EDF
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=Division of the perfect fifth (3/2) into n equal parts=
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 [[equivalence]] 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 [[Wendy Carlos]]. ( http://www.wendycarlos.com/resources/pitch.html ) [[Carlo Serafini]] 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. ("Microdiatonic" might be a good term for it, but in any case its what this is in: http://www.youtube.com/watch?v=x_HSMND6RnA )
[[4edf]]
[[5edf]]
[[6edf]]
[[7edf]]
[[8edf]] ([[88cET]])
[[Carlos Alpha|9 (Carlos Alpha)]]
[[10edf]]
[[Carlos Beta|11 (Carlos Beta)]]
[[12edf]]
[[13edf]]
[[14edf]]
[[15edf]]
[[16edf]]
[[17edf]]
[[18edf]]
[[19edf]]
[[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. ||Original HTML content:
<html><head><title>EDF</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Division of the perfect fifth (3/2) into n equal parts"></a><!-- ws:end:WikiTextHeadingRule:0 -->Division of the perfect fifth (3/2) into n equal parts</h1>
<br />
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.<br />
<br />
Perhaps the first to divide the perfect fifth was <a class="wiki_link" href="/Wendy%20Carlos">Wendy Carlos</a>. ( <!-- ws:start:WikiTextUrlRule:507:http://www.wendycarlos.com/resources/pitch.html --><a class="wiki_link_ext" href="http://www.wendycarlos.com/resources/pitch.html" rel="nofollow">http://www.wendycarlos.com/resources/pitch.html</a><!-- ws:end:WikiTextUrlRule:507 --> ) <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> has also made much use of the alpha beta and gamma scales.<br />
<br />
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. ("Microdiatonic" might be a good term for it, but in any case its what this is in: <!-- ws:start:WikiTextUrlRule:508:http://www.youtube.com/watch?v=x_HSMND6RnA --><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><!-- ws:end:WikiTextUrlRule:508 --> )<br />
<br />
<br />
<a class="wiki_link" href="/4edf">4edf</a><br />
<a class="wiki_link" href="/5edf">5edf</a><br />
<a class="wiki_link" href="/6edf">6edf</a><br />
<a class="wiki_link" href="/7edf">7edf</a><br />
<a class="wiki_link" href="/8edf">8edf</a> (<a class="wiki_link" href="/88cET">88cET</a>)<br />
<a class="wiki_link" href="/Carlos%20Alpha">9 (Carlos Alpha)</a><br />
<a class="wiki_link" href="/10edf">10edf</a><br />
<a class="wiki_link" href="/Carlos%20Beta">11 (Carlos Beta)</a><br />
<a class="wiki_link" href="/12edf">12edf</a><br />
<a class="wiki_link" href="/13edf">13edf</a><br />
<a class="wiki_link" href="/14edf">14edf</a><br />
<a class="wiki_link" href="/15edf">15edf</a><br />
<a class="wiki_link" href="/16edf">16edf</a><br />
<a class="wiki_link" href="/17edf">17edf</a><br />
<a class="wiki_link" href="/18edf">18edf</a><br />
<a class="wiki_link" href="/19edf">19edf</a><br />
<a class="wiki_link" href="/Carlos%20Gamma">20 (Carlos Gamma)</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="EDO-EDF correspondence"></a><!-- ws:end:WikiTextHeadingRule:2 -->EDO-EDF correspondence</h1>
<table class="wiki_table">
<tr>
<th>EDO<br />
</th>
<th>EDF<br />
</th>
<th>Comments<br />
</th>
</tr>
<tr>
<td><a class="wiki_link" href="/7edo">7edo</a><br />
</td>
<td><a class="wiki_link" href="/4edf">4edf</a><br />
</td>
<td>4edf is 7edo with 28.5 cent stretched octaves.<br />
Equivalently, 7edo is 4edf with 3/2s compressed by ~16 cents.<br />
Patent vals match through the 5 limit. Only a rough correspondence.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/8edo">8edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/9edo">9edo</a><br />
</td>
<td><a class="wiki_link" href="/5edf">5edf</a><br />
</td>
<td>Very rough correspondence - patent vals disagree in the 5 limit.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/10edo">10edo</a><br />
</td>
<td><a class="wiki_link" href="/6edf">6edf</a><br />
</td>
<td>Also very rough.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/11edo">11edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/12edo">12edo</a><br />
</td>
<td><a class="wiki_link" href="/7edf">7edf</a><br />
</td>
<td>7edf is 12edo with 3.4 cent stretched octaves.<br />
Equivalently, 12edo is 7edf with 2.0 cent compressed 3/2s.<br />
With the exception of 11 (which falls almost exactly halfway between steps in both cases),<br />
the patent vals match through the 31 limit, so the agreement is excellent.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/13edo">13edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/8edf">8edf</a><br />
</td>
<td>Since 88cET/octacot is well known to approximate some intervals quite accurately,<br />
it would be wrong to lump this in with 14edo.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/14edo">14edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/15edo">15edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/9edf">9edf</a><br />
</td>
<td>The Carlos alpha scale is neither 15edo nor 16edo.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/16edo">16edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/17edo">17edo</a><br />
</td>
<td><a class="wiki_link" href="/10edf">10edf</a><br />
</td>
<td>10edf is 17edo with 6.6 cent compressed octaves.<br />
Patent vals match through the 13 limit, with the exception of 5 (as expected).<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/18edo">18edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/19edo">19edo</a><br />
</td>
<td><a class="wiki_link" href="/11edf">11edf</a><br />
</td>
<td>11edf is 19edo with 12.5 cent stretched octaves.<br />
Patent vals match through the 7 limit.<br />
If you don't think Carlos beta is accurately represented by 19edo then ignore this correspondence.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/20edo">20edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/12edf">12edf</a><br />
</td>
<td>12edf entirely misses 2/1, but nails the "double octave" 4/1,<br />
so it strongly resembles the scale with generator 2\41 of an octave.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/21edo">21edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/22edo">22edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/13edf">13edf</a><br />
</td>
<td>Perhaps surprisingly, this is not very similar to 22edo. Patent vals differ in the 5 limit.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/23edo">23edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/24edo">24edo</a><br />
</td>
<td><a class="wiki_link" href="/14edf">14edf</a><br />
</td>
<td>14edf is 24edo with 3.4 cent stretched octaves. Patent vals agree through the 19 limit.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/25edo">25edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/26edo">26edo</a><br />
</td>
<td><a class="wiki_link" href="/15edf">15edf</a><br />
</td>
<td>Fairly rough correspondence. 15edf is 26edo with ~17 cent stretched octaves.<br />
Patent vals agree through the 5 limit, but not through the 7 limit.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/27edo">27edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/16edf">16edf</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/28edo">28edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/29edo">29edo</a><br />
</td>
<td><a class="wiki_link" href="/17edf">17edf</a><br />
</td>
<td>17edf is 29edo with 2.5 cent compressed octaves. Patent vals disagree in the 7 limit.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/30edo">30edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/18edf">18edf</a><br />
</td>
<td>Perhaps surprisingly, this is not very similar to 31edo. Patent vals differ in the 5 limit.<br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/31edo">31edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/32edo">32edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><a class="wiki_link" href="/19edf">19edf</a><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/33edo">33edo</a><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/34edo">34edo</a><br />
</td>
<td><a class="wiki_link" href="/20edf">20edf</a><br />
</td>
<td>20edf is 34edo with 6.6 cent compressed octaves.<br />
Patent vals match through the 5 limit, but not the 7 limit.<br />
If you don't think Carlos gamma is accurately represented by 34edo then ignore this correspondence.<br />
</td>
</tr>
</table>
</body></html>