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:<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-03 16:03:31 UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-03 16:42:16 UTC</tt>.<br>

: The original revision id was <tt>~~584770275~~</tt>.<br>

: The original revision id was <tt>584772157</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 761:

==__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 part V of my 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.</pre></div>

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).

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

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:

K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)

K(#) = X, K(b) = -X (X = keyspan of the sharp symbol, i.e., how many keys wide it is. For 22-tone, X = 3)

K(#vX) = K(#) + X * K(v) = 0 (going up X keys using a sharp, then going down X keys using X downs, must cancel out)

"v3" means three downs. "#vX" means one sharp plus X downs. Nonzero keyspans in the genchain always occur every N steps for a N-tone framework. E.g., 12-tone keyspans:

|| C || G || D || A || E || B || F# || C# || G# || D# || A# || E# || B# ||

|| 0 || 7 || 2 || 9 || 4 || 11 || 6 || 1 || 8 || 3 || 10 || 5 || 0 ||

Thus the final equation means that the genspan resulting from going up a sharp and down X downs must be either zero, N, -N, 2N, -2N, etc.

G(#) = 7 (by definition, the sharp's genspan = 7, assuming heptatonic notation)

G(#vX) = G(#) + X * G(v) = G(#) - X * G(^) = 7 - X * G(^)

G(#vX) mod N = 0, thus G(#vX) = i * N for some integer i

7 - X * G(^) = i * N

G(^) = - (i * N - 7) / X

For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. Thus i = 1, G(^) = -5, and ^ = min 2nd.

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>

<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 855:

Line 880:

fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢<br />

<br />

<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;" /><br />

<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;" /><br />

<br />

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.<br />

Line 1,041:

Line 1,066:

<br />

<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;" /><br />

<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;" /><br />

<br />

<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 2,821:

Line 2,846:

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

<br />

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 part V of my book for more on frameworks.) <br />

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.) <br />

<br />

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).<br />

<br />

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

<br />

To extend ups and downs to rank-2 tunings, the up symbol is given not only a <strong>keyspan</strong> (always +1) but also a <strong>genspan</strong>, which indicates how many steps forward or backwards along the generator chain, or <strong>genchain</strong>, 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:<br />

<br />

K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)<br />

K(#) = X, K(b) = -X (X = keyspan of the sharp symbol, i.e., how many keys wide it is. For 22-tone, X = 3)<br />

K(#vX) = K(#) + X * K(v) = 0 (going up X keys using a sharp, then going down X keys using X downs, must cancel out)<br />

<br />

&quot;v3&quot; means three downs. &quot;#vX&quot; means one sharp plus X downs. Nonzero keyspans in the genchain always occur every N steps for a N-tone framework. E.g., 12-tone keyspans:<br />

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

</table>

Thus the final equation means that the genspan resulting from going up a sharp and down X downs must be either zero, N, -N, 2N, -2N, etc.<br />

<br />

G(#) = 7 (by definition, the sharp's genspan = 7, assuming heptatonic notation)<br />

G(#vX) = G(#) + X * G(v) = G(#) - X * G(^) = 7 - X * G(^)<br />

G(#vX) mod N = 0, thus G(#vX) = i * N for some integer i<br />

7 - X * G(^) = i * N<br />

G(^) = - (i * N - 7) / X<br />

<br />

For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. Thus i = 1, G(^) = -5, and ^ = min 2nd.<br />

<br />

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

<br />

For ~~rank~~-2 ~~scales to work with a given framework~~, ~~the keyspans of the generator~~ and ~~the period must be coprime~~.</body></html></pre></div>

For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd.</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>2016-06-03 16:03:31 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-03 16:42:16 UTC</tt>.<br>
-: The original revision id was <tt>584770275</tt>.<br>
+: The original revision id was <tt>584772157</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 761: / Line 761: @@
 ==__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 part V of my 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.</pre></div>
+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).
+note to self: if K(#) is 1 or -1, ups and downs aren't needed to notate rank-2.
+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:
+K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)
+K(#) = X, K(b) = -X (X = keyspan of the sharp symbol, i.e., how many keys wide it is. For 22-tone, X = 3)
+K(#vX) = K(#) + X * K(v) = 0 (going up X keys using a sharp, then going down X keys using X downs, must cancel out)
+"v3" means three downs. "#vX" means one sharp plus X downs. Nonzero keyspans in the genchain always occur every N steps for a N-tone framework. E.g., 12-tone keyspans:
+|| C || G || D || A || E || B || F# || C# || G# || D# || A# || E# || B# ||
+|| 0 || 7 || 2 || 9 || 4 || 11 || 6 || 1 || 8 || 3 || 10 || 5 || 0 ||
+Thus the final equation means that the genspan resulting from going up a sharp and down X downs must be either zero, N, -N, 2N, -2N, etc.
+G(#) = 7 (by definition, the sharp's genspan = 7, assuming heptatonic notation)
+G(#vX) = G(#) + X * G(v) = G(#) - X * G(^) = 7 - X * G(^)
+G(#vX) mod N = 0, thus G(#vX) = i * N for some integer i
+- X * G(^) = i * N
+G(^) = - (i * N - 7) / X
+For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. Thus i = 1, G(^) = -5, and ^ = min 2nd.
+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>
 <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 855: / Line 880: @@
 fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:1384:&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:1384 --&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;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 1,041: / Line 1,066: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:1385:&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:1385 --&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;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:1386:&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:1386 --&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: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: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 2,821: / Line 2,846: @@
 &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 part V of my 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. 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;
+&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;
+&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;
+&lt;br /&gt;
+K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)&lt;br /&gt;
+K(#) = X, K(b) = -X (X = keyspan of the sharp symbol, i.e., how many keys wide it is. For 22-tone, X = 3)&lt;br /&gt;
+K(#vX) = K(#) + X * K(v) = 0 (going up X keys using a sharp, then going down X keys using X downs, must cancel out)&lt;br /&gt;
+&lt;br /&gt;
+&amp;quot;v3&amp;quot; means three downs. &amp;quot;#vX&amp;quot; means one sharp plus X downs. Nonzero keyspans in the genchain always occur every N steps for a N-tone framework. E.g., 12-tone keyspans:&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;C&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;G&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;D&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;A&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;E&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;B&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;F#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;G#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;D#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;A#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;E#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;B#&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+Thus the final equation means that the genspan resulting from going up a sharp and down X downs must be either zero, N, -N, 2N, -2N, etc.&lt;br /&gt;
+&lt;br /&gt;
+G(#) = 7 (by definition, the sharp's genspan = 7, assuming heptatonic notation)&lt;br /&gt;
+G(#vX) = G(#) + X * G(v) = G(#) - X * G(^) = 7 - X * G(^)&lt;br /&gt;
+G(#vX) mod N = 0, thus G(#vX) = i * N for some integer i&lt;br /&gt;
+- X * G(^) = i * N&lt;br /&gt;
+G(^) = - (i * N - 7) / X&lt;br /&gt;
+&lt;br /&gt;
+For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. Thus i = 1, G(^) = -5, and ^ = min 2nd.&lt;br /&gt;
+&lt;br /&gt;
+For 17-tone, X = 2, i = 1, G(^) = -5, and ^ = min 2nd&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.&lt;/body&gt;&lt;/html&gt;</pre></div>
+For 31-tone, X = 2, i = 1, G(^) = -12, and ^ = dim 2nd.&lt;/body&gt;&lt;/html&gt;</pre></div>