EDO: 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:genewardsmith|genewardsmith]] and made on <tt>2011-08-14 00:21:35 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-14 00:24:12 UTC</tt>.

: The original revision id was <tt>~~245828887~~</tt>.

: The original revision id was <tt>245829091</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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Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). This really amounts to the consideration of adding the associated //vals//, which are the mappings to primes larger than 2. An EDO is defined by a certain number of steps equating to 2; if we have more steps equating to 3, we get a 3-limit val, and so forth. So, for example, 12edo can be written <12|, saying that twelve steps maps to 2, but the 3-limit val for 12 is <12 19|, telling us that 19 steps maps to 3, and the 5-limit val is <12 19 28|, telling us that 28 steps maps to 5.

If we add 12 and 19 we get another good division, 12 + 19 = 31. We can understand why this works if we look at it as adding vals; <12 19 28| + <19 30 44| = <31 49 72|. The relative error in terms of [[Relative cent|relative cents]] is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is [-1.955 13.686] (the same as absolute cents) and the error of 19edo is [-11.429 -11.663], and this sums to [-13.384 2.023]. In relative cents the error of the fifth is not much increased, and on converting to absolute cents even shrinks, and the error of the fifth is much smaller. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 13) can have a much smaller error due to the cancellation.

If we add 12 and 19 we get another good division, 12 + 19 = 31. We can understand why this works if we look at it as adding vals; <12 19 28| + <19 30 44| = <31 49 72|. The relative error in terms of [[Relative cent|relative cents]] is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is [-1.955 13.686] (the same as absolute cents) and the error of 19edo is [-11.429 -11.663], and this sums to [-13.384 2.023]. In relative cents the error of the fifth is not much increased, and on converting to absolute cents even shrinks, and the error of the fifth is much smaller. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 31edo) can have a much smaller error due to the cancellation.

We may also look at addition of EDOs in terms of MOS; if a\n is a generator for an n-edo MOS, and b\m for an m-EDO MOS, where both of these are generators for the same linear temperament, then the mediant, (a+b)\(n+m), will be a generator for a MOS for the same temperament, this time in (n+m)-edo.

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Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). This really amounts to the consideration of adding the associated vals, which are the mappings to primes larger than 2. An EDO is defined by a certain number of steps equating to 2; if we have more steps equating to 3, we get a 3-limit val, and so forth. So, for example, 12edo can be written &lt;12|, saying that twelve steps maps to 2, but the 3-limit val for 12 is &lt;12 19|, telling us that 19 steps maps to 3, and the 5-limit val is &lt;12 19 28|, telling us that 28 steps maps to 5.

If we add 12 and 19 we get another good division, 12 + 19 = 31. We can understand why this works if we look at it as adding vals; &lt;12 19 28| + &lt;19 30 44| = &lt;31 49 72|. The relative error in terms of <a class="wiki_link" href="/Relative%20cent">relative cents</a> is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is [-1.955 13.686] (the same as absolute cents) and the error of 19edo is [-11.429 -11.663], and this sums to [-13.384 2.023]. In relative cents the error of the fifth is not much increased, and on converting to absolute cents even shrinks, and the error of the fifth is much smaller. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 13) can have a much smaller error due to the cancellation.

If we add 12 and 19 we get another good division, 12 + 19 = 31. We can understand why this works if we look at it as adding vals; &lt;12 19 28| + &lt;19 30 44| = &lt;31 49 72|. The relative error in terms of <a class="wiki_link" href="/Relative%20cent">relative cents</a> is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is [-1.955 13.686] (the same as absolute cents) and the error of 19edo is [-11.429 -11.663], and this sums to [-13.384 2.023]. In relative cents the error of the fifth is not much increased, and on converting to absolute cents even shrinks, and the error of the fifth is much smaller. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 31edo) can have a much smaller error due to the cancellation.

We may also look at addition of EDOs in terms of MOS; if a\n is a generator for an n-edo MOS, and b\m for an m-EDO MOS, where both of these are generators for the same linear temperament, then the mediant, (a+b)\(n+m), will be a generator for a MOS for the same temperament, this time in (n+m)-edo.

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2011-08-14 00:21:35 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-14 00:24:12 UTC</tt>.<br>
-: The original revision id was <tt>245828887</tt>.<br>
+: The original revision id was <tt>245829091</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 40: / Line 40: @@
 Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). This really amounts to the consideration of adding the associated //vals//, which are the mappings to primes larger than 2. An EDO is defined by a certain number of steps equating to 2; if we have more steps equating to 3, we get a 3-limit val, and so forth. So, for example, 12edo can be written &lt;12|, saying that twelve steps maps to 2, but the 3-limit val for 12 is &lt;12 19|, telling us that 19 steps maps to 3, and the 5-limit val is &lt;12 19 28|, telling us that 28 steps maps to 5.
-If we add 12 and 19 we get another good division, 12 + 19 = 31. We can understand why this works if we look at it as adding vals; &lt;12 19 28| + &lt;19 30 44| = &lt;31 49 72|. The relative error in terms of [[Relative cent|relative cents]] is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is [-1.955 13.686] (the same as absolute cents) and the error of 19edo is [-11.429 -11.663], and this sums to [-13.384 2.023]. In relative cents the error of the fifth is not much increased, and on converting to absolute cents even shrinks, and the error of the fifth is much smaller. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 13) can have a much smaller error due to the cancellation.
+If we add 12 and 19 we get another good division, 12 + 19 = 31. We can understand why this works if we look at it as adding vals; &lt;12 19 28| + &lt;19 30 44| = &lt;31 49 72|. The relative error in terms of [[Relative cent|relative cents]] is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is [-1.955 13.686] (the same as absolute cents) and the error of 19edo is [-11.429 -11.663], and this sums to [-13.384 2.023]. In relative cents the error of the fifth is not much increased, and on converting to absolute cents even shrinks, and the error of the fifth is much smaller. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 31edo) can have a much smaller error due to the cancellation.
 We may also look at addition of EDOs in terms of MOS; if a\n is a generator for an n-edo MOS, and b\m for an m-EDO MOS, where both of these are generators for the same linear temperament, then the mediant, (a+b)\(n+m), will be a generator for a MOS for the same temperament, this time in (n+m)-edo.
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 Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). This really amounts to the consideration of adding the associated &lt;em&gt;vals&lt;/em&gt;, which are the mappings to primes larger than 2. An EDO is defined by a certain number of steps equating to 2; if we have more steps equating to 3, we get a 3-limit val, and so forth. So, for example, 12edo can be written &amp;lt;12|, saying that twelve steps maps to 2, but the 3-limit val for 12 is &amp;lt;12 19|, telling us that 19 steps maps to 3, and the 5-limit val is &amp;lt;12 19 28|, telling us that 28 steps maps to 5. &lt;br /&gt;
 &lt;br /&gt;
-If we add 12 and 19 we get another good division, 12 + 19 = 31. We can understand why this works if we look at it as adding vals; &amp;lt;12 19 28| + &amp;lt;19 30 44| = &amp;lt;31 49 72|. The relative error in terms of &lt;a class="wiki_link" href="/Relative%20cent"&gt;relative cents&lt;/a&gt; is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is [-1.955 13.686] (the same as absolute cents) and the error of 19edo is [-11.429 -11.663], and this sums to [-13.384 2.023]. In relative cents the error of the fifth is not much increased, and on converting to absolute cents even shrinks, and the error of the fifth is much smaller. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 13) can have a much smaller error due to the cancellation.&lt;br /&gt;
+If we add 12 and 19 we get another good division, 12 + 19 = 31. We can understand why this works if we look at it as adding vals; &amp;lt;12 19 28| + &amp;lt;19 30 44| = &amp;lt;31 49 72|. The relative error in terms of &lt;a class="wiki_link" href="/Relative%20cent"&gt;relative cents&lt;/a&gt; is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is [-1.955 13.686] (the same as absolute cents) and the error of 19edo is [-11.429 -11.663], and this sums to [-13.384 2.023]. In relative cents the error of the fifth is not much increased, and on converting to absolute cents even shrinks, and the error of the fifth is much smaller. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 31edo) can have a much smaller error due to the cancellation.&lt;br /&gt;
 &lt;br /&gt;
 We may also look at addition of EDOs in terms of MOS; if a\n is a generator for an n-edo MOS, and b\m for an m-EDO MOS, where both of these are generators for the same linear temperament, then the mediant, (a+b)\(n+m), will be a generator for a MOS for the same temperament, this time in (n+m)-edo.&lt;br /&gt;