Kite's ups and downs notation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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=__"Ups and Downs" Notation for 22-EDO__=

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

Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works 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".

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

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The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.

Relative notation for 22-EDO is P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - d5 - ^d5 - vP5 - P5 etc. That's pronounced "upminor 2nd, downmajor 3rd", etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The ~~mid~~ notes always form a chain of fifths.

Relative notation 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 notes without ups or downs always form a chain of fifths.

You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See [[Kite's color notation]] for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.

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The mid symbol "~" means "exactly midway between major and minor", hence neutral. In other words, mid is a quality like major or perfect. This only applies to certain "neutral EDOs" in which the sharp equals an even number of EDOsteps. For example, in every seventh EDO (10edo, 17edo, 24edo, 31edo, etc.), a sharp is two EDOsteps, upminor equals downmajor, and "mid" replaces both terms. In 20edo, 27edo, 34edo, 41edo, etc., a sharp is four EDOsteps, and mid replaces both double-upminor and double-downmajor. In 11-edo and 18b-edo, mid replaces upmajor and downminor.

There are three special cases to be addressed. The first special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. The scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect. There are no major or minor intervals.

The chain of fifths in "perfect" EDOs (3/2 maps to 4\7):

P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.

F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.

21edo: P1 - ^P1 - vP2 - P2 - ^P2 - vP3 - P3 - ^P3 - vP4 - P4 - ^P4 - vP5 - P5 - ^P5 - vP6 - P6 - ^P6 - vP7 - P7 - ^P7 - vP8 - P8

Because everything is perfect, the quality can be omitted:

21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8

21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C

Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing, because the upfifth B - F# looks like a perfect fifth.

The second special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo and 18edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. 13edo and 18edo can be notated by using the 2nd best fifth.

The third special case is when the edo's 5th is narrower than 4\7, as in 16edo. There are two approaches. One preserves the harmonic (chain-of-fifths) meaning of sharp/flat, major/minor and aug/dim, and the other preserves the melodic meaning.

In the first approach, major is still fifthwards, which makes it narrower than minor. Aug is narrower than dim. This makes interval arithmetic and chord names unaffected. M2 + M2 is still M3, and a C minor chord is still C Eb G.

In the 2nd approach, major is still wider than minor, so major is not fifthwards but fourthwards. The chain of fifths runs backwards:

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

Interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.

M2 + M2 --> m2 + m2 = dim3 --> aug3

D to F# --> D to Fb = dim3 --> aug3

D to F --> D to F = m3 --> M3 (natural notes don't flip)

Eb + m3 --> E# + M3 = G## --> Gbb

Eb + P5 --> E# + P5 = B# --> Bb (perfect intervals don't flip)

Both approaches have their merit, but the first one will be used here, because it makes naming chords easier.

**__17-EDO__:** (2 keys per sharp/flat)

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=~~__Naming Chords__~~=

=__22edo Chord Names__=

Ups and downs allow us to name any chord easily. The quality of an interval (major, minor, perfect, etc.) 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.

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

There are three special cases to be addressed. The first special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. The scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect. There are no major or minor intervals.

Let's review the 22edo interval names:

0\22 = P1

~~The chain of fifths in "perfect" EDOs (3/2 maps to 4\7):~~

1\22 = m2

~~P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.~~

2\22 = ^m2

~~F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.~~

3\22 = vM2

~~21edo: P1 - ^P1 - vP2 - P2 - ^P2 - vP3 - P3 - ^P3 - vP4 - P4 - ^P4 - vP5 - P5 - ^P5 - vP6 - P6 - ^P6 - vP7 - P7 - ^P7 - vP8 - P8~~

4\22 = M2

~~Because everything is perfect, the quality can be omitted:~~

5\22 = m3

~~21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8~~

6\22 = ^m3

~~21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C~~

Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing, because the upfifth B - F# looks like a perfect fifth.

The second special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo and 18edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. 13edo and 18edo can be notated by using the 2nd best fifth.

The third special case is when the edo's 5th is narrower than 4\7, as in 16edo. There are two approaches. One preserves the harmonic (chain-of-fifths) meaning of sharp/flat, major/minor and aug/dim, and the other preserves the melodic meaning.

In the first approach, major is still fifthwards, which makes it narrower than minor. Aug is narrower than dim. This makes interval arithmetic and chord names unaffected. M2 + M2 is still M3, and a C minor chord is still C Eb G.

~~In the 2nd approach, major is still wider than minor, so major is not fifthwards but fourthwards. The chain of fifths runs backwards:~~

~~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~~

~~Interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.~~

~~M2 + M2 --> m2 + m2 = dim3 --> aug3~~

~~D to F# --> D to Fb = dim3 --> aug3~~

~~D to F --> D to F = m3 --> M3 (natural notes don't flip)~~

~~Eb + m3 --> E# + M3 = G## --> Gbb~~

~~Eb + P5 --> E# + P5 = B# --> Bb (perfect intervals don't flip)~~

~~C minor = C + m3 + P5 --> C major = C + M3 + P5 = C E G -->~~

~~Both approaches have their merit, but the first one will be used from here on.~~

~~=__22edo chord names__=~~

Let's review the 22edo interval names:

0\22 = P1

1\22 = m2

2\22 = ^m2

3\22 = vM2

4\22 = M2

5\22 = m3

6\22 = ^m3

7\22 = vM3

8\22 = M3

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

There are some alternate names. However double-ups and double-downs are to be avoided in 22edo. Thus 7\22 would never be written ^^m3. In larger edos, ~~they~~ would be necessary.

There are some alternate names. However double-ups and double-downs are to be avoided in 22edo. Thus 7\22 would never be written ^^m3. In larger edos, double-ups and double-downs would be necessary.

0-8-13 in C has C E & G, and is written "C" and pronounced "C" or "C major".

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=__Chord names in other EDOs__=

~~When applied to notes~~, the mid symbol "~"~~means "neither up nor down". But in chord names it~~ means "exactly midway between major and minor", hence neutral. This only applies to ~~certain "neutral EDOs" in which the sharp equals an~~ even ~~number of EDOsteps~~. ~~For example, in every seventh EDO~~ (~~10edo~~, ~~17edo~~, ~~24edo, 31edo~~, etc.)~~, a sharp is two EDOsteps~~, upminor equals downmajor, and "mid" replaces both terms. In ~~20edo~~, ~~27edo~~, ~~34edo, 41edo~~, etc., ~~a sharp is four EDOsteps, and~~ mid replaces both double-upminor and double-downmajor.

In chord names, the mid symbol "~" means "exactly midway between major and minor", hence neutral. This only applies to even-chroma edos. In chroma-2 edos (10, 17, 24, etc.), upminor equals downmajor, and "mid" replaces both terms. In chroma-4 edos (20, 27, 34, etc.), mid replaces both double-upminor and double-downmajor. In 11-edo and 18b-edo, mid replaces both upmajor and downminor.

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

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C Eb Gbv = Cdim(v5) = "C dim down-five"

C Eb Gb^ = Cdim(^5) = "C dim up-five"

C Eb^ Gb = Cdim(^3) = "C dim up-three" (in certain EDOs, Cdim(~3) = "C dim mid-three~~", or C~(b5) = "C mid flat-five~~")

C Eb^ Gb = Cdim(^3) = "C dim up-three" (in certain EDOs, Cdim(~3) = "C dim mid-three")

(note that here "up-three" means upminor 3rd, not upmajor 3rd, because "dim" indicates a minor 3rd)

C Eb^ Gb^ = Cdim(^3,^5) = "C dim up-three up-five" (in certain EDOs, C~(^b5) = "C mid upflat-five")

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C Dbv Ev G Bbv = C.v7(vb9) = "C dot down seven downflat-nine", or C.vb9(v9)

~~====~~__**Example EDOs:**__~~====~~

__**Example EDOs:**__

14edo: D * E * F * G * A * B * C * D, zero keys per #/b.

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Another example: I7 - bVII7 - IV7 - I7. To make this work, the 7th in the I7 chord must be a minor 7th. in 22edo, that 7th sounds blue. In 19edo, it sounds green. If you want a blue 7th in 19edo, you have to use the downminor 7th, which will cause shifts or drifts in the progression.

=__**Scale Fragments**__=

~~Besides the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO, there are five EDO categories, based on the size of the fifth:~~

~~"Fifth-less" EDOs (8, 11, 13 and 18)~~

~~"Fourthward" EDOs (9, 16 and 23)~~

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

~~"Pentatonic" EDOs (5, 10, 15, 20, 25 and 30)~~

~~"Regular" EDOs (all others)~~

~~The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The regular EDOs may or may not.~~

To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. Examples:

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Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: "wide-fifth" 35edo (which uses 21\35 as a fifth) has a different scale fragment than "narrow-fifth" 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute M2 for m2.)

~~In the chart below, 13edo and 18edo use the narrower fifth.~~

||= 5edo ||= pentatonic ||= ||= C/Db ||= C#/D ||= ||= ||= ||= ||= ||= ||= ||= ||

||= 6edo ||= fifthless ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||

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||= 11edo ||= fourthward ||= ||= C ||= D ||= C# ||= D# ||= ||= ||= ||= ||= ||= ||

||= 12edo ||= regular ||= ||= C ||= C#/Db ||= D ||= ||= ||= ||= ||= ||= ||= ||

||= ~~13edo~~ ||= fourthward ||= ||= C ||= D ||= * ||= C# ||= D# ||= ||= ||= ||= ||= ||

||= 13b-edo ||= fourthward ||= ||= C ||= D ||= * ||= C# ||= D# ||= ||= ||= ||= ||= ||

||= 14edo ||= perfect ||= ||= C/C# ||= * ||= Db/D ||= ||= ||= ||= ||= ||= ||= ||

||= 15edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= ||= ||

||= 16edo ||= fourthward ||= ||= C ||= C#/Db ||= D ||= D# ||= ||= ||= ||= ||= ||= ||

||= 17edo ||= regular ||= ||= C ||= Db ||= C# ||= D ||= ||= ||= ||= ||= ||= ||

||= ~~18edo~~ ||= fourthward ||= ||= C/Db ||= * ||= C#/D ||= * ||= D# ||= ||= ||= ||= ||= ||

||= 18b-edo ||= fourthward ||= ||= C/Db ||= * ||= C#/D ||= * ||= D# ||= ||= ||= ||= ||= ||

||= 19edo ||= regular ||= ||= C ||= C# ||= Db ||= D ||= ||= ||= ||= ||= ||= ||

||= 20edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||= ||= ||

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||= 30edo ||= pentatonic ||= ||= C/Db ||= * ||= * ||= * ||= * ||= * ||= C#/D ||= ||= ||= ||

||= 31edo ||= regular ||= ||= C ||= * ||= C# ||= Db ||= * ||= D ||= ||= ||= ||= ||

||= 32edo ||= ~~regular~~ ||= ||= C ||= Db ||= * ||= * ||= * ||= C# ||= D ||= ||= ||= ||

||= 32edo ||= " ||= ||= C ||= Db ||= * ||= * ||= * ||= C# ||= D ||= ||= ||= ||

||= 33edo ||= ~~regular~~ ||= ||= C ||= C# ||= * ||= * ||= Db ||= D ||= ||= ||= ||= ||

||= 33edo ||= " ||= ||= C ||= C# ||= * ||= * ||= Db ||= D ||= ||= ||= ||= ||

||= 34edo ||= ~~regular~~ ||= ||= C ||= * ||= Db ||= * ||= C# ||= * ||= D ||= ||= ||= ||

||= 34edo ||= " ||= ||= C ||= * ||= Db ||= * ||= C# ||= * ||= D ||= ||= ||= ||

||= 35edo ||= perfect ||= ||= C/C# ||= * ||= * ||= * ||= * ||= Db/D ||= ||= ||= ||= ||

||= 36edo ||= regular ||= ||= C ||= * ||= * ||= C#/Db ||= * ||= * ||= D ||= ||= ||= ||

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Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.

**__9edo__:** C/D# Cb/D, # = v

**__9edo__:** C/D# Cb/D (# = v)

D E * F G A B * C D

D - E - Eb/F# - F - G - A - B - Bb/C# - C - D

P1 - M2 - m2/M3 - m3 - P4 - P5 - M6 - m6/M7 - m7 - P8

**__11edo__:** C D Cb Db, # = vv

**__11edo__:** C D Cb Db (# = vv)

D E * * F G A B * * C D

D - E - E^/F# - Eb/Fv - F - G - A - B - B^/C# - Bb/Cv - C - D

P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8

problematic because M3 ~~= 2\11~~ is narrower than m2 ~~= 3\11~~

problematic because M3 is narrower than m2

**__13b-edo__:** C D * Cb Db, # = vvv

**__13b-edo__:** C D * Cb Db (# = vvv)

D E * * * F G A B * * * C D

D - E - E^/F# - Ebv/F#^ - Eb/Fv - F - G - A - B - B^/C# - Bbv/C#^ - Bb/Cv - C - D

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

problematic because M3 is narrower than m2

**__16edo__:** C Cb/D# D, # = v

**__16edo__:** C Cb/D# D (# = v)

D * E * * F * G * A * B * * C * D

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

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

**__18b-edo__:** ~~# = vv,~~ C/D# * Cb/D

**__18b-edo__:** C/D# * Cb/D (# = vv)

D * E * * * F * G * A * B * * * C * D

D - D^/Ev - E - E^ - Eb/F# - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - Bb/C# - Cv - C - C^/Dv - D

P1 - ^P1/vM2 - M2 - ~2 - m2/M3 - ~3 - m3 - ^m3/vP4 - P4 - ^P4/vP5 - P5 - ^P5/vM6 - M6 - ~6 - m6/M7 - ~7 - m7 - ^m2/d8 - P8

Mid "~" is midway between major and minor, ~~which equates it to~~ upmajor and downminor.

Mid "~" is midway between major and minor, and replaces both upmajor and downminor.

**__23edo__:** C Cb * D# D, # = v

**__23edo__:** C Cb * D# D (# = v)

D * * E * * * F * * G * * A * * B * * * C * * D

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

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P1/m2 - ^m2/vM2 - M2/m3 - ^m3/vM3 - M3/P4 - ^P4/vP5 - P5/m6 - ^m6/vM6 - M6/m7 - ^m7/vM7 - P8

**__15edo__:** 3 keys per sharp/flat: C/Db [*~~] [~~*] C#/D

**__15edo__:** 3 keys per sharp/flat: C/Db * * C#/D

D * * E/F * * G * * A * * B/C * * D

D - D^ - Ev - E/F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B/C - C^ - Dv - D

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P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8

Alternatively, pentatonic notation can be used:

Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.

C# - G# - D# - A# - E# - C - G - D - A - E - Cb - Gb - Db - Ab - Eb etc.

All intervals are perfect, so quality can be omitted.

s3 = subthird, 4d = fourthoid, 5d = fifthoid, s7 = subseventh, 8d = octoid.

~~===~~__**~~"Fifth-less" EDOs~~**__~~===~~

__**5edo**__**:** zero keys per sharp/flat: C/C# Db/D

~~(8, 11, 13 and 18)~~

D E G A C D

~~These use an alternate generator.~~

1 - s3 - 4d - 5d - s7 - 8d

**__8edo__:** (generator = 1\8 = perfect 2nd = 150¢)

__**10edo**__**:** zero keys per sharp/flat: C/C# * Db/D

~~D E F G * A B C D~~

D * E * G * A * C * D

~~D - E - F - G - G#/Ab - A -B - C - D~~

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

~~P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8~~

~~seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.~~

~~A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc.~~

~~__**11-edo**__: (generator = 3\11 = perfect 3rd)~~

~~D * E F * G A * B C * D~~

~~D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D~~

~~P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8~~

~~sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.~~

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

~~__**13edo**__**:** (generator = 2\13 = perfect 2nd)~~

~~D * E * F * G A * B * C * D~~

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

~~P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8~~

~~secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.~~

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

**__18edo__:** (generator = 5\18 = perfect 3rd)

~~D * * E * F * * G * A * * B * C * * D~~

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

~~P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8~~

~~sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.~~

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

~~==__Pentatonic notation__==~~

~~**__Alternative pentatonic notation for pentatonic EDOs__**~~

~~Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.~~

~~C# - G# - D# - A# - E# - C - G - D - A - E - Cb - Gb - Db - Ab - Eb etc.~~

~~All intervals are perfect, so quality can be omitted.~~

~~s3 = subthird, 4d = fourthoid, 5d = fifthoid, s7 = subseventh, 8d = octoid.~~

~~__**5edo**__**:** zero keys per sharp/flat: C/C# Db/D~~

~~D E G A C D~~

~~1 - s3 - 4d - 5d - s7 - 8d~~

__**10edo**__**:** zero keys per sharp/flat: C/C# * Db/D

D * E * G * A * C * D

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d

Line 907:

Line 861:

etc.

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

===__**"Fifth-less" EDOs**__===

(8, 11b, 13 and 18)

There are three strategies for notating these EDOs. One is to convert them to fourthwards EDOs by using an alternate fifth. This doesn't work for 8edo.

Another is to switch from heptatonic notation to some other type. Pentatonic notation is a natural fit, in the sense that no ups or downs are needed, for 8edo, 13edo and 18edo, but not 11edo.

The third approach is to use some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But these EDOs don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like.

__**Pentatonic notation for EDOs 8, 13 and 18**__

All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.

Line 916:

Line 880:

D - D#/Eb - E - G - G#/Ab - A - C - C#/Db - D

P1 - ms3 - Ms3 - P4d - A4d/d5d - P5d - ms7 - Ms7 - P8d

__**11edo**__**:** (generator = 7\11 = perfect 5thoid) C Db C# D, # is ^^

D * * E G * * A C * * D

P1 - ms3 - ^ms3/vMs3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ^ms7/vMs7 - Ms7 - P8d

__**13edo**__**:** (generator = 8\13 = perfect 5thoid) C C# Db D

Line 927:

Line 895:

P1 - A1 - ds3 - ms3 - Ms3 - As3 - d4d - P4d - A4d - AA4d/dd5d - d5d - P5d - A5d - ds7 - ms7 - Ms7 - As7 - d8d - P8d

__**23b-edo**__**:** (generator = 14\23 = perfect 5thoid) C C# * * Db D

D * * * * E * * * G * * * * A * * * C * * * * D

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

__**Other non-heptatonic notations for 8edo, 11edo, 13edo and 18edo**__

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

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

The other approach is to use some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But these EDOs don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like.

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

8edo octatonic (every note is a generator)

Line 948:

Line 909:

P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8

requires learning nonotonic interval arithmetic and staff notation

~~11edo pentatonic wide-fifth-based, fifthwards, # is ^^ (3/2 maps to 7\11 6th):~~

~~D * * E G * * A C * * D~~

~~pentatonic fifthwards chain of fifthoids: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7~~

~~P1 - ms3 - ^ms3/vMs3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ^ms7/vMs7 - Ms7 - P8d~~

~~pentatonic plus ups and downs is doubly confusing!~~

11edo octatonic wide-fifth-based, fifthwards, no ^/v (3/2 maps to 7\11 = perfect 6th):

Line 959:

Line 914:

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

requires learning octatonic interval arithmetic and notation

~~13edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^^ (3/2 maps to 7\13 perfect 5th):~~

~~D E * * * F G A B * * * C D~~

~~fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7~~

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

~~problematic because m3 = 2\13 is narrower than M2 = 4\13~~

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

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Line 931:

__**Alternate ~~notation~~ for ~~other edos~~:**__

__**Alternate generators for 8edo, 11edo, 13edo and 18edo**__

~~23b~~-~~edo pentatonic wide~~-~~fifth~~-~~based, fifthwards, 3~~/~~2 maps to 14\23 = perfect fifthoid~~:

D * * * * E ~~* *~~ * G ~~* * * *~~ A * ~~* *~~ C ~~* * *~~ * D

**__8edo__:** (generator = 1\8 = perfect 2nd = 150¢)

~~35b~~-~~edo heptatonic wide~~-~~fifth~~-~~based, sharp = seven ups, 3~~/~~2 maps to 21\35 = perfect fifth~~:

D E F G * A B C D

D ~~* * * * * *~~ E/F * * * * * * G * * * * * * A * * * ~~* * *~~ B/C ~~* * * * * *~~ D

D - E - F - G - G#/Ab - A -B - C - D

~~42b~~-~~edo heptatonic narrow~~-~~fifth~~-~~based, sharp = six ups, 3~~/~~2 maps to 24\42 = perfect fifth~~:

P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8

D * * * * * E ~~* * * *~~ * F ~~* * *~~ * * G ~~* * * *~~ * A * * * * * B ~~* * * * *~~ C ~~* * * * *~~ D

chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.

A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc.

__**11-edo**__: (generator = 3\11 = perfect 3rd)

D * E F * G A * B C * D

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.

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

__**13edo**__**:** (generator = 2\13 = perfect 2nd)

D * E * F * G A * B * C * D

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

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8

secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.

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

**__18edo__:** (generator = 5\18 = perfect 3rd)

D * * E * F * * G * A * * B * C * * D

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

P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8

sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.

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

=__Ups and downs solfege__=

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||= 22 ||= 0 ||= D ||= ||= ||</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"><html><head><title>Ups and Downs Notation</title></head><body><div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#x&quot;Ups and Downs&quot; Notation for 22-EDO">&quot;Ups and Downs&quot; Notation for 22-EDO</a></div>

<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"><html><head><title>Ups and Downs Notation</title></head><body><div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#x&quot;Ups and Downs&quot; Notation for 22-EDO">&quot;Ups and Downs&quot; Notation for 22-EDO</a></div>

<div style="margin-left: 1em;"><a href="#Other EDOs">Other EDOs</a></div>

<div style="margin-left: 1em;"><a href="#Other EDOs">Other EDOs</a></div>

<div style="margin-left: 1em;"><a href="#~~Naming Chords~~">~~Naming Chords~~</a></div>

<div style="margin-left: 1em;"><a href="#x22edo Chord Names">22edo Chord Names</a></div>

<div style="margin-left: ~~1em~~;"><a href="#~~x22edo chord names~~">~~22edo chord names~~</a></div>

<div style="margin-left: 2em;"><a href="#toc3"></a></div>

<div style="margin-left: 2em;"><a href="#toc4"></a></div>

<div style="margin-left: 2em;"><a href="#toc4"> </a></div>

<div style="margin-left: 2em;"><a href="#toc5"> </a></div>

<div style="margin-left: 2em;"><a href="#toc5"> </a></div>

<div style="margin-left: ~~2em~~;"><a href="#~~toc6"&~~gt; </a></div>

<div style="margin-left: 1em;"><a href="#Chord names in other EDOs">Chord names in other EDOs</a></div>

<div style="margin-left: 1em;"><a href="#~~Chord names in other EDOs~~">~~Chord names in other EDOs~~</a></div>

<div style="margin-left: 1em;"><a href="#Cross-EDO considerations">Cross-EDO considerations</a></div>

<div style="margin-left: ~~4em~~;"><a href="#~~Chord names in other EDOs---Example EDOs:~~">~~Example EDOs:~~</a></div>

<div style="margin-left: 1em;"><a href="#Scale Fragments">Scale Fragments</a></div>

<div style="margin-left: 1em;"><a href="#~~Cross-~~EDO ~~considerations~~">~~Cross-~~EDO ~~considerations~~</a></div>

<div style="margin-left: 1em;"><a href="#Summary of EDO notation">Summary of EDO notation</a></div>

<div style="margin-left: ~~1em~~;"><a href="#~~Scale Fragments"~~>~~Scale Fragments~~</a></div>

<div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;Regular&quot; EDOs">&quot;Regular&quot; EDOs</a></div>

<div style="margin-left: ~~1em~~;"><a href="#Summary of EDO notation">~~Summary of EDO notation~~</a></div>

<div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;Perfect&quot; EDOs">&quot;Perfect&quot; EDOs</a></div>

<div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;~~Regular~~&quot; EDOs">&quot;~~Regular~~&quot; EDOs</a></div>

<div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;Fourthward&quot; EDOs">&quot;Fourthward&quot; EDOs</a></div>

<div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;~~Perfect~~&quot; EDOs">&quot;~~Perfect~~&quot; EDOs</a></div>

<div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;Pentatonic&quot; EDOs">&quot;Pentatonic&quot; EDOs</a></div>

<div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;~~Fourthward~~&quot; EDOs">&quot;~~Fourthward~~&quot; EDOs</a></div>

<div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;Fifth-less&quot; EDOs">&quot;Fifth-less&quot; EDOs</a></div>

<div style="margin-left: ~~3em~~;"><a href="#~~Summary of EDO notation--~~&~~amp~~;~~quot~~;~~Pentatonic~~&~~amp~~;~~quot~~; ~~EDOs"~~>&~~amp~~;~~quot;Pentatonic&quot; EDOs</a></div>~~

<div style="margin-left: 1em;"><a href="#Ups and downs solfege">Ups and downs solfege</a></div>

~~<!~~-- ws:end:WikiTextTocRule:62 --><div style="margin-left: ~~3em~~;"><a href="#~~Summary of EDO notation-~~-&~~amp~~;~~quot;Fifth~~-~~less~~&~~amp~~;~~quot; EDOs"~~>&~~amp;quot;Fifth-less&quot; EDOs</a>&lt~~;/div>

<div style="margin-left: 1em;"><a href="#Rank-2 Notation">Rank-2 Notation</a></div>

<div style="margin-left: 2em;"><a href="#~~Summary of EDO notation-Pentatonic notation~~">~~Pentatonic notation~~</a></div>

<div style="margin-left: 2em;"><a href="#toc17"> </a></div>

<div style="margin-left: 1em;"><a href="#~~EDOs with an inaccurate 3/2~~">~~EDOs with an inaccurate 3/2~~</a></div>

<div style="margin-left: 1em;"><a href="#Generators other than a fifth">Generators other than a fifth</a></div>

<div ~~style="margin-left: 1em;"><a href="#Ups and downs solfege">Ups and downs solfege</a></div>~~

</div>

~~<div style="margin-left: 1em;"><a href="#Rank-2 Notation">Rank-2 Notation</a></div>~~

<h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation for 22-EDO"></a><u>&quot;Ups and Downs&quot; Notation for 22-EDO</u></h1>

~~<div style="margin-left: 2em;"><a href="#toc21"> </a></div>~~

<div style="margin-left: 1em;"><a href="#Generators other than a fifth">Generators other than a fifth</a></div>

~~</div>~~

<h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation for 22-EDO"></a><u>&quot;Ups and Downs&quot; Notation for 22-EDO</u></h1>

<br />

Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol &quot;^&quot; and the down symbol &quot;v&quot;~~. There's also the optional mid symbol &quot;~&quot; which undoes ups and downs~~.<br />

Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol &quot;^&quot; and the down symbol &quot;v&quot;.<br />

<br />

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

Line 1,381:

Line 1,349:

The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.<br />

<br />

Relative notation for 22-EDO is P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - d5 - ^d5 - vP5 - P5 etc. That's pronounced &quot;upminor 2nd, downmajor 3rd&quot;, etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The ~~mid~~ notes always form a chain of fifths.<br />

Relative notation for 22-EDO is P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - d5 - ^d5 - vP5 - P5 etc. That's pronounced &quot;upminor 2nd, downmajor 3rd&quot;, etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The notes without ups or downs always form a chain of fifths.<br />

<br />

You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See <a class="wiki_link" href="/Kite%27s%20color%20notation">Kite's color notation</a> for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.<br />

Line 1,458:

Line 1,426:

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

<br />

<img src="/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png" alt="The Scale Tree.png" title="The Scale Tree.png" style="height: 1002px; width: 800px;" /><br />

<img src="/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png" alt="The Scale Tree.png" title="The Scale Tree.png" style="height: 1002px; width: 800px;" /><br />

The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. This version of the Stern-Brocot tree is the scale tree. The colored regions of the tree are what I call <strong>kites</strong>, and The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side on the right (7\12, 11\19, etc.) and a fourthward side on the left (5\9, 9\16, etc.). Every node on a spine is a <strong>spinal</strong> node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.<br />

<br />

Line 1,471:

Line 1,439:

The mid symbol &quot;~&quot; means &quot;exactly midway between major and minor&quot;, hence neutral. In other words, mid is a quality like major or perfect. This only applies to certain &quot;neutral EDOs&quot; in which the sharp equals an even number of EDOsteps. For example, in every seventh EDO (10edo, 17edo, 24edo, 31edo, etc.), a sharp is two EDOsteps, upminor equals downmajor, and &quot;mid&quot; replaces both terms. In 20edo, 27edo, 34edo, 41edo, etc., a sharp is four EDOsteps, and mid replaces both double-upminor and double-downmajor. In 11-edo and 18b-edo, mid replaces upmajor and downminor.<br />

<br />

<strong><u>17-EDO</u>:</strong> (2 keys per sharp/flat)<br />

There are three special cases to be addressed. The first special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. The scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect. There are no major or minor intervals.<br />

Black and white keys: C * * D * * E F * * G * * A * * B C<br />

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Relative notation: 1 m2 ~2 M2 m3 ~3 M3 4 d5 v5 5 m6 ~6 M6 m7 ~7 M7 8 (1, 4, 5 and 8 are assumed to be perfect)<br />

The chain of fifths in &quot;perfect&quot; EDOs (3/2 maps to 4\7):<br />

The d5 could instead be an A4: P4 ^P4 A4 P5 or P4 vA4 A4 P5<br />

P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.<br />

Many other variations are possible, much freedom of spelling.<br />

F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.<br />

In C, with downmajors: C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C<br />

21edo: P1 - ^P1 - vP2 - P2 - ^P2 - vP3 - P3 - ^P3 - vP4 - P4 - ^P4 - vP5 - P5 - ^P5 - vP6 - P6 - ^P6 - vP7 - P7 - ^P7 - vP8 - P8<br />

In B, with upminors: B C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B<br />

Because everything is perfect, the quality can be omitted:<br />

One can't associate ups and downs with yellow and green because of the poor approximation of the 5-limit. However major = red or fifthward white, minor = blue or fourthward white, and downmajor = upminor = jade or amber.<br />

21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8<br />

21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C<br />

Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing, because the upfifth B - F# looks like a perfect fifth.<br />

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The second special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo and 18edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. 13edo and 18edo can be notated by using the 2nd best fifth.<br />

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The third special case is when the edo's 5th is narrower than 4\7, as in 16edo. There are two approaches. One preserves the harmonic (chain-of-fifths) meaning of sharp/flat, major/minor and aug/dim, and the other preserves the melodic meaning.<br />

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In the first approach, major is still fifthwards, which makes it narrower than minor. Aug is narrower than dim. This makes interval arithmetic and chord names unaffected. M2 + M2 is still M3, and a C minor chord is still C Eb G.<br />

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In the 2nd approach, major is still wider than minor, so major is not fifthwards but fourthwards. The chain of fifths runs backwards:<br />

M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 - A1 etc.<br />

F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.<br />

16edo: P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8<br />

16edo: C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C<br />

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Interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.<br />

M2 + M2 --&gt; m2 + m2 = dim3 --&gt; aug3<br />

D to F# --&gt; D to Fb = dim3 --&gt; aug3<br />

D to F --&gt; D to F = m3 --&gt; M3 (natural notes don't flip)<br />

Eb + m3 --&gt; E# + M3 = G## --&gt; Gbb<br />

Eb + P5 --&gt; E# + P5 = B# --&gt; Bb (perfect intervals don't flip)<br />

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Both approaches have their merit, but the first one will be used here, because it makes naming chords easier.<br />