Kite's ups and downs notation: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 06:00:25 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 06:30:50 UTC</tt>.

: The original revision id was <tt>~~584788485~~</tt>.

: The original revision id was <tt>584789109</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Line 836:

If the sharp's keyspan is 1 or -1, as with 12-tone, 19-tone, and all fourthward frameworks, ups and downs aren't needed to notate rank-2. They also aren't needed for 5-tone and 7-tone. Since perfect, pentatonic and fifthless frameworks are incompatible, 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 assigned 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.

To extend ups and downs to rank-2 tunings, the up symbol is assigned 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. The sharp is always genspan +7, and the flat is always genspan -7. By adding sharps/flats

For example, in the 22-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. And C^^ = C^ + m2 = (C + m2)^, exactly equivalent to Db^. However, C^^ is not equivalent to Dvv, even though they occuy the same key on the keyboard, just as C# may not equal Db in 12-tone.

Line 877:

||= ||= || || || etc. ||= ||= ||

Each of the 22 keys, with alternate tunings for the black keys:

Each of the 22 keys, with alternate tunings for the black keys ("^3" means triple-up):

||= keyspan from C ||= genspan from C ||= note ||= genspan from C ||= note ||

||= 0 ||= 0 ||= C ||= ||= ||

Line 903:

||= 22 ||= 0 ||= C ||= ||= ||

~~"^3" means triple-up.~~ Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.

Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.

The genspan for the up symbol in 22-tone is calculated from the keyspans:

Line 928:

The other relevant frameworks of size 53 or less:

||= ||= Keyspan of #

||= ||= Keyspan of # || value of i ||= genspan of ^ ||= example ||= stepspan &

~~= K(A1)~~ ||= value of i ||= genspan of ^ ||= example ||= stepspan &

quality of ^ ||

||= 17-tone ||= 2 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||

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Line 937:

||= 32-tone ||= 5 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||

||= 37-tone ||= 6 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||

||= 39-tone ||= 5 ||= 3, -2 ||= -22, +17 ||= C-Fbbb, C-Ax ||= triple-dim 4th, desc double-dim 3rd ||

||= 39-tone ||= 5 ||= 3, -2 ||= -22, +17 ||= C^ = Fbbb, C^ = Ax ||= triple-dim 4th, desc double-dim 3rd ||

||= 41-tone ||= 4 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||

||= 42-tone ||= 7 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||

Line 944:

Line 943:

||= 45-tone ||= 2 ||= 1 ||= -19 || C^ = Dbbb

||= double-dim 2nd ||

||= 49-tone ||= 7 ||= 0, 1~~, 2~~, -1 ||= 1, -6~~, -13~~, 8 ||= C-G, C-Gb, C~~-Gbb, C-~~G# ||= perf 5th, ~~dim 5th, double-~~dim 5th, aug 5th ||

||= 49-tone ||= 7 ||= 0, 1, -1 ||= 1, -6, 8 ||= C^ = G, C^ = Gb, C^ = G# ||= perf 5th, dim 5th, aug 5th ||

||= 50-tone ||= 3 ||= -1 ||= +19 ||= C^ = Bx ||= desc double-dim 2nd ||

||= 53-tone ||= 5 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||

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Line 990:

||= ||= || || || etc. ||= || ||

~~Some of the~~ 29 ~~keys~~, with alternate tunings for the black keys:

The 29-tone keyboard, with alternate tunings for the black keys:

||= keyspan from C ||= genspan from C ||= note ||= genspan from C ||= note ||

||= 0 ||= 0 ||= C ||= ||= ||

Line 1,006:

Line 1,005:

||= 11 ||= -13 ||= Fv = E#vv ||= +16 ||= E^ = Fb^^ ||

||= 12 ||= -1 ||= F ||= ||= ||

|= 13 ||= -22 ||= Dvv = Cxv3 ||= +7 ||= C# = Db^ ||

||= 14 ||= -10 ||= Dv = Cxvv ||= +19 ||= C#^ = Db^^ ||

||= 15 ||= +2 ||= D ||= ||= ||

||= 16 ||= -15 ||= Ebv = D#vv ||= +14 ||= D^ = Ebb^^ ||

||= 17 ||= -3 ||= Eb = D#v ||= +26 ||= D^^ = Ebb^3 ||

||= 18 ||= -20 ||= Evv = Dxv3 ||= +9 ||= D# = Eb^ ||

||= 19 ||= -8 ||= Ev = Dxvv ||= +21 ||= D#^ = Eb^^ ||

||= 20 ||= +4 ||= E ||= ||= ||

||= 21 ||= -13 ||= Fv = E#vv ||= +16 ||= E^ = Fb^^ ||

||= 22 ||= -1 ||= F ||= ||= ||

|= 23 ||= -22 ||= Dvv = Cxv3 ||= +7 ||= C# = Db^ ||

||= 24 ||= -10 ||= Dv = Cxvv ||= +19 ||= C#^ = Db^^ ||

||= 25 ||= +2 ||= D ||= ||= ||

||= 26 ||= -15 ||= Ebv = D#vv ||= +14 ||= D^ = Ebb^^ ||

||= 27 ||= -3 ||= Eb = D#v ||= +26 ||= D^^ = Ebb^3 ||

||= 28 ||= -20 ||= Evv = Dxv3 ||= +9 ||= D# = Eb^ ||

||= 29 ||= -8 ||= Ev = Dxvv ||= +21 ||= D#^ = Eb^^ ||

|

~~The~~</pre></div>

39-tone and 49-tone are problematic:

39-tone: C * Db * * C# * D

49-tone: C * Db * * * * C# * D

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

Line 1,099:

Line 1,120:

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.

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<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 3,140:

Line 3,161:

If the sharp's keyspan is 1 or -1, as with 12-tone, 19-tone, and all fourthward frameworks, ups and downs aren't needed to notate rank-2. They also aren't needed for 5-tone and 7-tone. Since perfect, pentatonic and fifthless frameworks are incompatible, 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 assigned 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.

To extend ups and downs to rank-2 tunings, the up symbol is assigned 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. The sharp is always genspan +7, and the flat is always genspan -7. By adding sharps/flats

For example, in the 22-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. And C^^ = C^ + m2 = (C + m2)^, exactly equivalent to Db^. However, C^^ is not equivalent to Dvv, even though they occuy the same key on the keyboard, just as C# may not equal Db in 12-tone.

Line 3,681:

Line 3,702:

Each of the 22 keys, with alternate tunings for the black keys:

Each of the 22 keys, with alternate tunings for the black keys (&quot;^3&quot; means triple-up):

Line 3,976:

Line 3,997:

~~&quot;^3&quot; means triple-up.~~ Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.

Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.

The genspan for the up symbol in 22-tone is calculated from the keyspans:

Line 4,129:

Line 4,150:

</td>

<td style="text-align: center;">Keyspan of #

~~<h1 id="toc19"><a name="K(A1) ||"></a> K(A1)~~</td>

</td>

<td>value of i

~~value of i~~</td>

</td>

<td style="text-align: center;">genspan of ^

</td>

Line 4,247:

Line 4,268:

<td style="text-align: center;">-22, +17

</td>

<td style="text-align: center;">C-Fbbb, C-Ax

<td style="text-align: center;">C^ = Fbbb, C^ = Ax

</td>

<td style="text-align: center;">triple-dim 4th, desc double-dim 3rd

Line 4,314:

Line 4,335:

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

</td>

<td style="text-align: center;">0, 1~~, 2~~, -1

<td style="text-align: center;">0, 1, -1

</td>

<td style="text-align: center;">1, -6~~, -13~~, 8

<td style="text-align: center;">1, -6, 8

</td>

<td style="text-align: center;">C-G, C-Gb, C~~-Gbb, C-~~G#

<td style="text-align: center;">C^ = G, C^ = Gb, C^ = G#

</td>

<td style="text-align: center;">perf 5th, ~~dim 5th, double-~~dim 5th, aug 5th

<td style="text-align: center;">perf 5th, dim 5th, aug 5th

</td>

</tr>

Line 4,897:

Line 4,918:

~~Some of the~~ 29 ~~keys~~, with alternate tunings for the black keys:

The 29-tone keyboard, with alternate tunings for the black keys:

Line 5,069:

Line 5,090:

</td>

</tr>

</table>

|= 13 ||= -22 ||= Dvv = Cxv3 ||= +7 ||= C# = Db^ ||

<tr>

<td style="text-align: center;">14

</td>

<td style="text-align: center;">-10

</td>

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

</td>

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

</td>

<td style="text-align: center;">C#^ = Db^^

</td>

</tr>

<tr>

<td style="text-align: center;">15

</td>

<td style="text-align: center;">+2

</td>

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

</td>

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">16

</td>

<td style="text-align: center;">-15

</td>

<td style="text-align: center;">Ebv = D#vv

</td>

<td style="text-align: center;">+14

</td>

<td style="text-align: center;">D^ = Ebb^^

</td>

</tr>

<tr>

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

</td>

<td style="text-align: center;">-3

</td>

<td style="text-align: center;">Eb = D#v

</td>

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

</td>

<td style="text-align: center;">D^^ = Ebb^3

</td>

</tr>

<tr>

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

</td>

<td style="text-align: center;">-20

</td>

<td style="text-align: center;">Evv = Dxv3

</td>

<td style="text-align: center;">+9

</td>

<td style="text-align: center;">D# = Eb^

</td>

</tr>

<tr>

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

</td>

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

</td>

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

</td>

<td style="text-align: center;">+21

</td>

<td style="text-align: center;">D#^ = Eb^^

</td>

</tr>

<tr>

<td style="text-align: center;">20

</td>

<td style="text-align: center;">+4

</td>

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

</td>

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">21

</td>

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

</td>

<td style="text-align: center;">Fv = E#vv

</td>

<td style="text-align: center;">+16

</td>

<td style="text-align: center;">E^ = Fb^^

</td>

</tr>

<tr>

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

</td>

<td style="text-align: center;">-1

</td>

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

</td>

</td>

</td>

</tr>

|= 23 ||= -22 ||= Dvv = Cxv3 ||= +7 ||= C# = Db^ ||

<tr>

<td style="text-align: center;">24

</td>

<td style="text-align: center;">-10

</td>

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

</td>

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

</td>

<td style="text-align: center;">C#^ = Db^^

</td>

</tr>

<tr>

<td style="text-align: center;">25

</td>

<td style="text-align: center;">+2

</td>

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

</td>

</td>

</td>

</tr>

<tr>

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

</td>

<td style="text-align: center;">-15

</td>

<td style="text-align: center;">Ebv = D#vv

</td>

<td style="text-align: center;">+14

</td>

<td style="text-align: center;">D^ = Ebb^^

</td>

</tr>

<tr>

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

</td>

<td style="text-align: center;">-3

</td>

<td style="text-align: center;">Eb = D#v

</td>

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

</td>

<td style="text-align: center;">D^^ = Ebb^3

</td>

</tr>

<tr>

<td style="text-align: center;">28

</td>

<td style="text-align: center;">-20

</td>

<td style="text-align: center;">Evv = Dxv3

</td>

<td style="text-align: center;">+9

</td>

<td style="text-align: center;">D# = Eb^

</td>

</tr>

<tr>

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

</td>

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

</td>

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

</td>

<td style="text-align: center;">+21

</td>

<td style="text-align: center;">D#^ = Eb^^

</td>

</tr>

</table>

|

39-tone and 49-tone are problematic:

39-tone: C * Db * * C# * D

49-tone: C * Db * * * * C# * D

~~The~~</body></html></pre></div>

There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for these two frameworks.</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 06:00:25 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 06:30:50 UTC</tt>.<br>
-: The original revision id was <tt>584788485</tt>.<br>
+: The original revision id was <tt>584789109</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 836: / Line 836: @@
 If the sharp's keyspan is 1 or -1, as with 12-tone, 19-tone, and all fourthward frameworks, ups and downs aren't needed to notate rank-2. They also aren't needed for 5-tone and 7-tone. Since perfect, pentatonic and fifthless frameworks are incompatible, 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 assigned 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.
+To extend ups and downs to rank-2 tunings, the up symbol is assigned 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. The sharp is always genspan +7, and the flat is always genspan -7. By adding sharps/flats
 For example, in the 22-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. And C^^ = C^ + m2 = (C + m2)^, exactly equivalent to Db^. However, C^^ is not equivalent to Dvv, even though they occuy the same key on the keyboard, just as C# may not equal Db in 12-tone.
@@ Line 877: / Line 877: @@
 ||=   ||=   ||   ||   || etc. ||=   ||=   ||
-Each of the 22 keys, with alternate tunings for the black keys:
+Each of the 22 keys, with alternate tunings for the black keys ("^3" means triple-up):
 ||= keyspan from C ||= genspan from C ||= note ||= genspan from C ||= note ||
 ||= 0 ||= 0 ||= C ||=   ||=   ||
@@ Line 903: / Line 903: @@
 ||= 22 ||= 0 ||= C ||=   ||=   ||
-"^3" means triple-up. Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.
+Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.
 The genspan for the up symbol in 22-tone is calculated from the keyspans:
@@ Line 928: / Line 928: @@
 The other relevant frameworks of size 53 or less:
-||=   ||= Keyspan of #
+||=   ||= Keyspan of # || value of i ||= genspan of ^ ||= example ||= stepspan &amp;
-= K(A1) ||= value of i ||= genspan of ^ ||= example ||= stepspan &amp;
 quality of ^ ||
 ||= 17-tone ||= 2 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||
@@ Line 938: / Line 937: @@
 ||= 32-tone ||= 5 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||
 ||= 37-tone ||= 6 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||
-||= 39-tone ||= 5 ||= 3, -2 ||= -22, +17 ||= C-Fbbb, C-Ax ||= triple-dim 4th, desc double-dim 3rd ||
+||= 39-tone ||= 5 ||= 3, -2 ||= -22, +17 ||= C^ = Fbbb, C^ = Ax ||= triple-dim 4th, desc double-dim 3rd ||
 ||= 41-tone ||= 4 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||
 ||= 42-tone ||= 7 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||
@@ Line 944: / Line 943: @@
 ||= 45-tone ||= 2 ||= 1 ||= -19 || &lt;span style="display: block; text-align: center;"&gt;C^ = Dbbb
 &lt;/span&gt; ||= double-dim 2nd ||
-||= 49-tone ||= 7 ||= 0, 1, 2, -1 ||= 1, -6, -13, 8 ||= C-G, C-Gb, C-Gbb, C-G# ||= perf 5th, dim 5th, double-dim 5th, aug 5th ||
+||= 49-tone ||= 7 ||= 0, 1, -1 ||= 1, -6, 8 ||= C^ = G, C^ = Gb, C^ = G# ||= perf 5th, dim 5th, aug 5th ||
 ||= 50-tone ||= 3 ||= -1 ||= +19 ||= C^ = Bx ||= desc double-dim 2nd ||
 ||= 53-tone ||= 5 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||
@@ Line 991: / Line 990: @@
 ||=   ||=   ||   ||   || etc. ||=   ||   ||
-Some of the 29 keys, with alternate tunings for the black keys:
+The 29-tone keyboard, with alternate tunings for the black keys:
 ||= keyspan from C ||= genspan from C ||= note ||= genspan from C ||= note ||
 ||= 0 ||= 0 ||= C ||=   ||=   ||
@@ Line 1,006: / Line 1,005: @@
 ||= 11 ||= -13 ||= Fv = E#vv ||= +16 ||= E^ = Fb^^ ||
 ||= 12 ||= -1 ||= F ||=   ||=   ||
+|= 13 ||= -22 ||= Dvv = Cxv3 ||= +7 ||= C# = Db^ ||
+||= 14 ||= -10 ||= Dv = Cxvv ||= +19 ||= C#^ = Db^^ ||
+||= 15 ||= +2 ||= D ||=   ||=   ||
+||= 16 ||= -15 ||= Ebv = D#vv ||= +14 ||= D^ = Ebb^^ ||
+||= 17 ||= -3 ||= Eb = D#v ||= +26 ||= D^^ = Ebb^3 ||
+||= 18 ||= -20 ||= Evv = Dxv3 ||= +9 ||= D# = Eb^ ||
+||= 19 ||= -8 ||= Ev = Dxvv ||= +21 ||= D#^ = Eb^^ ||
+||= 20 ||= +4 ||= E ||=   ||=   ||
+||= 21 ||= -13 ||= Fv = E#vv ||= +16 ||= E^ = Fb^^ ||
+||= 22 ||= -1 ||= F ||=   ||=   ||
+|= 23 ||= -22 ||= Dvv = Cxv3 ||= +7 ||= C# = Db^ ||
+||= 24 ||= -10 ||= Dv = Cxvv ||= +19 ||= C#^ = Db^^ ||
+||= 25 ||= +2 ||= D ||=   ||=   ||
+||= 26 ||= -15 ||= Ebv = D#vv ||= +14 ||= D^ = Ebb^^ ||
+||= 27 ||= -3 ||= Eb = D#v ||= +26 ||= D^^ = Ebb^3 ||
+||= 28 ||= -20 ||= Evv = Dxv3 ||= +9 ||= D# = Eb^ ||
+||= 29 ||= -8 ||= Ev = Dxvv ||= +21 ||= D#^ = Eb^^ ||
+|
-The</pre></div>
+-tone and 49-tone are problematic:
+-tone: C * Db * * C# * D
+-tone: C * Db * * * * C# * D
+There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for these two frameworks.</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 1,099: / Line 1,120: @@
 fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:3254:&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:3254 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:3431:&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:3431 --&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;
@@ Line 1,315: / Line 1,336: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:3255:&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:3255 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:3432:&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:3432 --&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:3256:&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:3256 --&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:3433:&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:3433 --&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 3,140: / Line 3,161: @@
 If the sharp's keyspan is 1 or -1, as with 12-tone, 19-tone, and all fourthward frameworks, ups and downs aren't needed to notate rank-2. They also aren't needed for 5-tone and 7-tone. Since perfect, pentatonic and fifthless frameworks are incompatible, 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 assigned 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.&lt;br /&gt;
+To extend ups and downs to rank-2 tunings, the up symbol is assigned 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. The sharp is always genspan +7, and the flat is always genspan -7. By adding sharps/flats&lt;br /&gt;
 &lt;br /&gt;
 For example, in the 22-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. And C^^ = C^ + m2 = (C + m2)^, exactly equivalent to Db^. However, C^^ is not equivalent to Dvv, even though they occuy the same key on the keyboard, just as C# may not equal Db in 12-tone.&lt;br /&gt;
@@ Line 3,681: / Line 3,702: @@
 &lt;br /&gt;
-Each of the 22 keys, with alternate tunings for the black keys:&lt;br /&gt;
+Each of the 22 keys, with alternate tunings for the black keys (&amp;quot;^3&amp;quot; means triple-up):&lt;br /&gt;
@@ Line 3,976: / Line 3,997: @@
 &lt;br /&gt;
-&amp;quot;^3&amp;quot; means triple-up. Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.&lt;br /&gt;
+Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.&lt;br /&gt;
 &lt;br /&gt;
 The genspan for the up symbol in 22-tone is calculated from the keyspans:&lt;br /&gt;
@@ Line 4,129: / Line 4,150: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Keyspan of #&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc19"&gt;&lt;a name="K(A1) ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt; K(A1)&lt;/td&gt;
+&lt;/td&gt;
-         &lt;td&gt;&lt;/h1&gt;
+         &lt;td&gt;value of i&lt;br /&gt;
- value of i&lt;/td&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;genspan of ^&lt;br /&gt;
 &lt;/td&gt;
@@ Line 4,247: / Line 4,268: @@
          &lt;td style="text-align: center;"&gt;-22, +17&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C-Fbbb, C-Ax&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^ = Fbbb, C^ = Ax&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;triple-dim 4th, desc double-dim 3rd&lt;br /&gt;
@@ Line 4,314: / Line 4,335: @@
          &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;0, 1, 2, -1&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;0, 1, -1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;1, -6, -13, 8&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1, -6, 8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C-G, C-Gb, C-Gbb, C-G#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^ = G, C^ = Gb, C^ = G#&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;perf 5th, dim 5th, double-dim 5th, aug 5th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;perf 5th, dim 5th, aug 5th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 4,897: / Line 4,918: @@
 &lt;br /&gt;
-Some of the 29 keys, with alternate tunings for the black keys:&lt;br /&gt;
+The 29-tone keyboard, with alternate tunings for the black keys:&lt;br /&gt;
@@ Line 5,069: / Line 5,090: @@
 &lt;/td&gt;
      &lt;/tr&gt;
-&lt;/table&gt;
+|= 13 ||= -22 ||= Dvv = Cxv3 ||= +7 ||= C# = Db^ ||&lt;br /&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;14&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Dv = Cxvv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+19&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C#^ = Db^^&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;16&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Ebv = D#vv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+14&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D^ = Ebb^^&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Eb = D#v&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+26&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D^^ = Ebb^3&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;18&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-20&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Evv = Dxv3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D# = Eb^&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;19&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Ev = Dxvv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+21&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D#^ = Eb^^&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;20&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;21&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Fv = E#vv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+16&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E^ = Fb^^&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;22&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+|= 23 ||= -22 ||= Dvv = Cxv3 ||= +7 ||= C# = Db^ ||&lt;br /&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;24&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Dv = Cxvv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+19&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C#^ = Db^^&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;25&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;26&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Ebv = D#vv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+14&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D^ = Ebb^^&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;27&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Eb = D#v&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+26&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D^^ = Ebb^3&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;28&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-20&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Evv = Dxv3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D# = Eb^&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;29&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Ev = Dxvv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;+21&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D#^ = Eb^^&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+|&lt;br /&gt;
+&lt;br /&gt;
+-tone and 49-tone are problematic:&lt;br /&gt;
+-tone: C * Db * * C# * D&lt;br /&gt;
+-tone: C * Db * * * * C# * D&lt;br /&gt;
 &lt;br /&gt;
-The&lt;/body&gt;&lt;/html&gt;</pre></div>
+There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for these two frameworks.&lt;/body&gt;&lt;/html&gt;</pre></div>