EDF: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:~~Kosmorsky~~|~~Kosmorsky~~]] and made on <tt>2011-12-~~26 16~~:09:12 UTC</tt>.

: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-12-27 01:09:16 UTC</tt>.

: The original revision id was <tt>~~288515662~~</tt>.

: The original revision id was <tt>288541854</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.

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[[18edf]]

[[19edf]]

[[Carlos Gamma|20 (Carlos Gamma)]]</pre></div>

[[Carlos Gamma|20 (Carlos Gamma)]]

=EDO-EDF correspondence=

||~ EDO ||~ EDF ||~ Comments ||

|| [[7edo]] || [[4edf]] || 4edf is 7edo with 28.5 cent stretched octaves.

Equivalently, 7edo is 4edf with 3/2s compressed by ~16 cents.

Patent vals match through the 5 limit. Only a rough correspondence. ||

|| [[8edo]] || || ||

|| [[9edo]] || [[5edf]] || Very rough correspondence - patent vals disagree in the 5 limit. ||

|| [[10edo]] || [[6edf]] || Also very rough. ||

|| [[11edo]] || || ||

|| [[12edo]] || [[7edf]] || 7edf is 12edo with 3.4 cent stretched octaves.

Equivalently, 12edo is 7edf with 2.0 cent compressed 3/2s.

With the exception of 11 (which falls almost exactly halfway between steps in both cases),

the patent vals match through the 31 limit, so the agreement is excellent. ||

|| [[13edo]] || || ||

|| || [[8edf]] || Since 88cET/octacot is well known to approximate some intervals quite accurately,

it would be wrong to lump this in with 14edo. ||

|| [[14edo]] || || ||

|| [[15edo]] || || ||

|| || [[9edf]] || The Carlos alpha scale is neither 15edo nor 16edo. ||

|| [[16edo]] || || ||

|| [[17edo]] || [[10edf]] || 10edf is 17edo with 6.6 cent compressed octaves.

Patent vals match through the 13 limit, with the exception of 5 (as expected). ||

|| [[18edo]] || || ||

|| [[19edo]] || [[11edf]] || 11edf is 19edo with 12.5 cent stretched octaves.

Patent vals match through the 7 limit.

If you don't think Carlos beta is accurately represented by 19edo then ignore this correspondence. ||

|| [[20edo]] || || ||

|| || [[12edf]] || 12edf entirely misses 2/1, but nails the "double octave" 4/1,

so it strongly resembles the scale with generator 2\41 of an octave. ||

|| [[21edo]] || || ||

|| [[22edo]] || || ||

|| || [[13edf]] || Perhaps surprisingly, this is not very similar to 22edo. Patent vals differ in the 5 limit. ||

|| [[23edo]] || || ||

|| [[24edo]] || [[14edf]] || 14edf is 24edo with 3.4 cent stretched octaves. Patent vals agree through the 19 limit. ||

|| [[25edo]] || || ||

|| [[26edo]] || [[15edf]] || Fairly rough correspondence. 15edf is 26edo with ~17 cent stretched octaves.

Patent vals agree through the 5 limit, but not through the 7 limit. ||

|| [[27edo]] || || ||

|| || [[16edf]] || ||

|| [[28edo]] || || ||

|| [[29edo]] || [[17edf]] || 17edf is 29edo with 2.5 cent compressed octaves. Patent vals disagree in the 7 limit. ||

|| [[30edo]] || || ||

|| || [[18edf]] || Perhaps surprisingly, this is not very similar to 31edo. Patent vals differ in the 5 limit. ||

|| [[31edo]] || || ||

|| [[32edo]] || || ||

|| || [[19edf]] || ||

|| [[33edo]] || || ||

|| [[34edo]] || [[20edf]] || 20edf is 34edo with 6.6 cent compressed octaves.

Patent vals match through the 5 limit, but not the 7 limit.

If you don't think Carlos gamma is accurately represented by 34edo then ignore this correspondence. ||</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>EDF</title></head><body><h1 id="toc0"><a name="Division of the perfect fifth (3/2) into n equal parts"></a>Division of the perfect fifth (3/2) into n equal parts</h1>

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Division of the 3:2 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of <a class="wiki_link" href="/equivalence">equivalence</a> is still in its infancy. The utility of 3:2 as a base though, is apparent by being one of the strongest consonances after the octave. Many, if not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.

Perhaps the first to divide the perfect fifth was <a class="wiki_link" href="/Wendy%20Carlos">Wendy Carlos</a>. ( <a class="wiki_link_ext" href="http://www.wendycarlos.com/resources/pitch.html" rel="nofollow">http://www.wendycarlos.com/resources/pitch.html</a> ) <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> has also made much use of the alpha beta and gamma scales.

Perhaps the first to divide the perfect fifth was <a class="wiki_link" href="/Wendy%20Carlos">Wendy Carlos</a>. ( <a class="wiki_link_ext" href="http://www.wendycarlos.com/resources/pitch.html" rel="nofollow">http://www.wendycarlos.com/resources/pitch.html</a> ) <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> has also made much use of the alpha beta and gamma scales.

Incidentally, one way to treat 3/2 as an equivalence is the use of the 8:9:10:(12) chord as exactly analogous to the 4:5:6:(8) chord in meantone. Doing this yields an 11 note MOS which 11edf tempers to equal, and 20edf is particularly good for. Actually, 11+9=20 so 9edf is also a related equal tuning of it. (&quot;Microdiatonic&quot; might be a good term for it, but in any case its what this is in: <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=x_HSMND6RnA" rel="nofollow">http://www.youtube.com/watch?v=x_HSMND6RnA</a> )

Incidentally, one way to treat 3/2 as an equivalence is the use of the 8:9:10:(12) chord as exactly analogous to the 4:5:6:(8) chord in meantone. Doing this yields an 11 note MOS which 11edf tempers to equal, and 20edf is particularly good for. Actually, 11+9=20 so 9edf is also a related equal tuning of it. (&quot;Microdiatonic&quot; might be a good term for it, but in any case its what this is in: <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=x_HSMND6RnA" rel="nofollow">http://www.youtube.com/watch?v=x_HSMND6RnA</a> )

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<a class="wiki_link" href="/Carlos%20Gamma">20 (Carlos Gamma)</a></body></html></pre></div>

<a class="wiki_link" href="/Carlos%20Gamma">20 (Carlos Gamma)</a>

<h1 id="toc1"><a name="EDO-EDF correspondence"></a>EDO-EDF correspondence</h1>

<tr>

<th>EDO

</th>

<th>EDF

</th>

<th>Comments

</th>

</tr>

<tr>

</td>

</td>

<td>4edf is 7edo with 28.5 cent stretched octaves.

Equivalently, 7edo is 4edf with 3/2s compressed by ~16 cents.

Patent vals match through the 5 limit. Only a rough correspondence.

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

</td>

</td>

<td>Very rough correspondence - patent vals disagree in the 5 limit.

</td>

</tr>

<tr>

</td>

</td>

<td>Also very rough.

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

</td>

</td>

<td>7edf is 12edo with 3.4 cent stretched octaves.

Equivalently, 12edo is 7edf with 2.0 cent compressed 3/2s.

With the exception of 11 (which falls almost exactly halfway between steps in both cases),

the patent vals match through the 31 limit, so the agreement is excellent.

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>

</td>

</td>

<td>Since 88cET/octacot is well known to approximate some intervals quite accurately,

it would be wrong to lump this in with 14edo.

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>

</td>

</td>

<td>The Carlos alpha scale is neither 15edo nor 16edo.

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

</td>

</td>

<td>10edf is 17edo with 6.6 cent compressed octaves.

Patent vals match through the 13 limit, with the exception of 5 (as expected).

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

</td>

</td>

<td>11edf is 19edo with 12.5 cent stretched octaves.

Patent vals match through the 7 limit.

If you don't think Carlos beta is accurately represented by 19edo then ignore this correspondence.

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>

</td>

</td>

<td>12edf entirely misses 2/1, but nails the &quot;double octave&quot; 4/1,

so it strongly resembles the scale with generator 2\41 of an octave.

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>

</td>

</td>

<td>Perhaps surprisingly, this is not very similar to 22edo. Patent vals differ in the 5 limit.

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

</td>

</td>

<td>14edf is 24edo with 3.4 cent stretched octaves. Patent vals agree through the 19 limit.

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

</td>

</td>

<td>Fairly rough correspondence. 15edf is 26edo with ~17 cent stretched octaves.

Patent vals agree through the 5 limit, but not through the 7 limit.

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>

</td>

</td>

<td>

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

</td>

</td>

<td>17edf is 29edo with 2.5 cent compressed octaves. Patent vals disagree in the 7 limit.

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>

</td>

</td>

<td>Perhaps surprisingly, this is not very similar to 31edo. Patent vals differ in the 5 limit.

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>

</td>

</td>

<td>

</td>

</tr>

<tr>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

</td>

</td>

<td>20edf is 34edo with 6.6 cent compressed octaves.

Patent vals match through the 5 limit, but not the 7 limit.

If you don't think Carlos gamma is accurately represented by 34edo then ignore this correspondence.

</td>

</tr>

</table>

</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:Kosmorsky|Kosmorsky]] and made on <tt>2011-12-26 16:09:12 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-12-27 01:09:16 UTC</tt>.<br>
-: The original revision id was <tt>288515662</tt>.<br>
+: The original revision id was <tt>288541854</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 31: / Line 31: @@
 [[18edf]]
 [[19edf]]
-[[Carlos Gamma|20 (Carlos Gamma)]]</pre></div>
+[[Carlos Gamma|20 (Carlos Gamma)]]
+=EDO-EDF correspondence=
+||~ EDO ||~ EDF ||~ Comments ||
+|| [[7edo]] || [[4edf]] || 4edf is 7edo with 28.5 cent stretched octaves.
+Equivalently, 7edo is 4edf with 3/2s compressed by ~16 cents.
+Patent vals match through the 5 limit. Only a rough correspondence. ||
+|| [[8edo]] ||   ||   ||
+|| [[9edo]] || [[5edf]] || Very rough correspondence - patent vals disagree in the 5 limit. ||
+|| [[10edo]] || [[6edf]] || Also very rough. ||
+|| [[11edo]] ||   ||   ||
+|| [[12edo]] || [[7edf]] || 7edf is 12edo with 3.4 cent stretched octaves.
+Equivalently, 12edo is 7edf with 2.0 cent compressed 3/2s.
+With the exception of 11 (which falls almost exactly halfway between steps in both cases),
+the patent vals match through the 31 limit, so the agreement is excellent. ||
+|| [[13edo]] ||   ||   ||
+||   || [[8edf]] || Since 88cET/octacot is well known to approximate some intervals quite accurately,
+it would be wrong to lump this in with 14edo. ||
+|| [[14edo]] ||   ||   ||
+|| [[15edo]] ||   ||   ||
+||   || [[9edf]] || The Carlos alpha scale is neither 15edo nor 16edo. ||
+|| [[16edo]] ||   ||   ||
+|| [[17edo]] || [[10edf]] || 10edf is 17edo with 6.6 cent compressed octaves.
+Patent vals match through the 13 limit, with the exception of 5 (as expected). ||
+|| [[18edo]] ||   ||   ||
+|| [[19edo]] || [[11edf]] || 11edf is 19edo with 12.5 cent stretched octaves.
+Patent vals match through the 7 limit.
+If you don't think Carlos beta is accurately represented by 19edo then ignore this correspondence. ||
+|| [[20edo]] ||   ||   ||
+||   || [[12edf]] || 12edf entirely misses 2/1, but nails the "double octave" 4/1,
+so it strongly resembles the scale with generator 2\41 of an octave. ||
+|| [[21edo]] ||   ||   ||
+|| [[22edo]] ||   ||   ||
+||   || [[13edf]] || Perhaps surprisingly, this is not very similar to 22edo. Patent vals differ in the 5 limit. ||
+|| [[23edo]] ||   ||   ||
+|| [[24edo]] || [[14edf]] || 14edf is 24edo with 3.4 cent stretched octaves. Patent vals agree through the 19 limit. ||
+|| [[25edo]] ||   ||   ||
+|| [[26edo]] || [[15edf]] || Fairly rough correspondence. 15edf is 26edo with ~17 cent stretched octaves.
+Patent vals agree through the 5 limit, but not through the 7 limit. ||
+|| [[27edo]] ||   ||   ||
+||   || [[16edf]] ||   ||
+|| [[28edo]] ||   ||   ||
+|| [[29edo]] || [[17edf]] || 17edf is 29edo with 2.5 cent compressed octaves. Patent vals disagree in the 7 limit. ||
+|| [[30edo]] ||   ||   ||
+||   || [[18edf]] || Perhaps surprisingly, this is not very similar to 31edo. Patent vals differ in the 5 limit. ||
+|| [[31edo]] ||   ||   ||
+|| [[32edo]] ||   ||   ||
+||   || [[19edf]] ||   ||
+|| [[33edo]] ||   ||   ||
+|| [[34edo]] || [[20edf]] || 20edf is 34edo with 6.6 cent compressed octaves.
+Patent vals match through the 5 limit, but not the 7 limit.
+If you don't think Carlos gamma is accurately represented by 34edo then ignore this correspondence. ||</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;EDF&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="Division of the perfect fifth (3/2) into n equal parts"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Division of the perfect fifth (3/2) into n equal parts&lt;/h1&gt;
@@ Line 37: / Line 88: @@
 Division of the 3:2 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of &lt;a class="wiki_link" href="/equivalence"&gt;equivalence&lt;/a&gt; is still in its infancy. The utility of 3:2 as a base though, is apparent by being one of the strongest consonances after the octave. Many, if not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.&lt;br /&gt;
 &lt;br /&gt;
-Perhaps the first to divide the perfect fifth was &lt;a class="wiki_link" href="/Wendy%20Carlos"&gt;Wendy Carlos&lt;/a&gt;. ( &lt;!-- ws:start:WikiTextUrlRule:47:http://www.wendycarlos.com/resources/pitch.html --&gt;&lt;a class="wiki_link_ext" href="http://www.wendycarlos.com/resources/pitch.html" rel="nofollow"&gt;http://www.wendycarlos.com/resources/pitch.html&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:47 --&gt; ) &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; has also made much use of the alpha beta and gamma scales.&lt;br /&gt;
+Perhaps the first to divide the perfect fifth was &lt;a class="wiki_link" href="/Wendy%20Carlos"&gt;Wendy Carlos&lt;/a&gt;. ( &lt;!-- ws:start:WikiTextUrlRule:507:http://www.wendycarlos.com/resources/pitch.html --&gt;&lt;a class="wiki_link_ext" href="http://www.wendycarlos.com/resources/pitch.html" rel="nofollow"&gt;http://www.wendycarlos.com/resources/pitch.html&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:507 --&gt; ) &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; has also made much use of the alpha beta and gamma scales.&lt;br /&gt;
 &lt;br /&gt;
-Incidentally, one way to treat 3/2 as an equivalence is the use of the 8:9:10:(12) chord as exactly analogous to the 4:5:6:(8) chord in meantone. Doing this yields an 11 note MOS which 11edf tempers to equal, and 20edf is particularly good for. Actually, 11+9=20 so 9edf is also a related equal tuning of it. (&amp;quot;Microdiatonic&amp;quot; might be a good term for it, but in any case its what this is in: &lt;!-- ws:start:WikiTextUrlRule:48:http://www.youtube.com/watch?v=x_HSMND6RnA --&gt;&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=x_HSMND6RnA" rel="nofollow"&gt;http://www.youtube.com/watch?v=x_HSMND6RnA&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:48 --&gt; )&lt;br /&gt;
+Incidentally, one way to treat 3/2 as an equivalence is the use of the 8:9:10:(12) chord as exactly analogous to the 4:5:6:(8) chord in meantone. Doing this yields an 11 note MOS which 11edf tempers to equal, and 20edf is particularly good for. Actually, 11+9=20 so 9edf is also a related equal tuning of it. (&amp;quot;Microdiatonic&amp;quot; might be a good term for it, but in any case its what this is in: &lt;!-- ws:start:WikiTextUrlRule:508:http://www.youtube.com/watch?v=x_HSMND6RnA --&gt;&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=x_HSMND6RnA" rel="nofollow"&gt;http://www.youtube.com/watch?v=x_HSMND6RnA&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:508 --&gt; )&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
@@ Line 58: / Line 109: @@
 &lt;a class="wiki_link" href="/18edf"&gt;18edf&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/19edf"&gt;19edf&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link" href="/Carlos%20Gamma"&gt;20 (Carlos Gamma)&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;a class="wiki_link" href="/Carlos%20Gamma"&gt;20 (Carlos Gamma)&lt;/a&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="EDO-EDF correspondence"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;EDO-EDF correspondence&lt;/h1&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;EDO&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;EDF&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Comments&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/4edf"&gt;4edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;4edf is 7edo with 28.5 cent stretched octaves.&lt;br /&gt;
+Equivalently, 7edo is 4edf with 3/2s compressed by ~16 cents.&lt;br /&gt;
+Patent vals match through the 5 limit. Only a rough correspondence.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/8edo"&gt;8edo&lt;/a&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&gt;&lt;a class="wiki_link" href="/9edo"&gt;9edo&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/5edf"&gt;5edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Very rough correspondence - patent vals disagree in the 5 limit.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/10edo"&gt;10edo&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/6edf"&gt;6edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Also very rough.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/11edo"&gt;11edo&lt;/a&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&gt;&lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/7edf"&gt;7edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7edf is 12edo with 3.4 cent stretched octaves.&lt;br /&gt;
+Equivalently, 12edo is 7edf with 2.0 cent compressed 3/2s.&lt;br /&gt;
+With the exception of 11 (which falls almost exactly halfway between steps in both cases),&lt;br /&gt;
+the patent vals match through the 31 limit, so the agreement is excellent.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/13edo"&gt;13edo&lt;/a&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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/8edf"&gt;8edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Since 88cET/octacot is well known to approximate some intervals quite accurately,&lt;br /&gt;
+it would be wrong to lump this in with 14edo.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/14edo"&gt;14edo&lt;/a&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&gt;&lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/9edf"&gt;9edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;The Carlos alpha scale is neither 15edo nor 16edo.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/16edo"&gt;16edo&lt;/a&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&gt;&lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/10edf"&gt;10edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;10edf is 17edo with 6.6 cent compressed octaves.&lt;br /&gt;
+Patent vals match through the 13 limit, with the exception of 5 (as expected).&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/18edo"&gt;18edo&lt;/a&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&gt;&lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/11edf"&gt;11edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;11edf is 19edo with 12.5 cent stretched octaves.&lt;br /&gt;
+Patent vals match through the 7 limit.&lt;br /&gt;
+If you don't think Carlos beta is accurately represented by 19edo then ignore this correspondence.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/20edo"&gt;20edo&lt;/a&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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/12edf"&gt;12edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;12edf entirely misses 2/1, but nails the &amp;quot;double octave&amp;quot; 4/1,&lt;br /&gt;
+so it strongly resembles the scale with generator 2\41 of an octave.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/21edo"&gt;21edo&lt;/a&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&gt;&lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/13edf"&gt;13edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Perhaps surprisingly, this is not very similar to 22edo. Patent vals differ in the 5 limit.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&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&gt;&lt;a class="wiki_link" href="/24edo"&gt;24edo&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/14edf"&gt;14edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;14edf is 24edo with 3.4 cent stretched octaves. Patent vals agree through the 19 limit.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/25edo"&gt;25edo&lt;/a&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&gt;&lt;a class="wiki_link" href="/26edo"&gt;26edo&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/15edf"&gt;15edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Fairly rough correspondence. 15edf is 26edo with ~17 cent stretched octaves.&lt;br /&gt;
+Patent vals agree through the 5 limit, but not through the 7 limit.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/27edo"&gt;27edo&lt;/a&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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/16edf"&gt;16edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/28edo"&gt;28edo&lt;/a&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&gt;&lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/17edf"&gt;17edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;17edf is 29edo with 2.5 cent compressed octaves. Patent vals disagree in the 7 limit.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/30edo"&gt;30edo&lt;/a&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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/18edf"&gt;18edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Perhaps surprisingly, this is not very similar to 31edo. Patent vals differ in the 5 limit.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&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&gt;&lt;a class="wiki_link" href="/32edo"&gt;32edo&lt;/a&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&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/19edf"&gt;19edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/33edo"&gt;33edo&lt;/a&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&gt;&lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/20edf"&gt;20edf&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;20edf is 34edo with 6.6 cent compressed octaves.&lt;br /&gt;
+Patent vals match through the 5 limit, but not the 7 limit.&lt;br /&gt;
+If you don't think Carlos gamma is accurately represented by 34edo then ignore this correspondence.&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;/body&gt;&lt;/html&gt;</pre></div>