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>2016-06-03 20:49:11 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-03 20:59:08 UTC</tt>.

: The original revision id was <tt>~~584780549~~</tt>.

: The original revision id was <tt>584780827</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 795:

Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:

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

This genchain suffices to name each note, but the note names will often run out of order. For example, in 22-tone, we might have C Db B# C# D, or C Db Ebb C# D. So ups and downs are needed to allow alternate names for each note. The first example becomes C C^

Fifth-generated rank-2 tunings can be written without ups and downs in any EDO on the side of the 4\7 kite:

Line 823:

Line 821:

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.

For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is 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. Here's the 22-tone genchain:

For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is 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.

The usual genchain suffices to name each note, but the note names will often run out of order. For example, in 22-tone, we might have C Db B# C# D. So ups and downs are needed to allow alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D.

Here's the 22-tone genchain:

Fb = Eb^ = D^^

Line 839:

Line 841:

|| || || Gb || F^ || E^^ ||

|| || || Db || C^ || B^^ ||

|| || || Ab || G^ || ||

|| || || Ab || G^ || F#^^ ||

|| || || Eb || D^ || ||

|| || Fbv || Eb || D^ || C#^^ ||

|| || || Bb || A^ || ||

|| || Cbv || Bb || A^ || G#^^ ||

|| || || F || E^ || ||

|| || Gbv || F || E^ || A#^^ ||

|| || || C || B^ || ||

|| || Dbv || C || B^ || E#^^ ||

|| || || G || F#^ || ||

|| || Abv || G || F#^ || B#^^ ||

|| || || D || C#^ || ||

|| Fbvv || Ebv || D || C#^ || ||

|| || || A || G#^ || ||

|| Cbvv || Bbv || A || G#^ || ||

|| || || E || D#^ || ||

|| Gbvv || Fv || E || D#^ || ||

|| || || B || A#^ || ||

|| Dbvv || Cv || B || A#^ || ||

|| || || F# || E#^ || ||

|| Abvv || Gv || F# || E#^ || ||

|| || || C# || B#^ || ||

|| Ebvv || Dv || C# || B#^ || ||

|| || || G# || || ||

|| Bbvv || Av || G# || || ||

|| || || D# || || ||

|| Fvv || Ev || D# || || ||

|| || || A# || || ||

|| Cvv || Bv || A# || || ||

|| || || E# || || ||

|| Gvv || F#v || E# || || ||

|| || || B# || || ||

|| Dvv || C#v || B# || || ||

|| || || Fx || || ||

|| Avv || G#v || Fx || || ||

|| || || Cx || || ||

|| Evv || D#v || Cx || || ||

|| || || Gx || || ||

|| Bvv || A#v || Gx || || ||

|| || || Dx || || ||

|| F#vv || E#v || Dx || || ||

|| C#vv || B#v || Ax || || ||

Line 1,010:

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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.

Line 1,226:

Line 1,229:

<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 3,010:

Line 3,013:

Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:

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

~~ ~~

This genchain suffices to name each note, but the note names will often run out of order. For example, in 22-tone, we might have C Db B# C# D, or C Db Ebb C# D. So ups and downs are needed to allow alternate names for each note. The first example becomes C C^

Fifth-generated rank-2 tunings can be written without ups and downs in any EDO on the side of the 4\7 kite:

Line 3,038:

Line 3,039:

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.

For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is 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. Here's the 22-tone genchain:

For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is 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.

The usual genchain suffices to name each note, but the note names will often run out of order. For example, in 22-tone, we might have C Db B# C# D. So ups and downs are needed to allow alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D.

Here's the 22-tone genchain:

Fb = Eb^ = D^^

Line 3,110:

Line 3,115:

<td>G^

</td>

<td>

<td>F#^^

</td>

</tr>

Line 3,116:

Line 3,121:

<td>

</td>

<td>

<td>Fbv

</td>

<td>Eb

Line 3,122:

Line 3,127:

<td>D^

</td>

<td>

<td>C#^^

</td>

</tr>

Line 3,128:

Line 3,133:

<td>

</td>

<td>

<td>Cbv

</td>

<td>Bb

Line 3,134:

Line 3,139:

<td>A^

</td>

<td>

<td>G#^^

</td>

</tr>

Line 3,140:

Line 3,145:

<td>

</td>

<td>

<td>Gbv

</td>

<td>F

Line 3,146:

Line 3,151:

<td>E^

</td>

<td>

<td>A#^^

</td>

</tr>

Line 3,152:

Line 3,157:

<td>

</td>

<td>

<td>Dbv

</td>

<td>C

Line 3,158:

Line 3,163:

<td>B^

</td>

<td>

<td>E#^^

</td>

</tr>

Line 3,164:

Line 3,169:

<td>

</td>

<td>

<td>Abv

</td>

<td>G

Line 3,170:

Line 3,175:

<td>F#^

</td>

<td>

<td>B#^^

</td>

</tr>

<tr>

<td>

<td>Fbvv

</td>

<td>

<td>Ebv

</td>

<td>D

Line 3,186:

Line 3,191:

</tr>

<tr>

<td>

<td>Cbvv

</td>

<td>

<td>Bbv

</td>

<td>A

Line 3,198:

Line 3,203:

</tr>

<tr>

<td>

<td>Gbvv

</td>

<td>

<td>Fv

</td>

<td>E

Line 3,210:

Line 3,215:

</tr>

<tr>

<td>

<td>Dbvv

</td>

<td>

<td>Cv

</td>

<td>B

Line 3,222:

Line 3,227:

</tr>

<tr>

<td>

<td>Abvv

</td>

<td>

<td>Gv

</td>

<td>F#

Line 3,234:

Line 3,239:

</tr>

<tr>

<td>

<td>Ebvv

</td>

<td>

<td>Dv

</td>

<td>C#

Line 3,246:

Line 3,251:

</tr>

<tr>

<td>

<td>Bbvv

</td>

<td>

<td>Av

</td>

<td>G#

Line 3,258:

Line 3,263:

</tr>

<tr>

<td>

<td>Fvv

</td>

<td>

<td>Ev

</td>

<td>D#

Line 3,270:

Line 3,275:

</tr>

<tr>

<td>

<td>Cvv

</td>

<td>

<td>Bv

</td>

<td>A#

Line 3,282:

Line 3,287:

</tr>

<tr>

<td>

<td>Gvv

</td>

<td>

<td>F#v

</td>

<td>E#

Line 3,294:

Line 3,299:

</tr>

<tr>

<td>

<td>Dvv

</td>

<td>

<td>C#v

</td>

<td>B#

Line 3,306:

Line 3,311:

</tr>

<tr>

<td>

<td>Avv

</td>

<td>

<td>G#v

</td>

<td>Fx

Line 3,318:

Line 3,323:

</tr>

<tr>

<td>

<td>Evv

</td>

<td>

<td>D#v

</td>

<td>Cx

Line 3,330:

Line 3,335:

</tr>

<tr>

<td>

<td>Bvv

</td>

<td>

<td>A#v

</td>

<td>Gx

Line 3,342:

Line 3,347:

</tr>

<tr>

<td>F#vv

</td>

<td>E#v

</td>

<td>Dx

</td>

<td>

</td>

<td>

</td>

<td>Dx

</tr>

<tr>

<td>C#vv

</td>

<td>B#v

</td>

<td>Ax

</td>

<td>

@@ 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>2016-06-03 20:49:11 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-03 20:59:08 UTC</tt>.<br>
-: The original revision id was <tt>584780549</tt>.<br>
+: The original revision id was <tt>584780827</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 795: / Line 795: @@
 Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:
 ...Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# ...
-This genchain suffices to name each note, but the note names will often run out of order. For example, in 22-tone, we might have C Db B# C# D, or C Db Ebb C# D. So ups and downs are needed to allow alternate names for each note. The first example becomes C C^
 Fifth-generated rank-2 tunings can be written without ups and downs in any EDO on the side of the 4\7 kite:
@@ Line 823: / Line 821: @@
 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.
-For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is 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. Here's the 22-tone genchain:
+For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is 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.
+The usual genchain suffices to name each note, but the note names will often run out of order. For example, in 22-tone, we might have C Db B# C# D. So ups and downs are needed to allow alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D.
+Here's the 22-tone genchain:
 Fb = Eb^ = D^^
@@ Line 839: / Line 841: @@
 ||   ||   || Gb || F^ || E^^ ||
 ||   ||   || Db || C^ || B^^ ||
-||   ||   || Ab || G^ ||   ||
+||   ||   || Ab || G^ || F#^^ ||
-||   ||   || Eb || D^ ||   ||
+||   || Fbv || Eb || D^ || C#^^ ||
-||   ||   || Bb || A^ ||   ||
+||   || Cbv || Bb || A^ || G#^^ ||
-||   ||   || F || E^ ||   ||
+||   || Gbv || F || E^ || A#^^ ||
-||   ||   || C || B^ ||   ||
+||   || Dbv || C || B^ || E#^^ ||
-||   ||   || G || F#^ ||   ||
+||   || Abv || G || F#^ || B#^^ ||
-||   ||   || D || C#^ ||   ||
+|| Fbvv || Ebv || D || C#^ ||   ||
-||   ||   || A || G#^ ||   ||
+|| Cbvv || Bbv || A || G#^ ||   ||
-||   ||   || E || D#^ ||   ||
+|| Gbvv || Fv || E || D#^ ||   ||
-||   ||   || B || A#^ ||   ||
+|| Dbvv || Cv || B || A#^ ||   ||
-||   ||   || F# || E#^ ||   ||
+|| Abvv || Gv || F# || E#^ ||   ||
-||   ||   || C# || B#^ ||   ||
+|| Ebvv || Dv || C# || B#^ ||   ||
-||   ||   || G# ||   ||   ||
+|| Bbvv || Av || G# ||   ||   ||
-||   ||   || D# ||   ||   ||
+|| Fvv || Ev || D# ||   ||   ||
-||   ||   || A# ||   ||   ||
+|| Cvv || Bv || A# ||   ||   ||
-||   ||   || E# ||   ||   ||
+|| Gvv || F#v || E# ||   ||   ||
-||   ||   || B# ||   ||   ||
+|| Dvv || C#v || B# ||   ||   ||
-||   ||   || Fx ||   ||   ||
+|| Avv || G#v || Fx ||   ||   ||
-||   ||   || Cx ||   ||   ||
+|| Evv || D#v || Cx ||   ||   ||
-||   ||   || Gx ||   ||   ||
+|| Bvv || A#v || Gx ||   ||   ||
-||   ||   || Dx ||   ||   ||
+|| F#vv || E#v || Dx ||   ||   ||
+|| C#vv || B#v || Ax ||   ||   ||
@@ Line 1,010: / Line 1,013: @@
 fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:2130:&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:2130 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:2142:&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:2142 --&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 1,226: / Line 1,229: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:2131:&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:2131 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:2143:&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:2143 --&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:2132:&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:2132 --&gt;&lt;/h2&gt;
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@@ Line 3,010: / Line 3,013: @@
 Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:&lt;br /&gt;
 ...Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# ...&lt;br /&gt;
-&lt;br /&gt;
-This genchain suffices to name each note, but the note names will often run out of order. For example, in 22-tone, we might have C Db B# C# D, or C Db Ebb C# D. So ups and downs are needed to allow alternate names for each note. The first example becomes C C^&lt;br /&gt;
 &lt;br /&gt;
 Fifth-generated rank-2 tunings can be written without ups and downs in any EDO on the side of the 4\7 kite:&lt;br /&gt;
@@ Line 3,038: / Line 3,039: @@
 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;
 &lt;br /&gt;
-For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is 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. Here's the 22-tone genchain:&lt;br /&gt;
+For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is 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;
+&lt;br /&gt;
+The usual genchain suffices to name each note, but the note names will often run out of order. For example, in 22-tone, we might have C Db B# C# D. So ups and downs are needed to allow alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D.&lt;br /&gt;
+&lt;br /&gt;
+Here's the 22-tone genchain:&lt;br /&gt;
 &lt;br /&gt;
 Fb = Eb^ = D^^&lt;br /&gt;
@@ Line 3,110: / Line 3,115: @@
          &lt;td&gt;G^&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;F#^^&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 3,116: / Line 3,121: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Fbv&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Eb&lt;br /&gt;
@@ Line 3,122: / Line 3,127: @@
          &lt;td&gt;D^&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;C#^^&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 3,128: / Line 3,133: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Cbv&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Bb&lt;br /&gt;
@@ Line 3,134: / Line 3,139: @@
          &lt;td&gt;A^&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;G#^^&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 3,140: / Line 3,145: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Gbv&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;F&lt;br /&gt;
@@ Line 3,146: / Line 3,151: @@
          &lt;td&gt;E^&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;A#^^&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 3,152: / Line 3,157: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Dbv&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;C&lt;br /&gt;
@@ Line 3,158: / Line 3,163: @@
          &lt;td&gt;B^&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;E#^^&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 3,164: / Line 3,169: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Abv&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;G&lt;br /&gt;
@@ Line 3,170: / Line 3,175: @@
          &lt;td&gt;F#^&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;B#^^&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Fbvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Ebv&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;D&lt;br /&gt;
@@ Line 3,186: / Line 3,191: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Cbvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Bbv&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;A&lt;br /&gt;
@@ Line 3,198: / Line 3,203: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Gbvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Fv&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;E&lt;br /&gt;
@@ Line 3,210: / Line 3,215: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Dbvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Cv&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;B&lt;br /&gt;
@@ Line 3,222: / Line 3,227: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Abvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Gv&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;F#&lt;br /&gt;
@@ Line 3,234: / Line 3,239: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Ebvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Dv&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;C#&lt;br /&gt;
@@ Line 3,246: / Line 3,251: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Bbvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Av&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;G#&lt;br /&gt;
@@ Line 3,258: / Line 3,263: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Fvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Ev&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;D#&lt;br /&gt;
@@ Line 3,270: / Line 3,275: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Cvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Bv&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;A#&lt;br /&gt;
@@ Line 3,282: / Line 3,287: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Gvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;F#v&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;E#&lt;br /&gt;
@@ Line 3,294: / Line 3,299: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Dvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;C#v&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;B#&lt;br /&gt;
@@ Line 3,306: / Line 3,311: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Avv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;G#v&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Fx&lt;br /&gt;
@@ Line 3,318: / Line 3,323: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Evv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;D#v&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Cx&lt;br /&gt;
@@ Line 3,330: / Line 3,335: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Bvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;A#v&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Gx&lt;br /&gt;
@@ Line 3,342: / Line 3,347: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td&gt;F#vv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;E#v&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Dx&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;td&gt;Dx&lt;br /&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;C#vv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;B#v&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td&gt;Ax&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;