Kite's ups and downs notation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.

The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same "generation" occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call **kites**. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.

The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same "generation" occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call **kites**. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.

Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. Likewise the pentatonic kite, minus the spine, contains the EDOs that can be notated pentatonically without ups and downs.

Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs.

The diagram only shows part of the full Stern-Brocot tree~~. The cents at the top extend~~ from 0¢ (0\1) to 1200¢ (1\1). The full tree contains four pentatonic kites and six heptatonic kites. The blue kite is the 4\7 kite; the others are the 1\7, 2\7 3\7, 5\7 and 6\7 kites. The 3\7 kite is the mirror image of the 4\7 kite, 5\7 mirrors 2\7, and 6\7 mirrors 1\7. The 4\7 kite contains EDOs best notated by heptatonic notation generated by the fifth (i.e., to sharpen or augment means to add 7 fifths, octave-reduced). The octave inverse of the generator is also a generator, thus fourth-generated is equivalent to fifth-generated, and the 3\7 kite contains the exact same EDOs as the 4\7 kite. The 2\7 kite is for notation generated by thirds, and the 1\7 kite is for notation generated by seconds. Every EDO will appear on ~~two~~ of these ~~six~~ kites. This means that every ~~edo~~ has a "natural" (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.

The diagram only shows part of the full Stern-Brocot tree, which extends sideways from 0¢ (0\1) to 1200¢ (1\1). The full tree contains four pentatonic kites and six heptatonic kites. The blue kite is the 4\7 kite; the others are the 1\7, 2\7 3\7, 5\7 and 6\7 kites. The 3\7 kite is the mirror image of the 4\7 kite, 5\7 mirrors 2\7, and 6\7 mirrors 1\7. The 4\7 kite contains EDOs best notated by heptatonic notation generated by the fifth (i.e., to sharpen or augment means to add seven fifths, octave-reduced). The octave inverse of the generator is also a generator, thus fourth-generated is equivalent to fifth-generated, and the 3\7 kite contains the exact same EDOs as the 4\7 kite. The 2\7 kite is for notation generated by thirds, and the 1\7 kite is for notation generated by seconds.

Every EDO larger than 7edo will appear on only one of these three mirror-pairs of kites. The only exception is perfect EDOs, which appear on the spine of every heptatonic kite. This means that every non-perfect EDO above 7edo has a "natural" (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.

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

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22\22 = P8

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.

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

There are some alternate names. The dim7 of a dim7 chord would be three EDOsteps below a min7 = 15\22 = ^m6. 14\22 could be written as m6 or as vd7. However double-ups and double-downs are to be avoided in 22edo. In larger edos, they would be necessary. Thus 7\22 would never be written ^^m3.

0-8-13 in C is written "C" and pronounced "C" or "C major".

0-8-13 in C has C E & G, and is written "C" and pronounced "C" or "C major".

0-7-13 is written "C.v", spoken as "C downmajor" or possibly "C dot down".

0-7-13 = C Ev G is written "C.v", spoken as "C downmajor" or possibly "C dot down".

The period is needed because "Cv", spoken as "C down", is a note, not a chord.

0-6-13 is "C.^m", "C upminor"

0-6-13 = C Eb^ G is "C.^m", "C upminor"

0-5-13 is "Cm", "C minor"

0-5-13 = C Eb G is "Cm", "C minor"

The period isn't needed here because there's no ups or downs immediately after the note name.

0-8-13-18 is "C7", "C seven", a standard C7 chord with a M3 and a m7.

0-8-13-18 = C E G Bb is "C7", "C seven", a standard C7 chord with a M3 and a m7.

0-7-13-18 is "C7(v3)", "C seven, down third". The altered note or notes are in parentheses.

0-7-13-18 = C Ev G Bb is "C7(v3)", "C seven, down third". The altered note or notes are in parentheses.

0-8-13-21 on C is "CM7", "C major seven".

0-8-13-21 = C E G B is "CM7", "C major seven".

0-7-13-20 on C is "C.vM7", "C downmajor seven". The down symbol affects both the 3rd and the 7th.

0-7-13-20 = C Ev G Bv is "C.vM7", "C downmajor seven". The down symbol affects both the 3rd and the 7th.

0-8-13-21 on Cv is "Cv.M7", "C down, major seven", so one has to pronounce the period.

Often the root of a chord will not be a mid note. The root in the next two examples is Cv.

0-8-13-21 = Cv Ev Gv Bv is "Cv.M7", "C down, major seven"

To distinguish between C.vM7 and Cv.M7, one has to pronounce the period with a small pause.

0-7-13-20 = Cv Evv Gv Bvv is "Cv.vM7", "C down, downmajor seven".

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.

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"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. Likewise d3,P5 is a "dim three chord", and m3,d5 is a "dim" chord.

0-3-13 = ~~susvM2~~

0-3-13 = C Dv G = Csusv2

0-4-13 = ~~sus2~~

0-4-13 = C D G = Csus2

0-5-13 = m

0-5-13 = C Eb G = Cm

0-6-13 = ^m

0-6-13 = C Eb^ G = C.^m

0-7-13 = vM

0-7-13 = C Ev G = C.vM

0-8-13 = M

0-8-13 = C E G = C

0-9-13 = ~~sus4~~

0-9-13 = C F G = Csus4

0-10-13 = ~~sus~~^4

0-10-13 = C F^ G = Csus^4

0-5-11 = ~~dim~~(^5)

0-5-10 = C Eb Gb = Cdim

0-5-12 = m(v5)

0-5-11 = C Eb Gb^ = Cdim(^5)

0-5-12 = C Eb Gv = Cm(v5)

0-5-10-15 = ~~dim7~~

0-5-10-15 = C Eb Gb Bbb = Cdim7

0-5-11-14 = ~~dim7~~(^5,v7)

0-5-11-14 = C Eb Gb^ Bbbv = Cdim7(^5,v7)

0-6-11-15 = ~~dim7~~(^3,^5)

0-6-11-15 = C Eb^ Gb^ Bbb = Cdim7(^3,^5)

0-6-11-16 = ^dim7 (the up symbol applies to m3, d5 and d7)

0-6-11-16 = C Eb^ Gb^ Bbb^ = C.^dim7 (the up symbol applies to m3, d5 and d7)

0-5-13-17 = m6

0-5-13-17 = C Eb G A = Cm6

0-6-12-15 = m6(^3,v5,vv6) or ~~dim7~~(^3,^^5~~) (doubled ups/downs are unavoidable~~)

Sometimes doubled ups/downs are unavoidable:

0-7-13-16 = vM6

0-6-12-15 = C Eb^ Gv Avv = Cm6(^3,v5,vv6), or C Eb^ Gb^^ Bbb = Cdim7(^3,^^5)

0-8-13-17 = 6

0-7-13-16 = C Ev G Av = C.vM6 (the down symbol applies to both the 3rd and the 6th)

0-8-13-17 = C E G A = C6

0-7-13-16 = C Ev G Av = C.v6

0-5-13-18 = m7

0-5-13-18 = C Eb G Bb = Cm7

0-6-13-19 = ^m7

0-6-13-19 = C Eb^ G Bb^ = C.^m7

0-7-13-20 = vM7

0-7-13-20 = C Ev G Bv = C.vM7

0-8-13-21 = M7

0-8-13-21 = C E G B = CM7

0-5-13-16 = m6(v6)

0-5-13-16 = C Eb G Av = Cm6(v6)

0-8-13-19 = 7(^7)

0-8-13-19 = C E G Bb^ = C7(^7)

0-7-13-18-26 = 9(v3)

0-7-13-18-26 = C Ev G Bb D = C9(v3)

0-7-13-18-26-32 = 11(v3,^11)

0-7-13-18-26-32 = C Ev G Bb D F^ = C11(v3,^11)

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:

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(no5) = "G downmajor seven, no five"

Eb^.v(add9) = "E flat up, downmajor, add nine"

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A7(v3) = "A seven, down three"

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.

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

I

^I or bII

v#I or ^bII

#I or vII

II

^II or bIII

v#II or ^bIII

#II or vIII

III

IV

^IV or bV

v#IV or ^bV

#IV or vV

V

^V or bVI

v#V or ^bVI

#V or vVI

VI

^VI or bVII

v#VI or ^bVII

#VI or vVII

VII or vI

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

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

Here's the Tibia chords. Periods are never needed after the root in relative notation because ups and downs are always leading, never trailing.

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

^bVIv(add9) = "up-flat six downmajor, add nine"

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==__Rank-2 Notation__==

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

For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. For ~~esxample~~, meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the 5th's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).

For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. I.e. the framework's node must not be on the spine of a kite. For example, meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the 5th's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).

~~note to self: if K(#)~~ is 1 or -1, ups and downs aren't needed to notate rank-2.

All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone.

If the sharp's keyspan is 1 or -1, ups and downs aren't needed to notate rank-2. The keyspan is zero only for perfect EDOs. Thus we can omit fourthward frameworks, and assume for this discussion that K(#) > 1. We need only consider sweet frameworks, excluding those that lie on the side of the heptatonic kite and those that lie on the spine of any kite.

To extend ups and downs to rank-2 tunings, the up symbol is given not only a **keyspan** (always +1) but also a **genspan**, which indicates how many steps forward or backwards along the generator chain, or **genchain**, one must travel to find the interval. For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. The interval is always a 2nd. The genspan is calculated from the keyspans:

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For 17-tone, X = 2, i = 1, G(^) = -5, and ^ = min 2nd

For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd.</pre></div>

For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd.

||= 5edo ||= pentatonic ||= || || ||

||= 7edo ||= perfect ||= || || ||

||= 8edo ||= fifthless ||= || || ||

||= 11edo ||= fifthless ||= || || ||

||= 13edo ||= fifthless ||= || || ||

||= 17edo ||= sweet ||= || -5 || min 2nd ||

||= 18edo ||= fifthless ||= || || ||

||= 19edo ||= sweet ||= || || ||

||= 22edo ||= sweet ||= || -5 || min 2nd ||

||= 26edo ||= sweet ||= || || ||

||= 27edo ||= sweet ||= || || ||

||= 29edo ||= sweet ||= || || ||

||= 31edo ||= sweet ||= || -12 || dim 2nd ||

||= 32edo ||= sweet ||= || || ||

||= 33edo ||= sweet ||= || || ||

||= 34edo ||= sweet ||= || || ||

||= 37edo ||= sweet ||= || || ||

||= 38edo ||= sweet ||= || || ||

||= 39edo ||= sweet ||= || || ||

||= 40edo ||= sweet ||= || || ||

||= 41edo ||= sweet ||= || || ||

||= 42edo ||= sweet ||= || || ||

||= 43edo ||= sweet ||= || || ||

||= 44ddo ||= sweet ||= || || ||

||= 45edo ||= sweet ||= || || ||

||= 46edo ||= sweet ||= || || ||

||= 47edo ||= sweet ||= || || ||

||= 49edo ||= sweet ||= || || ||

||= 50edo ||= sweet ||= || || ||

||= 51edo ||= sweet ||= || || ||

||= 52edo ||= sweet ||= || || ||

||= 53edo ||= sweet ||= || || ||</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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fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢

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

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

This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.

The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call kites. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.

The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call kites. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.

Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. Likewise the pentatonic kite, minus the spine, contains the EDOs that can be notated pentatonically without ups and downs.

Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs.

The diagram only shows part of the full Stern-Brocot tree~~. The cents at the top extend~~ from 0¢ (0\1) to 1200¢ (1\1). The full tree contains four pentatonic kites and six heptatonic kites. The blue kite is the 4\7 kite; the others are the 1\7, 2\7 3\7, 5\7 and 6\7 kites. The 3\7 kite is the mirror image of the 4\7 kite, 5\7 mirrors 2\7, and 6\7 mirrors 1\7. The 4\7 kite contains EDOs best notated by heptatonic notation generated by the fifth (i.e., to sharpen or augment means to add 7 fifths, octave-reduced). The octave inverse of the generator is also a generator, thus fourth-generated is equivalent to fifth-generated, and the 3\7 kite contains the exact same EDOs as the 4\7 kite. The 2\7 kite is for notation generated by thirds, and the 1\7 kite is for notation generated by seconds. Every EDO will appear on ~~two~~ of these ~~six~~ kites. This means that every ~~edo~~ has a &quot;natural&quot; (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.

The diagram only shows part of the full Stern-Brocot tree, which extends sideways from 0¢ (0\1) to 1200¢ (1\1). The full tree contains four pentatonic kites and six heptatonic kites. The blue kite is the 4\7 kite; the others are the 1\7, 2\7 3\7, 5\7 and 6\7 kites. The 3\7 kite is the mirror image of the 4\7 kite, 5\7 mirrors 2\7, and 6\7 mirrors 1\7. The 4\7 kite contains EDOs best notated by heptatonic notation generated by the fifth (i.e., to sharpen or augment means to add seven fifths, octave-reduced). The octave inverse of the generator is also a generator, thus fourth-generated is equivalent to fifth-generated, and the 3\7 kite contains the exact same EDOs as the 4\7 kite. The 2\7 kite is for notation generated by thirds, and the 1\7 kite is for notation generated by seconds.

Every EDO larger than 7edo will appear on only one of these three mirror-pairs of kites. The only exception is perfect EDOs, which appear on the spine of every heptatonic kite. This means that every non-perfect EDO above 7edo has a &quot;natural&quot; (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.

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

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22\22 = P8

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.

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-fourth&quot; and vP5 = &quot;down-fifth&quot;. In larger edos there may be &quot;down-octave&quot;, &quot;up-unison&quot;, etc.

There are some alternate names. The dim7 of a dim7 chord would be three EDOsteps below a min7 = 15\22 = ^m6. 14\22 could be written as m6 or as vd7. However double-ups and double-downs are to be avoided in 22edo. In larger edos, they would be necessary. Thus 7\22 would never be written ^^m3.

0-8-13 in C is written &quot;C&quot; and pronounced &quot;C&quot; or &quot;C major&quot;.

0-8-13 in C has C E &amp; G, and is written &quot;C&quot; and pronounced &quot;C&quot; or &quot;C major&quot;.

0-7-13 is written &quot;C.v&quot;, spoken as &quot;C downmajor&quot; or possibly &quot;C dot down&quot;.

0-7-13 = C Ev G is written &quot;C.v&quot;, spoken as &quot;C downmajor&quot; or possibly &quot;C dot down&quot;.

The period is needed because &quot;Cv&quot;, spoken as &quot;C down&quot;, is a note, not a chord.

0-6-13 is &quot;C.^m&quot;, &quot;C upminor&quot;

0-6-13 = C Eb^ G is &quot;C.^m&quot;, &quot;C upminor&quot;

0-5-13 is &quot;Cm&quot;, &quot;C minor&quot;

0-5-13 = C Eb G is &quot;Cm&quot;, &quot;C minor&quot;

The period isn't needed here because there's no ups or downs immediately after the note name.

0-8-13-18 is &quot;C7&quot;, &quot;C seven&quot;, a standard C7 chord with a M3 and a m7.

0-8-13-18 = C E G Bb is &quot;C7&quot;, &quot;C seven&quot;, a standard C7 chord with a M3 and a m7.

0-7-13-18 is &quot;C7(v3)&quot;, &quot;C seven, down third&quot;. The altered note or notes are in parentheses.

0-7-13-18 = C Ev G Bb is &quot;C7(v3)&quot;, &quot;C seven, down third&quot;. The altered note or notes are in parentheses.

0-8-13-21 on C is &quot;CM7&quot;, &quot;C major seven&quot;.

0-8-13-21 = C E G B is &quot;CM7&quot;, &quot;C major seven&quot;.

0-7-13-20 on C is &quot;C.vM7&quot;, &quot;C downmajor seven&quot;. The down symbol affects both the 3rd and the 7th.

0-7-13-20 = C Ev G Bv is &quot;C.vM7&quot;, &quot;C downmajor seven&quot;. The down symbol affects both the 3rd and the 7th.

0-8-13-21 on Cv is &quot;Cv.M7&quot;, &quot;C down, major seven&quot;, so one has to pronounce the period.

Often the root of a chord will not be a mid note. The root in the next two examples is Cv.

0-8-13-21 = Cv Ev Gv Bv is &quot;Cv.M7&quot;, &quot;C down, major seven&quot;

To distinguish between C.vM7 and Cv.M7, one has to pronounce the period with a small pause.

0-7-13-20 = Cv Evv Gv Bvv is &quot;Cv.vM7&quot;, &quot;C down, downmajor seven&quot;.

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.

Line 1,017:

Line 1,087:

&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. Likewise d3,P5 is a &quot;dim three chord&quot;, and m3,d5 is a &quot;dim&quot; chord.

0-3-13 = ~~susvM2~~

0-3-13 = C Dv G = Csusv2

0-4-13 = ~~sus2~~

0-4-13 = C D G = Csus2

0-5-13 = m

0-5-13 = C Eb G = Cm

0-6-13 = ^m

0-6-13 = C Eb^ G = C.^m

0-7-13 = vM

0-7-13 = C Ev G = C.vM

0-8-13 = M

0-8-13 = C E G = C

0-9-13 = ~~sus4~~

0-9-13 = C F G = Csus4

0-10-13 = ~~sus~~^4

0-10-13 = C F^ G = Csus^4

0-5-11 = ~~dim~~(^5)

0-5-10 = C Eb Gb = Cdim

0-5-12 = m(v5)

0-5-11 = C Eb Gb^ = Cdim(^5)

0-5-12 = C Eb Gv = Cm(v5)

0-5-10-15 = ~~dim7~~

0-5-10-15 = C Eb Gb Bbb = Cdim7

0-5-11-14 = ~~dim7~~(^5,v7)

0-5-11-14 = C Eb Gb^ Bbbv = Cdim7(^5,v7)

0-6-11-15 = ~~dim7~~(^3,^5)

0-6-11-15 = C Eb^ Gb^ Bbb = Cdim7(^3,^5)

0-6-11-16 = ^dim7 (the up symbol applies to m3, d5 and d7)

0-6-11-16 = C Eb^ Gb^ Bbb^ = C.^dim7 (the up symbol applies to m3, d5 and d7)

0-5-13-17 = m6

0-5-13-17 = C Eb G A = Cm6

0-6-12-15 = m6(^3,v5,vv6) or ~~dim7~~(^3,^^5~~) (doubled ups/downs are unavoidable~~)

Sometimes doubled ups/downs are unavoidable:

0-7-13-16 = vM6

0-6-12-15 = C Eb^ Gv Avv = Cm6(^3,v5,vv6), or C Eb^ Gb^^ Bbb = Cdim7(^3,^^5)

0-8-13-17 = 6

0-7-13-16 = C Ev G Av = C.vM6 (the down symbol applies to both the 3rd and the 6th)

0-8-13-17 = C E G A = C6

0-7-13-16 = C Ev G Av = C.v6

0-5-13-18 = m7

0-5-13-18 = C Eb G Bb = Cm7

0-6-13-19 = ^m7

0-6-13-19 = C Eb^ G Bb^ = C.^m7

0-7-13-20 = vM7

0-7-13-20 = C Ev G Bv = C.vM7

0-8-13-21 = M7

0-8-13-21 = C E G B = CM7

0-5-13-16 = m6(v6)

0-5-13-16 = C Eb G Av = Cm6(v6)

0-8-13-19 = 7(^7)

0-8-13-19 = C E G Bb^ = C7(^7)

0-7-13-18-26 = 9(v3)

0-7-13-18-26 = C Ev G Bb D = C9(v3)

0-7-13-18-26-32 = 11(v3,^11)

0-7-13-18-26-32 = C Ev G Bb D F^ = C11(v3,^11)

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:

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 &quot;Tibia&quot;:

G.vM7(no5) = &quot;G downmajor seven, no five&quot;

Eb^.v(add9) = &quot;E flat up, downmajor, add nine&quot;

Line 1,055:

Line 1,128:

A7(v3) = &quot;A seven, down three&quot;

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.

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

I

^I or bII

v#I or ^bII

#I or vII

II

^II or bIII

v#II or ^bIII

#II or vIII

III

IV

^IV or bV

v#IV or ^bV

#IV or vV

V

^V or bVI

v#V or ^bVI

#V or vVI

VI

^VI or bVII

v#VI or ^bVII

#VI or vVII

VII or vI

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

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

Here's the Tibia chords. Periods are never needed after the root in relative notation because ups and downs are always leading, never trailing.

IvM7(no5) = &quot;one downmajor seven, no five&quot;

^bVIv(add9) = &quot;up-flat six downmajor, add nine&quot;

Line 1,066:

Line 1,161:

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

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

<h2 id="toc3"><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /></h2>

<h2 id="toc3"><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /></h2>

<h2 id="toc4"> </h2>

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

Line 2,846:

Line 2,941:

<h2 id="toc18"><a name="Summary of EDO notation-Rank-2 Notation"></a>Rank-2 Notation</h2>

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

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

For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. I.e. the framework's node must not be on the spine of a kite. For example, meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the 5th's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).

~~For~~ rank-2 ~~scales to work with a given framework~~, ~~the keyspans of the generator and the period must be coprime. For esxample, meantone and pythagorean are compatible with 12~~-tone ~~because the fifth's keyspan is 7,~~ and 7 ~~is coprime with 12. But neither are compatible with 15edo, because the 5th's keyspan is 9. Likewise a third~~-~~generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24)~~.

All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone.

~~note to self: if K(#)~~ is 1 or -1, ups and downs aren't needed to notate rank-2.

If the sharp's keyspan is 1 or -1, ups and downs aren't needed to notate rank-2. The keyspan is zero only for perfect EDOs. Thus we can omit fourthward frameworks, and assume for this discussion that K(#) &gt; 1. We need only consider sweet frameworks, excluding those that lie on the side of the heptatonic kite and those that lie on the spine of any kite.

To extend ups and downs to rank-2 tunings, the up symbol is given not only a keyspan (always +1) but also a genspan, which indicates how many steps forward or backwards along the generator chain, or genchain, one must travel to find the interval. For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. The interval is always a 2nd. The genspan is calculated from the keyspans:

Line 2,932:

Line 3,029:

For 17-tone, X = 2, i = 1, G(^) = -5, and ^ = min 2nd

For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd.</body></html></pre></div>

For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd.

<tr>

<td style="text-align: center;">5edo

</td>

<td style="text-align: center;">pentatonic

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">7edo

</td>

<td style="text-align: center;">perfect

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">8edo

</td>

<td style="text-align: center;">fifthless

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">11edo

</td>

<td style="text-align: center;">fifthless

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">13edo

</td>

<td style="text-align: center;">fifthless

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">17edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>-5

</td>

<td>min 2nd

</td>

</tr>

<tr>

<td style="text-align: center;">18edo

</td>

<td style="text-align: center;">fifthless

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">19edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">22edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>-5

</td>

<td>min 2nd

</td>

</tr>

<tr>

<td style="text-align: center;">26edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">27edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">29edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">31edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>-12

</td>

<td>dim 2nd

</td>

</tr>

<tr>

<td style="text-align: center;">32edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">33edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">34edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">37edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">38edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">39edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">40edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">41edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">42edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">43edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">44ddo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">45edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">46edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">47edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">49edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">50edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">51edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">52edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">53edo

</td>

<td style="text-align: center;">sweet

</td>

</td>

<td>

</td>

<td>

</td>

</tr>

</table>

</body></html></pre></div>

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-03 16:42:16 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-03 20:06:33 UTC</tt>.<br>
-: The original revision id was <tt>584772157</tt>.<br>
+: The original revision id was <tt>584779483</tt>.<br>
 : The revision comment was: <tt></tt><br>
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 This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.
-The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same "generation" occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call **kites**. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.
+The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same "generation" occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call **kites**. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.
-Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. Likewise the pentatonic kite, minus the spine, contains the EDOs that can be notated pentatonically without ups and downs.
+Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs.
-The diagram only shows part of the full Stern-Brocot tree. The cents at the top extend from 0¢ (0\1) to 1200¢ (1\1). The full tree contains four pentatonic kites and six heptatonic kites. The blue kite is the 4\7 kite; the others are the 1\7, 2\7 3\7, 5\7 and 6\7 kites. The 3\7 kite is the mirror image of the 4\7 kite, 5\7 mirrors 2\7, and 6\7 mirrors 1\7. The 4\7 kite contains EDOs best notated by heptatonic notation generated by the fifth (i.e., to sharpen or augment means to add 7 fifths, octave-reduced). The octave inverse of the generator is also a generator, thus fourth-generated is equivalent to fifth-generated, and the 3\7 kite contains the exact same EDOs as the 4\7 kite. The 2\7 kite is for notation generated by thirds, and the 1\7 kite is for notation generated by seconds. Every EDO will appear on two of these six kites. This means that every edo has a "natural" (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.
+The diagram only shows part of the full Stern-Brocot tree, which extends sideways from 0¢ (0\1) to 1200¢ (1\1). The full tree contains four pentatonic kites and six heptatonic kites. The blue kite is the 4\7 kite; the others are the 1\7, 2\7 3\7, 5\7 and 6\7 kites. The 3\7 kite is the mirror image of the 4\7 kite, 5\7 mirrors 2\7, and 6\7 mirrors 1\7. The 4\7 kite contains EDOs best notated by heptatonic notation generated by the fifth (i.e., to sharpen or augment means to add seven fifths, octave-reduced). The octave inverse of the generator is also a generator, thus fourth-generated is equivalent to fifth-generated, and the 3\7 kite contains the exact same EDOs as the 4\7 kite. The 2\7 kite is for notation generated by thirds, and the 1\7 kite is for notation generated by seconds.
+Every EDO larger than 7edo will appear on only one of these three mirror-pairs of kites. The only exception is perfect EDOs, which appear on the spine of every heptatonic kite. This means that every non-perfect EDO above 7edo has a "natural" (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.
 This section will cover sweet EDOs and the other categories will be covered in later sections.
@@ Line 211: / Line 213: @@
 \22 = P8
-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.
+These are pronounced "downmajor second", "upminor third", etc. For 4ths and 5ths, "perfect" is implied and can be omitted: ^P4 = "up-fourth" and vP5 = "down-fifth". In larger edos there may be "down-octave", "up-unison", etc.
 There are some alternate names. The dim7 of a dim7 chord would be three EDOsteps below a min7 = 15\22 = ^m6. 14\22 could be written as m6 or as vd7. However double-ups and double-downs are to be avoided in 22edo. In larger edos, they would be necessary. Thus 7\22 would never be written ^^m3.
--8-13 in C is written "C" and pronounced "C" or "C major".
+-8-13 in C has C E &amp; G, and is written "C" and pronounced "C" or "C major".
--7-13 is written "C.v", spoken as "C downmajor" or possibly "C dot down".
+-7-13 = C Ev G is written "C.v", spoken as "C downmajor" or possibly "C dot down".
 The period is needed because "Cv", spoken as "C down", is a note, not a chord.
--6-13 is "C.^m", "C upminor"
+-6-13 = C Eb^ G is "C.^m", "C upminor"
--5-13 is "Cm", "C minor"
+-5-13 = C Eb G is "Cm", "C minor"
 The period isn't needed here because there's no ups or downs immediately after the note name.
--8-13-18 is "C7", "C seven", a standard C7 chord with a M3 and a m7.
+-8-13-18 = C E G Bb is "C7", "C seven", a standard C7 chord with a M3 and a m7.
--7-13-18 is "C7(v3)", "C seven, down third". The altered note or notes are in parentheses.
+-7-13-18 = C Ev G Bb is "C7(v3)", "C seven, down third". The altered note or notes are in parentheses.
--8-13-21 on C is "CM7", "C major seven".
+-8-13-21 = C E G B is "CM7", "C major seven".
--7-13-20 on C is "C.vM7", "C downmajor seven". The down symbol affects both the 3rd and the 7th.
+-7-13-20 = C Ev G Bv is "C.vM7", "C downmajor seven". The down symbol affects both the 3rd and the 7th.
--8-13-21 on Cv is "Cv.M7", "C down, major seven", so one has to pronounce the period.
+Often the root of a chord will not be a mid note. The root in the next two examples is Cv.
+-8-13-21 = Cv Ev Gv Bv is "Cv.M7", "C down, major seven"
+To distinguish between C.vM7 and Cv.M7, one has to pronounce the period with a small pause.
+-7-13-20 = Cv Evv Gv Bvv is "Cv.vM7", "C down, downmajor seven".
 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.
@@ Line 233: / Line 238: @@
 "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. Likewise d3,P5 is a "dim three chord", and m3,d5 is a "dim" chord.
--3-13 = susvM2
+-3-13 = C Dv G = Csusv2
--4-13 = sus2
+-4-13 = C D G = Csus2
--5-13 = m
+-5-13 = C Eb G = Cm
--6-13 = ^m
+-6-13 = C Eb^ G = C.^m
--7-13 = vM
+-7-13 = C Ev G = C.vM
--8-13 = M
+-8-13 = C E G = C
--9-13 = sus4
+-9-13 = C F G = Csus4
--10-13 = sus^4
+-10-13 = C F^ G = Csus^4
--5-11 = dim(^5)
+-5-10 = C Eb Gb = Cdim
--5-12 = m(v5)
+-5-11 = C Eb Gb^ = Cdim(^5)
+-5-12 = C Eb Gv = Cm(v5)
--5-10-15 = dim7
+-5-10-15 = C Eb Gb Bbb = Cdim7
--5-11-14 = dim7(^5,v7)
+-5-11-14 = C Eb Gb^ Bbbv = Cdim7(^5,v7)
--6-11-15 = dim7(^3,^5)
+-6-11-15 = C Eb^ Gb^ Bbb = Cdim7(^3,^5)
--6-11-16 = ^dim7 (the up symbol applies to m3, d5 and d7)
+-6-11-16 = C Eb^ Gb^ Bbb^ = C.^dim7 (the up symbol applies to m3, d5 and d7)
--5-13-17 = m6
+-5-13-17 = C Eb G A = Cm6
--6-12-15 = m6(^3,v5,vv6) or dim7(^3,^^5) (doubled ups/downs are unavoidable)
+Sometimes doubled ups/downs are unavoidable:
--7-13-16 = vM6
+-6-12-15 = C Eb^ Gv Avv = Cm6(^3,v5,vv6), or C Eb^ Gb^^ Bbb = Cdim7(^3,^^5)
--8-13-17 = 6
+-7-13-16 = C Ev G Av = C.vM6 (the down symbol applies to both the 3rd and the 6th)
+-8-13-17 = C E G A = C6
+-7-13-16 = C Ev G Av = C.v6
--5-13-18 = m7
+-5-13-18 = C Eb G Bb = Cm7
--6-13-19 = ^m7
+-6-13-19 = C Eb^ G Bb^ = C.^m7
--7-13-20 = vM7
+-7-13-20 = C Ev G Bv = C.vM7
--8-13-21 = M7
+-8-13-21 = C E G B = CM7
--5-13-16 = m6(v6)
+-5-13-16 = C Eb G Av = Cm6(v6)
--8-13-19 = 7(^7)
+-8-13-19 = C E G Bb^ = C7(^7)
--7-13-18-26 = 9(v3)
+-7-13-18-26 = C Ev G Bb D = C9(v3)
--7-13-18-26-32 = 11(v3,^11)
+-7-13-18-26-32 = C Ev G Bb D F^ = C11(v3,^11)
-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:
+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(no5) = "G downmajor seven, no five"
 Eb^.v(add9) = "E flat up, downmajor, add nine"
@@ Line 271: / Line 279: @@
 A7(v3) = "A seven, down three"
-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.
+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 might be 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 V ^V/bVI v#V/^bVI #V/vVI VI ^VI/bVII v#VI/^bVII #VI/vVII VII/vI
+I
+^I or bII
+v#I or ^bII
+#I or vII
+II
+^II or bIII
+v#II or ^bIII
+#II or vIII
+III
+IV
+^IV or bV
+v#IV or ^bV
+#IV or vV
+V
+^V or bVI
+v#V or ^bVI
+#V or vVI
+VI
+^VI or bVII
+v#VI or ^bVII
+#VI or vVII
+VII or vI
 These are pronounced "down-two", "up-flat-three", "down-sharp-four", etc.
-Here's the Tibia chords. No period is ever needed after the root because ups and downs are always leading, never trailing.
+Here's the Tibia chords. Periods are never needed after the root in relative notation because ups and downs are always leading, never trailing.
 IvM7(no5) = "one downmajor seven, no five"
 ^bVIv(add9) = "up-flat six downmajor, add nine"
@@ Line 761: / Line 791: @@
 ==__Rank-2 Notation__==
 Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)
-For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. For esxample, meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the 5th's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).
+For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. I.e. the framework's node must not be on the spine of a kite. For example, meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the 5th's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).
-note to self: if K(#) is 1 or -1, ups and downs aren't needed to notate rank-2.
+All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone.
+If the sharp's keyspan is 1 or -1, ups and downs aren't needed to notate rank-2. The keyspan is zero only for perfect EDOs. Thus we can omit fourthward frameworks, and assume for this discussion that K(#) &gt; 1. We need only consider sweet frameworks, excluding those that lie on the side of the heptatonic kite and those that lie on the spine of any kite.
 To extend ups and downs to rank-2 tunings, the up symbol is given not only a **keyspan** (always +1) but also a **genspan**, which indicates how many steps forward or backwards along the generator chain, or **genchain**, one must travel to find the interval. For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. The interval is always a 2nd. The genspan is calculated from the keyspans:
@@ Line 788: / Line 820: @@
 For 17-tone, X = 2, i = 1, G(^) = -5, and ^ = min 2nd
-For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd.</pre></div>
+For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd.
+||= 5edo ||= pentatonic ||=   ||   ||   ||
+||= 7edo ||= perfect ||=   ||   ||   ||
+||= 8edo ||= fifthless ||=   ||   ||   ||
+||= 11edo ||= fifthless ||=   ||   ||   ||
+||= 13edo ||= fifthless ||=   ||   ||   ||
+||= 17edo ||= sweet ||=   || -5 || min 2nd ||
+||= 18edo ||= fifthless ||=   ||   ||   ||
+||= 19edo ||= sweet ||=   ||   ||   ||
+||= 22edo ||= sweet ||=   || -5 || min 2nd ||
+||= 26edo ||= sweet ||=   ||   ||   ||
+||= 27edo ||= sweet ||=   ||   ||   ||
+||= 29edo ||= sweet ||=   ||   ||   ||
+||= 31edo ||= sweet ||=   || -12 || dim 2nd ||
+||= 32edo ||= sweet ||=   ||   ||   ||
+||= 33edo ||= sweet ||=   ||   ||   ||
+||= 34edo ||= sweet ||=   ||   ||   ||
+||= 37edo ||= sweet ||=   ||   ||   ||
+||= 38edo ||= sweet ||=   ||   ||   ||
+||= 39edo ||= sweet ||=   ||   ||   ||
+||= 40edo ||= sweet ||=   ||   ||   ||
+||= 41edo ||= sweet ||=   ||   ||   ||
+||= 42edo ||= sweet ||=   ||   ||   ||
+||= 43edo ||= sweet ||=   ||   ||   ||
+||= 44ddo ||= sweet ||=   ||   ||   ||
+||= 45edo ||= sweet ||=   ||   ||   ||
+||= 46edo ||= sweet ||=   ||   ||   ||
+||= 47edo ||= sweet ||=   ||   ||   ||
+||= 49edo ||= sweet ||=   ||   ||   ||
+||= 50edo ||= sweet ||=   ||   ||   ||
+||= 51edo ||= sweet ||=   ||   ||   ||
+||= 52edo ||= sweet ||=   ||   ||   ||
+||= 53edo ||= sweet ||=   ||   ||   ||</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 880: / Line 945: @@
 fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:1442:&amp;lt;img src=&amp;quot;/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1035px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png" alt="The fifth of EDOs 5-53.png" title="The fifth of EDOs 5-53.png" style="height: 1035px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1442 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:1828:&amp;lt;img src=&amp;quot;/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1035px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png" alt="The fifth of EDOs 5-53.png" title="The fifth of EDOs 5-53.png" style="height: 1035px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1828 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.&lt;br /&gt;
 &lt;br /&gt;
-The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &amp;quot;generation&amp;quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call &lt;strong&gt;kites&lt;/strong&gt;. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.&lt;br /&gt;
+The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &amp;quot;generation&amp;quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call &lt;strong&gt;kites&lt;/strong&gt;. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.&lt;br /&gt;
 &lt;br /&gt;
-Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. Likewise the pentatonic kite, minus the spine, contains the EDOs that can be notated pentatonically without ups and downs.&lt;br /&gt;
+Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs.&lt;br /&gt;
 &lt;br /&gt;
-The diagram only shows part of the full Stern-Brocot tree. The cents at the top extend from 0¢ (0\1) to 1200¢ (1\1). The full tree contains four pentatonic kites and six heptatonic kites. The blue kite is the 4\7 kite; the others are the 1\7, 2\7 3\7, 5\7 and 6\7 kites. The 3\7 kite is the mirror image of the 4\7 kite, 5\7 mirrors 2\7, and 6\7 mirrors 1\7. The 4\7 kite contains EDOs best notated by heptatonic notation generated by the fifth (i.e., to sharpen or augment means to add 7 fifths, octave-reduced). The octave inverse of the generator is also a generator, thus fourth-generated is equivalent to fifth-generated, and the 3\7 kite contains the exact same EDOs as the 4\7 kite. The 2\7 kite is for notation generated by thirds, and the 1\7 kite is for notation generated by seconds. Every EDO will appear on two of these six kites. This means that every edo has a &amp;quot;natural&amp;quot; (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.&lt;br /&gt;
+The diagram only shows part of the full Stern-Brocot tree, which extends sideways from 0¢ (0\1) to 1200¢ (1\1). The full tree contains four pentatonic kites and six heptatonic kites. The blue kite is the 4\7 kite; the others are the 1\7, 2\7 3\7, 5\7 and 6\7 kites. The 3\7 kite is the mirror image of the 4\7 kite, 5\7 mirrors 2\7, and 6\7 mirrors 1\7. The 4\7 kite contains EDOs best notated by heptatonic notation generated by the fifth (i.e., to sharpen or augment means to add seven fifths, octave-reduced). The octave inverse of the generator is also a generator, thus fourth-generated is equivalent to fifth-generated, and the 3\7 kite contains the exact same EDOs as the 4\7 kite. The 2\7 kite is for notation generated by thirds, and the 1\7 kite is for notation generated by seconds.&lt;br /&gt;
+&lt;br /&gt;
+Every EDO larger than 7edo will appear on only one of these three mirror-pairs of kites. The only exception is perfect EDOs, which appear on the spine of every heptatonic kite. This means that every non-perfect EDO above 7edo has a &amp;quot;natural&amp;quot; (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.&lt;br /&gt;
 &lt;br /&gt;
 This section will cover sweet EDOs and the other categories will be covered in later sections.&lt;br /&gt;
@@ Line 995: / Line 1,062: @@
 \22 = P8&lt;br /&gt;
 &lt;br /&gt;
-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;
+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-fourth&amp;quot; and vP5 = &amp;quot;down-fifth&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;
 There are some alternate names. The dim7 of a dim7 chord would be three EDOsteps below a min7 = 15\22 = ^m6. 14\22 could be written as m6 or as vd7. However double-ups and double-downs are to be avoided in 22edo. In larger edos, they would be necessary. Thus 7\22 would never be written ^^m3.&lt;br /&gt;
 &lt;br /&gt;
--8-13 in C is written &amp;quot;C&amp;quot; and pronounced &amp;quot;C&amp;quot; or &amp;quot;C major&amp;quot;.&lt;br /&gt;
+-8-13 in C has C E &amp;amp; G, and is written &amp;quot;C&amp;quot; and pronounced &amp;quot;C&amp;quot; or &amp;quot;C major&amp;quot;.&lt;br /&gt;
--7-13 is written &amp;quot;C.v&amp;quot;, spoken as &amp;quot;C downmajor&amp;quot; or possibly &amp;quot;C dot down&amp;quot;.&lt;br /&gt;
+-7-13 = C Ev G is written &amp;quot;C.v&amp;quot;, spoken as &amp;quot;C downmajor&amp;quot; or possibly &amp;quot;C dot down&amp;quot;.&lt;br /&gt;
 The period is needed because &amp;quot;Cv&amp;quot;, spoken as &amp;quot;C down&amp;quot;, is a note, not a chord.&lt;br /&gt;
--6-13 is &amp;quot;C.^m&amp;quot;, &amp;quot;C upminor&amp;quot;&lt;br /&gt;
+-6-13 = C Eb^ G is &amp;quot;C.^m&amp;quot;, &amp;quot;C upminor&amp;quot;&lt;br /&gt;
--5-13 is &amp;quot;Cm&amp;quot;, &amp;quot;C minor&amp;quot;&lt;br /&gt;
+-5-13 = C Eb G is &amp;quot;Cm&amp;quot;, &amp;quot;C minor&amp;quot;&lt;br /&gt;
 The period isn't needed here because there's no ups or downs immediately after the note name.&lt;br /&gt;
 &lt;br /&gt;
--8-13-18 is &amp;quot;C7&amp;quot;, &amp;quot;C seven&amp;quot;, a standard C7 chord with a M3 and a m7.&lt;br /&gt;
+-8-13-18 = C E G Bb is &amp;quot;C7&amp;quot;, &amp;quot;C seven&amp;quot;, a standard C7 chord with a M3 and a m7.&lt;br /&gt;
--7-13-18 is &amp;quot;C7(v3)&amp;quot;, &amp;quot;C seven, down third&amp;quot;. The altered note or notes are in parentheses.&lt;br /&gt;
+-7-13-18 = C Ev G Bb is &amp;quot;C7(v3)&amp;quot;, &amp;quot;C seven, down third&amp;quot;. The altered note or notes are in parentheses.&lt;br /&gt;
 &lt;br /&gt;
--8-13-21 on C is &amp;quot;CM7&amp;quot;, &amp;quot;C major seven&amp;quot;.&lt;br /&gt;
+-8-13-21 = C E G B is &amp;quot;CM7&amp;quot;, &amp;quot;C major seven&amp;quot;.&lt;br /&gt;
--7-13-20 on C is &amp;quot;C.vM7&amp;quot;, &amp;quot;C downmajor seven&amp;quot;. The down symbol affects both the 3rd and the 7th.&lt;br /&gt;
+-7-13-20 = C Ev G Bv is &amp;quot;C.vM7&amp;quot;, &amp;quot;C downmajor seven&amp;quot;. The down symbol affects both the 3rd and the 7th.&lt;br /&gt;
--8-13-21 on Cv is &amp;quot;Cv.M7&amp;quot;, &amp;quot;C down, major seven&amp;quot;, so one has to pronounce the period.&lt;br /&gt;
+Often the root of a chord will not be a mid note. The root in the next two examples is Cv.&lt;br /&gt;
+-8-13-21 = Cv Ev Gv Bv is &amp;quot;Cv.M7&amp;quot;, &amp;quot;C down, major seven&amp;quot;&lt;br /&gt;
+To distinguish between C.vM7 and Cv.M7, one has to pronounce the period with a small pause.&lt;br /&gt;
+-7-13-20 = Cv Evv Gv Bvv is &amp;quot;Cv.vM7&amp;quot;, &amp;quot;C down, downmajor seven&amp;quot;.&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;
@@ Line 1,017: / Line 1,087: @@
 &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. Likewise d3,P5 is a &amp;quot;dim three chord&amp;quot;, and m3,d5 is a &amp;quot;dim&amp;quot; chord.&lt;br /&gt;
 &lt;br /&gt;
--3-13 = susvM2&lt;br /&gt;
+-3-13 = C Dv G = Csusv2&lt;br /&gt;
--4-13 = sus2&lt;br /&gt;
+-4-13 = C D G = Csus2&lt;br /&gt;
--5-13 = m&lt;br /&gt;
+-5-13 = C Eb G = Cm&lt;br /&gt;
--6-13 = ^m&lt;br /&gt;
+-6-13 = C Eb^ G = C.^m&lt;br /&gt;
--7-13 = vM&lt;br /&gt;
+-7-13 = C Ev G = C.vM&lt;br /&gt;
--8-13 = M&lt;br /&gt;
+-8-13 = C E G = C&lt;br /&gt;
--9-13 = sus4&lt;br /&gt;
+-9-13 = C F G = Csus4&lt;br /&gt;
--10-13 = sus^4&lt;br /&gt;
+-10-13 = C F^ G = Csus^4&lt;br /&gt;
 &lt;br /&gt;
--5-11 = dim(^5)&lt;br /&gt;
+-5-10 = C Eb Gb = Cdim&lt;br /&gt;
--5-12 = m(v5)&lt;br /&gt;
+-5-11 = C Eb Gb^ = Cdim(^5)&lt;br /&gt;
+-5-12 = C Eb Gv = Cm(v5)&lt;br /&gt;
 &lt;br /&gt;
--5-10-15 = dim7&lt;br /&gt;
+-5-10-15 = C Eb Gb Bbb = Cdim7&lt;br /&gt;
--5-11-14 = dim7(^5,v7)&lt;br /&gt;
+-5-11-14 = C Eb Gb^ Bbbv = Cdim7(^5,v7)&lt;br /&gt;
--6-11-15 = dim7(^3,^5)&lt;br /&gt;
+-6-11-15 = C Eb^ Gb^ Bbb = Cdim7(^3,^5)&lt;br /&gt;
--6-11-16 = ^dim7 (the up symbol applies to m3, d5 and d7)&lt;br /&gt;
+-6-11-16 = C Eb^ Gb^ Bbb^ = C.^dim7 (the up symbol applies to m3, d5 and d7)&lt;br /&gt;
--5-13-17 = m6&lt;br /&gt;
+-5-13-17 = C Eb G A = Cm6&lt;br /&gt;
 &lt;br /&gt;
--6-12-15 = m6(^3,v5,vv6) or dim7(^3,^^5) (doubled ups/downs are unavoidable)&lt;br /&gt;
+Sometimes doubled ups/downs are unavoidable:&lt;br /&gt;
--7-13-16 = vM6&lt;br /&gt;
+-6-12-15 = C Eb^ Gv Avv = Cm6(^3,v5,vv6), or C Eb^ Gb^^ Bbb = Cdim7(^3,^^5)&lt;br /&gt;
--8-13-17 = 6&lt;br /&gt;
+-7-13-16 = C Ev G Av = C.vM6 (the down symbol applies to both the 3rd and the 6th)&lt;br /&gt;
+-8-13-17 = C E G A = C6&lt;br /&gt;
+-7-13-16 = C Ev G Av = C.v6&lt;br /&gt;
 &lt;br /&gt;
--5-13-18 = m7&lt;br /&gt;
+-5-13-18 = C Eb G Bb = Cm7&lt;br /&gt;
--6-13-19 = ^m7&lt;br /&gt;
+-6-13-19 = C Eb^ G Bb^ = C.^m7&lt;br /&gt;
--7-13-20 = vM7&lt;br /&gt;
+-7-13-20 = C Ev G Bv = C.vM7&lt;br /&gt;
--8-13-21 = M7&lt;br /&gt;
+-8-13-21 = C E G B = CM7&lt;br /&gt;
 &lt;br /&gt;
--5-13-16 = m6(v6)&lt;br /&gt;
+-5-13-16 = C Eb G Av = Cm6(v6)&lt;br /&gt;
--8-13-19 = 7(^7)&lt;br /&gt;
+-8-13-19 = C E G Bb^ = C7(^7)&lt;br /&gt;
--7-13-18-26 = 9(v3)&lt;br /&gt;
+-7-13-18-26 = C Ev G Bb D = C9(v3)&lt;br /&gt;
--7-13-18-26-32 = 11(v3,^11)&lt;br /&gt;
+-7-13-18-26-32 = C Ev G Bb D F^ = C11(v3,^11)&lt;br /&gt;
 &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;
+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 &amp;quot;Tibia&amp;quot;:&lt;br /&gt;
 G.vM7(no5) = &amp;quot;G downmajor seven, no five&amp;quot;&lt;br /&gt;
 Eb^.v(add9) = &amp;quot;E flat up, downmajor, add nine&amp;quot;&lt;br /&gt;
@@ Line 1,055: / Line 1,128: @@
 A7(v3) = &amp;quot;A seven, down three&amp;quot;&lt;br /&gt;
 &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;
+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 might be 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 V ^V/bVI v#V/^bVI #V/vVI VI ^VI/bVII v#VI/^bVII #VI/vVII VII/vI&lt;br /&gt;
+&lt;br /&gt;
+I&lt;br /&gt;
+^I or bII&lt;br /&gt;
+v#I or ^bII&lt;br /&gt;
+#I or vII&lt;br /&gt;
+II&lt;br /&gt;
+^II or bIII&lt;br /&gt;
+v#II or ^bIII&lt;br /&gt;
+#II or vIII&lt;br /&gt;
+III&lt;br /&gt;
+IV&lt;br /&gt;
+^IV or bV&lt;br /&gt;
+v#IV or ^bV&lt;br /&gt;
+#IV or vV&lt;br /&gt;
+V&lt;br /&gt;
+^V or bVI&lt;br /&gt;
+v#V or ^bVI&lt;br /&gt;
+#V or vVI&lt;br /&gt;
+VI&lt;br /&gt;
+^VI or bVII&lt;br /&gt;
+v#VI or ^bVII&lt;br /&gt;
+#VI or vVII&lt;br /&gt;
+VII or 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;
-Here's the Tibia chords. No period is ever needed after the root because ups and downs are always leading, never trailing.&lt;br /&gt;
+Here's the Tibia chords. Periods are never needed after the root in relative notation because ups and downs are always leading, never trailing.&lt;br /&gt;
 IvM7(no5) = &amp;quot;one downmajor seven, no five&amp;quot;&lt;br /&gt;
 ^bVIv(add9) = &amp;quot;up-flat six downmajor, add nine&amp;quot;&lt;br /&gt;
@@ Line 1,066: / Line 1,161: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:1443:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1035px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1443 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:1829:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1035px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1829 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;!-- ws:start:WikiTextLocalImageRule:1444:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 957px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1444 --&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;!-- ws:start:WikiTextLocalImageRule:1830:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 957px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1830 --&gt;&lt;/h2&gt;
   &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt; &lt;/h2&gt;
   &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt; &lt;/h2&gt;
@@ Line 2,846: / Line 2,941: @@
 &lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Summary of EDO notation-Rank-2 Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;&lt;u&gt;Rank-2 Notation&lt;/u&gt;&lt;/h2&gt;
   &lt;br /&gt;
-Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.) &lt;br /&gt;
+Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)&lt;br /&gt;
+&lt;br /&gt;
+For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. I.e. the framework's node must not be on the spine of a kite. For example, meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the 5th's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).&lt;br /&gt;
 &lt;br /&gt;
-For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. For esxample, meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the 5th's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).&lt;br /&gt;
+All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone.&lt;br /&gt;
 &lt;br /&gt;
-note to self: if K(#) is 1 or -1, ups and downs aren't needed to notate rank-2.&lt;br /&gt;
+If the sharp's keyspan is 1 or -1, ups and downs aren't needed to notate rank-2. The keyspan is zero only for perfect EDOs. Thus we can omit fourthward frameworks, and assume for this discussion that K(#) &amp;gt; 1. We need only consider sweet frameworks, excluding those that lie on the side of the heptatonic kite and those that lie on the spine of any kite.&lt;br /&gt;
 &lt;br /&gt;
 To extend ups and downs to rank-2 tunings, the up symbol is given not only a &lt;strong&gt;keyspan&lt;/strong&gt; (always +1) but also a &lt;strong&gt;genspan&lt;/strong&gt;, which indicates how many steps forward or backwards along the generator chain, or &lt;strong&gt;genchain&lt;/strong&gt;, one must travel to find the interval. For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. The interval is always a 2nd. The genspan is calculated from the keyspans:&lt;br /&gt;
@@ Line 2,932: / Line 3,029: @@
 For 17-tone, X = 2, i = 1, G(^) = -5, and ^ = min 2nd&lt;br /&gt;
 &lt;br /&gt;
-For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd.&lt;/body&gt;&lt;/html&gt;</pre></div>
+For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd.&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;5edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;7edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;perfect&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;8edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;fifthless&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;11edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;fifthless&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;13edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;fifthless&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;17edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;-5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;min 2nd&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;18edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;fifthless&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;19edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;22edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;-5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;min 2nd&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;26edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;27edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;29edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;31edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;-12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;dim 2nd&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;32edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;33edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;34edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;37edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;38edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;39edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;40edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;41edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;42edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;43edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;44ddo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;45edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;46edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;47edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;49edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;50edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;51edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;52edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;53edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;sweet&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;/body&gt;&lt;/html&gt;</pre></div>