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>2015-09-07 18:20:06 UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-07 20:37:32 UTC</tt>.<br>

: The original revision id was <tt>~~558477179~~</tt>.<br>

: The original revision id was <tt>558488833</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>

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

0-7-13-18 in C is "C,vM,m7", pronounced "C downmajor, minor seventh". The comma between the C and the down symbol is needed because Cv is a note, and "Cv,M,m7" is a different chord. That chord is pronounced "C down, major, minor 7th", so one has to "speak the comma". The extra comma isn't needed ~~when~~ there's no ups or downs immediately after the note name, e.g. Cm.

0-7-13-18 in C is "C,vM,m7", pronounced "C downmajor, minor seventh". The comma between the C and the down symbol is needed because Cv is a note, and "Cv,M,m7" is a different chord. That chord is pronounced "C down, major, minor 7th", so one has to "speak the comma". The extra comma isn't needed if there's no ups or downs immediately after the note name, e.g. Cm.

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

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Thus the shorthand should be largely abandoned and all the components of the chord should be explicitly spelled out, with a few exceptions: 1) The root, obviously. 2) The perfect 5th is assumed present unless otherwise specified. Thus 0-7-18 is C,vM,m7,no5 and 0-6-11 is C,^m,^d5. 3) The 3rd is also assumed to be present, and to be major, and is implied by a quality with no degree. Thus 0-8-13 is C and 0-7-13 is C,vM. 4) The 3rd isn't spelled out if the 6th or 7th has the same quality as the 3rd. Thus 0-7-13-16 is C,vM6, but 0-7-13-17 is C,vM,M6. Thirdless chords: 0-13-18 is either Cm7,no3 or C5,m7.

The 6th, the 7th, the 9th, the 11th, etc. are each explicitly written out, and assumed to be major or perfect. Thus the presence of a 9th doesn't imply the presence of a 7th.

The 6th, the 7th, the 9th, the 11th, etc. are each explicitly written out, and assumed to be major or perfect, except that the 7th is assumed to be minor. Thus the presence of a 9th doesn't imply the presence of a 7th.

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. An A3,P5 chord is A3 = "aug three chord", not "aug chord", to distinguish it from the conventional aug chord M3,A5. That chord ~~could be~~ called a "major, aug five" chord. Likewise d3,P5 is a "dim three chord", and m3,d5 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. An A3,P5 chord is A3 = "aug three chord", not "aug chord", to distinguish it from the conventional aug chord M3,A5. That chord is still called an aug chord, or more exactly a "major, aug five" chord. Likewise d3,P5 is a "dim three chord", and m3,d5 is a "dim" chord, or "minor, dim five" chord.

0-5-13 = m

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0-6-11-15 = ^m6,^d5

0-7-13-16 = vM6

0-8-13-17 = M6

0-8-13-17 = 6

0-5-13-18 = m7

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You can write out chord progressions using the ups/downs notation for note names. Here's the first 4 chords of Paul Erlich's 22edo composition Tibia:

G,vM7,-5 = "G downmajor seven, no five""

G,vM7,no5 = "G downmajor seven, no five""

Eb^,vM,M9 = "E flat up, downmajor, major nine"

Gm7,-5 (no comma needed) = "G minor seven, no five"

Gm7,no5 (no comma needed) = "G minor seven, no five"

A,vM,m7 = "A downmajor, minor seven"

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Here's the Tibia chords. No comma is needed after the root because ups and downs are always leading, never trailing.

IvM7,-5 = "one downmajor seven, no five"

IvM7,no5 = "one downmajor seven, no five"

^bVIvM,M9 = "up-flat six downmajor, major nine"

Im7,-5 = "one minor seven, no five"

Im7,no5 = "one minor seven, no five"

IIvM,m7 = "two downmajor, minor seven"

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0-4-9 = m

0-5-9 = M

0-5-10 = M,A5 (the conventional aug chord)

0-5-10 = aug or M,A5 (the conventional aug chord)

0-6-9 = A (aug 3rd, perfect 5th)

0-6-9 = A3 (aug 3rd, perfect 5th)

0-7-9 = sus4

0-4-8-12 = m,d5,d7 (the conventional dim tetrad)

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0-5-10-15 = vM7

0-6-10-16 = M7

~~Alternatively, one could replace downmajor with n = neutral or somesuch.~~

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

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".~~

0-4-11 = d3 (dim 3rd, perfect 5th)

0-4-11 = d (dim 3rd, perfect 5th~~, not a conventional dim chord~~)

0-4-10 = d3,d5 or dim,d3

0-4-10 = d,d5

0-5-11 = m

0-5-10 = m,d5 (conventional dim chord)

0-5-10 = dim or m,d5 (conventional dim chord)

0-6-11 = M

0-7-11 = A (aug 3rd, perfect 5th~~, not a conventional aug chord~~)

0-7-11 = A3 (aug 3rd, perfect 5th)

0-6-12 = M,A5 (conventional aug chord)

0-6-12 = aug or M,A5 (conventional aug chord)

0-7-12 = A,A5

0-7-12 = A3,A5 or aug,A3

0-8-11 = sus4

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0-5-14 = vm

0-6-14 = m

0-7-14 = vM

0-7-14 = ^m or vM

0-8-14 = M

0-9-14 = ^M

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0-7-18 = vm

0-8-18 = m

0-9-18 = vM

0-9-18 = ^m or vM

0-10-18 = M

0-11-18 = ^M

0-12-18 = ~~sus-v4~~

0-12-18 = susv4

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

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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 = perfect 5th):

11edo heptatonic narrow-fifth-based, fourthwards, # = ^^ (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 = perfect 6th):

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

nonotonic ~~fourthwards~~ chain of sixths:

nonotonic fifthwards 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 staff notation

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

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

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

pentatonic plus ups and downs is doubly confusing!

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requires learning octatonic interval arithmetic and notation

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

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

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

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

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(13edo octatonic wide-fifth-based, fourthwards)

18edo heptatonic narrow-fifth-based, fourthwards, ~~2 keys per~~ #/b (3/2 maps to 10\18 = perfect 5th)

18edo heptatonic narrow-fifth-based, fourthwards, # = ^^ (3/2 maps to 10\18 = perfect 5th)

P1 - vm2 - m2 - vM2 - M2/m3 - vM3 - M3 - ^M3 - P4 - ^P4/vP5 - P5 - vm6 - m6 - vM6 - M6/m7 - vM7 - M7 - ^M7 - P8

fourthwards plus ups and downs is doubly confusing!

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These are pronounced &quot;downmajor second&quot;, &quot;upminor third&quot;, etc. For 4ths and 5ths, &quot;perfect&quot; is implied and can be omitted: ^P4 = &quot;up-four&quot; and vP5 = &quot;down-five&quot;. In larger edos there may be &quot;down-octave&quot;, &quot;up-unison&quot;, etc.<br />

<br />

0-7-13-18 in C is &quot;C,vM,m7&quot;, pronounced &quot;C downmajor, minor seventh&quot;. The comma between the C and the down symbol is needed because Cv is a note, and &quot;Cv,M,m7&quot; is a different chord. That chord is pronounced &quot;C down, major, minor 7th&quot;, so one has to &quot;speak the comma&quot;. The extra comma isn't needed ~~when~~ there's no ups or downs immediately after the note name, e.g. Cm.<br />

0-7-13-18 in C is &quot;C,vM,m7&quot;, pronounced &quot;C downmajor, minor seventh&quot;. The comma between the C and the down symbol is needed because Cv is a note, and &quot;Cv,M,m7&quot; is a different chord. That chord is pronounced &quot;C down, major, minor 7th&quot;, so one has to &quot;speak the comma&quot;. The extra comma isn't needed if there's no ups or downs immediately after the note name, e.g. Cm.<br />

<br />

The conventional chord naming system uses a lot of &quot;shorthand&quot; like dom7 for M3,m7 and min6 for m3,M6. This causes problems in 22edo where there are so many choices for the 3rd, the 6th, the 7th and the 9th. For example, min6 could mean m3,vM6 = approximate 6:7:9:10 chord, or it could mean ^m3,M6 = approximate 1/1-6/5-3/2-12/7 chord. Larger edos would present even greater problems. Furthermore there's some ambiguity in the shorthand, e.g. in 12edo, both 0-3-6 and 0-3-6-9 are called dim chords.<br />

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Thus the shorthand should be largely abandoned and all the components of the chord should be explicitly spelled out, with a few exceptions: 1) The root, obviously. 2) The perfect 5th is assumed present unless otherwise specified. Thus 0-7-18 is C,vM,m7,no5 and 0-6-11 is C,^m,^d5. 3) The 3rd is also assumed to be present, and to be major, and is implied by a quality with no degree. Thus 0-8-13 is C and 0-7-13 is C,vM. 4) The 3rd isn't spelled out if the 6th or 7th has the same quality as the 3rd. Thus 0-7-13-16 is C,vM6, but 0-7-13-17 is C,vM,M6. Thirdless chords: 0-13-18 is either Cm7,no3 or C5,m7.<br />

<br />

The 6th, the 7th, the 9th, the 11th, etc. are each explicitly written out, and assumed to be major or perfect. Thus the presence of a 9th doesn't imply the presence of a 7th.<br />

The 6th, the 7th, the 9th, the 11th, etc. are each explicitly written out, and assumed to be major or perfect, except that the 7th is assumed to be minor. Thus the presence of a 9th doesn't imply the presence of a 7th.<br />

<br />

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.<br />

<br />

&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. An A3,P5 chord is A3 = &quot;aug three chord&quot;, not &quot;aug chord&quot;, to distinguish it from the conventional aug chord M3,A5. That chord ~~could be~~ called a &quot;major, aug five&quot; chord. Likewise d3,P5 is a &quot;dim three chord&quot;, and m3,d5 is a &quot;minor, dim five&quot; chord.<br />

&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. An A3,P5 chord is A3 = &quot;aug three chord&quot;, not &quot;aug chord&quot;, to distinguish it from the conventional aug chord M3,A5. That chord is still called an aug chord, or more exactly a &quot;major, aug five&quot; chord. Likewise d3,P5 is a &quot;dim three chord&quot;, and m3,d5 is a &quot;dim&quot; chord, or &quot;minor, dim five&quot; chord.<br />

<br />

0-5-13 = m<br />

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0-6-11-15 = ^m6,^d5<br />

0-7-13-16 = vM6<br />

0-8-13-17 = M6<br />

0-8-13-17 = 6<br />

<br />

0-5-13-18 = m7<br />

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<br />

You can write out chord progressions using the ups/downs notation for note names. Here's the first 4 chords of Paul Erlich's 22edo composition Tibia:<br />

G,vM7,-5 = &quot;G downmajor seven, no five&quot;&quot;<br />

G,vM7,no5 = &quot;G downmajor seven, no five&quot;&quot;<br />

Eb^,vM,M9 = &quot;E flat up, downmajor, major nine&quot;<br />

Gm7,-5 (no comma needed) = &quot;G minor seven, no five&quot;<br />

Gm7,no5 (no comma needed) = &quot;G minor seven, no five&quot;<br />

A,vM,m7 = &quot;A downmajor, minor seven&quot;<br />

<br />

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<br />

Here's the Tibia chords. No comma is needed after the root because ups and downs are always leading, never trailing.<br />

IvM7,-5 = &quot;one downmajor seven, no five&quot;<br />

IvM7,no5 = &quot;one downmajor seven, no five&quot;<br />

^bVIvM,M9 = &quot;up-flat six downmajor, major nine&quot;<br />

Im7,-5 = &quot;one minor seven, no five&quot;<br />

Im7,no5 = &quot;one minor seven, no five&quot;<br />

IIvM,m7 = &quot;two downmajor, minor seven&quot;<br />

<br />

<img src="/file/view/Tibia%20in%20G%20using%20ups%20and%20downs.jpg/558356591/800x1130/Tibia%20in%20G%20using%20ups%20and%20downs.jpg" alt="Tibia in G using ups and downs.jpg" title="Tibia in G using ups and downs.jpg" style="height: 1130px; width: 800px;" /><br />

<img src="/file/view/Tibia%20in%20G%20using%20ups%20and%20downs.jpg/558356591/800x1130/Tibia%20in%20G%20using%20ups%20and%20downs.jpg" alt="Tibia in G using ups and downs.jpg" title="Tibia in G using ups and downs.jpg" style="height: 1130px; width: 800px;" /><br />

<br />

<img src="/file/view/Tibia%20in%20G%20using%20ups%20and%20downs-2.jpg/558356607/800x1130/Tibia%20in%20G%20using%20ups%20and%20downs-2.jpg" alt="Tibia in G using ups and downs-2.jpg" title="Tibia in G using ups and downs-2.jpg" style="height: 1130px; width: 800px;" /><br />

<img src="/file/view/Tibia%20in%20G%20using%20ups%20and%20downs-2.jpg/558356607/800x1130/Tibia%20in%20G%20using%20ups%20and%20downs-2.jpg" alt="Tibia in G using ups and downs-2.jpg" title="Tibia in G using ups and downs-2.jpg" style="height: 1130px; width: 800px;" /><br />

<br />

<h2 id="toc3"><a name="Naming Chords-Chord names in other EDOs"></a><u>Chord names in other EDOs</u></h2>

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0-4-9 = m<br />

0-5-9 = M<br />

0-5-10 = M,A5 (the conventional aug chord)<br />

0-5-10 = aug or M,A5 (the conventional aug chord)<br />

0-6-9 = A (aug 3rd, perfect 5th)<br />

0-6-9 = A3 (aug 3rd, perfect 5th)<br />

0-7-9 = sus4<br />

0-4-8-12 = m,d5,d7 (the conventional dim tetrad)<br />

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0-5-10-15 = vM7<br />

0-6-10-16 = M7<br />

~~Alternatively, one could replace downmajor with n = neutral or somesuch.<br />~~

<br />

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

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

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<br />

~~The possibility of a dim 3rd or an aug 3rd changes the meaning of &quot;dim chord&quot; and &quot;aug chord&quot;.<br />~~

0-4-11 = d3 (dim 3rd, perfect 5th)<br />

0-4-11 = d (dim 3rd, perfect 5th~~, not a conventional dim chord~~)<br />

0-4-10 = d3,d5 or dim,d3<br />

0-4-10 = d,d5<br />

0-5-11 = m<br />

0-5-10 = m,d5 (conventional dim chord)<br />

0-5-10 = dim or m,d5 (conventional dim chord)<br />

0-6-11 = M<br />

0-7-11 = A (aug 3rd, perfect 5th~~, not a conventional aug chord~~)<br />

0-7-11 = A3 (aug 3rd, perfect 5th)<br />

0-6-12 = M,A5 (conventional aug chord)<br />

0-6-12 = aug or M,A5 (conventional aug chord)<br />

0-7-12 = A,A5<br />

0-7-12 = A3,A5 or aug,A3<br />

0-8-11 = sus4<br />

<br />

Line 969:

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0-5-14 = vm<br />

0-6-14 = m<br />

0-7-14 = vM<br />

0-7-14 = ^m or vM<br />

0-8-14 = M<br />

0-9-14 = ^M<br />

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0-7-18 = vm<br />

0-8-18 = m<br />

0-9-18 = vM<br />

0-9-18 = ^m or vM<br />

0-10-18 = M<br />

0-11-18 = ^M<br />

0-12-18 = ~~sus-v4~~<br />

0-12-18 = susv4<br />

<br />

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

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Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in this diagram, they are the ones to the left of the central line in the light blue region, plus the ones to the right of the central line in the orange region. The ones on the left edge of the blue region 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 />

<br />

<img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1002/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: 1002px; width: 800px;" /><br />

<img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1002/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: 1002px; width: 800px;" /><br />

<br />

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

<br />

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

11edo heptatonic narrow-fifth-based, fourthwards, # <h1 id="toc6"><a name="x^^ (3/2 maps to 6\11"></a> ^^ (3/2 maps to 6\11 </h1>

perfect 5th):<br />

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

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

<br />

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

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

nonotonic ~~fourthwards~~ chain of sixths:<br />

nonotonic fifthwards chain of sixths:<br />

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

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

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

<br />

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

11edo pentatonic wide-fifth-based, fifthwards, # = ^^ (3/2 maps to 7\11 6th):<br />

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

pentatonic plus ups and downs is doubly confusing!<br />

Line 1,039:

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requires learning octatonic interval arithmetic and notation<br />

<br />

13edo heptatonic narrow-fifth-based, fourthwards, 3 ~~keys per #~~/b, (3/2 maps to 7\13 = perfect 5th):<br />

13edo heptatonic narrow-fifth-based, fourthwards, # <h1 id="toc7"><a name="x^^^ (3/2 maps to 7\13"></a> ^^^ (3/2 maps to 7\13 </h1>

perfect 5th):<br />

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

problematic because m3 = 2\13 is narrower than M2 = 4\13<br />

Line 1,047:

Line 1,045:

(13edo octatonic wide-fifth-based, fourthwards)<br />

<br />

18edo heptatonic narrow-fifth-based, fourthwards, 2 ~~keys per #~~/b (3/2 maps to 10\18 = perfect 5th)<br />

18edo heptatonic narrow-fifth-based, fourthwards, # <h1 id="toc8"><a name="x^^ (3/2 maps to 10\18"></a> ^^ (3/2 maps to 10\18 </h1>

perfect 5th)<br />

P1 - vm2 - m2 - vM2 - M2/m3 - vM3 - M3 - ^M3 - P4 - ^P4/vP5 - P5 - vm6 - m6 - vM6 - M6/m7 - vM7 - M7 - ^M7 - P8<br />

fourthwards plus ups and downs is doubly confusing!<br />

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<br />

<h1 id="~~toc6~~"><a name="Summary of EDO notation"></a><u><strong>Summary of EDO notation</strong></u></h1>

<h1 id="toc9"><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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The C-D interval is the major 2nd (P2 for perfect EDOs, m2 for fourthward EDOs).<br />

<br />

<h3 id="~~toc7~~"><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="toc10"><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="~~toc8~~"><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="toc11"><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="~~toc9~~"><a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"></a><u>Fourthward EDOs (9, 16 and 23)</u></h3>

<h3 id="toc12"><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="~~toc10~~"><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="toc13"><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="~~toc11~~"><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="toc14"><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="~~toc12~~"><a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"></a><u>Alternative pentatonic notation for pentatonic EDOs:</u></h3>

<h3 id="toc15"><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="~~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><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="toc16"><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 />

@@ 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>2015-09-07 18:20:06 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-07 20:37:32 UTC</tt>.<br>
-: The original revision id was <tt>558477179</tt>.<br>
+: The original revision id was <tt>558488833</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 205: / Line 205: @@
 These are pronounced "downmajor second", "upminor third", etc. For 4ths and 5ths, "perfect" is implied and can be omitted: ^P4 = "up-four" and vP5 = "down-five". In larger edos there may be "down-octave", "up-unison", etc.
--7-13-18 in C is "C,vM,m7", pronounced "C downmajor, minor seventh". The comma between the C and the down symbol is needed because Cv is a note, and "Cv,M,m7" is a different chord. That chord is pronounced "C down, major, minor 7th", so one has to "speak the comma". The extra comma isn't needed when there's no ups or downs immediately after the note name, e.g. Cm.
+-7-13-18 in C is "C,vM,m7", pronounced "C downmajor, minor seventh". The comma between the C and the down symbol is needed because Cv is a note, and "Cv,M,m7" is a different chord. That chord is pronounced "C down, major, minor 7th", so one has to "speak the comma". The extra comma isn't needed if there's no ups or downs immediately after the note name, e.g. Cm.
 The conventional chord naming system uses a lot of "shorthand" like dom7 for M3,m7 and min6 for m3,M6. This causes problems in 22edo where there are so many choices for the 3rd, the 6th, the 7th and the 9th. For example, min6 could mean m3,vM6 = approximate 6:7:9:10 chord, or it could mean ^m3,M6 = approximate 1/1-6/5-3/2-12/7 chord. Larger edos would present even greater problems. Furthermore there's some ambiguity in the shorthand, e.g. in 12edo, both 0-3-6 and 0-3-6-9 are called dim chords.
@@ Line 211: / Line 211: @@
 Thus the shorthand should be largely abandoned and all the components of the chord should be explicitly spelled out, with a few exceptions: 1) The root, obviously. 2) The perfect 5th is assumed present unless otherwise specified. Thus 0-7-18 is C,vM,m7,no5 and 0-6-11 is C,^m,^d5. 3) The 3rd is also assumed to be present, and to be major, and is implied by a quality with no degree. Thus 0-8-13 is C and 0-7-13 is C,vM. 4) The 3rd isn't spelled out if the 6th or 7th has the same quality as the 3rd. Thus 0-7-13-16 is C,vM6, but 0-7-13-17 is C,vM,M6. Thirdless chords: 0-13-18 is either Cm7,no3 or C5,m7.
-The 6th, the 7th, the 9th, the 11th, etc. are each explicitly written out, and assumed to be major or perfect. Thus the presence of a 9th doesn't imply the presence of a 7th.
+The 6th, the 7th, the 9th, the 11th, etc. are each explicitly written out, and assumed to be major or perfect, except that the 7th is assumed to be minor. Thus the presence of a 9th doesn't imply the presence of a 7th.
 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. An A3,P5 chord is A3 = "aug three chord", not "aug chord", to distinguish it from the conventional aug chord M3,A5. That chord could be called a "major, aug five" chord. Likewise d3,P5 is a "dim three chord", and m3,d5 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. An A3,P5 chord is A3 = "aug three chord", not "aug chord", to distinguish it from the conventional aug chord M3,A5. That chord is still called an aug chord, or more exactly a "major, aug five" chord. Likewise d3,P5 is a "dim three chord", and m3,d5 is a "dim" chord, or "minor, dim five" chord.
 -5-13 = m
@@ Line 232: / Line 232: @@
 -6-11-15 = ^m6,^d5
 -7-13-16 = vM6
--8-13-17 = M6
+-8-13-17 = 6
 -5-13-18 = m7
@@ Line 245: / Line 245: @@
 You can write out chord progressions using the ups/downs notation for note names. Here's the first 4 chords of Paul Erlich's 22edo composition Tibia:
-G,vM7,-5 = "G downmajor seven, no five""
+G,vM7,no5 = "G downmajor seven, no five""
 Eb^,vM,M9 = "E flat up, downmajor, major nine"
-Gm7,-5 (no comma needed) = "G minor seven, no five"
+Gm7,no5 (no comma needed) = "G minor seven, no five"
 A,vM,m7 = "A downmajor, minor seven"
@@ Line 255: / Line 255: @@
 Here's the Tibia chords. No comma is needed after the root because ups and downs are always leading, never trailing.
-IvM7,-5 = "one downmajor seven, no five"
+IvM7,no5 = "one downmajor seven, no five"
 ^bVIvM,M9 = "up-flat six downmajor, major nine"
-Im7,-5 = "one minor seven, no five"
+Im7,no5 = "one minor seven, no five"
 IIvM,m7 = "two downmajor, minor seven"
@@ Line 283: / Line 283: @@
 -4-9 = m
 -5-9 = M
--5-10 = M,A5 (the conventional aug chord)
+-5-10 = aug or M,A5 (the conventional aug chord)
--6-9 = A (aug 3rd, perfect 5th)
+-6-9 = A3 (aug 3rd, perfect 5th)
 -7-9 = sus4
 -4-8-12 = m,d5,d7 (the conventional dim tetrad)
@@ Line 298: / Line 298: @@
 -5-10-15 = vM7
 -6-10-16 = M7
-Alternatively, one could replace downmajor with n = neutral or somesuch.
 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
 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".
+-4-11 = d3 (dim 3rd, perfect 5th)
--4-11 = d (dim 3rd, perfect 5th, not a conventional dim chord)
+-4-10 = d3,d5 or dim,d3
--4-10 = d,d5
 -5-11 = m
--5-10 = m,d5 (conventional dim chord)
+-5-10 = dim or m,d5 (conventional dim chord)
 -6-11 = M
--7-11 = A (aug 3rd, perfect 5th, not a conventional aug chord)
+-7-11 = A3 (aug 3rd, perfect 5th)
--6-12 = M,A5 (conventional aug chord)
+-6-12 = aug or M,A5 (conventional aug chord)
--7-12 = A,A5
+-7-12 = A3,A5 or aug,A3
 -8-11 = sus4
@@ Line 341: / Line 339: @@
 -5-14 = vm
 -6-14 = m
--7-14 = vM
+-7-14 = ^m or vM
 -8-14 = M
 -9-14 = ^M
@@ Line 351: / Line 349: @@
 -7-18 = vm
 -8-18 = m
--9-18 = vM
+-9-18 = ^m or vM
 -10-18 = M
 -11-18 = ^M
--12-18 = sus-v4
+-12-18 = susv4
 ==**__Cross-EDO considerations__**==
@@ Line 392: / Line 390: @@
 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 = perfect 5th):
+edo heptatonic narrow-fifth-based, fourthwards, # = ^^ (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 = perfect 6th):
+edo nonotonic narrow-fifth-based, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):
-nonotonic fourthwards chain of sixths:
+nonotonic fifthwards 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 staff notation
-edo pentatonic wide-fifth-based, fifthwards using ^/v, 2 keys per #/b (3/2 maps to 7\11 6th):
+edo pentatonic wide-fifth-based, fifthwards, # = ^^ (3/2 maps to 7\11 6th):
 P1 - ms3 - ^ms3/vMs3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ^ms7/vMs7 - Ms7 - P8d
 pentatonic plus ups and downs is doubly confusing!
@@ Line 411: / Line 409: @@
 requires learning octatonic interval arithmetic and notation
-edo heptatonic narrow-fifth-based, fourthwards, 3 keys per #/b, (3/2 maps to 7\13 = perfect 5th):
+edo heptatonic narrow-fifth-based, fourthwards, # = ^^^ (3/2 maps to 7\13 = perfect 5th):
 P1 - m2 - m3 - vM2/^m3 - M2 - M3 - P4 - P5 - m6 - m7 - vM6/^m7 - M6 - M7 - P8
 problematic because m3 = 2\13 is narrower than M2 = 4\13
@@ Line 419: / Line 417: @@
 (13edo octatonic wide-fifth-based, fourthwards)
-edo heptatonic narrow-fifth-based, fourthwards, 2 keys per #/b (3/2 maps to 10\18 = perfect 5th)
+edo heptatonic narrow-fifth-based, fourthwards, # = ^^ (3/2 maps to 10\18 = perfect 5th)
 P1 - vm2 - m2 - vM2 - M2/m3 - vM3 - M3 - ^M3 - P4 - ^P4/vP5 - P5 - vm6 - m6 - vM6 - M6/m7 - vM7 - M7 - ^M7 - P8
 fourthwards plus ups and downs is doubly confusing!
@@ Line 833: / Line 831: @@
 These are pronounced &amp;quot;downmajor second&amp;quot;, &amp;quot;upminor third&amp;quot;, etc. For 4ths and 5ths, &amp;quot;perfect&amp;quot; is implied and can be omitted: ^P4 = &amp;quot;up-four&amp;quot; and vP5 = &amp;quot;down-five&amp;quot;. In larger edos there may be &amp;quot;down-octave&amp;quot;, &amp;quot;up-unison&amp;quot;, etc.&lt;br /&gt;
 &lt;br /&gt;
--7-13-18 in C is &amp;quot;C,vM,m7&amp;quot;, pronounced &amp;quot;C downmajor, minor seventh&amp;quot;. The comma between the C and the down symbol is needed because Cv is a note, and &amp;quot;Cv,M,m7&amp;quot; is a different chord. That chord is pronounced &amp;quot;C down, major, minor 7th&amp;quot;, so one has to &amp;quot;speak the comma&amp;quot;. The extra comma isn't needed when there's no ups or downs immediately after the note name, e.g. Cm.&lt;br /&gt;
+-7-13-18 in C is &amp;quot;C,vM,m7&amp;quot;, pronounced &amp;quot;C downmajor, minor seventh&amp;quot;. The comma between the C and the down symbol is needed because Cv is a note, and &amp;quot;Cv,M,m7&amp;quot; is a different chord. That chord is pronounced &amp;quot;C down, major, minor 7th&amp;quot;, so one has to &amp;quot;speak the comma&amp;quot;. The extra comma isn't needed if there's no ups or downs immediately after the note name, e.g. Cm.&lt;br /&gt;
 &lt;br /&gt;
 The conventional chord naming system uses a lot of &amp;quot;shorthand&amp;quot; like dom7 for M3,m7 and min6 for m3,M6. This causes problems in 22edo where there are so many choices for the 3rd, the 6th, the 7th and the 9th. For example, min6 could mean m3,vM6 = approximate 6:7:9:10 chord, or it could mean ^m3,M6 = approximate 1/1-6/5-3/2-12/7 chord. Larger edos would present even greater problems. Furthermore there's some ambiguity in the shorthand, e.g. in 12edo, both 0-3-6 and 0-3-6-9 are called dim chords.&lt;br /&gt;
@@ Line 839: / Line 837: @@
 Thus the shorthand should be largely abandoned and all the components of the chord should be explicitly spelled out, with a few exceptions: 1) The root, obviously. 2) The perfect 5th is assumed present unless otherwise specified. Thus 0-7-18 is C,vM,m7,no5 and 0-6-11 is C,^m,^d5. 3) The 3rd is also assumed to be present, and to be major, and is implied by a quality with no degree. Thus 0-8-13 is C and 0-7-13 is C,vM. 4) The 3rd isn't spelled out if the 6th or 7th has the same quality as the 3rd. Thus 0-7-13-16 is C,vM6, but 0-7-13-17 is C,vM,M6. Thirdless chords: 0-13-18 is either Cm7,no3 or C5,m7.&lt;br /&gt;
 &lt;br /&gt;
-The 6th, the 7th, the 9th, the 11th, etc. are each explicitly written out, and assumed to be major or perfect. Thus the presence of a 9th doesn't imply the presence of a 7th.&lt;br /&gt;
+The 6th, the 7th, the 9th, the 11th, etc. are each explicitly written out, and assumed to be major or perfect, except that the 7th is assumed to be minor. Thus the presence of a 9th doesn't imply the presence of a 7th.&lt;br /&gt;
 &lt;br /&gt;
 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. An A3,P5 chord is A3 = &amp;quot;aug three chord&amp;quot;, not &amp;quot;aug chord&amp;quot;, to distinguish it from the conventional aug chord M3,A5. That chord could be called a &amp;quot;major, aug five&amp;quot; chord. Likewise d3,P5 is a &amp;quot;dim three chord&amp;quot;, and m3,d5 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. An A3,P5 chord is A3 = &amp;quot;aug three chord&amp;quot;, not &amp;quot;aug chord&amp;quot;, to distinguish it from the conventional aug chord M3,A5. That chord is still called an aug chord, or more exactly a &amp;quot;major, aug five&amp;quot; chord. Likewise d3,P5 is a &amp;quot;dim three chord&amp;quot;, and m3,d5 is a &amp;quot;dim&amp;quot; chord, or &amp;quot;minor, dim five&amp;quot; chord.&lt;br /&gt;
 &lt;br /&gt;
 -5-13 = m&lt;br /&gt;
@@ Line 860: / Line 858: @@
 -6-11-15 = ^m6,^d5&lt;br /&gt;
 -7-13-16 = vM6&lt;br /&gt;
--8-13-17 = M6&lt;br /&gt;
+-8-13-17 = 6&lt;br /&gt;
 &lt;br /&gt;
 -5-13-18 = m7&lt;br /&gt;
@@ Line 873: / Line 871: @@
 &lt;br /&gt;
 You can write out chord progressions using the ups/downs notation for note names. Here's the first 4 chords of Paul Erlich's 22edo composition Tibia:&lt;br /&gt;
-G,vM7,-5 = &amp;quot;G downmajor seven, no five&amp;quot;&amp;quot;&lt;br /&gt;
+G,vM7,no5 = &amp;quot;G downmajor seven, no five&amp;quot;&amp;quot;&lt;br /&gt;
 Eb^,vM,M9 = &amp;quot;E flat up, downmajor, major nine&amp;quot;&lt;br /&gt;
-Gm7,-5 (no comma needed) = &amp;quot;G minor seven, no five&amp;quot;&lt;br /&gt;
+Gm7,no5 (no comma needed) = &amp;quot;G minor seven, no five&amp;quot;&lt;br /&gt;
 A,vM,m7 = &amp;quot;A downmajor, minor seven&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
@@ Line 883: / Line 881: @@
 &lt;br /&gt;
 Here's the Tibia chords. No comma is needed after the root because ups and downs are always leading, never trailing.&lt;br /&gt;
-IvM7,-5 = &amp;quot;one downmajor seven, no five&amp;quot;&lt;br /&gt;
+IvM7,no5 = &amp;quot;one downmajor seven, no five&amp;quot;&lt;br /&gt;
 ^bVIvM,M9 = &amp;quot;up-flat six downmajor, major nine&amp;quot;&lt;br /&gt;
-Im7,-5 = &amp;quot;one minor seven, no five&amp;quot;&lt;br /&gt;
+Im7,no5 = &amp;quot;one minor seven, no five&amp;quot;&lt;br /&gt;
 IIvM,m7 = &amp;quot;two downmajor, minor seven&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:28:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20using%20ups%20and%20downs.jpg/558356591/800x1130/Tibia%20in%20G%20using%20ups%20and%20downs.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1130px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20using%20ups%20and%20downs.jpg/558356591/800x1130/Tibia%20in%20G%20using%20ups%20and%20downs.jpg" alt="Tibia in G using ups and downs.jpg" title="Tibia in G using ups and downs.jpg" style="height: 1130px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:28 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:34:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20using%20ups%20and%20downs.jpg/558356591/800x1130/Tibia%20in%20G%20using%20ups%20and%20downs.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1130px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20using%20ups%20and%20downs.jpg/558356591/800x1130/Tibia%20in%20G%20using%20ups%20and%20downs.jpg" alt="Tibia in G using ups and downs.jpg" title="Tibia in G using ups and downs.jpg" style="height: 1130px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:34 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:29:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20using%20ups%20and%20downs-2.jpg/558356607/800x1130/Tibia%20in%20G%20using%20ups%20and%20downs-2.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1130px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20using%20ups%20and%20downs-2.jpg/558356607/800x1130/Tibia%20in%20G%20using%20ups%20and%20downs-2.jpg" alt="Tibia in G using ups and downs-2.jpg" title="Tibia in G using ups and downs-2.jpg" style="height: 1130px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:29 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:35:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20using%20ups%20and%20downs-2.jpg/558356607/800x1130/Tibia%20in%20G%20using%20ups%20and%20downs-2.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1130px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20using%20ups%20and%20downs-2.jpg/558356607/800x1130/Tibia%20in%20G%20using%20ups%20and%20downs-2.jpg" alt="Tibia in G using ups and downs-2.jpg" title="Tibia in G using ups and downs-2.jpg" style="height: 1130px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:35 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Naming Chords-Chord names in other EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;u&gt;Chord names in other EDOs&lt;/u&gt;&lt;/h2&gt;
@@ Line 911: / Line 909: @@
 -4-9 = m&lt;br /&gt;
 -5-9 = M&lt;br /&gt;
--5-10 = M,A5 (the conventional aug chord)&lt;br /&gt;
+-5-10 = aug or M,A5 (the conventional aug chord)&lt;br /&gt;
--6-9 = A (aug 3rd, perfect 5th)&lt;br /&gt;
+-6-9 = A3 (aug 3rd, perfect 5th)&lt;br /&gt;
 -7-9 = sus4&lt;br /&gt;
 -4-8-12 = m,d5,d7 (the conventional dim tetrad)&lt;br /&gt;
@@ Line 926: / Line 924: @@
 -5-10-15 = vM7&lt;br /&gt;
 -6-10-16 = M7&lt;br /&gt;
-Alternatively, one could replace downmajor with n = neutral or somesuch.&lt;br /&gt;
 &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&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;
+-4-11 = d3 (dim 3rd, perfect 5th)&lt;br /&gt;
--4-11 = d (dim 3rd, perfect 5th, not a conventional dim chord)&lt;br /&gt;
+-4-10 = d3,d5 or dim,d3&lt;br /&gt;
--4-10 = d,d5&lt;br /&gt;
 -5-11 = m&lt;br /&gt;
--5-10 = m,d5 (conventional dim chord)&lt;br /&gt;
+-5-10 = dim or m,d5 (conventional dim chord)&lt;br /&gt;
 -6-11 = M&lt;br /&gt;
--7-11 = A (aug 3rd, perfect 5th, not a conventional aug chord)&lt;br /&gt;
+-7-11 = A3 (aug 3rd, perfect 5th)&lt;br /&gt;
--6-12 = M,A5 (conventional aug chord)&lt;br /&gt;
+-6-12 = aug or M,A5 (conventional aug chord)&lt;br /&gt;
--7-12 = A,A5&lt;br /&gt;
+-7-12 = A3,A5 or aug,A3&lt;br /&gt;
 -8-11 = sus4&lt;br /&gt;
 &lt;br /&gt;
@@ Line 969: / Line 965: @@
 -5-14 = vm&lt;br /&gt;
 -6-14 = m&lt;br /&gt;
--7-14 = vM&lt;br /&gt;
+-7-14 = ^m or vM&lt;br /&gt;
 -8-14 = M&lt;br /&gt;
 -9-14 = ^M&lt;br /&gt;
@@ Line 979: / Line 975: @@
 -7-18 = vm&lt;br /&gt;
 -8-18 = m&lt;br /&gt;
--9-18 = vM&lt;br /&gt;
+-9-18 = ^m or vM&lt;br /&gt;
 -10-18 = M&lt;br /&gt;
 -11-18 = ^M&lt;br /&gt;
--12-18 = sus-v4&lt;br /&gt;
+-12-18 = susv4&lt;br /&gt;
 &lt;br /&gt;
 &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;
@@ Line 999: / Line 995: @@
 Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in this diagram, they are the ones to the left of the central line in the light blue region, plus the ones to the right of the central line in the orange region. The ones on the left edge of the blue region 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;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:30:&amp;lt;img src=&amp;quot;/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1002/The%20fifth%20of%20EDOs%205-53.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1002px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1002/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: 1002px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:30 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:36:&amp;lt;img src=&amp;quot;/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1002/The%20fifth%20of%20EDOs%205-53.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1002px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1002/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: 1002px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:36 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,020: / Line 1,016: @@
 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 = perfect 5th):&lt;br /&gt;
+edo heptatonic narrow-fifth-based, fourthwards, #  &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="x^^ (3/2 maps to 6\11"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt; ^^ (3/2 maps to 6\11 &lt;/h1&gt;
+ 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 = perfect 6th):&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;
-nonotonic fourthwards chain of sixths:&lt;br /&gt;
+nonotonic fifthwards 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 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;
+edo pentatonic wide-fifth-based, fifthwards, # = ^^ (3/2 maps to 7\11 6th):&lt;br /&gt;
 P1 - ms3 - ^ms3/vMs3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ^ms7/vMs7 - Ms7 - P8d&lt;br /&gt;
 pentatonic plus ups and downs is doubly confusing!&lt;br /&gt;
@@ Line 1,039: / Line 1,036: @@
 requires learning octatonic interval arithmetic and notation&lt;br /&gt;
 &lt;br /&gt;
-edo heptatonic narrow-fifth-based, fourthwards, 3 keys per #/b, (3/2 maps to 7\13 = perfect 5th):&lt;br /&gt;
+edo heptatonic narrow-fifth-based, fourthwards, #  &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="x^^^ (3/2 maps to 7\13"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt; ^^^ (3/2 maps to 7\13 &lt;/h1&gt;
+ perfect 5th):&lt;br /&gt;
 P1 - m2 - m3 - vM2/^m3 - M2 - M3 - P4 - P5 - m6 - m7 - vM6/^m7 - M6 - M7 - P8&lt;br /&gt;
 problematic because m3 = 2\13 is narrower than M2 = 4\13&lt;br /&gt;
@@ Line 1,047: / Line 1,045: @@
 (13edo octatonic wide-fifth-based, fourthwards)&lt;br /&gt;
 &lt;br /&gt;
-edo heptatonic narrow-fifth-based, fourthwards, 2 keys per #/b (3/2 maps to 10\18 = perfect 5th)&lt;br /&gt;
+edo heptatonic narrow-fifth-based, fourthwards, #  &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="x^^ (3/2 maps to 10\18"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt; ^^ (3/2 maps to 10\18 &lt;/h1&gt;
+ perfect 5th)&lt;br /&gt;
 P1 - vm2 - m2 - vM2 - M2/m3 - vM3 - M3 - ^M3 - P4 - ^P4/vP5 - P5 - vm6 - m6 - vM6 - M6/m7 - vM7 - M7 - ^M7 - P8&lt;br /&gt;
 fourthwards plus ups and downs is doubly confusing!&lt;br /&gt;
@@ Line 1,056: / Line 1,055: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Summary of EDO notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;u&gt;&lt;strong&gt;Summary of EDO notation&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Summary of EDO notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&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 1,069: / Line 1,068: @@
 The C-D interval is the major 2nd (P2 for perfect EDOs, m2 for fourthward EDOs).&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&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:14 --&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:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&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:20 --&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 1,099: / Line 1,098: @@
 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:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&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:16 --&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:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&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:22 --&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 1,120: / Line 1,119: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;&lt;u&gt;Fourthward EDOs (9, 16 and 23)&lt;/u&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&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 1,141: / Line 1,140: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&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:20 --&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:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&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:26 --&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 1,172: / Line 1,171: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&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:22 --&gt;&lt;u&gt;Pentatonic EDOs (5, 10, 15, 20, 25 and 30)&lt;/u&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&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:28 --&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 1,207: / Line 1,206: @@
 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: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;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&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 1,230: / Line 1,229: @@
 &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, 31-34, 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;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&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:32 --&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;