EDO: Difference between revisions
Wikispaces>genewardsmith **Imported revision 250106274 - Original comment: ** |
Wikispaces>Kosmorsky **Imported revision 250153792 - Original comment: 7 note diatonicism for instance** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2011-09-01 17:15:27 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>250153792</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>7 note diatonicism for instance</tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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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. | 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. A visual way of putting this is that through this addition of n and m, | 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. A visual way of putting this is that through this addition of n and m, one becomes the accidentals or black keys, and the other the naturals or white keys. The choice of accidental/natural or black keys/white keys is a question of emphasis on the composer or designers part. Furthermore, one may add more than two numbers, hierarchically expanding the possibilities to double flats and sharps and beyond. This can be useful in designing keyboards and systems of notation. | ||
==Size of an EDO== | ==Size of an EDO== | ||
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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.<br /> | 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.<br /> | ||
<br /> | <br /> | ||
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. A visual way of putting this is that through this addition of n and m, | 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. A visual way of putting this is that through this addition of n and m, one becomes the accidentals or black keys, and the other the naturals or white keys. The choice of accidental/natural or black keys/white keys is a question of emphasis on the composer or designers part. Furthermore, one may add more than two numbers, hierarchically expanding the possibilities to double flats and sharps and beyond. This can be useful in designing keyboards and systems of notation.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="EDO FAQ-Size of an EDO"></a><!-- ws:end:WikiTextHeadingRule:12 -->Size of an EDO</h2> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="EDO FAQ-Size of an EDO"></a><!-- ws:end:WikiTextHeadingRule:12 -->Size of an EDO</h2> | ||