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:<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-08-01 02:19:22 UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-08-01 03:53:08 UTC</tt>.<br>

: The original revision id was <tt>~~588456270~~</tt>.<br>

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In general, the period is pronounced as "dot". For example, C.v is "C dot down", because "C down" means Cv major = Cv Ev Gv. However sometimes a slight pause suffices, e.g. C.vm = "C downminor" and Cv.m = "C-down minor". This is analogous to saying "A-flat nine" for Ab C Eb Gb Bb and "A flat-nine" for A C# E G Bb. However C.v7 must be "C dot down-seven" because "C down-seven" is C(v7) = C E G Bbv. Even if the period doesn't need to be pronounced, it's always acceptable to do so.

Applying "dot up" or "dot down" to a chord raises or lowers ~~only two chord components:~~ the 3rd, and ~~either~~ the 6th or the 7th, ~~whichever is~~ present. Thus "C dot down nine" = C.v9 = C Ev G Bbv D. The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd (e.g. the maj6, min7, maj7, 6/9, min9 and maj9 chords). Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the fifth.

Applying "dot up" or "dot down" to a chord raises or lowers the 3rd, and also the 6th or the 7th, if present. Thus "C dot down nine" = C.v9 = C Ev G Bbv D. The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd (e.g. the maj6, min7, maj7, 6/9, min9 and maj9 chords). Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the fifth.

__**Various triads:**__

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C Fv G = Csusv4 = "C sus down-four"

C Fvv G = Csusvv4 = "C sus double-down four"

C D# G = Csus#2 = "C sus aug-two"

C Ebb G = C(bb3) = "C dim-three"

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

C Fb G = Csusb4 = "C sus dim-four"

__**Altered fifths:**__

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C E^ G# is Caug(^3) = "C aug up-three" (note that here "up-three" means upmajor 3rd, not upminor 3rd)

C E^ G#^ = Caug(^3,^5) = "C aug up-three up-five"

C E# G# is Caug(#3)

__**Seventh chords:**__

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C Eb Gb^ Bb^ = Cdim(^5,^7) = "C dim up-five up-seven" or "C half-dim up-five up-seven"

C Eb^ Gb^ Bb^ = C.^m7(^b5) = "C dot upminor seven upflat-five" or "C half-dim up-three up-five up-seven"

C E G B# = C(#7) = "C aug-seven"

C Eb G B# = Cm(#7) = "C minor sharp-seven"

__**Sixth chords:**__

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C Ev G A = C6(v3) = "C six down-three" (in certain EDOs, C6(~3) = "C six mid-three")

C E G Av = C(v6) = "C down-six" (in certain EDOs, C(~6) = "C mid-six")

C Ev G Av = C.v6 = "C dot down~~-six"~~

C Ev G Av = C.v6 = "C dot down-six" (in certain EDOs, C.~6 = "C dot mid-six")

~~C E G Ab^ = C(^m6) = "C upminor~~-six" (in certain EDOs, C(~6) = "C mid-six")

C Eb G A = Cm6 = "C minor six" (in perfect EDOs, C6 = "C six")

C Eb^ G A = Cm6(^3) = "C minor six up-three" (in certain EDOs, C6(~3) = "C six mid three")

C Eb G Av = Cm(v6) = "C minor down-six" (in certain EDOs, Cm(~6) = "C minor mid-six")

C Eb^ G A^ = C.^m6 = "C dot ~~upminor~~-six"

C Eb^ G A^ = C.^m6 = "C dot up minor-six"

C Eb^ G A = Cm6(^3) = "C minor six up-three" (in certain EDOs, C6(~3) = "C six mid-three")

C Eb G Ab = Cm(b6) = "C minor flat-six"

C E G Ab = C(b6) = "C flat-six"

C Eb G A# = Cm(#6) = "C minor sharp six"

C E G A# = C(#6) = "C sharp-six"

__**Ninth chords:**__

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

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

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

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

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

0-3-9 = D F## A = D(A3) = "D aug-three"

0-3-9 = D F## A = D(#3) = "D aug-three" (or possibly D Eb A = Dsusb2 = "D sus dim-two")

0-4-9 = D F# A = D = "D" or "D major"

0-5-9 = D F A = Dm = "D minor"

0-6-9 = D Fb A = D(d3) = "D dim-three" (~~dim 3rd, perfect 5th~~)

0-6-9 = D Fb A = D(bb3) = "D dim-three" (or possibly D G# A = Dsus#4 = "D sus aug-four")

0-5-10 = D F Ab = Ddim = "D dim" ~~(the conventional dim triad)~~

0-5-10 = D F Ab = Ddim = "D dim"

0-6-10 = D Fb Ab = Ddim(d3) = "D dim dim-three"

0-6-10 = D Fb Ab = Ddim(bb3) = "D dim dim-three"

0-7-9 = D G A = Dsus4 = "D sus four"

0-5-9-13 = D F A C# = DmM7

0-5-9-13 = D F A C# = DmM7 = " D minor major seven"

0-4-8-12 = D F# A# C## = Daug(A7)

0-4-8-12 = D F# A# C## = Daug(#7) = "D aug aug-seven"

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

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0-5-10-15 = D F^ A C^ = D.~7

0-6-10-15 = D F# A C^ = D(~7)

18edo:

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

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0-14-28 = D F# A# = Daug = "D aug"

11edo: C# * * C D * * Db

D E F# Fv F G A B B^ Bb C D, -2 keys per sharp, upmajor = downminor = mid.

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

m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7

3 9 4 10 5 0 6 1 7 2 8

3 12 5 14 7 0 9 2 11 4

0-1-6 = D E A = Dsus2

0-2-6 = D F# A = D = "D major"

0-3-6 = D Fv A = D.~ = "D mid"

0-4-6 = D F A = Dm = "D minor"

0-5-6 = D G A = Dsus4

0-2-5 = D F# Av = D(v5) = "D down-five"

0-3-5 = D Fv Av = D.~(v5) = "D mid down-five"

0-3-7 = D Fv A^ = D.~(^5)

0-4-7 = D F A^ = Dm(^5)

0-2-6-7 = D F# A B = D6

0-3-6-7 = D Fv A B = D6(~3)

0-4-6-7 = D F A B = Dm6

0-2-6-8 = D F# A C# = DM7, or D F# A B^ = D(~6) = "D mid-six"

0-3-6-8 = D F#^ A C# = DM7(^3), or D F#^ A B^ = D.^6 = "D dot up-six"

0-3-6-9 = D Fv A Cv = D.~7, or D Fv A Bb = D.~(b6) = "D mid flat-six"

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

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requires learning octatonic interval arithmetic and staff notation

11edo heptatonic narrow-fifth-based, fourthwards, # is ^^ (3/2 maps to 6\11 perfect 5th):

11edo heptatonic narrow-fifth-based, fourthwards, # is vv (3/2 maps to 6\11 perfect 5th):

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

D E * * F G A B * * C D = D E F# F~ F G A B B~ Bb C D

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

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

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

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 = 2\11 is narrower than m2 = 3\11

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

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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.<br />

<br />

<img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png" alt="The fifth of EDOs 5-53.png" title="The fifth of EDOs 5-53.png" style="height: 1035px; width: 800px;" /><br />

<img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png" alt="The fifth of EDOs 5-53.png" title="The fifth of EDOs 5-53.png" style="height: 1035px; 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. The colored regions of the tree are what I call <strong>kites</strong>, and this version of the Stern-Brocot tree I call the <strong>Tree of Kites</strong>. 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 />

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<br />

<img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" /><br />

<img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" /><br />

<br />

<h2 id="toc4"><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /></h2>

<h2 id="toc4"><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /></h2>

<h2 id="toc5"> </h2>

<h2 id="toc6"> </h2>

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In general, the period is pronounced as &quot;dot&quot;. For example, C.v is &quot;C dot down&quot;, because &quot;C down&quot; means Cv major = Cv Ev Gv. However sometimes a slight pause suffices, e.g. C.vm = &quot;C downminor&quot; and Cv.m = &quot;C-down minor&quot;. This is analogous to saying &quot;A-flat nine&quot; for Ab C Eb Gb Bb and &quot;A flat-nine&quot; for A C# E G Bb. However C.v7 must be &quot;C dot down-seven&quot; because &quot;C down-seven&quot; is C(v7) = C E G Bbv. Even if the period doesn't need to be pronounced, it's always acceptable to do so.<br />

<br />

Applying &quot;dot up&quot; or &quot;dot down&quot; to a chord raises or lowers ~~only two chord components:~~ the 3rd, and ~~either~~ the 6th or the 7th, ~~whichever is~~ present. Thus &quot;C dot down nine&quot; = C.v9 = C Ev G Bbv D. The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd (e.g. the maj6, min7, maj7, 6/9, min9 and maj9 chords). Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the fifth.<br />

Applying &quot;dot up&quot; or &quot;dot down&quot; to a chord raises or lowers the 3rd, and also the 6th or the 7th, if present. Thus &quot;C dot down nine&quot; = C.v9 = C Ev G Bbv D. The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd (e.g. the maj6, min7, maj7, 6/9, min9 and maj9 chords). Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the fifth.<br />

<br />

<u><strong>Various triads:</strong></u><br />

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C Fv G = Csusv4 = &quot;C sus down-four&quot;<br />

C Fvv G = Csusvv4 = &quot;C sus double-down four&quot;<br />

<br />

C D# G = Csus#2 = &quot;C sus aug-two&quot;<br />

C Ebb G = C(bb3) = &quot;C dim-three&quot;<br />

C E# G = C(#3) = &quot;C aug-three&quot;<br />

C Fb G = Csusb4 = &quot;C sus dim-four&quot;<br />

<br />

<u><strong>Altered fifths:</strong></u><br />

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C E^ G# is Caug(^3) = &quot;C aug up-three&quot; (note that here &quot;up-three&quot; means upmajor 3rd, not upminor 3rd)<br />

C E^ G#^ = Caug(^3,^5) = &quot;C aug up-three up-five&quot;<br />

C E# G# is Caug(#3)<br />

<br />

<u><strong>Seventh chords:</strong></u><br />

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C Eb Gb^ Bb^ = Cdim(^5,^7) = &quot;C dim up-five up-seven&quot; or &quot;C half-dim up-five up-seven&quot;<br />

C Eb^ Gb^ Bb^ = C.^m7(^b5) = &quot;C dot upminor seven upflat-five&quot; or &quot;C half-dim up-three up-five up-seven&quot;<br />

<br />

C E G B# <h1 id="toc8"><a name="C(#7)"></a> C(#7) </h1>

&quot;C aug-seven&quot;<br />

C Eb G B# <h1 id="toc9"><a name="Cm(#7)"></a> Cm(#7) </h1>

&quot;C minor sharp-seven&quot;<br />

<br />

<u><strong>Sixth chords:</strong></u><br />

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C Ev G A = C6(v3) = &quot;C six down-three&quot; (in certain EDOs, C6(~3) = &quot;C six mid-three&quot;)<br />

C E G Av = C(v6) = &quot;C down-six&quot; (in certain EDOs, C(~6) = &quot;C mid-six&quot;)<br />

C Ev G Av = C.v6 = &quot;C dot down~~-six&quot;<br />~~

C Ev G Av = C.v6 = &quot;C dot down-six&quot; (in certain EDOs, C.~6 = &quot;C dot mid-six&quot;)<br />

~~C E G Ab^ = C(^m6) = &quot;C upminor~~-six&quot; (in certain EDOs, C(~6) = &quot;C mid-six&quot;)<br />

<br />

C Eb G A = Cm6 = &quot;C minor six&quot; (in perfect EDOs, C6 = &quot;C six&quot;)<br />

C Eb^ G A = Cm6(^3) = &quot;C minor six up-three&quot; (in certain EDOs, C6(~3) = &quot;C six mid three&quot;)<br />

C Eb G Av = Cm(v6) = &quot;C minor down-six&quot; (in certain EDOs, Cm(~6) = &quot;C minor mid-six&quot;)<br />

C Eb^ G A^ = C.^m6 = &quot;C dot ~~upminor~~-six&quot;<br />

C Eb^ G A^ = C.^m6 = &quot;C dot up minor-six&quot;<br />

C Eb^ G A = Cm6(^3) = &quot;C minor six up-three&quot; (in certain EDOs, C6(~3) = &quot;C six mid-three&quot;)<br />

<br />

C Eb G Ab = Cm(b6) = &quot;C minor flat-six&quot;<br />

C E G Ab = C(b6) = &quot;C flat-six&quot;<br />

C Eb G A# <h1 id="toc10"><a name="Cm(#6)"></a> Cm(#6) </h1>

&quot;C minor sharp six&quot;<br />

C E G A# <h1 id="toc11"><a name="C(#6)"></a> C(#6) </h1>

&quot;C sharp-six&quot;<br />

<br />

<u><strong>Ninth chords:</strong></u><br />

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C Dbv Ev G Bbv = C.v7(vb9) = &quot;C dot down seven downflat-nine&quot;<br />

<br />

<h4 id="toc12"><a name="C(#6)---Example EDOs:"></a><u><strong>Example EDOs:</strong></u></h4>

<br />

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

(the chain of fifths is always centered on D, for symmetry)<br />

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chord components: P1 A2 M2 m2/A3 M3 m3 d3/A4 P4 d4/A5 P5 d5/A6 M6 m6/A7 M7 m7 d7<br />

chord roots: I bI/#II II bII III bIII bbIII/#IV IV bIV/#V V bV/#VI VI bVI VII bVII bbVII/#I<br />

0-3-9 = D F## A = D(A3) = &quot;D aug-three&quot;<br />

0-3-9 = D F## A = D(#3) = &quot;D aug-three&quot; (or possibly D Eb A = Dsusb2 = &quot;D sus dim-two&quot;)<br />

0-4-9 = D F# A = D = &quot;D&quot; or &quot;D major&quot;<br />

0-5-9 = D F A = Dm = &quot;D minor&quot;<br />

0-6-9 = D Fb A = D(d3) = &quot;D dim-three&quot; (~~dim 3rd, perfect 5th~~)<br />

0-6-9 = D Fb A = D(bb3) = &quot;D dim-three&quot; (or possibly D G# A = Dsus#4 = &quot;D sus aug-four&quot;)<br />

0-5-10 = D F Ab = Ddim = &quot;D dim&quot; ~~(the conventional dim triad)~~<br />

0-5-10 = D F Ab = Ddim = &quot;D dim&quot;<br />

0-6-10 = D Fb Ab = Ddim(d3) = &quot;D dim dim-three&quot;<br />

0-6-10 = D Fb Ab = Ddim(bb3) = &quot;D dim dim-three&quot;<br />

0-7-9 = D G A = Dsus4 = &quot;D sus four&quot;<br />

0-5-9-13 = D F A C# = DmM7<br />

0-5-9-13 = D F A C# <h1 id="toc13"><a name="DmM7"></a> DmM7 </h1>

0-4-8-12 = D F# A# C## = Daug(A7)<br />

&quot; D minor major seven&quot;<br />

0-4-8-12 = D F# A# C## <h1 id="toc14"><a name="Daug(#7)"></a> Daug(#7) </h1>

&quot;D aug aug-seven&quot;<br />

<br />

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

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0-5-10-15 = D F^ A C^ = D.~7<br />

0-6-10-15 = D F# A C^ = D(~7)<br />

<br />

18edo:<br />

<br />

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

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0-4-10 = D Fb Ab = Ddim(bb3) = &quot;D dim dim-three&quot;<br />

0-7-11 = D F## A = D(#3) = &quot;D aug-three&quot; (aug 3rd, perfect 5th)<br />

0-7-12 = D F## A# <h1 id="~~toc8~~"><a name="Daug(#3)"></a> Daug(#3) </h1>

0-7-12 = D F## A# <h1 id="toc15"><a name="Daug(#3)"></a> Daug(#3) </h1>

&quot;D aug aug-three&quot;<br />

0-5-10-15 = D F Ab Cb = Ddim7 = &quot;D dim-seven&quot;<br />

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0-14-26 = D F# A^^ = D(^^5) = &quot;D double-up-five&quot;<br />

0-14-27 = D F# A#v = Daug(v5) = &quot;D aug down-five&quot;<br />

0-14-28 = D F# A# <h1 id="~~toc9~~"><a name="Daug"></a> Daug </h1>

0-14-28 = D F# A# <h1 id="toc16"><a name="Daug"></a> Daug </h1>

&quot;D aug&quot;<br />

<br />

11edo: C# * * C D * * Db<br />

D E F# Fv F G A B B^ Bb C D, -2 keys per sharp, upmajor = downminor = mid.<br />

P1 M2 ~2/M3 ~3 m3 P4 P5 M6 ~6/M7 m6/~7 m7 P8 <br />

m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7<br />

3 9 4 10 5 0 6 1 7 2 8<br />

3 12 5 14 7 0 9 2 11 4<br />

0-1-6 = D E A = Dsus2<br />

0-2-6 = D F# A = D = &quot;D major&quot;<br />

0-3-6 = D Fv A = D.~ = &quot;D mid&quot;<br />

0-4-6 = D F A = Dm = &quot;D minor&quot;<br />

0-5-6 = D G A = Dsus4<br />

<br />

<h2 id="~~toc10~~"><a name="~~Daug~~-Cross-EDO considerations"></a><strong><u>Cross-EDO considerations</u></strong></h2>

0-2-5 = D F# Av = D(v5) = &quot;D down-five&quot;<br />

0-3-5 = D Fv Av = D.~(v5) = &quot;D mid down-five&quot;<br />

0-3-7 = D Fv A^ = D.~(^5)<br />

0-4-7 = D F A^ = Dm(^5)<br />

<br />

0-2-6-7 = D F# A B = D6<br />

0-3-6-7 = D Fv A B = D6(~3)<br />

0-4-6-7 = D F A B = Dm6<br />

0-2-6-8 = D F# A C# <h1 id="toc17"><a name="DM7, or D F# A B^"></a> DM7, or D F# A B^ </h1>

D(~6) = &quot;D mid-six&quot;<br />

0-3-6-8 = D F#^ A C# <h1 id="toc18"><a name="DM7(^3), or D F#^ A B^"></a> DM7(^3), or D F#^ A B^ </h1>

D.^6 = &quot;D dot up-six&quot;<br />

0-3-6-9 = D Fv A Cv = D.~7, or D Fv A Bb = D.~(b6) = &quot;D mid flat-six&quot;<br />

<br />

<h2 id="toc19"><a name="DM7(^3), or D F#^ A B^-Cross-EDO considerations"></a><strong><u>Cross-EDO considerations</u></strong></h2>

<br />

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.<br />

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<br />

<h2 id="~~toc11~~"><a name="~~Daug~~-EDOs with an inaccurate 3/2"></a><u>EDOs with an inaccurate 3/2</u></h2>

<h2 id="toc20"><a name="DM7(^3), or D F#^ A B^-EDOs with an inaccurate 3/2"></a><u>EDOs with an inaccurate 3/2</u></h2>

<br />

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.<br />

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requires learning octatonic interval arithmetic and staff notation<br />

<br />

11edo heptatonic narrow-fifth-based, fourthwards, # is ^^ (3/2 maps to 6\11 perfect 5th):<br />

11edo heptatonic narrow-fifth-based, fourthwards, # is vv (3/2 maps to 6\11 perfect 5th):<br />

D E * * F G A B * * C D<br />

D E * * F G A B * * C D = D E F# F~ F G A B B~ Bb C D<br />

fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7<br />

fourthwards chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7<br />

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

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

problematic because m3 = 2\11 is narrower than M2 = 3\11<br />

problematic because M3 = 2\11 is narrower than m2 = 3\11<br />

<br />

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

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<br />

<h1 id="~~toc12~~"><a name="Summary of EDO notation"></a><u><strong>Summary of EDO notation</strong></u></h1>

<h1 id="toc21"><a name="Summary of EDO notation"></a><u><strong>Summary of EDO notation</strong></u></h1>

<br />

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:<br />

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<br />

<h3 id="~~toc13~~"><a name="Summary of EDO notation--&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)"></a><u><strong>&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)</strong></u></h3>

<h3 id="toc22"><a name="Summary of EDO notation--&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)"></a><u><strong>&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)</strong></u></h3>

<br />

<strong><u>8edo</u>:</strong> (generator = 1\8 = perfect 2nd = 150¢)<br />

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

<br />

<h3 id="~~toc14~~"><a name="Summary of EDO notation--Alternate pentatonic notation for EDOs 8, 13 and 18"></a><u><strong>Alternate pentatonic notation for EDOs 8, 13 and 18</strong></u></h3>

<h3 id="toc23"><a name="Summary of EDO notation--Alternate pentatonic notation for EDOs 8, 13 and 18"></a><u><strong>Alternate pentatonic notation for EDOs 8, 13 and 18</strong></u></h3>

<br />

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

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<br />

<h3 id="~~toc15~~"><a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"></a><u>Fourthward EDOs (9, 16 and 23)</u></h3>

<h3 id="toc24"><a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"></a><u>Fourthward EDOs (9, 16 and 23)</u></h3>

<br />

All fourthwards EDOs use the same chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 etc.<br />

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<br />

<h3 id="~~toc16~~"><a name="Summary of EDO notation--&quot;Perfect&quot; EDOs (7, 14, 21, 28 and 35)"></a><u>&quot;Perfect&quot; EDOs (7, 14, 21, 28 and 35)</u></h3>

<h3 id="toc25"><a name="Summary of EDO notation--&quot;Perfect&quot; EDOs (7, 14, 21, 28 and 35)"></a><u>&quot;Perfect&quot; EDOs (7, 14, 21, 28 and 35)</u></h3>

<br />

All perfect EDOs use the same chain of fifths: P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.<br />

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<br />

<h3 id="~~toc17~~"><a name="Summary of EDO notation--Pentatonic EDOs (5, 10, 15, 20, 25 and 30)"></a><u>Pentatonic EDOs (5, 10, 15, 20, 25 and 30)</u></h3>

<h3 id="toc26"><a name="Summary of EDO notation--Pentatonic EDOs (5, 10, 15, 20, 25 and 30)"></a><u>Pentatonic EDOs (5, 10, 15, 20, 25 and 30)</u></h3>

<br />

All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br />

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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<br />

<br />

<h3 id="~~toc18~~"><a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"></a><u>Alternative pentatonic notation for pentatonic EDOs:</u></h3>

<h3 id="toc27"><a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"></a><u>Alternative pentatonic notation for pentatonic EDOs:</u></h3>

<br />

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

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<br />

<h3 id="~~toc19~~"><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><u>&quot;Sweet&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)</u></h3>

<h3 id="toc28"><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><u>&quot;Sweet&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)</u></h3>

<br />

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

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<br />

<h2 id="~~toc20~~"><a name="Summary of EDO notation-Ups and downs solfege"></a><u>Ups and downs solfege</u></h2>

<h2 id="toc29"><a name="Summary of EDO notation-Ups and downs solfege"></a><u>Ups and downs solfege</u></h2>

<br />

Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:<br />

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<br />

<h2 id="~~toc21~~"><a name="Summary of EDO notation-Rank-2 Notation"></a><u>Rank-2 Notation</u></h2>

<h2 id="toc30"><a name="Summary of EDO notation-Rank-2 Notation"></a><u>Rank-2 Notation</u></h2>

<br />

Ups and downs can be used to notate rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)<br />

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There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for rank-2 fifth-generated tunings in these two frameworks.<br />

<br />

<h2 id="~~toc22~~"> </h2>

<h2 id="toc31"> </h2>

<h2 id="~~toc23~~"><a name="Summary of EDO notation-Generators other than a fifth"></a><u>Generators other than a fifth</u></h2>

<h2 id="toc32"><a name="Summary of EDO notation-Generators other than a fifth"></a><u>Generators other than a fifth</u></h2>

<br />

The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation.<br />

@@ 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>2016-08-01 02:19:22 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-08-01 03:53:08 UTC</tt>.<br>
-: The original revision id was <tt>588456270</tt>.<br>
+: The original revision id was <tt>588457358</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 326: / Line 326: @@
 In general, the period is pronounced as "dot". For example, C.v is "C dot down", because "C down" means Cv major = Cv Ev Gv. However sometimes a slight pause suffices, e.g. C.vm = "C downminor" and Cv.m = "C-down minor". This is analogous to saying "A-flat nine" for Ab C Eb Gb Bb and "A flat-nine" for A C# E G Bb. However C.v7 must be "C dot down-seven" because "C down-seven" is C(v7) = C E G Bbv. Even if the period doesn't need to be pronounced, it's always acceptable to do so.
-Applying "dot up" or "dot down" to a chord raises or lowers only two chord components: the 3rd, and either the 6th or the 7th, whichever is present. Thus "C dot down nine" = C.v9 = C Ev G Bbv D. The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd (e.g. the maj6, min7, maj7, 6/9, min9 and maj9 chords). Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the fifth.
+Applying "dot up" or "dot down" to a chord raises or lowers the 3rd, and also the 6th or the 7th, if present. Thus "C dot down nine" = C.v9 = C Ev G Bbv D. The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd (e.g. the maj6, min7, maj7, 6/9, min9 and maj9 chords). Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the fifth.
 __**Various triads:**__
@@ Line 348: / Line 348: @@
 C Fv G = Csusv4 = "C sus down-four"
 C Fvv G = Csusvv4 = "C sus double-down four"
+C D# G = Csus#2 = "C sus aug-two"
+C Ebb G = C(bb3) = "C dim-three"
+C E# G = C(#3) = "C aug-three"
+C Fb G = Csusb4 = "C sus dim-four"
 __**Altered fifths:**__
@@ Line 369: / Line 374: @@
 C E^ G# is Caug(^3) = "C aug up-three" (note that here "up-three" means upmajor 3rd, not upminor 3rd)
 C E^ G#^ = Caug(^3,^5) = "C aug up-three up-five"
+C E# G# is Caug(#3)
 __**Seventh chords:**__
@@ Line 407: / Line 413: @@
 C Eb Gb^ Bb^ = Cdim(^5,^7) = "C dim up-five up-seven" or "C half-dim up-five up-seven"
 C Eb^ Gb^ Bb^ = C.^m7(^b5) = "C dot upminor seven upflat-five" or "C half-dim up-three up-five up-seven"
+C E G B# = C(#7) = "C aug-seven"
+C Eb G B# = Cm(#7) = "C minor sharp-seven"
 __**Sixth chords:**__
@@ Line 412: / Line 421: @@
 C Ev G A = C6(v3) = "C six down-three" (in certain EDOs, C6(~3) = "C six mid-three")
 C E G Av = C(v6) = "C down-six" (in certain EDOs, C(~6) = "C mid-six")
-C Ev G Av = C.v6 = "C dot down-six"
+C Ev G Av = C.v6 = "C dot down-six" (in certain EDOs, C.~6 = "C dot mid-six")
-C E G Ab^ = C(^m6) = "C upminor-six" (in certain EDOs, C(~6) = "C mid-six")
 C Eb G A = Cm6 = "C minor six" (in perfect EDOs, C6 = "C six")
 C Eb^ G A = Cm6(^3) = "C minor six up-three" (in certain EDOs, C6(~3) = "C six mid three")
 C Eb G Av = Cm(v6) = "C minor down-six" (in certain EDOs, Cm(~6) = "C minor mid-six")
-C Eb^ G A^ = C.^m6 = "C dot upminor-six"
+C Eb^ G A^ = C.^m6 = "C dot up minor-six"
 C Eb^ G A = Cm6(^3) = "C minor six up-three" (in certain EDOs, C6(~3) = "C six mid-three")
+C Eb G Ab = Cm(b6) = "C minor flat-six"
+C E G Ab = C(b6) = "C flat-six"
+C Eb G A# = Cm(#6) = "C minor sharp six"
+C E G A# = C(#6) = "C sharp-six"
 __**Ninth chords:**__
@@ Line 455: / Line 469: @@
 C Dbv Ev G Bbv = C.v7(vb9) = "C dot down seven downflat-nine"
+====__**Example EDOs:**__====
 edo: D * E * F * G * A * B * C * D, zero keys per #/b.
@@ Line 505: / Line 519: @@
 chord components: P1 A2 M2 m2/A3 M3 m3 d3/A4 P4 d4/A5 P5 d5/A6 M6 m6/A7 M7 m7 d7
 chord roots: I bI/#II II bII III bIII bbIII/#IV IV bIV/#V V bV/#VI VI bVI VII bVII bbVII/#I
--3-9 = D F## A = D(A3) = "D aug-three"
+-3-9 = D F## A = D(#3) = "D aug-three" (or possibly D Eb A = Dsusb2 = "D sus dim-two")
 -4-9 = D F# A = D = "D" or "D major"
 -5-9 = D F A = Dm = "D minor"
--6-9 = D Fb A = D(d3) = "D dim-three" (dim 3rd, perfect 5th)
+-6-9 = D Fb A = D(bb3) = "D dim-three" (or possibly D G# A = Dsus#4 = "D sus aug-four")
--5-10 = D F Ab = Ddim = "D dim" (the conventional dim triad)
+-5-10 = D F Ab = Ddim = "D dim"
--6-10 = D Fb Ab = Ddim(d3) = "D dim dim-three"
+-6-10 = D Fb Ab = Ddim(bb3) = "D dim dim-three"
 -7-9 = D G A = Dsus4 = "D sus four"
--5-9-13 = D F A C# = DmM7
+-5-9-13 = D F A C# = DmM7 = " D minor major seven"
--4-8-12 = D F# A# C## = Daug(A7)
+-4-8-12 = D F# A# C## = Daug(#7) = "D aug aug-seven"
 edo: D * * E F * * G * * A * * B C * * D, 2 keys per #/b.
@@ Line 525: / Line 539: @@
 -5-10-15 = D F^ A C^ = D.~7
 -6-10-15 = D F# A C^ = D(~7)
+edo:
 edo: D * * E * F * * G * * A * * B * C * * D, ups and downs not needed.
@@ Line 579: / Line 595: @@
 -14-28 = D F# A# = Daug = "D aug"
+edo: C# * * C D * * Db
+D E F# Fv F G A B B^ Bb C D, -2 keys per sharp, upmajor = downminor = mid.
+P1 M2 ~2/M3 ~3 m3 P4 P5 M6 ~6/M7 m6/~7 m7 P8
+m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7
+9 4 10 5 0 6 1 7 2 8
+12 5 14 7 0 9 2 11 4
+-1-6 = D E A = Dsus2
+-2-6 = D F# A = D = "D major"
+-3-6 = D Fv A = D.~ = "D mid"
+-4-6 = D F A = Dm = "D minor"
+-5-6 = D G A = Dsus4
+-2-5 = D F# Av = D(v5) = "D down-five"
+-3-5 = D Fv Av = D.~(v5) = "D mid down-five"
+-3-7 = D Fv A^ = D.~(^5)
+-4-7 = D F A^ = Dm(^5)
+-2-6-7 = D F# A B = D6
+-3-6-7 = D Fv A B = D6(~3)
+-4-6-7 = D F A B = Dm6
+-2-6-8 = D F# A C# = DM7, or D F# A B^ = D(~6) = "D mid-six"
+-3-6-8 = D F#^ A C# = DM7(^3), or D F#^ A B^ = D.^6 = "D dot up-six"
+-3-6-9 = D Fv A Cv = D.~7, or D Fv A Bb = D.~(b6) = "D mid flat-six"
 ==**__Cross-EDO considerations__**==
@@ Line 607: / Line 646: @@
 requires learning octatonic interval arithmetic and staff notation
-edo heptatonic narrow-fifth-based, fourthwards, # is ^^ (3/2 maps to 6\11 perfect 5th):
+edo heptatonic narrow-fifth-based, fourthwards, # is vv (3/2 maps to 6\11 perfect 5th):
-D E * * F G A B * * C D
+D E * * F G A B * * C D = D E F# F~ F G A B B~ Bb C D
-fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7
+fourthwards chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7
-P1 - m2 - vM2/m3 - M2/^m3 - M3 - P4 - P5 - m6 - vM6/m7 - M6/^m7 - M7 - P8
+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 = 2\11 is narrower than m2 = 3\11
 edo nonotonic narrow-fifth-based, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):
@@ Line 1,370: / Line 1,409: @@
 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;
-&lt;!-- ws:start:WikiTextLocalImageRule:4040:&amp;lt;img src=&amp;quot;/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1035px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png" alt="The fifth of EDOs 5-53.png" title="The fifth of EDOs 5-53.png" style="height: 1035px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4040 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:4058:&amp;lt;img src=&amp;quot;/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1035px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png" alt="The fifth of EDOs 5-53.png" title="The fifth of EDOs 5-53.png" style="height: 1035px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4058 --&gt;&lt;br /&gt;
 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 &amp;quot;generation&amp;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. The colored regions of the tree are what I call &lt;strong&gt;kites&lt;/strong&gt;, and this version of the Stern-Brocot tree I call the &lt;strong&gt;Tree of Kites&lt;/strong&gt;. 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 &lt;strong&gt;spinal&lt;/strong&gt; node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,586: / Line 1,625: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:4041:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1035px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4041 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:4059:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1035px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4059 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;!-- ws:start:WikiTextLocalImageRule:4042:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 957px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4042 --&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;!-- ws:start:WikiTextLocalImageRule:4060:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 957px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4060 --&gt;&lt;/h2&gt;
   &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt; &lt;/h2&gt;
   &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt; &lt;/h2&gt;
@@ Line 1,599: / Line 1,638: @@
 In general, the period is pronounced as &amp;quot;dot&amp;quot;. For example, C.v is &amp;quot;C dot down&amp;quot;, because &amp;quot;C down&amp;quot; means Cv major = Cv Ev Gv. However sometimes a slight pause suffices, e.g. C.vm = &amp;quot;C downminor&amp;quot; and Cv.m = &amp;quot;C-down minor&amp;quot;. This is analogous to saying &amp;quot;A-flat nine&amp;quot; for Ab C Eb Gb Bb and &amp;quot;A flat-nine&amp;quot; for A C# E G Bb. However C.v7 must be &amp;quot;C dot down-seven&amp;quot; because &amp;quot;C down-seven&amp;quot; is C(v7) = C E G Bbv. Even if the period doesn't need to be pronounced, it's always acceptable to do so.&lt;br /&gt;
 &lt;br /&gt;
-Applying &amp;quot;dot up&amp;quot; or &amp;quot;dot down&amp;quot; to a chord raises or lowers only two chord components: the 3rd, and either the 6th or the 7th, whichever is present. Thus &amp;quot;C dot down nine&amp;quot; = C.v9 = C Ev G Bbv D. The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd (e.g. the maj6, min7, maj7, 6/9, min9 and maj9 chords). Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the fifth.&lt;br /&gt;
+Applying &amp;quot;dot up&amp;quot; or &amp;quot;dot down&amp;quot; to a chord raises or lowers the 3rd, and also the 6th or the 7th, if present. Thus &amp;quot;C dot down nine&amp;quot; = C.v9 = C Ev G Bbv D. The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd (e.g. the maj6, min7, maj7, 6/9, min9 and maj9 chords). Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the fifth.&lt;br /&gt;
 &lt;br /&gt;
 &lt;u&gt;&lt;strong&gt;Various triads:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
@@ Line 1,621: / Line 1,660: @@
 C Fv G = Csusv4 = &amp;quot;C sus down-four&amp;quot;&lt;br /&gt;
 C Fvv G = Csusvv4 = &amp;quot;C sus double-down four&amp;quot;&lt;br /&gt;
+&lt;br /&gt;
+C D# G = Csus#2 = &amp;quot;C sus aug-two&amp;quot;&lt;br /&gt;
+C Ebb G = C(bb3) = &amp;quot;C dim-three&amp;quot;&lt;br /&gt;
+C E# G = C(#3) = &amp;quot;C aug-three&amp;quot;&lt;br /&gt;
+C Fb G = Csusb4 = &amp;quot;C sus dim-four&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
 &lt;u&gt;&lt;strong&gt;Altered fifths:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
@@ Line 1,642: / Line 1,686: @@
 C E^ G# is Caug(^3) = &amp;quot;C aug up-three&amp;quot; (note that here &amp;quot;up-three&amp;quot; means upmajor 3rd, not upminor 3rd)&lt;br /&gt;
 C E^ G#^ = Caug(^3,^5) = &amp;quot;C aug up-three up-five&amp;quot;&lt;br /&gt;
+C E# G# is Caug(#3)&lt;br /&gt;
 &lt;br /&gt;
 &lt;u&gt;&lt;strong&gt;Seventh chords:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
@@ Line 1,680: / Line 1,725: @@
 C Eb Gb^ Bb^ = Cdim(^5,^7) = &amp;quot;C dim up-five up-seven&amp;quot; or &amp;quot;C half-dim up-five up-seven&amp;quot;&lt;br /&gt;
 C Eb^ Gb^ Bb^ = C.^m7(^b5) = &amp;quot;C dot upminor seven upflat-five&amp;quot; or &amp;quot;C half-dim up-three up-five up-seven&amp;quot;&lt;br /&gt;
+&lt;br /&gt;
+C E G B#  &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="C(#7)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt; C(#7) &lt;/h1&gt;
+ &amp;quot;C aug-seven&amp;quot;&lt;br /&gt;
+C Eb G B#  &lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Cm(#7)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt; Cm(#7) &lt;/h1&gt;
+ &amp;quot;C minor sharp-seven&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
 &lt;u&gt;&lt;strong&gt;Sixth chords:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
@@ Line 1,685: / Line 1,735: @@
 C Ev G A = C6(v3) = &amp;quot;C six down-three&amp;quot; (in certain EDOs, C6(~3) = &amp;quot;C six mid-three&amp;quot;)&lt;br /&gt;
 C E G Av = C(v6) = &amp;quot;C down-six&amp;quot; (in certain EDOs, C(~6) = &amp;quot;C mid-six&amp;quot;)&lt;br /&gt;
-C Ev G Av = C.v6 = &amp;quot;C dot down-six&amp;quot;&lt;br /&gt;
+C Ev G Av = C.v6 = &amp;quot;C dot down-six&amp;quot; (in certain EDOs, C.~6 = &amp;quot;C dot mid-six&amp;quot;)&lt;br /&gt;
-C E G Ab^ = C(^m6) = &amp;quot;C upminor-six&amp;quot; (in certain EDOs, C(~6) = &amp;quot;C mid-six&amp;quot;)&lt;br /&gt;
 &lt;br /&gt;
 C Eb G A = Cm6 = &amp;quot;C minor six&amp;quot; (in perfect EDOs, C6 = &amp;quot;C six&amp;quot;)&lt;br /&gt;
 C Eb^ G A = Cm6(^3) = &amp;quot;C minor six up-three&amp;quot; (in certain EDOs, C6(~3) = &amp;quot;C six mid three&amp;quot;)&lt;br /&gt;
 C Eb G Av = Cm(v6) = &amp;quot;C minor down-six&amp;quot; (in certain EDOs, Cm(~6) = &amp;quot;C minor mid-six&amp;quot;)&lt;br /&gt;
-C Eb^ G A^ = C.^m6 = &amp;quot;C dot upminor-six&amp;quot;&lt;br /&gt;
+C Eb^ G A^ = C.^m6 = &amp;quot;C dot up minor-six&amp;quot;&lt;br /&gt;
 C Eb^ G A = Cm6(^3) = &amp;quot;C minor six up-three&amp;quot; (in certain EDOs, C6(~3) = &amp;quot;C six mid-three&amp;quot;)&lt;br /&gt;
+&lt;br /&gt;
+C Eb G Ab = Cm(b6) = &amp;quot;C minor flat-six&amp;quot;&lt;br /&gt;
+C E G Ab = C(b6) = &amp;quot;C flat-six&amp;quot;&lt;br /&gt;
+C Eb G A#  &lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Cm(#6)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt; Cm(#6) &lt;/h1&gt;
+ &amp;quot;C minor sharp six&amp;quot;&lt;br /&gt;
+C E G A#  &lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="C(#6)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt; C(#6) &lt;/h1&gt;
+ &amp;quot;C sharp-six&amp;quot;&lt;br /&gt;
+&lt;br /&gt;
 &lt;br /&gt;
 &lt;u&gt;&lt;strong&gt;Ninth chords:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
@@ Line 1,728: / Line 1,785: @@
 C Dbv Ev G Bbv = C.v7(vb9) = &amp;quot;C dot down seven downflat-nine&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
-&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc12"&gt;&lt;a name="C(#6)---Example EDOs:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;&lt;u&gt;&lt;strong&gt;Example EDOs:&lt;/strong&gt;&lt;/u&gt;&lt;/h4&gt;
-&lt;br /&gt;
+ &lt;br /&gt;
 edo: D * E * F * G * A * B * C * D, zero keys per #/b.&lt;br /&gt;
 (the chain of fifths is always centered on D, for symmetry)&lt;br /&gt;
@@ Line 1,778: / Line 1,835: @@
 chord components: P1 A2 M2 m2/A3 M3 m3 d3/A4 P4 d4/A5 P5 d5/A6 M6 m6/A7 M7 m7 d7&lt;br /&gt;
 chord roots: I bI/#II II bII III bIII bbIII/#IV IV bIV/#V V bV/#VI VI bVI VII bVII bbVII/#I&lt;br /&gt;
--3-9 = D F## A = D(A3) = &amp;quot;D aug-three&amp;quot;&lt;br /&gt;
+-3-9 = D F## A = D(#3) = &amp;quot;D aug-three&amp;quot; (or possibly D Eb A = Dsusb2 = &amp;quot;D sus dim-two&amp;quot;)&lt;br /&gt;
 -4-9 = D F# A = D = &amp;quot;D&amp;quot; or &amp;quot;D major&amp;quot;&lt;br /&gt;
 -5-9 = D F A = Dm = &amp;quot;D minor&amp;quot;&lt;br /&gt;
--6-9 = D Fb A = D(d3) = &amp;quot;D dim-three&amp;quot; (dim 3rd, perfect 5th)&lt;br /&gt;
+-6-9 = D Fb A = D(bb3) = &amp;quot;D dim-three&amp;quot; (or possibly D G# A = Dsus#4 = &amp;quot;D sus aug-four&amp;quot;)&lt;br /&gt;
--5-10 = D F Ab = Ddim = &amp;quot;D dim&amp;quot; (the conventional dim triad)&lt;br /&gt;
+-5-10 = D F Ab = Ddim = &amp;quot;D dim&amp;quot;&lt;br /&gt;
--6-10 = D Fb Ab = Ddim(d3) = &amp;quot;D dim dim-three&amp;quot;&lt;br /&gt;
+-6-10 = D Fb Ab = Ddim(bb3) = &amp;quot;D dim dim-three&amp;quot;&lt;br /&gt;
 -7-9 = D G A = Dsus4 = &amp;quot;D sus four&amp;quot;&lt;br /&gt;
--5-9-13 = D F A C# = DmM7&lt;br /&gt;
+-5-9-13 = D F A C#  &lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="DmM7"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt; DmM7 &lt;/h1&gt;
--4-8-12 = D F# A# C## = Daug(A7)&lt;br /&gt;
+ &amp;quot; D minor major seven&amp;quot;&lt;br /&gt;
+-4-8-12 = D F# A# C##  &lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Daug(#7)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt; Daug(#7) &lt;/h1&gt;
+ &amp;quot;D aug aug-seven&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
 edo: D * * E F * * G * * A * * B C * * D, 2 keys per #/b.&lt;br /&gt;
@@ Line 1,798: / Line 1,857: @@
 -5-10-15 = D F^ A C^ = D.~7&lt;br /&gt;
 -6-10-15 = D F# A C^ = D(~7)&lt;br /&gt;
+&lt;br /&gt;
+edo:&lt;br /&gt;
 &lt;br /&gt;
 edo: D * * E * F * * G * * A * * B * C * * D, ups and downs not needed.&lt;br /&gt;
@@ Line 1,805: / Line 1,866: @@
 -4-10 = D Fb Ab = Ddim(bb3) = &amp;quot;D dim dim-three&amp;quot;&lt;br /&gt;
 -7-11 = D F## A = D(#3) = &amp;quot;D aug-three&amp;quot; (aug 3rd, perfect 5th)&lt;br /&gt;
--7-12 = D F## A#  &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Daug(#3)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt; Daug(#3) &lt;/h1&gt;
+-7-12 = D F## A#  &lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc15"&gt;&lt;a name="Daug(#3)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt; Daug(#3) &lt;/h1&gt;
   &amp;quot;D aug aug-three&amp;quot;&lt;br /&gt;
 -5-10-15 = D F Ab Cb = Ddim7 = &amp;quot;D dim-seven&amp;quot;&lt;br /&gt;
@@ Line 1,851: / Line 1,912: @@
 -14-26 = D F# A^^ = D(^^5) = &amp;quot;D double-up-five&amp;quot;&lt;br /&gt;
 -14-27 = D F# A#v = Daug(v5) = &amp;quot;D aug down-five&amp;quot;&lt;br /&gt;
--14-28 = D F# A#  &lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Daug"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt; Daug &lt;/h1&gt;
+-14-28 = D F# A#  &lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc16"&gt;&lt;a name="Daug"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt; Daug &lt;/h1&gt;
   &amp;quot;D aug&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
+edo: C# * * C D * * Db&lt;br /&gt;
+D E F# Fv F G A B B^ Bb C D, -2 keys per sharp, upmajor = downminor = mid.&lt;br /&gt;
+P1 M2 ~2/M3 ~3 m3 P4 P5 M6 ~6/M7 m6/~7 m7 P8 &lt;br /&gt;
+m2 m6 m3 m7 P4 P1 P5 M2 M6 M3 M7&lt;br /&gt;
+9 4 10 5 0 6 1 7 2 8&lt;br /&gt;
+12 5 14 7 0 9 2 11 4&lt;br /&gt;
+-1-6 = D E A = Dsus2&lt;br /&gt;
+-2-6 = D F# A = D = &amp;quot;D major&amp;quot;&lt;br /&gt;
+-3-6 = D Fv A = D.~ = &amp;quot;D mid&amp;quot;&lt;br /&gt;
+-4-6 = D F A = Dm = &amp;quot;D minor&amp;quot;&lt;br /&gt;
+-5-6 = D G A = Dsus4&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Daug-Cross-EDO considerations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;strong&gt;&lt;u&gt;Cross-EDO considerations&lt;/u&gt;&lt;/strong&gt;&lt;/h2&gt;
+-2-5 = D F# Av = D(v5) = &amp;quot;D down-five&amp;quot;&lt;br /&gt;
+-3-5 = D Fv Av = D.~(v5) = &amp;quot;D mid down-five&amp;quot;&lt;br /&gt;
+-3-7 = D Fv A^ = D.~(^5)&lt;br /&gt;
+-4-7 = D F A^ = Dm(^5)&lt;br /&gt;
+&lt;br /&gt;
+-2-6-7 = D F# A B = D6&lt;br /&gt;
+-3-6-7 = D Fv A B = D6(~3)&lt;br /&gt;
+-4-6-7 = D F A B = Dm6&lt;br /&gt;
+-2-6-8 = D F# A C#  &lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc17"&gt;&lt;a name="DM7, or D F# A B^"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt; DM7, or D F# A B^ &lt;/h1&gt;
+ D(~6) = &amp;quot;D mid-six&amp;quot;&lt;br /&gt;
+-3-6-8 = D F#^ A C#  &lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc18"&gt;&lt;a name="DM7(^3), or D F#^ A B^"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt; DM7(^3), or D F#^ A B^ &lt;/h1&gt;
+ D.^6 = &amp;quot;D dot up-six&amp;quot;&lt;br /&gt;
+-3-6-9 = D Fv A Cv = D.~7, or D Fv A Bb = D.~(b6) = &amp;quot;D mid flat-six&amp;quot;&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="DM7(^3), or D F#^ A B^-Cross-EDO considerations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&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 0-8-13 = 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;
@@ Line 1,866: / Line 1,952: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Daug-EDOs with an inaccurate 3/2"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;&lt;u&gt;EDOs with an inaccurate 3/2&lt;/u&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="DM7(^3), or D F#^ A B^-EDOs with an inaccurate 3/2"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;&lt;u&gt;EDOs with an inaccurate 3/2&lt;/u&gt;&lt;/h2&gt;
   &lt;br /&gt;
 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.&lt;br /&gt;
@@ Line 1,882: / Line 1,968: @@
 requires learning octatonic interval arithmetic and staff notation&lt;br /&gt;
 &lt;br /&gt;
-edo heptatonic narrow-fifth-based, fourthwards, # is ^^ (3/2 maps to 6\11 perfect 5th):&lt;br /&gt;
+edo heptatonic narrow-fifth-based, fourthwards, # is vv (3/2 maps to 6\11 perfect 5th):&lt;br /&gt;
-D E * * F G A B * * C D&lt;br /&gt;
+D E * * F G A B * * C D = D E F# F~ F G A B B~ Bb C D&lt;br /&gt;
-fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7&lt;br /&gt;
+fourthwards chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7&lt;br /&gt;
-P1 - m2 - vM2/m3 - M2/^m3 - M3 - P4 - P5 - m6 - vM6/m7 - M6/^m7 - M7 - P8&lt;br /&gt;
+P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8&lt;br /&gt;
-problematic because m3 = 2\11 is narrower than M2 = 3\11&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, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):&lt;br /&gt;
@@ Line 1,940: / Line 2,026: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Summary of EDO notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;&lt;u&gt;&lt;strong&gt;Summary of EDO notation&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc21"&gt;&lt;a name="Summary of EDO notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;&lt;u&gt;&lt;strong&gt;Summary of EDO notation&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 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:&lt;br /&gt;
@@ Line 3,312: / Line 3,398: @@
 &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;Fifth-less&amp;quot; EDOs (8, 11, 13 and 18)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;&lt;u&gt;&lt;strong&gt;&amp;quot;Fifth-less&amp;quot; EDOs (8, 11, 13 and 18)&lt;/strong&gt;&lt;/u&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc22"&gt;&lt;a name="Summary of EDO notation--&amp;quot;Fifth-less&amp;quot; EDOs (8, 11, 13 and 18)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;&lt;u&gt;&lt;strong&gt;&amp;quot;Fifth-less&amp;quot; EDOs (8, 11, 13 and 18)&lt;/strong&gt;&lt;/u&gt;&lt;/h3&gt;
   &lt;br /&gt;
 &lt;strong&gt;&lt;u&gt;8edo&lt;/u&gt;:&lt;/strong&gt; (generator = 1\8 = perfect 2nd = 150¢)&lt;br /&gt;
@@ Line 3,342: / Line 3,428: @@
 E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Summary of EDO notation--Alternate pentatonic notation for EDOs 8, 13 and 18"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;&lt;u&gt;&lt;strong&gt;Alternate pentatonic notation for EDOs 8, 13 and 18&lt;/strong&gt;&lt;/u&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc23"&gt;&lt;a name="Summary of EDO notation--Alternate pentatonic notation for EDOs 8, 13 and 18"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;&lt;u&gt;&lt;strong&gt;Alternate pentatonic notation for EDOs 8, 13 and 18&lt;/strong&gt;&lt;/u&gt;&lt;/h3&gt;
   &lt;br /&gt;
 All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.&lt;br /&gt;
@@ Line 3,363: / Line 3,449: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;&lt;u&gt;Fourthward EDOs (9, 16 and 23)&lt;/u&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc24"&gt;&lt;a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;&lt;u&gt;Fourthward EDOs (9, 16 and 23)&lt;/u&gt;&lt;/h3&gt;
   &lt;br /&gt;
 All fourthwards EDOs use the same chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 etc.&lt;br /&gt;
@@ Line 3,384: / Line 3,470: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Summary of EDO notation--&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;&lt;u&gt;&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)&lt;/u&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc25"&gt;&lt;a name="Summary of EDO notation--&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;&lt;u&gt;&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)&lt;/u&gt;&lt;/h3&gt;
   &lt;br /&gt;
 All perfect EDOs use the same chain of fifths: P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.&lt;br /&gt;
@@ Line 3,415: / Line 3,501: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="Summary of EDO notation--Pentatonic EDOs (5, 10, 15, 20, 25 and 30)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;&lt;u&gt;Pentatonic EDOs (5, 10, 15, 20, 25 and 30)&lt;/u&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc26"&gt;&lt;a name="Summary of EDO notation--Pentatonic EDOs (5, 10, 15, 20, 25 and 30)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;&lt;u&gt;Pentatonic EDOs (5, 10, 15, 20, 25 and 30)&lt;/u&gt;&lt;/h3&gt;
   &lt;br /&gt;
 All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.&lt;br /&gt;
@@ Line 3,450: / Line 3,536: @@
 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&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc18"&gt;&lt;a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;&lt;u&gt;Alternative pentatonic notation for pentatonic EDOs:&lt;/u&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc27"&gt;&lt;a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&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;
@@ Line 3,473: / Line 3,559: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc19"&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:38 --&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;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc28"&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:56 --&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 3,506: / Line 3,592: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Summary of EDO notation-Ups and downs solfege"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;&lt;u&gt;Ups and downs solfege&lt;/u&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc29"&gt;&lt;a name="Summary of EDO notation-Ups and downs solfege"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;&lt;u&gt;Ups and downs solfege&lt;/u&gt;&lt;/h2&gt;
   &lt;br /&gt;
 Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:&lt;br /&gt;
@@ Line 3,537: / Line 3,623: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Summary of EDO notation-Rank-2 Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;&lt;u&gt;Rank-2 Notation&lt;/u&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc30"&gt;&lt;a name="Summary of EDO notation-Rank-2 Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;&lt;u&gt;Rank-2 Notation&lt;/u&gt;&lt;/h2&gt;
   &lt;br /&gt;
 Ups and downs can be used to notate rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)&lt;br /&gt;
@@ Line 5,697: / Line 5,783: @@
 There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for rank-2 fifth-generated tunings in these two frameworks.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt; &lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc31"&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt; &lt;/h2&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="Summary of EDO notation-Generators other than a fifth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;&lt;u&gt;Generators other than a fifth&lt;/u&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc32"&gt;&lt;a name="Summary of EDO notation-Generators other than a fifth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;&lt;u&gt;Generators other than a fifth&lt;/u&gt;&lt;/h2&gt;
   &lt;br /&gt;
 The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation.&lt;br /&gt;