowspan="4" | [[13-edo|13b-edo]]<br />{{normal|flat-3}}
! rowspan="2" | Sharp lowers the pitch,<br />major narrower than minor 
| '''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
|-
! rowspan="2" | Sharp raises the pitch,<br />major wider than minor 
| '''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
|}

Because every 14edo interval is perfect, the quality can be omitted. Sharps and flats can also be omitted. 14edo contains 2 rings of 7edo: an up/down-ring and a plain-ring. Enharmonic unisons: A1 and vvm2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[14-edo|14edo]]<br />{{normal|sharp-0}}
| '''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
|}

15edo contains 3 rings of 5edo: an up-ring, a down-ring, and a plain-ring. Enharmonic unisons: v<sup>3</sup>A1 and m2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[15-edo|15edo]]<br />{{normal|sharp-3}}  
| '''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
|}

16edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic unison: either AA2 or dd2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="4" | [[16-edo|16edo]]<br />{{normal|flat-1}}
! rowspan="2" | Sharp lowers the pitch,<br />major narrower than minor
| '''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
|-
! rowspan="2" | Sharp raises the pitch,<br />major wider than minor
| '''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 is sharp-2 and thus has mid intervals. Enharmonic unisons: vvA1 and vm2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[17edo]]<br />{{normal|sharp-2}}
| '''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/~4/d5
| A4/v5/~5
| P5
| m6
| ~6
| M6
| m7
| ~7
| M7
| P8
|}

18b-edo contains 2 rings of 9edo: an up/down-ring and a plain-ring. There are two ways to notate it. Enharmonic unisons: either ^^A1 and vvM2, or vvA1 and vvm2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="4" | '''[[18-edo|18b-edo]]'''<br />flat-2
! rowspan="2" | Sharp lowers,<br />major is narrower
| '''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
| ^1/vM2
| M2
| ~2
| m2/M3
| ~3
| m3
| ^m3/v4
| P4
| ^4/v5
| P5
| ^5/vM6
| M6
| ~6
| m6/M7
| ~7
| m7
| ^m2/d8
| P8
|-
! rowspan="2" | Sharp raises,<br />major is wider
| '''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
| ^1/vm2
| m2
| ~2
| M2/m3
| ~3
| M3
| ^M3/v4
| P4
| ^4/v5
| P5
| ^5/vm6
| m6
| ~6
| M6/m7
| ~7
| M7
| ^M7/d8
| P8
|}

19edo is sharp-1, thus doesn't need ups and downs. Enharmonic unison: dd2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[19-edo|19edo]]<br />{{normal|sharp-1}}
| '''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
|}

20edo contains 4 rings of 5edo: an up-ring, a down-ring, a dup/dud-ring, and a plain-ring. Enharmonic unisons: v<sup>4</sup>A1 and m2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[20-edo|20edo]]<br />{{normal|sharp-4}}
| '''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/~5
| v5
| P5/m6
| ^m6
| ~6
| vM6
| M6/m7
| ^m7
| ~7
| vM7
| P8
|}

Because every 21edo interval is perfect, the quality can be omitted. 21edo contains 3 rings of 7edo: an up-ring, a down-ring and a plain-ring. Enharmonic unisons: A1 and v<sup>3</sup>m2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[21-edo|21edo]]<br />{{normal|sharp-0}}
| '''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
|}

22edo is sharp-3. Enharmonic unisons: v<sup>3</sup>A1 and vm2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[22-edo|22edo]]<br />{{normal|sharp-3}}
| '''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
| vA1/^m2
| vM2
| M2
| m3
| ^m3
| vM3
| M3
| P4
| ^4/d5
| vA4/^d5
| A4/v5
| P5
| etc.
|}

23edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic unison: either A<sup>3</sup>2 or d<sup>3</sup>2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="4" | [[23-edo|23edo]]<br />{{normal|flat-1}}
! rowspan="2" | Sharp lowers,<br />major is narrower
| '''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
|-
! rowspan="2" | Sharp raises,<br />major is wider
| '''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
|}

24edo contains 2 rings of 12edo: an up/down-ring and a plain-ring. Enharmonic unisons: vvA1 and d2.

{| class="wikitable" style="text-align: center;"
|-
! rowspan="2" | [[24-edo|24edo]]<br />{{normal|sharp-2}}
| '''D'''
| ^D/vEb
| D#/Eb
| ^D#/vE
| '''E'''
| ^E/vF
| '''F'''
| ^F
| F#/Gb
| vG
| '''G'''
| ^G/vAb
| G#/Ab
| ^G#/vA
| '''A'''
| etc.
|-
| P1
| ^1/vm2
| A1/m2
| ~2
| M2
| ^M2/vm3
| m3
| ~3
| M3
| ^M3/v4
| P4
| ^4/~4
| A4/d5
| v5/~5
| P5
| 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. Alterations always come last in the chord name. Examples:

*[[19edo chords]]
*[[22edo chords]]
*[[24edo chord names]]
*[[31edo chord names]]
*[[41edo chord names]]
*[[Kite Guitar chord shapes (downmajor tuning)]]

In [[Sharpness|sharp-0]] edos aka perfect edos (7, 14, 21, 28 and 35), every interval is perfect, and there is no major or minor. In the following lists 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".

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, <u>'''never use lower case roman numerals'''</u> for minor chords, because both vIIm and VIIm would be written vii. 

=== Triads ===
<span style="display: block; text-align: left;">The major chord and various alterations of it:</span>
*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 sharp-2 edos, C~ = "C mid")
* C vvE G = Cvv = "C dud" or "C dudmajor" (in sharp-4 edos, C~ = "C mid", in sharp-6 edos, C^~ = "C upmid")
This table shows how altering the 3rd or the 5th affects the name of the triad. The conventional abbreviations for aug and dim are + and <sup>o</sup>. These are rather cryptic, and can be replaced with the more obvious and intuitive a and d. Likewise the symbols Δ and − can be replaced with M and m.

{| class="wikitable" style="text-align:center;" 
|-
! 
! Major
! Minor
! sus4
! sus2
! colspan="2" | Augmented
! colspan="2" | Diminished
|-
! what's downed
! C E G
! C Eb G
! C F G
! C D G
! colspan="2" | C E G#
! colspan="2" | C Eb Gb
|-
! nothing
| C
| Cm
| C4
| C2
| Ca
| C+
| Cd
| C<sup>o</sup>
|-
! 3rd
| Cv
| Cvm
| Cv4
| Cv2
| Cva
| Cv+
| Cvd
| Cv<sup>o</sup>
|-
! 5th
| C(v5)
| Cm(v5)
| C4(v5)
| C2(v5)
| Ca(v5)
| C+(v5)
| Cd(v5)
| C<sup>o</sup>(v5)
|-
! 3rd, 5th
| Cv(v5)
| Cvm(v5)
| Cv4(v5)
| Cv2(v5)
| Cva(v5)
| Cv+(v5)
| Cvd(v5)
| Cv<sup>o</sup>(v5)
|}

Note that the dim chord is a triad, not a tetrad. A dim tetrad should always be written C<sup>o</sup>7, never C<sup>o</sup>. In jazz, the 7 is omitted because dim triads are so much rarer than dim tetrads. But ups and downs notation is meant to work for all genres, not just jazz. So the dim triad and the dim tetrad need different names.

Many edos have notes between the major 3rd and the perfect 4th, creating triads impossible in 12edo, such as:
*C Fb G = C(d4) or C(b4) = "C dim-four" or "C sus-flat-four"
*C E# G = C(a3) or C(#3) = "C aug-three" or "C sus-sharp-three"
*C Ebb G = C(d3) or C(bb3) = "C dim-three" or "C sus-double-flat-three"
*C D# G = C(a2) or C(#2) =  "C aug-two" or "C sus-sharp-two"
The "sus" is needed so that C(#2) doesn't sound like C#2, which is C# D# G#. 

=== Global arrows ===
A global arrow occurs between the chord root and the conventional chord type (e.g. C^m7). It 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. Mnemonic: every other note of a stacked-3rds chord is affected: '''<u>6th</u>''' - root - '''<u>3rd</u>''' - 5th - '''<u>7th</u>''' - 9th - '''<u>11th</u>''' - 13th. Note that the 6th is affected, but the 13th is not. 

The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd. Furthermore, in larger 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.

A 2nd or 4th in a sus chord is also affected: C4 = C F G but Cv4 = C vF G = "C down-four" or "C sus-down-four". But Cv7(4) = C F G vBb 

Every conventional chord can accept a global arrow, with one exception: it's pointless for a C5 chord, because there is no 3rd, 6th or 7th to alter. Thus Cv5 is invalid. But C(v5) is valid, and if someone says "C down five", it means C(v5) = C E vG.

=== 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. An added 7th is usually preceded by a comma (the actual punctuation mark, not an interval), which is spoken as "add":
*C E G Bb = C7 = "C seven" (conventional chord)
*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" (global down)
All 7th chords follow this same pattern. Likewise, if the 6th is not a perfect 4th or aug 4th above the 3rd, it's an add-6 chord. Permitting add-7 chords has the added benefit that the wordy "minor-7 flat-5" and the illogical "half-dim" can both be replaced with "dim add-7", written Cd,7.  

In the table below, if a chord is '''bolded''', the comma punctuation is <u>not</u> spoken as "add".   
{| class="wikitable" style="text-align:center;" 
|-
! 
! maj7
! dom7
! min7
! colspan="3" | dim-add-7 or min7(b5) or half-dim
! colspan="2" | dim7
! maj6
! min6
|-
! what's downed
! C E G B
! C E G Bb
! C Eb G Bb
! colspan="3" | C Eb Gb Bb
! colspan="2" | C Eb Gb Bbb
! C E G A
! C Eb G A
|-
! nothing
| CM7
| C7
| Cm7
| Cd,7
| Cm7(b5)
| C<sup>ø</sup>
| Cd7
| C<sup>o</sup>7
| C6
| Cm6
|-
! 3rd
| Cv,M7
| Cv,7
| Cvm,7
| Cvd,7
| Cvm,7(b5)
| C<sup>ø</sup>(v3)
| '''Cvd,d7'''
| '''Cv<sup>o</sup>,d7'''
| Cv,6
| Cvm,6
|-
! 5th
| CM7(v5)
| C7(v5)
| Cm7(v5)
| Cd,7(v5)
| Cm7(vb5)
| C<sup>ø</sup>(v5)
| Cd7(v5)
| C<sup>o</sup>7(v5)
| C6(v5)
| Cm6(v5)
|-
! 6th/7th
| C,vM7
| C,v7
| Cmv7
| Cdv7
| Cmv7(b5)
| C<sup>ø</sup>(v7)
| Cdvd7
| C<sup>o</sup>vd7
| C,v6
| Cmv6
|-
! 3rd, 5th
| Cv,M7(v5)
| Cv,7(v5)
| Cvm,7(v5)
| Cvd,7(v5)
| Cvm,7(vb5)
| C<sup>ø</sup>(v3v5)
| '''Cvd,d7(v5)'''
| '''Cv<sup>o</sup>,d7(v5)'''
| Cv,6(v5)
| Cvm,6(v5)
|-
! 3rd, 6th/7th
| CvM7
| Cv7
| Cvm7
| Cvdv7
| Cvm7(b5)
| Cv<sup>ø</sup>
| Cvd7
| Cv<sup>o</sup>7
| Cv6
| Cvm6
|-
! 5th, 6th/7th
| C,vM7(v5)
| C,v7(v5)
| Cmv7(v5)
| Cdv7(v5)
| Cmv7(vb5)
| C<sup>ø</sup>(v5v7)
| Cdvd7(v5)
| C<sup>o</sup>vd7(v5)
| C,v6(v5)
| Cm,v6(v5)
|-
! 3rd, 5th, 6th/7th
| CvM7(v5)
| Cv7(v5)
| Cvm7(v5)
| Cvdv7(v5)
| Cvm7(vb5)
| Cv<sup>ø</sup>(v5)
| Cvd7(v5)
| Cv<sup>o</sup>7(v5)
| Cv6(v5)
| Cvm6(v5)
|}

Various unusual tetrads:
*C vE G ^Bb = Cv^7 = "C down up-seven" (in sharp-2 edos 17, 24, 31, etc. C~7 = "C mid-seven")
*C E G A# = C,#6 or C,A6 = "C add sharp-six" or "C add aug-six"
*C E G Ab = C,b6 or C,m6 = "C add flat-six" or "C add minor-six"
*C E G Bbb = C,bb7 or C,d7 = "C add double-flat-seven" or "C add dim-seven" (19edo's 4:5:6:7 chord)
*C E G B# = 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 ===
As in conventional chord naming, a sharp-9 or flat-9 chord is always named as a 7th chord with an added 9th. Thus B D# F# A C is named B7b9 (not Bb9 which would be Bb D F A C). Likewise C#7b9 not C#b9, even thought the latter is clearly the same flat-9 chord as the former. Likewise Cm7b9 not Cmb9, etc.

Double alterations need only a single pair of parentheses, e.g. C E vG vB D is named CM9(v5v7). Double additions mostly need only a single comma, e.g. C E G vBb vD is named C,v7v9. But certain 6/9 chords require two commas. In '''bolded''' 6/9 chords, the comma between the 6 and the 9 is <u>not</u> spoken as "add". However any comma before "6" is, e.g. Cv,6,9 is "C down add six nine".

{| class="wikitable" style="text-align:center;" 
|-
! 
! 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
| C7b9
| '''C6,9'''
| '''Cm6,9'''
|-
! 3rd
| Cv,9
| CM9(v3)
| C9(v3)
| Cm9(v3)
| Cv,7b9
| '''Cv,6,9'''
| '''Cvm,6,9'''
|-
! 5th
| C,9(v5)
| CM9(v5)
| C9(v5)
| Cm9(v5)
| C7b9(v5)
| '''C6,9(v5)'''
| '''Cm6,9(v5)'''
|-
! 6th/7th
|  ------
| CM9(v7)
| C9(v7)
| Cm9(v7)
| C,v7b9
| '''C,v6,9'''
| '''Cmv6,9'''
|-
! 9th
| C,v9
| CM7v9
| C7v9
| Cm7v9
| C7vb9
| C6v9
| Cm6v9
|-
! 3rd, 5th
| Cv,9(v5)
| CM9(v3v5)
| C9(v3v5)
| Cm9(v3v5)
| Cv,7b9(v5)
| '''Cv,6,9(v5)'''
| '''Cvm,6,9(v5)'''
|-
! 3rd, 6th/7th
|  ------
| CvM9
| Cv9
| Cvm9
| Cv7b9
| '''Cv6,9'''
| '''Cvm6,9'''
|-
! 3rd, 9th
| Cv,v9
| Cv,M7v9 or<br>CM7v9(v3)
| Cv,7v9 or<br>C7v9(v3)
| Cvm,7v9 or<br>Cm7v9(v3)
| Cv,7vb9 or<br>C7vb9(v3)
| Cv,6v9 or<br>C6v9(v3)
| Cvm,6v9 or<br>Cm6v9(v3)
|-
! 5th, 6th/7th
|  ------
| CM9(v5v7)
| C9(v5v7)
| Cm9(v5v7)
| C,v7b9(v5)
| '''C,v6,9(v5)'''
| '''Cm,v6,9(v5)'''
|-
! 5th, 9th
| C,v9(v5)
| CM7v9(v5)
| C7v9(v5)
| Cm7v9(v5)
| C7vb9(v5)
| C6v9(v5)
| Cm6v9(v5)
|-
! 6th/7th, 9th
|  ------
| C,vM7v9
| C,v7v9
| Cmv7v9
| C,v7vb9
| C,v6v9
| Cmv6v9
|-
! 3rd, 5th, 6th/7th
|  ------
| CvM9(v5)
| Cv9(v5)
| Cvm9(v5)
| Cv7b9(v5)
| '''Cv6,9(v5)'''
| '''Cvm6,9(v5)'''
|-
! 3rd, 5th, 9th
| Cv,v9(v5)
| Cv,M7v9(v5) or<br>CM7v9(v3v5)
| Cv,7v9(v5) or<br>C7v9(v3v5)
| Cvm,7v9(v5) or<br>Cm7v9(v3v5)
| Cv,7vb9(v5) or<br>C7vb9(v3v5)
| Cv,6v9(v5) or<br>C6v9(v3v5)
| Cvm,6v9(v5) or<br>Cm6v9(v3v5)
|-
! 3rd, 6th/7th, 9th
|  ------
| CvM7v9
| Cv7v9
| Cvm7v9
| Cv7vb9
| Cv6v9
| Cvm6v9
|-
! 5th, 6th/7th, 9th
|  ------
| C,vM7v9(v5)
| C,v7v9(v5)
| Cmv7v9(v5)
| C,v7vb9(v5)
| C,v6v9(v5)
| Cmv6v9(v5)
|-
! 3rd, 5th, 6th/7th, 9th
|  ------
| CvM7v9(v5)
| Cv7v9(v5)
| Cvm7v9(v5)
| Cv7vb9(v5)
| Cv6v9(v5)
| Cvm6v9(v5)
|}

=== Rules for punctuation usage ===
Tetrads, pentads, etc. often require a comma (the actual punctuation mark) to ensure correct parsing of the chord name. Only use a comma when needed, to reduce clutter and standardize chord names. A comma is needed in Cv,7 = C vE G Bb because omitting it makes Cv7 = C vE G vBb, a different chord. But C7,v9 is incorrect because C7v9 is the same chord.

The rule is, omit the comma unless doing so changes the chord. This simple rule suffices in most situations. What follows is a detailed analysis, designed to aid in writing computer code that automates chord naming.

A comma separates an added note and prevents it from merging with what comes before it. The comma is unneeded in C7v9 because the 7 can't merge with the down to make a 7v. But Cm,7 is incorrect even though the m and the 7 can merge, because Cm7 is the same chord.

The various components of a chord name are either numbers (for the 6th, 7th, 9th, etc.) or adjectives (up, down, mid, sharp, flat, major, minor, aug and dim). These adjectives usually modify the following number, but they sometimes modify the preceding root, e.g. Caug or C#7. Up, down and mid can't modify the preceding root. 

A comma is always needed to separate a number from a number (Cv6,9). It's usually needed to separate an adjective from a number (Cv,7). The only exception is for certain conventional chords like Cm7 where separation is unneeded. A comma is always needed to separate the root of a plain major chord from an adjective (D,v7) or a number (Eb,9). It's never needed to separate a number from an adjective (C7^9). It's needed to separate an adjective from an adjective only if the two adjectives could apply to a single noun. There are six types of such adjective pairs.

*up followed by any adjective except down (C^,^9 or C^,~7 or C^,#9 or C^,b9 or C^,M7 or C^,m6 or C^,a7 or C^,d7)
*down followed by any adjective except up
*sharp followed by sharp (C#,#9)
*flat followed by flat (Bb,b9)
*aug followed by aug (Ca,a7)
*dim followed by dim (Cd,d9)

No other adjective pair can apply to a single noun, thus the comma is omitted:

* Cv^9 = C vE G ^D (an interval can't be both upped and downed)
* CmM7 = C Eb G B (an interval can't be both minor and major) *
* Cma7 = C Eb Gb B# (an interval can't be both minor and aug) **
* Cm#11 = C Eb G F# (an interval can't be both minor and sharp)
* Cvmm6 = C vEb G Ab (an interval can't be doubly minor)
* Cmv7 = C Eb G vBb (an interval can be downminor, but it can't be minordown)
* C~v7 = C vvE G vBb in a sharp-4 edo (an interval can be downmid, but it can't be middown)
* C~~9 = C vvE G vvD in a sharp-4 edo (an interval can't be doubly mid)
<nowiki>*</nowiki>But beware of the minor-major chord. CvmM7 means C vEb G vB and Cvm,M7 means C vEb G B.

<nowiki>**</nowiki>But since Cma7 can be read as an alternate spelling of Cmaj7, adding a comma is wise: Cm,a7.

In the spoken name, a comma is almost always pronounced as "add". The only exceptions are:

* a comma separating two numbers: C6,9 is spoken as "C six nine"
* a comma separating two ups or two downs: Cv,v9 is spoken as "C-down down-nine", since Cvv9 would be "C dud-nine"
* a comma separating two sharps or two flats: C#,#9 is "C sharp sharp-nine" since C##9 would be "C double-sharp nine"
* a comma separating two augs or two dims: Cvd,d7 is "C down-dim dim-seven", since Cvdd7 would be "C down-double-dim-seven"

Of course, there's no great harm in saying "add" when it isn't strictly needed.
{| class="wikitable"
|+ style="font-size: 105%;" | When to write a comma or say "add"
|-
! colspan="2" rowspan="2" |
! colspan="2" | Component after the possible comma
|-
! adjective
! number
|-
! rowspan="3" | Component<br />before the<br />possible<br />comma
! root
| comma always<br />"add" always
| comma always<br />"add" always
|-
! adjective
| comma sometimes<br />"add" sometimes if comma,<br>never if no comma
| comma usually<br />"add" always if comma,<br>never if no comma
|-
! number
| comma never<br />"add" never
| comma always<br />"add" never
|}

More examples, in which the comma is almost always spoken as "add":

*B9 = B D# F# AvC#
*B,9 = B D# F# C#
*Bb9 = Bb D F Ab C
* Bb,9 = Bb D F C
*B,b9 = B D# F# C
*B7b9 = B D# F# A C
* Bbb9 = Bbb Db Fb Abb Cb
*Bbb,9 = Bbb Db Fb Cb
*Bb,b9 = Bb D F Cb (no "add", "B flat flat-nine")
* B,bb9 = B D# F# Cbb

== 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 ru (7-under) and 19edo 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 22edo, 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 22edo's yo chord 0-7-13 = 0¢-382¢-709¢ is <u>down</u>major, it doesn't work in that progression.

Another example: I7 - bVII7 - IV7 - I7. To play this progression without shifts or drifts, the 7th in the I7 chord must be a minor 7th. in 22edo, that 7th sounds zo (7-over, thus 7/4). In 19edo, it sounds gu (5-under, thus 9/5).

== Ups and downs solfege ==
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down. See [[Uniform solfege|Uniform Solfege]].

== See also ==
* [[Enharmonic unisons in ups and downs notation]]
* [[Alternative symbols for ups and downs notation]]
* [[Lambda ups and downs notation]]

Ups and downs notation was invented by [[Kite Giedraitis]] in early 2014.

{{Navbox notation}}
{{todo|intro}}

[[Category:Ups and downs notation| ]] <!-- main article -->
[[Category:Notation]]