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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
== Definition ==
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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.
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-08-31 19:54:31 UTC</tt>.<br>
: The original revision id was <tt>557847281</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">="Ups and Downs" Notation=


Ups and Downs is a notation system developed by Kite that works very well with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol "^" and the down symbol "v". There's also the mid symbol "~" which undoes ups and downs.
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.  


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.
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]].  


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 too 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!
'''<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.


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.
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.  


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.
== 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.
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 (tempered) pythagorean chain of fifths.
=== 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.
|}


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.
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.


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.
{| 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
|}


__**Interval arithmetic**__
24edo contains 2 rings of 12edo: an up/down-ring and a plain-ring. Enharmonic unisons: vvA1 and d2.
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:
{| class="wikitable" style="text-align: center;"
m3-m7-P4-P1
|-
P1-P5-M2
! rowspan="2" | [[24-edo|24edo]]<br />{{normal|sharp-2}}
Line up the lower P1 with m3 and look for what lies above M2.
| '''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.
|}


22-EDO interval arithmetic works out very neatly. Ups and downs are just added in:
== Chords and chord progressions==
C + M3 = E, C + vM3 = Ev, C^ + M3 = E^
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:
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:
*[[19edo chords]]
C Db Db^ Dv D Eb Eb^ Ev E F Gb Gb^ Gv G Ab Ab^ Av A Bb Bb^ Bv B C
*[[22edo chords]]
Here's our fifths: C-G, Db-Ab, Db^-Ab^, Dv-Av, D-A, etc. Most fifths *look* like fifths and are easy to find. So do the 4ths. Our 4\22 maj 2nds are C-D, Db-Eb, Db^-Eb^, Dv-Ev, D-E, Eb-F, good until we reach Eb^-Gb, which looks like a min 3rd. Here's this scale's chain of 5ths:
*[[24edo chord names]]
*[[31edo chord names]]
*[[41edo chord names]]
*[[Kite Guitar chord shapes (downmajor tuning)]]


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


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. Because 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary. Here's 22-EDO in pentatonic notation:
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.  


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


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.
{| 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)
|}


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


__**Staff Notation**__
Many edos have notes between the major 3rd and the perfect 4th, creating triads impossible in 12edo, such as:
For staff notation, just put an up or down to the left of the note and any standard accidental it might have. To write Db^ followed by Db in the same measure, use the mid sign: Db^ Db~. All 22 possible keys can be written out. The tonic is always a mid note, i.e. not up or down. Just as conventionally each black key produces both a sharp key and a flat key (Db major and C# minor), each of the 15 black keys of 22-EDO produces both, and there are 37 possible keys. The 2 most remote are Bbbb and F###, and triple-sharps and triple-flat keys seem rather extreme. Avoiding those, we have 35 possible tonics that run from Fbb to Bx. Some of the key signatures will have double-sharps or double-flats in them, or even triple-sharps.
*C Fb G = C(d4) or C(b4) = "C dim-four" or "C sus-flat-four"
C: no sharps
*C E# G = C(a3) or C(#3) = "C aug-three" or "C sus-sharp-three"
C#: 7 sharps
*C Ebb G = C(d3) or C(bb3) = "C dim-three" or "C sus-double-flat-three"
G#: 6 sharps, 1 double-sharp on F
*C D# G = C(a2) or C(#2) =  "C aug-two" or "C sus-sharp-two"
D#: 5 sharps, 2 double-sharps on F and C
The "sus" is needed so that C(#2) doesn't sound like C#2, which is C# D# G#.
B#: 2 sharps, 5 double-sharps on F , C, G, D and A
Bx: 2 double-sharps on E and B, 5 triple-sharps on F, C, G, D and A


__**Other EDOs**__
=== Global arrows ===
So that's 22-EDO. This notation works for almost every EDO. 9, 11, 16, and 23 have weird interval arithmetic because of the narrow fifth, but they can be notated. 13 and 18 are best notated using the narrower of the 2 possible fifths, which makes them like 9, 11, 16 and 23. 8-EDO is hard. It works with pentatonic notation, if you don't mind learning pentatonic interval arithmetic. (Big if!)
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.  


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


This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy. This section will cover sweet EDOs and the other categories will be covered in other sections.
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


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.
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.


**__17-EDO__:**
=== Sixth and seventh chords ===
Black and white keys: C _ _ D _ _ E F _ _ G _ _ A _ _ B C
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":
Relative notation: P1 m2 vM2 M2 m3 vM3 M3 P4 d5 vP5 P5 m6 vM6 M6 m7 vM7 M7 P8
*C E G Bb = C7 = "C seven" (conventional chord)
or with upminors instead of downmajors: P1 m2 ^m2 M2 m3 ^m3 M3 P4 d5 ^d5 P5 m6 ^m6 M6 m7 ^m7 M7 P8
*C vE G Bb = Cv,7 = "C down add-seven"
The d5 could instead be an A4: P4 ^P4 A4 P5 or P4 vA4 A4 P5
*C E G vBb = C,v7 = "C add down-seven"
Many other variations are possible, much freedom of spelling.
*C vE G vBb = Cv7 = "C down seven" (global down)
In C, with downmajors: C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C
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 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 JI as easily because of the poor approximation of the 5-limit. However major = red or fifthward white and minor = blue or fourthward white.
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)
|}


**__24-EDO__:**
Various unusual tetrads:
black and white keys: C _ _ _ D _ _ _ E _ F _ _ _ G _ _ _ A _ _ _ B _ C
*C vE G ^Bb = Cv^7 = "C down up-seven" (in sharp-2 edos 17, 24, 31, etc. C~7 = "C mid-seven")
Relative notation: P1 vm2 m2 vM2 M2 vm3 m3 vM3 M3 vP4 P4 ^P4 d5 vP5 P5 etc.
*C E G A# = C,#6 or C,A6 = "C add sharp-six" or "C add aug-six"
Many alternate spellings available, for example vm3 = ^M2, vM3 = ^m3, ^P4 = vd5, etc.
*C E G Ab = C,b6 or C,m6 = "C add flat-six" or "C add minor-six"
In C: C Dbv Db Dv D Ebv Eb Ev E Fv F F^ Gb Gv G etc.
*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"


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


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


JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber.
{| 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)
|}


**__31-EDO__:**
=== Rules for punctuation usage ===
Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C
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.
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).


=__Naming Chords__=
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.


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.
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 quality of an interval is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc. Major is always wider than minor, so if the edo's 5th is narrower than 4\7, as is in 16edo, major is not fifthwards but fourthwards:
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.


The chain of fifths in fourthwards EDOs:
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.
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 are flipped harmonically. Perfect and natural are unaffected. Interval arithmetic in fourthwards edos: First flip the meaning, then perform normal arithmetic, then flip the meaning again:
*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)
M2 + M2 --&gt; m2 + m2 = dim3 --&gt; aug3
*down followed by any adjective except up
D to F# --&gt; D to Fb = dim3 --&gt; aug3
*sharp followed by sharp (C#,#9)
Eb + m3 --&gt; E# + M3 = Gx --&gt; Gbb
*flat followed by flat (Bb,b9)
*aug followed by aug (Ca,a7)
*dim followed by dim (Cd,d9)


==__22edo chord names__==
No other adjective pair can apply to a single noun, thus the comma is omitted:


Chord names are based entirely on the ups/downs interval names:
* 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.


0\22 = P1
<nowiki>**</nowiki>But since Cma7 can be read as an alternate spelling of Cmaj7, adding a comma is wise: Cm,a7.
1\22 = m2
2\22 = ^m2
3\22 = vM2
4\22 = M2
5\22 = m3
6\22 = ^m3
7\22 = vM3
8\22 = M3
9\22 = P4
10\22 = ^P4, d5
11\22 = vA4, ^d5
12\22 = A4, vP5
13\22 = P5
14\22 = m6
15\22 = ^m6
16\22 = vM6
17\22 = M6
18\22 = m7
19\22 = ^m7
20\22 = vM7
21\22 = M7
22\22 = P8


These are pronounced "downmajor second", "upminor third", etc. For 4ths and 5ths, "perfect" is implied and can be omitted: ^P4 = "up-four" and vP5 = "down-five". In larger edos there may be "down-octave", "up-unison", etc.
In the spoken name, a comma is almost always pronounced as "add". The only exceptions are:


0-7-13-18 in C is "C vM,m7", pronounced "C downmajor, minor seventh". The space between the C and the down symbol is needed because Cv is a note, and "Cv M,m7" is a different chord. That chord is pronounced "C down, major, minor 7th", so you have to "speak the space". I see the need for a space as a small drawback, but can't think of a good way to avoid it. Alternatively, a comma could be used: C,vM,m7 vs. Cv,M,m7. The extra space/comma isn't needed when there's no usp or downs immediately after the note name, e.g. Cm.
* 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"


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


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


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


Sus chords: "sus" means the 3rd is replaced by the named note, a 2nd or 4th. "Sus4" means a perfect 4th, and sus^4 means an up-perfect 4th. Some edos would have susv4, susvv4, etc. "Sus2" means a major 2nd. In most edos, M2 is always a perfect 4th below the perfect 5th, see 16edo below for an exception.
== 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).


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


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


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


0-5-13-18 = m7
== Ups and downs solfege ==
0-6-13-19 = ^m7
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down. See [[Uniform solfege|Uniform Solfege]].
0-7-13-20 = vM7
0-8-13-21 = M7


0-5-13-16 = m,vM6
== See also ==
0-8-13-19 = M,^m7
* [[Enharmonic unisons in ups and downs notation]]
0-7-13-18-26 = vM,m7,M9
* [[Alternative symbols for ups and downs notation]]
0-7-13-18-26-32 = vM,m7,M9,^P11
* [[Lambda ups and downs notation]]


You can write out chord progressions using the ups/downs notation for note names. Here's the first 2 bars of Tibia:
Ups and downs notation was invented by [[Kite Giedraitis]] in early 2014.
G vM7,-5 = "G downmajor seven, no five""
Eb^ vM,M9 = "E flat up, downmajor, major nine"
Gm7,-5 (no space needed) = "G minor seven, no five"
A vM,m7 = "A downmajor, minor seven"


To use relative notation, first write out all possible 22edo chord roots relatively. This is just the interval notation with Roman numerals instead of Arabic, # for aug, and b for minor. Dim from perfect is b, but dim from minor is bb. I've also included more enharmonic equivalents like ^I = bII.
{{Navbox notation}}
I ^I/bII v#I/^bII #I/vII II ^II/bIII v#II/^bIII #II/vIII III IV ^IV/bV v#IV/^bV #IV/vV
{{todo|intro}}
V ^V/bVI v#V/^bVI #V/vVI VI ^VI/bVII v#VI/^bVII #VI/vVII VII
These are pronounced "down-two", "up-flat-three", "down-sharp-four", etc.


Here's the Tibia chords. No spaces are needed because ups and downs are always leading, never trailing.
[[Category:Ups and downs notation| ]] <!-- main article -->
IvM7,-5 = "one downmajor seven, no five"
[[Category:Notation]]
^bVIvM,M9 = "up-flat six downmajor, major nine"
Im7,-5 = "one minor seven, no five"
IIvM,m7 = "two downmajor, minor seven"</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Ups and Downs Notation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x&amp;quot;Ups and Downs&amp;quot; Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&amp;quot;Ups and Downs&amp;quot; Notation&lt;/h1&gt;
&lt;br /&gt;
Ups and Downs is a notation system developed by 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 &amp;quot;^&amp;quot; and the down symbol &amp;quot;v&amp;quot;. There's also the mid symbol &amp;quot;~&amp;quot; which undoes ups and downs.&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
In contrast, 22-EDO is hard to notate because 7 fifths are &lt;u&gt;three&lt;/u&gt; 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 too 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!&lt;br /&gt;
&lt;br /&gt;
The solution is to use the sharp symbol to mean &amp;quot;raised by 7 fifths&amp;quot;, and to use the up symbol &amp;quot;^&amp;quot; to mean &amp;quot;sharpened by one EDO-step&amp;quot;. 22-EDO can be written C-Db-Db^-Dv-D-Eb-Eb^-Ev-E-F etc. The notes are pronounced &amp;quot;D-flat-up, D-down&amp;quot;, etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp.&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
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 &amp;quot;upminor 2nd, downmajor 3rd&amp;quot;, etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The mid notes always form a (tempered) pythagorean chain of fifths.&lt;br /&gt;
&lt;br /&gt;
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 &lt;a class="wiki_link" href="/Kite%27s%20color%20notation"&gt;Kite's color notation&lt;/a&gt; for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.&lt;br /&gt;
&lt;br /&gt;
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^^ (&amp;quot;B double-up&amp;quot;). However avoid using both C# and Db, as the ascending Db-C# looks descending.&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;Interval arithmetic&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
In ups and downs notation, as in conventional notation, the chain of fifths runs:&lt;br /&gt;
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.&lt;br /&gt;
This chain can be expressed in relative notation:&lt;br /&gt;
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.&lt;br /&gt;
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 &amp;quot;raised by 4 fifths&amp;quot; and E is 4 fifths away from C. Likewise, C + M3 = E.&lt;br /&gt;
C - G - D - A - E&lt;br /&gt;
P1-P5-M2-M6-M3&lt;br /&gt;
&lt;br /&gt;
To add any two intervals, superimpose two copies of the relative chain. m3 + M2 = P4:&lt;br /&gt;
m3-m7-P4-P1&lt;br /&gt;
P1-P5-M2&lt;br /&gt;
Line up the lower P1 with m3 and look for what lies above M2.&lt;br /&gt;
&lt;br /&gt;
22-EDO interval arithmetic works out very neatly. Ups and downs are just added in:&lt;br /&gt;
C + M3 = E, C + vM3 = Ev, C^ + M3 = E^&lt;br /&gt;
D-F# is a M3, Dv-F#v = M3&lt;br /&gt;
M2 + m2 = m3, M2 + ^m2 = ^m3, vM2 + m2 = vm3&lt;br /&gt;
&lt;br /&gt;
There are some exceptions. Take this scale:&lt;br /&gt;
C Db Db^ Dv D Eb Eb^ Ev E F Gb Gb^ Gv G Ab Ab^ Av A Bb Bb^ Bv B C&lt;br /&gt;
Here's our fifths: C-G, Db-Ab, Db^-Ab^, Dv-Av, D-A, etc. Most fifths *look* like fifths and are easy to find. So do the 4ths. Our 4\22 maj 2nds are C-D, Db-Eb, Db^-Eb^, Dv-Ev, D-E, Eb-F, good until we reach Eb^-Gb, which looks like a min 3rd. Here's this scale's chain of 5ths:&lt;br /&gt;
&lt;br /&gt;
Gb^ Db^ Ab^ Eb^ Bb^ Gb Db Ab Eb Bb F C G D A E B Gv Dv Av Ev Bv&lt;br /&gt;
&lt;br /&gt;
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. Because 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary. Here's 22-EDO in pentatonic notation:&lt;br /&gt;
&lt;br /&gt;
Gx Dx Ax F# C# G# D# A# F C G D A Fb Cb Gb Db Ab Fbb Cbb Gbb Dbb&lt;br /&gt;
C C# Dbb Db D D# Dx Fbb Fb F F# Gbb Gb G G# Gx Ab A A# Ax Cbb Cb C&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
So the chain of fifths has a few spots to watch out for. You have to remember that B-something to G-something is sometimes a fifth, sometimes a sixth. A little tricky, but manageable. Analogous to 12-ET, where G# to Eb is a fifth that looks like a sixth.&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;Staff Notation&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
For staff notation, just put an up or down to the left of the note and any standard accidental it might have. To write Db^ followed by Db in the same measure, use the mid sign: Db^ Db~. All 22 possible keys can be written out. The tonic is always a mid note, i.e. not up or down. Just as conventionally each black key produces both a sharp key and a flat key (Db major and C# minor), each of the 15 black keys of 22-EDO produces both, and there are 37 possible keys. The 2 most remote are Bbbb and F###, and triple-sharps and triple-flat keys seem rather extreme. Avoiding those, we have 35 possible tonics that run from Fbb to Bx. Some of the key signatures will have double-sharps or double-flats in them, or even triple-sharps.&lt;br /&gt;
C: no sharps&lt;br /&gt;
C#: 7 sharps&lt;br /&gt;
G#: 6 sharps, 1 double-sharp on F&lt;br /&gt;
D#: 5 sharps, 2 double-sharps on F and C&lt;br /&gt;
B#: 2 sharps, 5 double-sharps on F , C, G, D and A&lt;br /&gt;
Bx: 2 double-sharps on E and B, 5 triple-sharps on F, C, G, D and A&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;Other EDOs&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
So that's 22-EDO. This notation works for almost every EDO. 9, 11, 16, and 23 have weird interval arithmetic because of the narrow fifth, but they can be notated. 13 and 18 are best notated using the narrower of the 2 possible fifths, which makes them like 9, 11, 16 and 23. 8-EDO is hard. It works with pentatonic notation, if you don't mind learning pentatonic interval arithmetic. (Big if!)&lt;br /&gt;
&lt;br /&gt;
EDOs come in 5 categories, based on the size of the fifth:&lt;br /&gt;
supersharp EDOs, with fifths wider than 720¢&lt;br /&gt;
pentatonic EDOs, with a fifth = 720¢&lt;br /&gt;
&amp;quot;sweet&amp;quot; EDOs, so-called because the fifth hits the &amp;quot;sweet spot&amp;quot; between 720¢ and 686¢&lt;br /&gt;
heptatonic EDOs, with a fifth = four sevenths of an octave = 686¢&lt;br /&gt;
superflat EDOs or Mavila EDOs, with a fifth less than 686¢&lt;br /&gt;
&lt;br /&gt;
This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy. This section will cover sweet EDOs and the other categories will be covered in other sections.&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;17-EDO&lt;/u&gt;:&lt;/strong&gt;&lt;br /&gt;
Black and white keys: C _ _ D _ _ E F _ _ G _ _ A _ _ B C&lt;br /&gt;
Relative notation: P1 m2 vM2 M2 m3 vM3 M3 P4 d5 vP5 P5 m6 vM6 M6 m7 vM7 M7 P8&lt;br /&gt;
or with upminors instead of downmajors: P1 m2 ^m2 M2 m3 ^m3 M3 P4 d5 ^d5 P5 m6 ^m6 M6 m7 ^m7 M7 P8&lt;br /&gt;
The d5 could instead be an A4: P4 ^P4 A4 P5 or P4 vA4 A4 P5&lt;br /&gt;
Many other variations are possible, much freedom of spelling.&lt;br /&gt;
In C, with downmajors: C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C&lt;br /&gt;
In B, with upminors: B C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B&lt;br /&gt;
&lt;br /&gt;
One can't associate ups and downs with JI as easily because of the poor approximation of the 5-limit. However major = red or fifthward white and minor = blue or fourthward white.&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;24-EDO&lt;/u&gt;:&lt;/strong&gt;&lt;br /&gt;
black and white keys: C _ _ _ D _ _ _ E _ F _ _ _ G _ _ _ A _ _ _ B _ C&lt;br /&gt;
Relative notation: P1 vm2 m2 vM2 M2 vm3 m3 vM3 M3 vP4 P4 ^P4 d5 vP5 P5 etc.&lt;br /&gt;
Many alternate spellings available, for example vm3 = ^M2, vM3 = ^m3, ^P4 = vd5, etc.&lt;br /&gt;
In C: C Dbv Db Dv D Ebv Eb Ev E Fv F F^ Gb Gv G etc.&lt;br /&gt;
&lt;br /&gt;
24-EDO is an example of a closed EDO. An EDO is closed if the keyspan of the fifth isn't coprime with the keyspan of the octave, and open if it is. 24-EDO has a fifth of 14 steps, and 14 isn't coprime with 24, because they have a common divisor of 2. 24-EDO is said to close at 12 (1/2 of 24), because the circle of fifths has only 12 notes. There are actually 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one:&lt;br /&gt;
Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#&lt;br /&gt;
Eb^-Bb^-F^-C^-G^-D^-A^-E^-B^-F#^-C#^-G#^&lt;br /&gt;
Just as G# could be written as Ab, all the up notes could be written as down notes.&lt;br /&gt;
&lt;br /&gt;
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:&lt;br /&gt;
C^ Db Db^ D D^ Eb Eb^ E E^ F F^ F^^ Gb^ G G^ etc.&lt;br /&gt;
&lt;br /&gt;
JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber.&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;31-EDO&lt;/u&gt;:&lt;/strong&gt;&lt;br /&gt;
Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C&lt;br /&gt;
relative notation: P1 ^P1 vm2 m2 ^m2 M2 ^M2 vm3 m3 ^m3 M3 ^M3 vP4 P4 ^P4 A4 d5 ^d5 P5 etc.&lt;br /&gt;
alternate spellings: A1=vm2, ^m2=vM2, ^M3=vP4, ^P4=vA4, etc.&lt;br /&gt;
In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.&lt;br /&gt;
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).&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Naming Chords"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;u&gt;Naming Chords&lt;/u&gt;&lt;/h1&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
The quality of an interval is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc. Major is always wider than minor, so if the edo's 5th is narrower than 4\7, as is in 16edo, major is not fifthwards but fourthwards:&lt;br /&gt;
&lt;br /&gt;
The chain of fifths in fourthwards EDOs:&lt;br /&gt;
M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 - A1 etc.&lt;br /&gt;
F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.&lt;br /&gt;
16edo: P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8&lt;br /&gt;
16edo: C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C&lt;br /&gt;
&lt;br /&gt;
In other words, sharp/flat, major/minor, and aug/dim all retain their melodic meaning but are flipped harmonically. Perfect and natural are unaffected. Interval arithmetic in fourthwards edos: First flip the meaning, then perform normal arithmetic, then flip the meaning again:&lt;br /&gt;
M2 + M2 --&amp;gt; m2 + m2 = dim3 --&amp;gt; aug3&lt;br /&gt;
D to F# --&amp;gt; D to Fb = dim3 --&amp;gt; aug3&lt;br /&gt;
Eb + m3 --&amp;gt; E# + M3 = Gx --&amp;gt; Gbb&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Naming Chords-22edo chord names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;u&gt;22edo chord names&lt;/u&gt;&lt;/h2&gt;
&lt;br /&gt;
Chord names are based entirely on the ups/downs interval names:&lt;br /&gt;
&lt;br /&gt;
0\22 = P1&lt;br /&gt;
1\22 = m2&lt;br /&gt;
2\22 = ^m2&lt;br /&gt;
3\22 = vM2&lt;br /&gt;
4\22 = M2&lt;br /&gt;
5\22 = m3&lt;br /&gt;
6\22 = ^m3&lt;br /&gt;
7\22 = vM3&lt;br /&gt;
8\22 = M3&lt;br /&gt;
9\22 = P4&lt;br /&gt;
10\22 = ^P4, d5&lt;br /&gt;
11\22 = vA4, ^d5&lt;br /&gt;
12\22 = A4, vP5&lt;br /&gt;
13\22 = P5&lt;br /&gt;
14\22 = m6&lt;br /&gt;
15\22 = ^m6&lt;br /&gt;
16\22 = vM6&lt;br /&gt;
17\22 = M6&lt;br /&gt;
18\22 = m7&lt;br /&gt;
19\22 = ^m7&lt;br /&gt;
20\22 = vM7&lt;br /&gt;
21\22 = M7&lt;br /&gt;
22\22 = P8&lt;br /&gt;
&lt;br /&gt;
These are pronounced &amp;quot;downmajor second&amp;quot;, &amp;quot;upminor third&amp;quot;, etc. For 4ths and 5ths, &amp;quot;perfect&amp;quot; is implied and can be omitted: ^P4 = &amp;quot;up-four&amp;quot; and vP5 = &amp;quot;down-five&amp;quot;. In larger edos there may be &amp;quot;down-octave&amp;quot;, &amp;quot;up-unison&amp;quot;, etc.&lt;br /&gt;
&lt;br /&gt;
0-7-13-18 in C is &amp;quot;C vM,m7&amp;quot;, pronounced &amp;quot;C downmajor, minor seventh&amp;quot;. The space between the C and the down symbol is needed because Cv is a note, and &amp;quot;Cv M,m7&amp;quot; is a different chord. That chord is pronounced &amp;quot;C down, major, minor 7th&amp;quot;, so you have to &amp;quot;speak the space&amp;quot;. I see the need for a space as a small drawback, but can't think of a good way to avoid it. Alternatively, a comma could be used: C,vM,m7 vs. Cv,M,m7. The extra space/comma isn't needed when there's no usp or downs immediately after the note name, e.g. Cm.&lt;br /&gt;
&lt;br /&gt;
The conventional chord naming system uses a lot of &amp;quot;shorthand&amp;quot; like dom7 for M3,m7 and min6 for m3,M6. I think this would cause problems in 22edo where there are so many choices for the 3rd, the 6th, the 7th and the 9th. For example, min6 could mean m3,vM6 = approximate 6:7:9:10 chord, or it could mean ^m3,M6 = approximate 1/1-6/5-3/2-12/7 utonal chord. And larger edos would present even greater problems. Plus there's some ambiguity in the shorthand, e.g. in 12edo, both 0-3-6 and 0-3-6-9 are called dim chords.&lt;br /&gt;
&lt;br /&gt;
So I propose abandoning the shorthand and explicitly spelling out all the components of the chord, with a few exceptions: 1) The root, obviously. 2) The perfect 5th is assumed present unless otherwise specified. Thus 0-7-18 is &amp;quot;C vM,m7,-5&amp;quot; and 0-6-11 is &amp;quot;C ^m,^d5&amp;quot;. 3) The 3rd is also assumed to be present, and is implied by a quality with no degree. Thus 0-7-13 is &amp;quot;C vM&amp;quot;. 4) The 3rd isn't spelled out if the 6th or 7th has the same quality as the 3rd. Thus 0-7-13-16 is &amp;quot;C vM6&amp;quot;, but 0-7-13-17 is &amp;quot;C vM,M6&amp;quot;. Thirdless chords: 0-13-18 is either &amp;quot;Cm7,-3&amp;quot; or &amp;quot;C5,m7&amp;quot;.&lt;br /&gt;
&lt;br /&gt;
The 6th, the 7th, the 9th, the 11th, etc. are explicitly written out, including their qualities. Thus the 9th isn't assumed to be major, and the presence of a 9th doesn't imply the presence of a 7th.&lt;br /&gt;
&lt;br /&gt;
Sus chords: &amp;quot;sus&amp;quot; means the 3rd is replaced by the named note, a 2nd or 4th. &amp;quot;Sus4&amp;quot; means a perfect 4th, and sus^4 means an up-perfect 4th. Some edos would have susv4, susvv4, etc. &amp;quot;Sus2&amp;quot; means a major 2nd. In most edos, M2 is always a perfect 4th below the perfect 5th, see 16edo below for an exception.&lt;br /&gt;
&lt;br /&gt;
0-5-13 = m&lt;br /&gt;
0-6-13 = ^m&lt;br /&gt;
0-7-13 = vM&lt;br /&gt;
0-8-13 = M&lt;br /&gt;
0-9-13 = sus4&lt;br /&gt;
0-10-13 = sus^4&lt;br /&gt;
0-4-13 = sus2&lt;br /&gt;
0-3-13 = susvM2&lt;br /&gt;
&lt;br /&gt;
0-5-11 = m,^d5&lt;br /&gt;
0-5-12 = m,vP5 (or possibly m,A4)&lt;br /&gt;
&lt;br /&gt;
0-5-11-14 = m6,^d5&lt;br /&gt;
0-6-11-15 = ^m6,^d5&lt;br /&gt;
0-7-13-16 = vM6&lt;br /&gt;
0-8-13-17 = M6&lt;br /&gt;
&lt;br /&gt;
0-5-13-18 = m7&lt;br /&gt;
0-6-13-19 = ^m7&lt;br /&gt;
0-7-13-20 = vM7&lt;br /&gt;
0-8-13-21 = M7&lt;br /&gt;
&lt;br /&gt;
0-5-13-16 = m,vM6&lt;br /&gt;
0-8-13-19 = M,^m7&lt;br /&gt;
0-7-13-18-26 = vM,m7,M9&lt;br /&gt;
0-7-13-18-26-32 = vM,m7,M9,^P11&lt;br /&gt;
&lt;br /&gt;
You can write out chord progressions using the ups/downs notation for note names. Here's the first 2 bars of Tibia:&lt;br /&gt;
G vM7,-5 = &amp;quot;G downmajor seven, no five&amp;quot;&amp;quot;&lt;br /&gt;
Eb^ vM,M9 = &amp;quot;E flat up, downmajor, major nine&amp;quot;&lt;br /&gt;
Gm7,-5 (no space needed) = &amp;quot;G minor seven, no five&amp;quot;&lt;br /&gt;
A vM,m7 = &amp;quot;A downmajor, minor seven&amp;quot;&lt;br /&gt;
&lt;br /&gt;
To use relative notation, first write out all possible 22edo chord roots relatively. This is just the interval notation with Roman numerals instead of Arabic, # for aug, and b for minor. Dim from perfect is b, but dim from minor is bb. I've also included more enharmonic equivalents like ^I = bII.&lt;br /&gt;
I ^I/bII v#I/^bII #I/vII II ^II/bIII v#II/^bIII #II/vIII III IV ^IV/bV v#IV/^bV #IV/vV&lt;br /&gt;
V ^V/bVI v#V/^bVI #V/vVI VI ^VI/bVII v#VI/^bVII #VI/vVII VII&lt;br /&gt;
These are pronounced &amp;quot;down-two&amp;quot;, &amp;quot;up-flat-three&amp;quot;, &amp;quot;down-sharp-four&amp;quot;, etc.&lt;br /&gt;
&lt;br /&gt;
Here's the Tibia chords. No spaces are needed because ups and downs are always leading, never trailing.&lt;br /&gt;
IvM7,-5 = &amp;quot;one downmajor seven, no five&amp;quot;&lt;br /&gt;
^bVIvM,M9 = &amp;quot;up-flat six downmajor, major nine&amp;quot;&lt;br /&gt;
Im7,-5 = &amp;quot;one minor seven, no five&amp;quot;&lt;br /&gt;
IIvM,m7 = &amp;quot;two downmajor, minor seven&amp;quot;&lt;/body&gt;&lt;/html&gt;</pre></div>
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