Kite's ups and downs notation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-08-~~31 23~~:10:21 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-01 02:57:17 UTC</tt>.

: The original revision id was <tt>~~557860773~~</tt>.

: The original revision id was <tt>557870423</tt>.

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=__Naming Chords__=

Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos.

Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.

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

There are 3 special cases to be addressed. The first is when the edo's 5th is narrower than 4\7, as in 16edo. Major is defined as always wider than minor, so major is not fifthwards but fourthwards:

The chain of fifths in ~~fourthwards~~ EDOs:

The fourthwards chain of fifths in superflat aka Mavila EDOs (3/2 maps to less than 4\7):

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

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

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

In other words, sharp/flat, major/minor, and aug/dim all retain their melodic meaning but the ~~harmonic~~ meaning is reversed. Perfect and natural are unaffected. Interval arithmetic in fourthwards edos is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.

In other words, sharp/flat, major/minor, and aug/dim all retain their melodic meaning but the chain-of-fifths meaning is reversed. Perfect and natural are unaffected. Interval arithmetic in fourthwards edos is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.

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

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

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

~~In edos that are a multiple of~~ 7, ~~such~~ as 14, 21, 28, etc. there are 0 keys per sharp/flat, and all intervals are perfect.

The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. 42edo, 49edo, etc. have a fifth wider than 4\7. In these edos, there are zero keys per sharp/flat, and all intervals are perfect.

The chain of fifths in ~~multiple-of-~~7 ~~EDOs~~:

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

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

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

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

21edo: P1 - A1 - d2 - P2 - A2 - D3 - P3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - P6 - A6 - D7 - P7 - A7 - d8 - P8

21edo: P1 - A1 - d2 - P2 - A2 - d3 - P3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - P6 - A6 - D7 - P7 - A7 - d8 - P8

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

Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. However they can be used for familiarity's sake: an A major chord can be written A - C#^ - E.

The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. Heptatonic fifth-based notation is impossible in these cases, because the chain of 7 fifths isn't a MOS scale. Such EDOs are dealt with below.

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

==__22edo chord names__==

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

Let's review the 22edo interval names:

0\22 = P1

1\22 = m2

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0-7-13-18 in C is "C vM,m7", pronounced "C downmajor, minor seventh". The space between the C and the down symbol is needed because Cv is a note, and "Cv M,m7" is a different chord. That chord is pronounced "C down, major, minor 7th", so one has to "speak the space". Alternatively, a comma could be used: C,vM,m7 vs. Cv,M,m7. The extra space/comma isn't needed when 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 ~~utonal~~ 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.

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.

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,-5" and 0-6-11 is "C ^m,^d5". 3) The 3rd is also assumed to be present, and is implied by a quality with no degree. Thus 0-7-13 is "C vM". 4) The 3rd isn't spelled out if the 6th or 7th has the same quality as the 3rd. Thus 0-7-13-16 is "C vM6", but 0-7-13-17 is "C vM,M6". Thirdless chords: 0-13-18 is either "Cm7,-3" or "C5,m7".

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

To use relative notation, first write out all possible 22edo chord roots relatively. This is ~~just~~ the interval notation with Roman numerals ~~instead of~~ Arabic, # for aug, and b for minor. Dim from perfect is b, but dim from minor is bb. Enharmonic equivalents like ^I = bII are used in certain chord progressions like Im - ^IIIM - ^VIIM - ^IVm - ^Im.

To use relative notation, first write out all possible 22edo chord roots relatively. This is equivalent to the interval notation with Roman numerals substituted for Arabic, # for aug, and b for minor. Dim from perfect is b, but dim from minor is bb. Enharmonic equivalents like ^I = bII are used in certain chord progressions like Im - ^IIIM - ^VIIM - ^IVm - ^Im.

I ^I/bII v#I/^bII #I/vII II ^II/bIII v#II/^bIII #II/vIII III IV ^IV/bV v#IV/^bV #IV/vV

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

V ^V/bVI v#V/^bVI #V/vVI VI ^VI/bVII v#VI/^bVII #VI/vVII VII/vI

These are pronounced "down-two", "up-flat-three", "down-sharp-four", etc.

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

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

0-3-9 = m

0-3-9 = m or sus2

0-4-9 = ^m

0-5-9 = vM

0-6-9 = sus4 ~~(or possibly M)~~

0-6-9 = M or sus4

0-5-9-12 = vM,m7

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

0-2-9 = susm2

0-3-9 = ~~susM2~~

0-3-9 = sus2

0-4-9 = m

0-5-9 = M

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0-12-18 = sus-v4

==__EDOs with inaccurate 3/2__==

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

Not counting the trivial edos 2, 3, 4 and 6, there are only a few such edos: 8, 9, 11, 13, and 18. As seen in this diagram, they are the ones to the left of the light blue region, plus the ones to the right of the central line in the orange region. 23edo can be notated similarly to 16edo, using 13\23 instead of 14\23.

[[image:The 5th of EDOs 5-53.png width="800" height="1004"]]

~~Such~~ EDOs may be better notated with a notation not generated by the fifth.</pre></div>

There are several strategies for notating these EDOs. Since heptatonic fifth-based may be better notated with a notation not generated by the fifth.</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a>&quot;Ups and Downs&quot; Notation</h1>

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<h1 id="toc1"><a name="Naming Chords"></a>Naming Chords</h1>

Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos.

Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.

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

There are 3 special cases to be addressed. The first is when the edo's 5th is narrower than 4\7, as in 16edo. Major is defined as always wider than minor, so major is not fifthwards but fourthwards:

The chain of fifths in ~~fourthwards~~ EDOs:

The fourthwards chain of fifths in superflat aka Mavila EDOs (3/2 maps to less than 4\7):

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

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

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

In other words, sharp/flat, major/minor, and aug/dim all retain their melodic meaning but the ~~harmonic~~ meaning is reversed. Perfect and natural are unaffected. Interval arithmetic in fourthwards edos is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.

In other words, sharp/flat, major/minor, and aug/dim all retain their melodic meaning but the chain-of-fifths meaning is reversed. Perfect and natural are unaffected. Interval arithmetic in fourthwards edos is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.

M2 + M2 --&gt; m2 + m2 = dim3 --&gt; aug3

D to F# --&gt; D to Fb = dim3 --&gt; aug3

Eb + m3 --&gt; E# + M3 = G## --&gt; Gbb

~~In edos that are a multiple of~~ 7, ~~such~~ as 14, 21, 28, etc. there are 0 keys per sharp/flat, and all intervals are perfect.

The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. 42edo, 49edo, etc. have a fifth wider than 4\7. In these edos, there are zero keys per sharp/flat, and all intervals are perfect.

The chain of fifths in ~~multiple-of-~~7 ~~EDOs~~:

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

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

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

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

21edo: P1 - A1 - d2 - P2 - A2 - D3 - P3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - P6 - A6 - D7 - P7 - A7 - d8 - P8

21edo: P1 - A1 - d2 - P2 - A2 - d3 - P3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - P6 - A6 - D7 - P7 - A7 - d8 - P8

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

Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. However they can be used for familiarity's sake: an A major chord can be written A - C#^ - E.

The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. Heptatonic fifth-based notation is impossible in these cases, because the chain of 7 fifths isn't a MOS scale. Such EDOs are dealt with below.

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

<h2 id="toc2"><a name="Naming Chords-22edo chord names"></a>22edo chord names</h2>

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

Let's review the 22edo interval names:

0\22 = P1

1\22 = m2

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0-7-13-18 in C is &quot;C vM,m7&quot;, pronounced &quot;C downmajor, minor seventh&quot;. The space 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 space&quot;. Alternatively, a comma could be used: C,vM,m7 vs. Cv,M,m7. The extra space/comma isn't needed when there's no ups or downs immediately after the note name, e.g. Cm.

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 ~~utonal~~ 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.

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.

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 &quot;C vM,m7,-5&quot; and 0-6-11 is &quot;C ^m,^d5&quot;. 3) The 3rd is also assumed to be present, and is implied by a quality with no degree. Thus 0-7-13 is &quot;C vM&quot;. 4) The 3rd isn't spelled out if the 6th or 7th has the same quality as the 3rd. Thus 0-7-13-16 is &quot;C vM6&quot;, but 0-7-13-17 is &quot;C vM,M6&quot;. Thirdless chords: 0-13-18 is either &quot;Cm7,-3&quot; or &quot;C5,m7&quot;.

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A vM,m7 = &quot;A downmajor, minor seven&quot;

To use relative notation, first write out all possible 22edo chord roots relatively. This is ~~just~~ the interval notation with Roman numerals ~~instead of~~ Arabic, # for aug, and b for minor. Dim from perfect is b, but dim from minor is bb. Enharmonic equivalents like ^I = bII are used in certain chord progressions like Im - ^IIIM - ^VIIM - ^IVm - ^Im.

To use relative notation, first write out all possible 22edo chord roots relatively. This is equivalent to the interval notation with Roman numerals substituted for Arabic, # for aug, and b for minor. Dim from perfect is b, but dim from minor is bb. Enharmonic equivalents like ^I = bII are used in certain chord progressions like Im - ^IIIM - ^VIIM - ^IVm - ^Im.

I ^I/bII v#I/^bII #I/vII II ^II/bIII v#II/^bIII #II/vIII III IV ^IV/bV v#IV/^bV #IV/vV~~ ~~

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

V ^V/bVI v#V/^bVI #V/vVI VI ^VI/bVII v#VI/^bVII #VI/vVII VII/vI

These are pronounced &quot;down-two&quot;, &quot;up-flat-three&quot;, &quot;down-sharp-four&quot;, etc.

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

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

0-3-9 = m

0-3-9 = m or sus2

0-4-9 = ^m

0-5-9 = vM

0-6-9 = sus4 ~~(or possibly M)~~

0-6-9 = M or sus4

0-5-9-12 = vM,m7

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

0-2-9 = susm2

0-3-9 = ~~susM2~~

0-3-9 = sus2

0-4-9 = m

0-5-9 = M

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0-12-18 = sus-v4

<h2 id="toc4"><a name="Naming Chords-EDOs with inaccurate 3/2"></a>EDOs with inaccurate 3/2</h2>

<h2 id="toc4"><a name="Naming Chords-EDOs with an inaccurate 3/2"></a>EDOs with an inaccurate 3/2</h2>

~~Such~~ EDOs may be better notated with a notation not generated by the fifth.</body></html></pre></div>

Not counting the trivial edos 2, 3, 4 and 6, there are only a few such edos: 8, 9, 11, 13, and 18. As seen in this diagram, they are the ones to the left of the light blue region, plus the ones to the right of the central line in the orange region. 23edo can be notated similarly to 16edo, using 13\23 instead of 14\23.

<img src="/file/view/The%205th%20of%20EDOs%205-53.png/558356515/800x1004/The%205th%20of%20EDOs%205-53.png" alt="The 5th of EDOs 5-53.png" title="The 5th of EDOs 5-53.png" style="height: 1004px; width: 800px;" />

There are several strategies for notating these EDOs. Since heptatonic fifth-based may be better notated with a notation not generated by the fifth.</body></html></pre></div>

@@ 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>2015-08-31 23:10:21 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-01 02:57:17 UTC</tt>.<br>
-: The original revision id was <tt>557860773</tt>.<br>
+: The original revision id was <tt>557870423</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 119: / Line 119: @@
 =__Naming Chords__=
-Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos.
+Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.
-The quality of an interval is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc. Major is always wider than minor, so if the edo's 5th is narrower than 4\7, as is in 16edo, major is not fifthwards but fourthwards:
+There are 3 special cases to be addressed. The first is when the edo's 5th is narrower than 4\7, as in 16edo. Major is defined as always wider than minor, so major is not fifthwards but fourthwards:
-The chain of fifths in fourthwards EDOs:
+The fourthwards chain of fifths in superflat aka Mavila EDOs (3/2 maps to less than 4\7):
 M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 - A1 etc.
 F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.
@@ Line 129: / Line 129: @@
 edo: C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C
-In other words, sharp/flat, major/minor, and aug/dim all retain their melodic meaning but the harmonic meaning is reversed. Perfect and natural are unaffected. Interval arithmetic in fourthwards edos is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.
+In other words, sharp/flat, major/minor, and aug/dim all retain their melodic meaning but the chain-of-fifths meaning is reversed. Perfect and natural are unaffected. Interval arithmetic in fourthwards edos is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.
 M2 + M2 --&gt; m2 + m2 = dim3 --&gt; aug3
 D to F# --&gt; D to Fb = dim3 --&gt; aug3
 Eb + m3 --&gt; E# + M3 = G## --&gt; Gbb
-In edos that are a multiple of 7, such as 14, 21, 28, etc. there are 0 keys per sharp/flat, and all intervals are perfect.
+The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. 42edo, 49edo, etc. have a fifth wider than 4\7. In these edos, there are zero keys per sharp/flat, and all intervals are perfect.
-The chain of fifths in multiple-of-7 EDOs:
+The chain of fifths in heptatonic EDOs (3/2 maps to 4\7):
 P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.
-F - C - G - D - A - E - B - F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.
+F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.
-edo: P1 - A1 - d2 - P2 - A2 - D3 - P3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - P6 - A6 - D7 - P7 - A7 - d8 - P8
+edo: P1 - A1 - d2 - P2 - A2 - d3 - P3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - P6 - A6 - D7 - P7 - A7 - d8 - P8
 edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C
 Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. However they can be used for familiarity's sake: an A major chord can be written A - C#^ - E.
+The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. Heptatonic fifth-based notation is impossible in these cases, because the chain of 7 fifths isn't a MOS scale. Such EDOs are dealt with below.
+Chord names are based entirely on the ups/downs interval names, not on JI ratios. This avoids identifying one EDOstep with multiple ratios, as happens in 22edo when 0-7-18 implies 4:5:7 but 0-9-18 implies 9:12:16. 18\22 is neither 7/4 nor 16/9, it's 18\22!
 ==__22edo chord names__==
-Chord names are based entirely on the ups/downs interval names, not on JI ratios. This avoids identifying one EDOstep with multiple ratios, as happens when 0-7-18 implies 4:5:7 but 0-9-18 implies 9:12:16. 18\22 is neither 7/4 nor 16/9, it's 18\22!
+Let's review the 22edo interval names:
 \22 = P1
 \22 = m2
@@ Line 175: / Line 178: @@
 -7-13-18 in C is "C vM,m7", pronounced "C downmajor, minor seventh". The space between the C and the down symbol is needed because Cv is a note, and "Cv M,m7" is a different chord. That chord is pronounced "C down, major, minor 7th", so one has to "speak the space". Alternatively, a comma could be used: C,vM,m7 vs. Cv,M,m7. The extra space/comma isn't needed when 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 utonal 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.
+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.
 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,-5" and 0-6-11 is "C ^m,^d5". 3) The 3rd is also assumed to be present, and is implied by a quality with no degree. Thus 0-7-13 is "C vM". 4) The 3rd isn't spelled out if the 6th or 7th has the same quality as the 3rd. Thus 0-7-13-16 is "C vM6", but 0-7-13-17 is "C vM,M6". Thirdless chords: 0-13-18 is either "Cm7,-3" or "C5,m7".
@@ Line 216: / Line 219: @@
 A vM,m7 = "A downmajor, minor seven"
-To use relative notation, first write out all possible 22edo chord roots relatively. This is just the interval notation with Roman numerals instead of Arabic, # for aug, and b for minor. Dim from perfect is b, but dim from minor is bb. Enharmonic equivalents like ^I = bII are used in certain chord progressions like Im - ^IIIM - ^VIIM - ^IVm - ^Im.
+To use relative notation, first write out all possible 22edo chord roots relatively. This is equivalent to the interval notation with Roman numerals substituted for Arabic, # for aug, and b for minor. Dim from perfect is b, but dim from minor is bb. Enharmonic equivalents like ^I = bII are used in certain chord progressions like Im - ^IIIM - ^VIIM - ^IVm - ^Im.
-I ^I/bII v#I/^bII #I/vII II ^II/bIII v#II/^bIII #II/vIII III IV ^IV/bV v#IV/^bV #IV/vV
+I ^I/bII v#I/^bII #I/vII II ^II/bIII v#II/^bIII #II/vIII III IV ^IV/bV v#IV/^bV #IV/vV V ^V/bVI v#V/^bVI #V/vVI VI ^VI/bVII v#VI/^bVII #VI/vVII VII/vI
-V ^V/bVI v#V/^bVI #V/vVI VI ^VI/bVII v#VI/^bVII #VI/vVII VII/vI
 These are pronounced "down-two", "up-flat-three", "down-sharp-four", etc.
@@ Line 234: / Line 236: @@
 chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7 P8
 chord roots: I ^bII vII II/bIII ^bIII vIII IV ^IV vV V ^bVI vVI VI/bVII ^bVII vVII
--3-9 = m
+-3-9 = m or sus2
 -4-9 = ^m
 -5-9 = vM
--6-9 = sus4 (or possibly M)
+-6-9 = M or sus4
 -5-9-12 = vM,m7
@@ Line 244: / Line 246: @@
 chord roots: I #I/bbII bII II bIII III #III/vIV IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI
 -2-9 = susm2
--3-9 = susM2
+-3-9 = sus2
 -4-9 = m
 -5-9 = M
@@ Line 327: / Line 329: @@
 -12-18 = sus-v4
-==__EDOs with inaccurate 3/2__==
+==__EDOs with an inaccurate 3/2__==
+Not counting the trivial edos 2, 3, 4 and 6, there are only a few such edos: 8, 9, 11, 13, and 18. As seen in this diagram, they are the ones to the left of the light blue region, plus the ones to the right of the central line in the orange region. 23edo can be notated similarly to 16edo, using 13\23 instead of 14\23.
+[[image:The 5th of EDOs 5-53.png width="800" height="1004"]]
-Such EDOs may be better notated with a notation not generated by the fifth.</pre></div>
+There are several strategies for notating these EDOs. Since heptatonic fifth-based may be better notated with a notation not generated by the fifth.</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Ups and Downs Notation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x&amp;quot;Ups and Downs&amp;quot; Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&amp;quot;Ups and Downs&amp;quot; Notation&lt;/h1&gt;
@@ Line 444: / Line 451: @@
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Naming Chords"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;u&gt;Naming Chords&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
-Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos.&lt;br /&gt;
+Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc. &lt;br /&gt;
 &lt;br /&gt;
-The quality of an interval is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc. Major is always wider than minor, so if the edo's 5th is narrower than 4\7, as is in 16edo, major is not fifthwards but fourthwards:&lt;br /&gt;
+There are 3 special cases to be addressed. The first is when the edo's 5th is narrower than 4\7, as in 16edo. Major is defined as always wider than minor, so major is not fifthwards but fourthwards:&lt;br /&gt;
 &lt;br /&gt;
-The chain of fifths in fourthwards EDOs:&lt;br /&gt;
+The fourthwards chain of fifths in superflat aka Mavila EDOs (3/2 maps to less than 4\7):&lt;br /&gt;
 M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 - A1 etc.&lt;br /&gt;
 F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.&lt;br /&gt;
@@ Line 454: / Line 461: @@
 edo: C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C&lt;br /&gt;
 &lt;br /&gt;
-In other words, sharp/flat, major/minor, and aug/dim all retain their melodic meaning but the harmonic meaning is reversed. Perfect and natural are unaffected. Interval arithmetic in fourthwards edos is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.&lt;br /&gt;
+In other words, sharp/flat, major/minor, and aug/dim all retain their melodic meaning but the chain-of-fifths meaning is reversed. Perfect and natural are unaffected. Interval arithmetic in fourthwards edos is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.&lt;br /&gt;
 M2 + M2 --&amp;gt; m2 + m2 = dim3 --&amp;gt; aug3&lt;br /&gt;
 D to F# --&amp;gt; D to Fb = dim3 --&amp;gt; aug3&lt;br /&gt;
 Eb + m3 --&amp;gt; E# + M3 = G## --&amp;gt; Gbb&lt;br /&gt;
 &lt;br /&gt;
-In edos that are a multiple of 7, such as 14, 21, 28, etc. there are 0 keys per sharp/flat, and all intervals are perfect.&lt;br /&gt;
+The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. 42edo, 49edo, etc. have a fifth wider than 4\7. In these edos, there are zero keys per sharp/flat, and all intervals are perfect.&lt;br /&gt;
 &lt;br /&gt;
-The chain of fifths in multiple-of-7 EDOs:&lt;br /&gt;
+The chain of fifths in heptatonic EDOs (3/2 maps to 4\7):&lt;br /&gt;
 P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.&lt;br /&gt;
-F - C - G - D - A - E - B - F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.&lt;br /&gt;
+F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.&lt;br /&gt;
-edo: P1 - A1 - d2 - P2 - A2 - D3 - P3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - P6 - A6 - D7 - P7 - A7 - d8 - P8&lt;br /&gt;
+edo: P1 - A1 - d2 - P2 - A2 - d3 - P3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - P6 - A6 - D7 - P7 - A7 - d8 - P8&lt;br /&gt;
 edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C&lt;br /&gt;
 Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. However they can be used for familiarity's sake: an A major chord can be written A - C#^ - E.&lt;br /&gt;
+&lt;br /&gt;
+The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. Heptatonic fifth-based notation is impossible in these cases, because the chain of 7 fifths isn't a MOS scale. Such EDOs are dealt with below.&lt;br /&gt;
+&lt;br /&gt;
+Chord names are based entirely on the ups/downs interval names, not on JI ratios. This avoids identifying one EDOstep with multiple ratios, as happens in 22edo when 0-7-18 implies 4:5:7 but 0-9-18 implies 9:12:16. 18\22 is neither 7/4 nor 16/9, it's 18\22!&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Naming Chords-22edo chord names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;u&gt;22edo chord names&lt;/u&gt;&lt;/h2&gt;
   &lt;br /&gt;
-Chord names are based entirely on the ups/downs interval names, not on JI ratios. This avoids identifying one EDOstep with multiple ratios, as happens when 0-7-18 implies 4:5:7 but 0-9-18 implies 9:12:16. 18\22 is neither 7/4 nor 16/9, it's 18\22!&lt;br /&gt;
+Let's review the 22edo interval names:&lt;br /&gt;
-&lt;br /&gt;
 \22 = P1&lt;br /&gt;
 \22 = m2&lt;br /&gt;
@@ Line 500: / Line 510: @@
 -7-13-18 in C is &amp;quot;C vM,m7&amp;quot;, pronounced &amp;quot;C downmajor, minor seventh&amp;quot;. The space between the C and the down symbol is needed because Cv is a note, and &amp;quot;Cv M,m7&amp;quot; is a different chord. That chord is pronounced &amp;quot;C down, major, minor 7th&amp;quot;, so one has to &amp;quot;speak the space&amp;quot;. Alternatively, a comma could be used: C,vM,m7 vs. Cv,M,m7. The extra space/comma isn't needed when 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 utonal 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;
+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;
 &lt;br /&gt;
 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 &amp;quot;C vM,m7,-5&amp;quot; and 0-6-11 is &amp;quot;C ^m,^d5&amp;quot;. 3) The 3rd is also assumed to be present, and is implied by a quality with no degree. Thus 0-7-13 is &amp;quot;C vM&amp;quot;. 4) The 3rd isn't spelled out if the 6th or 7th has the same quality as the 3rd. Thus 0-7-13-16 is &amp;quot;C vM6&amp;quot;, but 0-7-13-17 is &amp;quot;C vM,M6&amp;quot;. Thirdless chords: 0-13-18 is either &amp;quot;Cm7,-3&amp;quot; or &amp;quot;C5,m7&amp;quot;.&lt;br /&gt;
@@ Line 541: / Line 551: @@
 A vM,m7 = &amp;quot;A downmajor, minor seven&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
-To use relative notation, first write out all possible 22edo chord roots relatively. This is just the interval notation with Roman numerals instead of Arabic, # for aug, and b for minor. Dim from perfect is b, but dim from minor is bb. Enharmonic equivalents like ^I = bII are used in certain chord progressions like Im - ^IIIM - ^VIIM - ^IVm - ^Im.&lt;br /&gt;
+To use relative notation, first write out all possible 22edo chord roots relatively. This is equivalent to the interval notation with Roman numerals substituted for Arabic, # for aug, and b for minor. Dim from perfect is b, but dim from minor is bb. Enharmonic equivalents like ^I = bII are used in certain chord progressions like Im - ^IIIM - ^VIIM - ^IVm - ^Im.&lt;br /&gt;
-I ^I/bII v#I/^bII #I/vII II ^II/bIII v#II/^bIII #II/vIII III IV ^IV/bV v#IV/^bV #IV/vV&lt;br /&gt;
+I ^I/bII v#I/^bII #I/vII II ^II/bIII v#II/^bIII #II/vIII III IV ^IV/bV v#IV/^bV #IV/vV V ^V/bVI v#V/^bVI #V/vVI VI ^VI/bVII v#VI/^bVII #VI/vVII VII/vI&lt;br /&gt;
-V ^V/bVI v#V/^bVI #V/vVI VI ^VI/bVII v#VI/^bVII #VI/vVII VII/vI&lt;br /&gt;
 These are pronounced &amp;quot;down-two&amp;quot;, &amp;quot;up-flat-three&amp;quot;, &amp;quot;down-sharp-four&amp;quot;, etc.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 559: / Line 568: @@
 chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7 P8&lt;br /&gt;
 chord roots: I ^bII vII II/bIII ^bIII vIII IV ^IV vV V ^bVI vVI VI/bVII ^bVII vVII&lt;br /&gt;
--3-9 = m&lt;br /&gt;
+-3-9 = m or sus2&lt;br /&gt;
 -4-9 = ^m&lt;br /&gt;
 -5-9 = vM&lt;br /&gt;
--6-9 = sus4 (or possibly M)&lt;br /&gt;
+-6-9 = M or sus4&lt;br /&gt;
 -5-9-12 = vM,m7&lt;br /&gt;
 &lt;br /&gt;
@@ Line 569: / Line 578: @@
 chord roots: I #I/bbII bII II bIII III #III/vIV IV #IV/bV V #V/bbVI bVI VI bVII VII #VII/bI&lt;br /&gt;
 -2-9 = susm2&lt;br /&gt;
--3-9 = susM2&lt;br /&gt;
+-3-9 = sus2&lt;br /&gt;
 -4-9 = m&lt;br /&gt;
 -5-9 = M&lt;br /&gt;
@@ Line 652: / Line 661: @@
 -12-18 = sus-v4&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-EDOs with inaccurate 3/2"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;u&gt;EDOs with inaccurate 3/2&lt;/u&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Naming Chords-EDOs with an inaccurate 3/2"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;u&gt;EDOs with an inaccurate 3/2&lt;/u&gt;&lt;/h2&gt;
   &lt;br /&gt;
-Such EDOs may be better notated with a notation not generated by the fifth.&lt;/body&gt;&lt;/html&gt;</pre></div>
+Not counting the trivial edos 2, 3, 4 and 6, there are only a few such edos: 8, 9, 11, 13, and 18. As seen in this diagram, they are the ones to the left of the light blue region, plus the ones to the right of the central line in the orange region. 23edo can be notated similarly to 16edo, using 13\23 instead of 14\23.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:10:&amp;lt;img src=&amp;quot;/file/view/The%205th%20of%20EDOs%205-53.png/558356515/800x1004/The%205th%20of%20EDOs%205-53.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1004px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/The%205th%20of%20EDOs%205-53.png/558356515/800x1004/The%205th%20of%20EDOs%205-53.png" alt="The 5th of EDOs 5-53.png" title="The 5th of EDOs 5-53.png" style="height: 1004px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:10 --&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+There are several strategies for notating these EDOs. Since heptatonic fifth-based may be better notated with a notation not generated by the fifth.&lt;/body&gt;&lt;/html&gt;</pre></div>