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 20~~:37:32 UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-09 19:21:42 UTC</tt>.<br>

: The original revision id was <tt>~~558488833~~</tt>.<br>

: The original revision id was <tt>558786899</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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There are two strategies for notating these "oddball" EDOs, besides heptatonic fifth-based notation with ups and downs. One is to switch from heptatonic notation to some other type. The orange region contains edos for which pentatonic notation is a natural fit, in the sense that no ups or downs are needed. This includes 8edo, 13edo and 18edo.

The other approach is to use some interval other than the fifth to generate the notation. Above I said 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C.

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

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requires learning octatonic interval arithmetic and staff notation

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

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

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

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

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

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

nonotonic fifthwards chain of sixths:

nonotonic fifthwards chain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9

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, # = ^^ (3/2 maps to 7\11 6th):

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

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

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

pentatonic plus ups and downs is doubly confusing!

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requires learning octatonic interval arithmetic and notation

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

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

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

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

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

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

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

undecatonic sixthwards chain of sevenths:

M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11

P1 - m2 - M2/m3 - M3/m4 - M4/m5 - M5 - P6 - P7 - m8 - M8/m9 - M9/m10 - M10/m11 - M11 - P12

requires learning undecatonic interval arithmetic and notation

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

13edo octatonic wide-fifth-based, fourthwards, 3/2 maps to 8\13 = perfect 6th

octotonic chain of sixths: M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7

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

requires learning octatonic interval arithmetic and notation

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

18edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^ (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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P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10

requires learning nonotonic interval arithmetic and staff notation

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

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

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

42edo heptatonic narrow-fifth-based, 3/2 maps to 24\42 = perfect fifth

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

To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. The C-C# interval is the augmented unison, and if ~~they are not adjacent~~, ups and downs are required.The C-Db interval is the minor 2nd

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

~~The~~ C-D interval is the major 2nd (P2 ~~for perfect~~ EDOs, m2 ~~for fourthward~~ EDOs).

C C#/Db D (12edo)

C Db C# D (17edo)

C C# Db D (19edo)

C C# _ Db D (26edo)

C _ C# Db _ D (31edo)

The scale fragment concisely conveys the "flavor" of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For fourthward EDOs, C-Db = d2 and C-D = m2.

Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: "wide-fifth" 35edo (which uses 21\35 as a fifth) has a different scale fragment than "narrow-fifth" 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute d2 for m2.)

===__**"Fifth-less" EDOs (8, 11, 13 and 18)**__===

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

<br />

The other approach is to use some interval other than the fifth to generate the notation. Above I said 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C.<br />

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

<br />

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

<br />

11edo heptatonic narrow-fifth-based, fourthwards, # ~~<h1 id="toc6"><a name="x~~^^ (3/2 maps to 6\11~~">~~</a>~~<!~~-- ~~ws:end:WikiTextHeadingRule:12~~ --~~> ^^ (3/2 maps to 6\11 </h1>~~

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

~~perfect 5th):~~<br />

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

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

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

<br />

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

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

nonotonic fifthwards chain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9<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, # = ^^ (3/2 maps to 7\11 6th):<br />

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

pentatonic fifthwards chain of fifthoids: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7<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 1,036:

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requires learning octatonic interval arithmetic and notation<br />

<br />

13edo heptatonic narrow-fifth-based, fourthwards, # ~~<h1 id="toc7"><a name="x~~^^^ (3/2 maps to 7\13~~">~~</a>~~<!~~-- ~~ws:end:WikiTextHeadingRule:14~~ --~~> ^^^ (3/2 maps to 7\13 </h1>~~

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

~~perfect 5th):~~<br />

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

<br />

(13edo undecatonic narrow-fifth-based, 3/2 maps to 7\13 = perfect 7th)<br />

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

undecatonic sixthwards chain of sevenths: <br />

M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11<br />

P1 - m2 - M2/m3 - M3/m4 - M4/m5 - M5 - P6 - P7 - m8 - M8/m9 - M9/m10 - M10/m11 - M11 - P12<br />

requires learning undecatonic interval arithmetic and notation<br />

<br />

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

13edo octatonic wide-fifth-based, fourthwards, 3/2 maps to 8\13 = perfect 6th<br />

octotonic chain of sixths: M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7<br />

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

requires learning octatonic interval arithmetic and notation<br />

<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>~~

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

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="~~toc9~~"><a name="Summary of EDO notation"></a><u><strong>Summary of EDO notation</strong></u></h1>

<u><strong>Alternate notation for other edos:</strong></u><br />

23edo pentatonic wide-fifth-based, fifthwards, 3/2 maps to 14\23 = perfect fifthoid<br />

35edo heptatonic wide-fifth-based, sharp = seven ups, 3/2 maps to 21\35 = perfect fifth<br />

42edo heptatonic narrow-fifth-based, 3/2 maps to 24\42 = perfect fifth<br />

<br />

<h1 id="toc6"><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 first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The sweet EDOs may or may not.<br />

<br />

To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. The C-C# interval is the augmented unison, and if ~~they are not adjacent~~, ups and downs are required.The C-Db interval is the minor 2nd<br />

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

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

C C#/Db D (12edo)<br />

C Db C# D (17edo)<br />

C C# Db D (19edo)<br />

C C# _ Db D (26edo)<br />

C _ C# Db _ D (31edo)<br />

<br />

The scale fragment concisely conveys the &quot;flavor&quot; of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For fourthward EDOs, C-Db = d2 and C-D = m2.<br />

<br />

Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: &quot;wide-fifth&quot; 35edo (which uses 21\35 as a fifth) has a different scale fragment than &quot;narrow-fifth&quot; 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute d2 for m2.)<br />

<br />

<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>

<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>

<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="~~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>

<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>

<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="~~toc12~~"><a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"></a><u>Fourthward EDOs (9, 16 and 23)</u></h3>

<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>

<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="~~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>

<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>

<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="~~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>

<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>

<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="~~toc15~~"><a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"></a><u>Alternative pentatonic notation for pentatonic EDOs:</u></h3>

<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>

<br />

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

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<br />

<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>

<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>

<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 20:37:32 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-09 19:21:42 UTC</tt>.<br>
-: The original revision id was <tt>558488833</tt>.<br>
+: The original revision id was <tt>558786899</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 381: / Line 381: @@
 There are two strategies for notating these "oddball" EDOs, besides heptatonic fifth-based notation with ups and downs. One is to switch from heptatonic notation to some other type. The orange region contains edos for which pentatonic notation is a natural fit, in the sense that no ups or downs are needed. This includes 8edo, 13edo and 18edo.
-The other approach is to use some interval other than the fifth to generate the notation. Above I said 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C.
+The other approach is to use some interval other than the fifth to generate the notation. Above I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But these EDOs don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like.
@@ Line 390: / Line 390: @@
 requires learning octatonic interval arithmetic and staff notation
-edo heptatonic narrow-fifth-based, fourthwards, # = ^^ (3/2 maps to 6\11 = perfect 5th):
+edo heptatonic narrow-fifth-based, fourthwards, # is ^^ (3/2 maps to 6\11 perfect 5th):
+fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7
 P1 - m2 - vM2/m3 - M2/^m3 - M3 - P4 - P5 - m6 - vM6/m7 - M6/^m7 - M7 - P8
 problematic because m3 = 2\11 is narrower than M2 = 3\11
 edo nonotonic narrow-fifth-based, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):
-nonotonic fifthwards chain of sixths:
+nonotonic fifthwards chain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9
-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, # = ^^ (3/2 maps to 7\11 6th):
+edo pentatonic wide-fifth-based, fifthwards, # is ^^ (3/2 maps to 7\11 6th):
+pentatonic fifthwards chain of fifthoids: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7
 P1 - ms3 - ^ms3/vMs3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ^ms7/vMs7 - Ms7 - P8d
 pentatonic plus ups and downs is doubly confusing!
@@ Line 409: / Line 410: @@
 requires learning octatonic interval arithmetic and notation
-edo heptatonic narrow-fifth-based, fourthwards, # = ^^^ (3/2 maps to 7\13 = perfect 5th):
+edo heptatonic narrow-fifth-based, fourthwards, # is ^^^ (3/2 maps to 7\13 perfect 5th):
+fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7
 P1 - m2 - m3 - vM2/^m3 - M2 - M3 - P4 - P5 - m6 - m7 - vM6/^m7 - M6 - M7 - P8
 problematic because m3 = 2\13 is narrower than M2 = 4\13
-(13edo undecatonic narrow-fifth-based, 3/2 maps to 7\13 = perfect 7th)
+edo undecatonic narrow-fifth-based, fourthwards, 3/2 maps to 7\13 = perfect 7th
+undecatonic sixthwards chain of sevenths:
+M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11
+P1 - m2 - M2/m3 - M3/m4 - M4/m5 - M5 - P6 - P7 - m8 - M8/m9 - M9/m10 - M10/m11 - M11 - P12
+requires learning undecatonic interval arithmetic and notation
-(13edo octatonic wide-fifth-based, fourthwards)
+edo octatonic wide-fifth-based, fourthwards, 3/2 maps to 8\13 = perfect 6th
+octotonic chain of sixths: M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7
+P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9
+requires learning octatonic interval arithmetic and notation
-edo heptatonic narrow-fifth-based, fourthwards, # = ^^ (3/2 maps to 10\18 = perfect 5th)
+edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^ (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 424: / Line 433: @@
 P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10
 requires learning nonotonic interval arithmetic and staff notation
+__**Alternate notation for other edos:**__
+edo pentatonic wide-fifth-based, fifthwards, 3/2 maps to 14\23 = perfect fifthoid
+edo heptatonic wide-fifth-based, sharp = seven ups, 3/2 maps to 21\35 = perfect fifth
+edo heptatonic narrow-fifth-based, 3/2 maps to 24\42 = perfect fifth
@@ Line 436: / Line 451: @@
 The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The sweet EDOs may or may not.
-To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. The C-C# interval is the augmented unison, and if they are not adjacent, ups and downs are required.The C-Db interval is the minor 2nd
+To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. Examples:
-The C-D interval is the major 2nd (P2 for perfect EDOs, m2 for fourthward EDOs).
+C C#/Db D (12edo)
+C Db C# D (17edo)
+C C# Db D (19edo)
+C C# _ Db D (26edo)
+C _ C# Db _ D (31edo)
+The scale fragment concisely conveys the "flavor" of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For fourthward EDOs, C-Db = d2 and C-D = m2.
+Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: "wide-fifth" 35edo (which uses 21\35 as a fifth) has a different scale fragment than "narrow-fifth" 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute d2 for m2.)
 ===__**"Fifth-less" EDOs (8, 11, 13 and 18)**__===
@@ Line 886: / Line 909: @@
 IIvM,m7 = &amp;quot;two downmajor, minor seven&amp;quot;&lt;br /&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;!-- 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;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;!-- 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;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 995: / Line 1,018: @@
 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: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;!-- 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;br /&gt;
 &lt;br /&gt;
@@ Line 1,007: / Line 1,030: @@
 There are two strategies for notating these &amp;quot;oddball&amp;quot; EDOs, besides heptatonic fifth-based notation with ups and downs. One is to switch from heptatonic notation to some other type. The orange region contains edos for which pentatonic notation is a natural fit, in the sense that no ups or downs are needed. This includes 8edo, 13edo and 18edo.&lt;br /&gt;
 &lt;br /&gt;
-The other approach is to use some interval other than the fifth to generate the notation. Above I said 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C.&lt;br /&gt;
+The other approach is to use some interval other than the fifth to generate the notation. Above I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But these EDOs don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like.&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,016: / Line 1,039: @@
 requires learning octatonic interval arithmetic and staff notation&lt;br /&gt;
 &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;
+edo heptatonic narrow-fifth-based, fourthwards, # is ^^ (3/2 maps to 6\11 perfect 5th):&lt;br /&gt;
- perfect 5th):&lt;br /&gt;
+fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7&lt;br /&gt;
 P1 - m2 - vM2/m3 - M2/^m3 - M3 - P4 - P5 - m6 - vM6/m7 - M6/^m7 - M7 - P8&lt;br /&gt;
 problematic because m3 = 2\11 is narrower than M2 = 3\11&lt;br /&gt;
 &lt;br /&gt;
 edo nonotonic narrow-fifth-based, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):&lt;br /&gt;
-nonotonic fifthwards chain of sixths:&lt;br /&gt;
+nonotonic fifthwards chain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9&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, # = ^^ (3/2 maps to 7\11 6th):&lt;br /&gt;
+edo pentatonic wide-fifth-based, fifthwards, # is ^^ (3/2 maps to 7\11 6th):&lt;br /&gt;
+pentatonic fifthwards chain of fifthoids: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7&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 1,036: / Line 1,059: @@
 requires learning octatonic interval arithmetic and notation&lt;br /&gt;
 &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;
+edo heptatonic narrow-fifth-based, fourthwards, # is ^^^ (3/2 maps to 7\13 perfect 5th):&lt;br /&gt;
- perfect 5th):&lt;br /&gt;
+fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7&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;
 &lt;br /&gt;
-(13edo undecatonic narrow-fifth-based, 3/2 maps to 7\13 = perfect 7th)&lt;br /&gt;
+edo undecatonic narrow-fifth-based, fourthwards, 3/2 maps to 7\13 = perfect 7th&lt;br /&gt;
+undecatonic sixthwards chain of sevenths: &lt;br /&gt;
+M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11&lt;br /&gt;
+P1 - m2 - M2/m3 - M3/m4 - M4/m5 - M5 - P6 - P7 - m8 - M8/m9 - M9/m10 - M10/m11 - M11 - P12&lt;br /&gt;
+requires learning undecatonic interval arithmetic and notation&lt;br /&gt;
 &lt;br /&gt;
-(13edo octatonic wide-fifth-based, fourthwards)&lt;br /&gt;
+edo octatonic wide-fifth-based, fourthwards, 3/2 maps to 8\13 = perfect 6th&lt;br /&gt;
+octotonic chain of sixths: M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7&lt;br /&gt;
+P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9&lt;br /&gt;
+requires learning octatonic interval arithmetic and notation&lt;br /&gt;
 &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;
+edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^ (3/2 maps to 10\18 perfect 5th)&lt;br /&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 1,055: / Line 1,084: @@
 &lt;br /&gt;
 &lt;br /&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;u&gt;&lt;strong&gt;Alternate notation for other edos:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
+edo pentatonic wide-fifth-based, fifthwards, 3/2 maps to 14\23 = perfect fifthoid&lt;br /&gt;
+edo heptatonic wide-fifth-based, sharp = seven ups, 3/2 maps to 21\35 = perfect fifth&lt;br /&gt;
+edo heptatonic narrow-fifth-based, 3/2 maps to 24\42 = perfect fifth&lt;br /&gt;
+&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;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 1,065: / Line 1,100: @@
 The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The sweet EDOs may or may not.&lt;br /&gt;
 &lt;br /&gt;
-To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. The C-C# interval is the augmented unison, and if they are not adjacent, ups and downs are required.The C-Db interval is the minor 2nd&lt;br /&gt;
+To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. Examples: &lt;br /&gt;
-The C-D interval is the major 2nd (P2 for perfect EDOs, m2 for fourthward EDOs).&lt;br /&gt;
+C C#/Db D (12edo)&lt;br /&gt;
+C Db C# D (17edo)&lt;br /&gt;
+C C# Db D (19edo)&lt;br /&gt;
+C C# _ Db D (26edo)&lt;br /&gt;
+C _ C# Db _ D (31edo)&lt;br /&gt;
+&lt;br /&gt;
+The scale fragment concisely conveys the &amp;quot;flavor&amp;quot; of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For fourthward EDOs, C-Db = d2 and C-D = m2.&lt;br /&gt;
+&lt;br /&gt;
+Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: &amp;quot;wide-fifth&amp;quot; 35edo (which uses 21\35 as a fifth) has a different scale fragment than &amp;quot;narrow-fifth&amp;quot; 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute d2 for m2.)&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;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;!-- 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;
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 &lt;strong&gt;&lt;u&gt;8edo&lt;/u&gt;:&lt;/strong&gt; (generator = 1\8 = perfect 2nd = 150¢)&lt;br /&gt;
@@ Line 1,098: / Line 1,141: @@
 E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb&lt;br /&gt;
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-&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;!-- 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;
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 All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.&lt;br /&gt;
@@ Line 1,119: / Line 1,162: @@
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-&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;!-- 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;
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 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 1,140: / Line 1,183: @@
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-&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;!-- 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;
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 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 1,171: / Line 1,214: @@
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-&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;!-- 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;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 1,206: / Line 1,249: @@
 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;
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-&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;!-- 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;
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 Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.&lt;br /&gt;
@@ Line 1,229: / Line 1,272: @@
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-&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;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Summary of EDO notation--&amp;quot;Sweet&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;&lt;u&gt;&amp;quot;Sweet&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)&lt;/u&gt;&lt;/h3&gt;
   &lt;br /&gt;
 All sweet EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.&lt;br /&gt;