Kite's ups and downs notation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-03 18:47:30 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-03 20:02:37 UTC</tt>.

: The original revision id was <tt>~~558216055~~</tt>.

: The original revision id was <tt>558220645</tt>.

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The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. By which is meant, 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary. Here's 22-EDO in pentatonic notation:

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

chain of "fifths": Gx Dx Ax F# C# G# D# A# F C G D A Fb Cb Gb Db Ab Fbb Cbb Gbb Dbb

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

scale in C: C C# Dbb Db D D# Dx Fbb Fb F F# Gbb Gb G G# Gx Ab A A# Ax Cbb Cb C

Now that's an awful lot of sharps and flats, but that does make a neat and tidy notation (except for the Gbb-Gx fifth). And it exists as an alternative, embedded within our standard notation, with a key signature with circled X's on the B and E spots.

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__**Other EDOs**__

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!)

EDOs come in 5 categories, based on the size of the fifth:

EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:

~~supersharp~~ EDOs, with fifths wider than 720¢

"fifth-less" EDOs, with fifths wider than 720¢

pentatonic EDOs, with a fifth = 720¢

"sweet" EDOs, so-called because the fifth hits the "sweet spot" between 720¢ and 686¢

"perfect" EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢

~~superflat~~ EDOs or Mavila EDOs, with a fifth less than 686¢

fourthwards EDOs aka 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.

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.

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.

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Sus chords: as usual, "sus" means the 3rd is replaced by the named note, a 2nd or 4th. "Sus4" implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Some larger edos would have susv4, susvv4, etc. "Sus2" implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the fourthwards EDOs below for an exception.

"Aug" and "dim" chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. Thus "aug chord" means ~~not M3,A5 but~~ A3,P5~~. An~~ M3,A5 ~~chord~~ is a "major, aug five" chord. Likewise "dim chord" means ~~not m3,d5 but~~ d3,P5~~. An~~ m3,d5 ~~chord~~ is a "minor, dim five" chord.

"Aug" and "dim" chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. Thus "aug chord" means A3,P5, not M3,A5, which is a "major, aug five" chord. Likewise "dim chord" means d3,P5, not m3,d5, which is a "minor, dim five" chord.

0-5-13 = m

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keyboard/fretboard: D * * E/F * * G * * A * * B/C * * D

(the chain of fifths is always centered on D)

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

chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 M3/P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7

chord roots: I ^bII vII II/bIII ^bIII vIII III/IV ^IV vV V ^bVI vVI VI/bVII ^bVII vVII

0-3-9 = m or sus2

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16edo: D * E * * F * G * A * B * * C * D, 1 key per #/b, ups and downs not needed. # is fourthward.

chord components: P1 d2 m2 M2 m3 M3 A3~~/d4~~ P4 A4/d5 P5 d6 m6 M6/d7 m7 M7 A7 P8

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

chord roots: I #I/bbII bII II bIII III #III/~~vIV~~ IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI

chord roots: I #I/bbII bII II bIII III #III/bIV IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI

0-3-9 = sus2

0-4-9 = m

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17edo: D * * E F * * G * * A * * B C * * D, 2 keys per #/b.

chord components: P1 m2 ^m2/vM2 M2 m3 ^m3/vM3 M3 P4 ^P4/d5 A4/vP5 P5 m6 ^m6/vM6 M6 m7 ^m7/vM7 M7 P8

chord components: P1 m2 ^m2/vM2 M2 m3 ^m3/vM3 M3 P4 ^P4/d5 A4/vP5 P5 m6 ^m6/vM6 M6 m7 ^m7/vM7 M7

chord roots: I v#I/bII #I/vII II bIII vIII III IV ^IV/bV #IV/vV V #V/bVI vVI VI bVII vVII VII

0-4-10 = m

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19edo: D * * E * F * * G * * A * * B * C * * D, ups and downs not needed.

chord components: P1 d2 m2 M2 d3 m3 M3 A3 P4 A4 d5 P5 d6 m6 M6 d7 m7 M7 A7 P8

chord components: P1 d2 m2 M2 d3 m3 M3 A3 P4 A4 d5 P5 d6 m6 M6 d7 m7 M7 A7

chord roots: I v#I/bII #I/vII II bIII vIII III IV ^IV/bV #IV/vV V #V/bVI vVI VI bVII vVII VII

The possibility of a dim 3rd or an aug 3rd changes the meaning of "dim chord" and "aug chord".

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==**__Cross-EDO considerations__**==

In 22edo, the major chord is 0¢-436¢-709¢. In 19edo, it's 0¢-379¢-695¢. The two chords sound quite different, because "major 3rd" is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.

In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because "major 3rd" is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.

The name "major" refers not to the sound but to the function of the chord. If you want to play a I - VIm - IIm - V - I progression without pitch shifts or tonic drift, you can do that in any edo, as long as you use major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22edo, the chord that you need sounds like a red chord.

In other words, I - VIm - IIm - V - I in JI implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in certain EDOs. The notation tells you which ones.

If 22edo's downmajor chord 0-7-13 = 0¢-382¢-709¢ were called "major", you wouldn't know that it dosn't work in that progression.

~~The name "major" refers not to~~ the ~~sound but to the function of~~ the chord. If you want ~~to pump an 81/80 with~~ a ~~I - VIm - IIm - V - I progression, you can do that~~ in ~~any edo~~, ~~as long as~~ you use ~~major and minor chords~~.

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.

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

8edo octatonic (every note is a generator)

P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9

requires learning octatonic interval arithmetic and staff notation

11edo heptatonic narrow-fifth-based, fourthwards with ^/v, 2 keys per #/b (3/2 maps to 6\11 5th):

11edo heptatonic narrow-fifth-based, fourthwards with ^/v, 2 keys per #/b (3/2 maps to 6\11 = perfect 5th):

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

problematic because m3 = 2\11 is narrower than M2 = 3\11

11edo nonotonic narrow-fifth-based, fourthwards with no ups and downs (3/2 maps to 6\11 6th):

11edo nonotonic narrow-fifth-based, fourthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):

nonotonic fourthwards chain of sixths:

M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9 - d5 etc.

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

requires learning nonotonic interval arithmetic and ~~devising a nonotonic~~ staff notation

requires learning nonotonic interval arithmetic and staff notation

11edo pentatonic wide-fifth-based, fifthwards using ^/v, 2 keys per #/b (3/2 maps to 7\11 6th):

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18edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th)

P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10

requires learning nonotonic interval arithmetic and ~~devising a nonotonic~~ staff notation

requires learning nonotonic interval arithmetic and staff notation

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===__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.

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===__"Sweet" EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31~~, 32, 33,~~ 34, 36 and all higher ~~ones~~)__===

===__"Sweet" EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)__===

All sweet EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.

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

P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8

perfect = white, major = red, yellow and fifthward white, minor = green, blue and fourthwards white

Chord names are compatible with conventional names

0-3-7 = m

0-4-7 = M

0-3-6 = m,d5

0-4-7-10 = M,m7

**__17edo__:** sharp = 2 keys: C Db C# C

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The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. By which is meant, 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary. Here's 22-EDO in pentatonic notation:

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

chain of &quot;fifths&quot;: Gx Dx Ax F# C# G# D# A# F C G D A Fb Cb Gb Db Ab Fbb Cbb Gbb Dbb

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

scale in C: C C# Dbb Db D D# Dx Fbb Fb F F# Gbb Gb G G# Gx Ab A A# Ax Cbb Cb C

Now that's an awful lot of sharps and flats, but that does make a neat and tidy notation (except for the Gbb-Gx fifth). And it exists as an alternative, embedded within our standard notation, with a key signature with circled X's on the B and E spots.

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Other EDOs

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!)

EDOs come in 5 categories, based on the size of the fifth:

EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:

~~supersharp~~ EDOs, with fifths wider than 720¢

&quot;fifth-less&quot; EDOs, with fifths wider than 720¢

pentatonic EDOs, with a fifth = 720¢

&quot;sweet&quot; EDOs, so-called because the fifth hits the &quot;sweet spot&quot; between 720¢ and 686¢

&quot;perfect&quot; EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢

~~superflat~~ EDOs or Mavila EDOs, with a fifth less than 686¢

fourthwards EDOs aka 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 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.

This section will cover sweet EDOs and the other categories will be covered in other sections.

As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.

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Sus chords: as usual, &quot;sus&quot; means the 3rd is replaced by the named note, a 2nd or 4th. &quot;Sus4&quot; implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Some larger edos would have susv4, susvv4, etc. &quot;Sus2&quot; implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the fourthwards EDOs below for an exception.

&quot;Aug&quot; and &quot;dim&quot; chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. Thus &quot;aug chord&quot; means ~~not M3,A5 but~~ A3,P5~~. An~~ M3,A5 ~~chord~~ is a &quot;major, aug five&quot; chord. Likewise &quot;dim chord&quot; means ~~not m3,d5 but~~ d3,P5~~. An~~ m3,d5 ~~chord~~ is a &quot;minor, dim five&quot; chord.

&quot;Aug&quot; and &quot;dim&quot; chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. Thus &quot;aug chord&quot; means A3,P5, not M3,A5, which is a &quot;major, aug five&quot; chord. Likewise &quot;dim chord&quot; means d3,P5, not m3,d5, which is a &quot;minor, dim five&quot; chord.

0-5-13 = m

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keyboard/fretboard: D * * E/F * * G * * A * * B/C * * D

(the chain of fifths is always centered on D)

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

chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 M3/P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7

chord roots: I ^bII vII II/bIII ^bIII vIII III/IV ^IV vV V ^bVI vVI VI/bVII ^bVII vVII

0-3-9 = m or sus2

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16edo: D * E * * F * G * A * B * * C * D, 1 key per #/b, ups and downs not needed. # is fourthward.

chord components: P1 d2 m2 M2 m3 M3 A3~~/d4~~ P4 A4/d5 P5 d6 m6 M6/d7 m7 M7 A7 P8

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

chord roots: I #I/bbII bII II bIII III #III/~~vIV~~ IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI

chord roots: I #I/bbII bII II bIII III #III/bIV IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI

0-3-9 = sus2

0-4-9 = m

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17edo: D * * E F * * G * * A * * B C * * D, 2 keys per #/b.

chord components: P1 m2 ^m2/vM2 M2 m3 ^m3/vM3 M3 P4 ^P4/d5 A4/vP5 P5 m6 ^m6/vM6 M6 m7 ^m7/vM7 M7 P8

chord components: P1 m2 ^m2/vM2 M2 m3 ^m3/vM3 M3 P4 ^P4/d5 A4/vP5 P5 m6 ^m6/vM6 M6 m7 ^m7/vM7 M7

chord roots: I v#I/bII #I/vII II bIII vIII III IV ^IV/bV #IV/vV V #V/bVI vVI VI bVII vVII VII

0-4-10 = m

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19edo: D * * E * F * * G * * A * * B * C * * D, ups and downs not needed.

chord components: P1 d2 m2 M2 d3 m3 M3 A3 P4 A4 d5 P5 d6 m6 M6 d7 m7 M7 A7 P8

chord components: P1 d2 m2 M2 d3 m3 M3 A3 P4 A4 d5 P5 d6 m6 M6 d7 m7 M7 A7

chord roots: I v#I/bII #I/vII II bIII vIII III IV ^IV/bV #IV/vV V #V/bVI vVI VI bVII vVII VII

The possibility of a dim 3rd or an aug 3rd changes the meaning of &quot;dim chord&quot; and &quot;aug chord&quot;.

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<h2 id="toc4"><a name="Naming Chords-Cross-EDO considerations"></a>Cross-EDO considerations</h2>

In 22edo, the major chord is 0¢-436¢-709¢. In 19edo, it's 0¢-379¢-695¢. The two chords sound quite different, because &quot;major 3rd&quot; is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.

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 &quot;major 3rd&quot; is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.

The name &quot;major&quot; refers not to the sound but to the function of the chord. If you want to play a I - VIm - IIm - V - I progression without pitch shifts or tonic drift, you can do that in any edo, as long as you use major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22edo, the chord that you need sounds like a red chord.

In other words, I - VIm - IIm - V - I in JI implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in certain EDOs. The notation tells you which ones.

If 22edo's downmajor chord 0-7-13 = 0¢-382¢-709¢ were called &quot;major&quot;, you wouldn't know that it dosn't work in that progression.

~~The name &quot;major&quot; refers not to~~ the ~~sound but to the function of~~ the chord. If you want ~~to pump an 81/80 with~~ a ~~I - VIm - IIm - V - I progression, you can do that~~ in ~~any edo~~, ~~as long as~~ you use ~~major and minor chords~~.

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.

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Theoretical alternatives for 8edo, 11edo, 13edo and 18edo

~~ ~~

8edo octatonic (every note is a generator)

P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9

requires learning octatonic interval arithmetic and staff notation

11edo heptatonic narrow-fifth-based, fourthwards with ^/v, 2 keys per #/b (3/2 maps to 6\11 5th):

11edo heptatonic narrow-fifth-based, fourthwards with ^/v, 2 keys per #/b (3/2 maps to 6\11 = perfect 5th):

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

problematic because m3 = 2\11 is narrower than M2 = 3\11

11edo nonotonic narrow-fifth-based, fourthwards with no ups and downs (3/2 maps to 6\11 6th):

11edo nonotonic narrow-fifth-based, fourthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):

nonotonic fourthwards chain of sixths:

M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9 - d5 etc.

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

requires learning nonotonic interval arithmetic and ~~devising a nonotonic~~ staff notation

requires learning nonotonic interval arithmetic and staff notation

11edo pentatonic wide-fifth-based, fifthwards using ^/v, 2 keys per #/b (3/2 maps to 7\11 6th):

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18edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th)

P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10

requires learning nonotonic interval arithmetic and ~~devising a nonotonic~~ staff notation~~ ~~

requires learning nonotonic interval arithmetic and staff notation

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<h3 id="toc12"><a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"></a>Alternative pentatonic notation for pentatonic EDOs:</h3>

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.

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<h3 id="toc13"><a name="Summary of EDO notation--&quot;Sweet&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31~~, 32, 33,~~ 34, 36 and all higher ~~ones~~)"></a>&quot;Sweet&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31~~, 32, 33,~~ 34, 36 and all higher ~~ones~~)</h3>

<h3 id="toc13"><a name="Summary of EDO notation--&quot;Sweet&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)"></a>&quot;Sweet&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)</h3>

All sweet EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.

Line 1,179:

Line 1,201:

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

P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8

perfect = white, major = red, yellow and fifthward white, minor = green, blue and fourthwards white

Chord names are compatible with conventional names

0-3-7 = m

0-4-7 = M

0-3-6 = m,d5

0-4-7-10 = M,m7

17edo: sharp = 2 keys: C Db C# C

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-03 18:47:30 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-03 20:02:37 UTC</tt>.<br>
-: The original revision id was <tt>558216055</tt>.<br>
+: The original revision id was <tt>558220645</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>
@@ Line 53: / Line 53: @@
 The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. By which is meant, 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary. Here's 22-EDO in pentatonic notation:
-Gx Dx Ax F# C# G# D# A# F C G D A Fb Cb Gb Db Ab Fbb Cbb Gbb Dbb
+chain of "fifths": Gx Dx Ax F# C# G# D# A# F C G D A Fb Cb Gb Db Ab Fbb Cbb Gbb Dbb
-C C# Dbb Db D D# Dx Fbb Fb F F# Gbb Gb G G# Gx Ab A A# Ax Cbb Cb C
+scale in C: C C# Dbb Db D D# Dx Fbb Fb F F# Gbb Gb G G# Gx Ab A A# Ax Cbb Cb C
 Now that's an awful lot of sharps and flats, but that does make a neat and tidy notation (except for the Gbb-Gx fifth). And it exists as an alternative, embedded within our standard notation, with a key signature with circled X's on the B and E spots.
@@ Line 76: / Line 76: @@
 __**Other EDOs**__
-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!)
-EDOs come in 5 categories, based on the size of the fifth:
+EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:
-supersharp EDOs, with fifths wider than 720¢
+"fifth-less" EDOs, with fifths wider than 720¢
 pentatonic EDOs, with a fifth = 720¢
 "sweet" EDOs, so-called because the fifth hits the "sweet spot" between 720¢ and 686¢
 "perfect" EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢
-superflat EDOs or Mavila EDOs, with a fifth less than 686¢
+fourthwards EDOs aka 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.
+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.
 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.
@@ Line 202: / Line 203: @@
 Sus chords: as usual, "sus" means the 3rd is replaced by the named note, a 2nd or 4th. "Sus4" implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Some larger edos would have susv4, susvv4, etc. "Sus2" implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the fourthwards EDOs below for an exception.
-"Aug" and "dim" chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. Thus "aug chord" means not M3,A5 but A3,P5. An M3,A5 chord is a "major, aug five" chord. Likewise "dim chord" means not m3,d5 but d3,P5. An m3,d5 chord is a "minor, dim five" chord.
+"Aug" and "dim" chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. Thus "aug chord" means A3,P5, not M3,A5, which is a "major, aug five" chord. Likewise "dim chord" means d3,P5, not m3,d5, which is a "minor, dim five" chord.
 -5-13 = m
@@ Line 252: / Line 253: @@
 keyboard/fretboard: D * * E/F * * G * * A * * B/C * * D
 (the chain of fifths is always centered on D)
-chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 M3/P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7 P8
+chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 M3/P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7
 chord roots: I ^bII vII II/bIII ^bIII vIII III/IV ^IV vV V ^bVI vVI VI/bVII ^bVII vVII
 -3-9 = m or sus2
@@ Line 261: / Line 262: @@
 edo: D * E * * F * G * A * B * * C * D, 1 key per #/b, ups and downs not needed. # is fourthward.
-chord components: P1 d2 m2 M2 m3 M3 A3/d4 P4 A4/d5 P5 d6 m6 M6/d7 m7 M7 A7 P8
+chord components: P1 d2 m2 M2 m3 M3 A3 P4 A4/d5 P5 d6 m6 M6/d7 m7 M7 A7
-chord roots: I #I/bbII bII II bIII III #III/vIV IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI
+chord roots: I #I/bbII bII II bIII III #III/bIV IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI
 -3-9 = sus2
 -4-9 = m
@@ Line 272: / Line 273: @@
 edo: D * * E F * * G * * A * * B C * * D, 2 keys per #/b.
-chord components: P1 m2 ^m2/vM2 M2 m3 ^m3/vM3 M3 P4 ^P4/d5 A4/vP5 P5 m6 ^m6/vM6 M6 m7 ^m7/vM7 M7 P8
+chord components: P1 m2 ^m2/vM2 M2 m3 ^m3/vM3 M3 P4 ^P4/d5 A4/vP5 P5 m6 ^m6/vM6 M6 m7 ^m7/vM7 M7
 chord roots: I v#I/bII #I/vII II bIII vIII III IV ^IV/bV #IV/vV V #V/bVI vVI VI bVII vVII VII
 -4-10 = m
@@ Line 284: / Line 285: @@
 edo: D * * E * F * * G * * A * * B * C * * D, ups and downs not needed.
-chord components: P1 d2 m2 M2 d3 m3 M3 A3 P4 A4 d5 P5 d6 m6 M6 d7 m7 M7 A7 P8
+chord components: P1 d2 m2 M2 d3 m3 M3 A3 P4 A4 d5 P5 d6 m6 M6 d7 m7 M7 A7
 chord roots: I v#I/bII #I/vII II bIII vIII III IV ^IV/bV #IV/vV V #V/bVI vVI VI bVII vVII VII
 The possibility of a dim 3rd or an aug 3rd changes the meaning of "dim chord" and "aug chord".
@@ Line 341: / Line 342: @@
 ==**__Cross-EDO considerations__**==
-In 22edo, the major chord is 0¢-436¢-709¢. In 19edo, it's 0¢-379¢-695¢. The two chords sound quite different, because "major 3rd" is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.
+In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because "major 3rd" is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.
+The name "major" refers not to the sound but to the function of the chord. If you want to play a I - VIm - IIm - V - I progression without pitch shifts or tonic drift, you can do that in any edo, as long as you use major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22edo, the chord that you need sounds like a red chord.
+In other words, I - VIm - IIm - V - I in JI implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in certain EDOs. The notation tells you which ones.
+If 22edo's downmajor chord 0-7-13 = 0¢-382¢-709¢ were called "major", you wouldn't know that it dosn't work in that progression.
-The name "major" refers not to the sound but to the function of the chord. If you want to pump an 81/80 with a I - VIm - IIm - V - I progression, you can do that in any edo, as long as you use major and minor chords.
+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.
@@ Line 359: / Line 366: @@
 __**Theoretical alternatives for 8edo, 11edo, 13edo and 18edo**__
 edo octatonic (every note is a generator)
 P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9
+requires learning octatonic interval arithmetic and staff notation
-edo heptatonic narrow-fifth-based, fourthwards with ^/v, 2 keys per #/b (3/2 maps to 6\11 5th):
+edo heptatonic narrow-fifth-based, fourthwards with ^/v, 2 keys per #/b (3/2 maps to 6\11 = perfect 5th):
 P1 - m2 - vM2/m3 - M2/^m3 - M3 - P4 - P5 - m6 - vM6/m7 - M6/^m7 - M7 - P8
 problematic because m3 = 2\11 is narrower than M2 = 3\11
-edo nonotonic narrow-fifth-based, fourthwards with no ups and downs (3/2 maps to 6\11 6th):
+edo nonotonic narrow-fifth-based, fourthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):
 nonotonic fourthwards chain of sixths:
 M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9 - d5 etc.
 P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8
-requires learning nonotonic interval arithmetic and devising a nonotonic staff notation
+requires learning nonotonic interval arithmetic and staff notation
 edo pentatonic wide-fifth-based, fifthwards using ^/v, 2 keys per #/b (3/2 maps to 7\11 6th):
@@ Line 397: / Line 404: @@
 edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th)
 P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10
-requires learning nonotonic interval arithmetic and devising a nonotonic staff notation
+requires learning nonotonic interval arithmetic and staff notation
@@ Line 554: / Line 560: @@
 ===__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.
@@ Line 573: / Line 581: @@
-===__"Sweet" EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31, 32, 33, 34, 36 and all higher ones)__===
+===__"Sweet" EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)__===
 All sweet EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.
@@ Line 582: / Line 590: @@
 D - D#/Eb - E - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - C - C#/Db - D
 P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8
+perfect = white, major = red, yellow and fifthward white, minor = green, blue and fourthwards white
+Chord names are compatible with conventional names
+-3-7 = m
+-4-7 = M
+-3-6 = m,d5
+-4-7-10 = M,m7
 **__17edo__:** sharp = 2 keys: C Db C# C
@@ Line 650: / Line 664: @@
 The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. By which is meant, 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary. Here's 22-EDO in pentatonic notation:&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;
+chain of &amp;quot;fifths&amp;quot;: 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;
+scale in C: C C# Dbb Db D D# Dx Fbb Fb F F# Gbb Gb G G# Gx Ab A A# Ax Cbb Cb C&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;
@@ Line 673: / Line 687: @@
 &lt;br /&gt;
 &lt;u&gt;&lt;strong&gt;Other EDOs&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
-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;
+EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:&lt;br /&gt;
-supersharp EDOs, with fifths wider than 720¢&lt;br /&gt;
+&amp;quot;fifth-less&amp;quot; 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;
 &amp;quot;perfect&amp;quot; EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢&lt;br /&gt;
-superflat EDOs or Mavila EDOs, with a fifth less than 686¢&lt;br /&gt;
+fourthwards EDOs aka 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. &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;
+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;
@@ Line 799: / Line 814: @@
 Sus chords: as usual, &amp;quot;sus&amp;quot; means the 3rd is replaced by the named note, a 2nd or 4th. &amp;quot;Sus4&amp;quot; implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Some larger edos would have susv4, susvv4, etc. &amp;quot;Sus2&amp;quot; implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the fourthwards EDOs below for an exception.&lt;br /&gt;
 &lt;br /&gt;
-&amp;quot;Aug&amp;quot; and &amp;quot;dim&amp;quot; chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. Thus &amp;quot;aug chord&amp;quot; means not M3,A5 but A3,P5. An M3,A5 chord is a &amp;quot;major, aug five&amp;quot; chord. Likewise &amp;quot;dim chord&amp;quot; means not m3,d5 but d3,P5. An m3,d5 chord is a &amp;quot;minor, dim five&amp;quot; chord.&lt;br /&gt;
+&amp;quot;Aug&amp;quot; and &amp;quot;dim&amp;quot; chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. Thus &amp;quot;aug chord&amp;quot; means A3,P5, not M3,A5, which is a &amp;quot;major, aug five&amp;quot; chord. Likewise &amp;quot;dim chord&amp;quot; means d3,P5, not m3,d5, which is a &amp;quot;minor, dim five&amp;quot; chord.&lt;br /&gt;
 &lt;br /&gt;
 -5-13 = m&lt;br /&gt;
@@ Line 849: / Line 864: @@
 keyboard/fretboard: D * * E/F * * G * * A * * B/C * * D&lt;br /&gt;
 (the chain of fifths is always centered on D)&lt;br /&gt;
-chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 M3/P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7 P8&lt;br /&gt;
+chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 M3/P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7&lt;br /&gt;
 chord roots: I ^bII vII II/bIII ^bIII vIII III/IV ^IV vV V ^bVI vVI VI/bVII ^bVII vVII&lt;br /&gt;
 -3-9 = m or sus2&lt;br /&gt;
@@ Line 858: / Line 873: @@
 &lt;br /&gt;
 edo: D * E * * F * G * A * B * * C * D, 1 key per #/b, ups and downs not needed. # is fourthward.&lt;br /&gt;
-chord components: P1 d2 m2 M2 m3 M3 A3/d4 P4 A4/d5 P5 d6 m6 M6/d7 m7 M7 A7 P8&lt;br /&gt;
+chord components: P1 d2 m2 M2 m3 M3 A3 P4 A4/d5 P5 d6 m6 M6/d7 m7 M7 A7&lt;br /&gt;
-chord roots: I #I/bbII bII II bIII III #III/vIV IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI&lt;br /&gt;
+chord roots: I #I/bbII bII II bIII III #III/bIV IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI&lt;br /&gt;
 -3-9 = sus2&lt;br /&gt;
 -4-9 = m&lt;br /&gt;
@@ Line 869: / Line 884: @@
 &lt;br /&gt;
 edo: D * * E F * * G * * A * * B C * * D, 2 keys per #/b.&lt;br /&gt;
-chord components: P1 m2 ^m2/vM2 M2 m3 ^m3/vM3 M3 P4 ^P4/d5 A4/vP5 P5 m6 ^m6/vM6 M6 m7 ^m7/vM7 M7 P8&lt;br /&gt;
+chord components: P1 m2 ^m2/vM2 M2 m3 ^m3/vM3 M3 P4 ^P4/d5 A4/vP5 P5 m6 ^m6/vM6 M6 m7 ^m7/vM7 M7&lt;br /&gt;
 chord roots: I v#I/bII #I/vII II bIII vIII III IV ^IV/bV #IV/vV V #V/bVI vVI VI bVII vVII VII&lt;br /&gt;
 -4-10 = m&lt;br /&gt;
@@ Line 881: / Line 896: @@
 &lt;br /&gt;
 edo: D * * E * F * * G * * A * * B * C * * D, ups and downs not needed.&lt;br /&gt;
-chord components: P1 d2 m2 M2 d3 m3 M3 A3 P4 A4 d5 P5 d6 m6 M6 d7 m7 M7 A7 P8&lt;br /&gt;
+chord components: P1 d2 m2 M2 d3 m3 M3 A3 P4 A4 d5 P5 d6 m6 M6 d7 m7 M7 A7&lt;br /&gt;
 chord roots: I v#I/bII #I/vII II bIII vIII III IV ^IV/bV #IV/vV V #V/bVI vVI VI bVII vVII VII&lt;br /&gt;
 The possibility of a dim 3rd or an aug 3rd changes the meaning of &amp;quot;dim chord&amp;quot; and &amp;quot;aug chord&amp;quot;.&lt;br /&gt;
@@ Line 938: / Line 953: @@
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Naming Chords-Cross-EDO considerations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;strong&gt;&lt;u&gt;Cross-EDO considerations&lt;/u&gt;&lt;/strong&gt;&lt;/h2&gt;
   &lt;br /&gt;
-In 22edo, the major chord is 0¢-436¢-709¢. In 19edo, it's 0¢-379¢-695¢. The two chords sound quite different, because &amp;quot;major 3rd&amp;quot; is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.&lt;br /&gt;
+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 &amp;quot;major 3rd&amp;quot; is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.&lt;br /&gt;
+&lt;br /&gt;
+The name &amp;quot;major&amp;quot; refers not to the sound but to the function of the chord. If you want to play a I - VIm - IIm - V - I progression without pitch shifts or tonic drift, you can do that in any edo, as long as you use major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22edo, the chord that you need sounds like a red chord. &lt;br /&gt;
+&lt;br /&gt;
+In other words, I - VIm - IIm - V - I in JI implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in certain EDOs. The notation tells you which ones.&lt;br /&gt;
+&lt;br /&gt;
+If 22edo's downmajor chord 0-7-13 = 0¢-382¢-709¢ were called &amp;quot;major&amp;quot;, you wouldn't know that it dosn't work in that progression.&lt;br /&gt;
 &lt;br /&gt;
-The name &amp;quot;major&amp;quot; refers not to the sound but to the function of the chord. If you want to pump an 81/80 with a I - VIm - IIm - V - I progression, you can do that in any edo, as long as you use major and minor chords.&lt;br /&gt;
+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.&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
@@ Line 956: / Line 977: @@
 &lt;br /&gt;
 &lt;u&gt;&lt;strong&gt;Theoretical alternatives for 8edo, 11edo, 13edo and 18edo&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
-&lt;br /&gt;
 &lt;br /&gt;
 edo octatonic (every note is a generator)&lt;br /&gt;
 P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9&lt;br /&gt;
+requires learning octatonic interval arithmetic and staff notation&lt;br /&gt;
 &lt;br /&gt;
-edo heptatonic narrow-fifth-based, fourthwards with ^/v, 2 keys per #/b (3/2 maps to 6\11 5th):&lt;br /&gt;
+edo heptatonic narrow-fifth-based, fourthwards with ^/v, 2 keys per #/b (3/2 maps to 6\11 = perfect 5th):&lt;br /&gt;
 P1 - m2 - vM2/m3 - M2/^m3 - M3 - P4 - P5 - m6 - vM6/m7 - M6/^m7 - M7 - P8&lt;br /&gt;
 problematic because m3 = 2\11 is narrower than M2 = 3\11&lt;br /&gt;
 &lt;br /&gt;
-edo nonotonic narrow-fifth-based, fourthwards with no ups and downs (3/2 maps to 6\11 6th):&lt;br /&gt;
+edo nonotonic narrow-fifth-based, fourthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):&lt;br /&gt;
 nonotonic fourthwards chain of sixths:&lt;br /&gt;
 M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9 - d5 etc.&lt;br /&gt;
 P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8&lt;br /&gt;
-requires learning nonotonic interval arithmetic and devising a nonotonic staff notation&lt;br /&gt;
+requires learning nonotonic interval arithmetic and staff notation&lt;br /&gt;
 &lt;br /&gt;
 edo pentatonic wide-fifth-based, fifthwards using ^/v, 2 keys per #/b (3/2 maps to 7\11 6th):&lt;br /&gt;
@@ Line 994: / Line 1,015: @@
 edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th)&lt;br /&gt;
 P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10&lt;br /&gt;
-requires learning nonotonic interval arithmetic and devising a nonotonic staff notation&lt;br /&gt;
+requires learning nonotonic interval arithmetic and staff notation&lt;br /&gt;
-&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,151: / Line 1,171: @@
 &lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;&lt;u&gt;Alternative pentatonic notation for pentatonic EDOs:&lt;/u&gt;&lt;/h3&gt;
   &lt;br /&gt;
+Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.&lt;br /&gt;
+C# - G# - D# - A# - E# - C - G - D - A - E - Cb - Gb - Db - Ab - Eb etc.&lt;br /&gt;
 All intervals are perfect, so quality can be omitted.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,170: / Line 1,192: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Summary of EDO notation--&amp;quot;Sweet&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31, 32, 33, 34, 36 and all higher ones)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;&lt;u&gt;&amp;quot;Sweet&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31, 32, 33, 34, 36 and all higher ones)&lt;/u&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Summary of EDO notation--&amp;quot;Sweet&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;&lt;u&gt;&amp;quot;Sweet&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)&lt;/u&gt;&lt;/h3&gt;
   &lt;br /&gt;
 All sweet EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.&lt;br /&gt;
@@ Line 1,179: / Line 1,201: @@
 D - D#/Eb - E - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - C - C#/Db - D&lt;br /&gt;
 P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8&lt;br /&gt;
+perfect = white, major = red, yellow and fifthward white, minor = green, blue and fourthwards white&lt;br /&gt;
+Chord names are compatible with conventional names&lt;br /&gt;
+-3-7 = m&lt;br /&gt;
+-4-7 = M&lt;br /&gt;
+-3-6 = m,d5&lt;br /&gt;
+-4-7-10 = M,m7&lt;br /&gt;
 &lt;br /&gt;
 &lt;strong&gt;&lt;u&gt;17edo&lt;/u&gt;:&lt;/strong&gt; sharp = 2 keys: C Db C# C&lt;br /&gt;