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:<br>

: This revision was by author [[User:~~Kosmorsky~~|~~Kosmorsky~~]] and made on <tt>2011-08-~~13 22~~:50:26 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-14 00:21:35 UTC</tt>.<br>

: The original revision id was <tt>~~245821255~~</tt>.<br>

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

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EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of [[RHM Bosanquet]].

==Non-tuning properties==

You will quickly find that the //factorization// of the total number of notes in each EDO has consequences for its structure and the way it relates to other EDOs. For example, 6 = 2 x 3, so 6-edo contains all of the intervals in both 2-edo and 3-edo. On the other hand, 7 is a prime number, so no 7-edo intervals are redundant with those of smaller EDOs. See [[prime numbers#prime numbers in EDOs|prime numbers in EDOs]] for more details.

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The [[MOSScales|Moments of Symmetry]] paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.

Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). ~~Most obviously~~, ~~an interval that they both approximate will also be approximated~~. ~~(Since there~~ is ~~only one sum~~, ~~you could call this "Chinese Parenting" har har har)~~. ~~Taking this further~~, ~~any MOS they both support~~, ~~may~~ be ~~found in~~ the ~~child~~ - this is ~~because~~ of the ~~nature~~ of the ~~Moment~~ of ~~SymmetryrtemmyS fo tnemoM~~ as a X(L)+~~Y(s~~)~~. The breakthrough here would~~ be ~~figuring out mathematically how multiple-limit harmony interacts rather than~~ the ~~dual~~-~~limit perspective of MOSes, but that is more important and less immediately obvious~~.

==Adding EDOs==

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.

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.

~~Dissimilar pairings have less mathematically well defined (yet) results. FFFF pointed out that 15+16~~=~~31, which triggered my insight. The tip~~ of the iceberg in this example is that 15edo, a multiple of 5, tunes the fifth sharp 20 cents and 16 edo tunes the fifth flat 25 cents. More importantly would be the higher limit implications, and the route of tempering between them (and thus, I suspect, the mood) are hybrid too, though certainly not chimeric. While many pairs add up to the same number, it seems that in each case that heritage could be seen as relevant.

==Size of an EDO==

When an edo divides the octave into fewer than 12 divisions (so that each step exceeds 100 cents), you might call it a [[macrotonal edos|macrotonal edo]]. Of these, 1, 2, 3, 4 and 6 divide 12 and so are already available to anyone wishing to explore them. The 5, 7 and 9 edos have arguably been used in various kinds of musical traditions in different parts of the world.

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* [[http://www.webcitation.org/5xZz8RtQB|Teen Tunes]] by [[Ivor Darreg]]</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>EDO</title></head><body><div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#EDO FAQ">EDO FAQ</a></div>

<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>EDO</title></head><body><div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#EDO FAQ">EDO FAQ</a></div>

<div style="margin-left: 2em;"><a href="#EDO FAQ-What are EDO scales like?">What are EDO scales like?</a></div>

<div style="margin-left: 2em;"><a href="#EDO FAQ-What are EDO scales like?">What are EDO scales like?</a></div>

<div style="margin-left: 2em;"><a href="#EDO FAQ-Why would I want to use an EDO?">Why would I want to use an EDO?</a></div>

<div style="margin-left: 2em;"><a href="#EDO FAQ-Why would I want to use an EDO?">Why would I want to use an EDO?</a></div>

<div style="margin-left: 2em;"><a href="#EDO FAQ-How do I explore so many?">How do I explore so many?</a></div>

<div style="margin-left: 2em;"><a href="#EDO FAQ-How do I explore so many?">How do I explore so many?</a></div>

<div style="margin-left: 2em;"><a href="#EDO FAQ-What's the difference between EDOs and Equal Temperaments?">What's the difference between EDOs and Equal Temperaments?</a></div>

<div style="margin-left: 2em;"><a href="#EDO FAQ-Non-tuning properties">Non-tuning properties</a></div>

<div style="margin-left: 1em;"><a href="#Individual pages for EDOs">Individual pages for EDOs</a></div>

<div style="margin-left: 2em;"><a href="#EDO FAQ-Adding EDOs">Adding EDOs</a></div>

<div style="margin-left: 1em;"><a href="#Articles">Articles</a></div>

<div style="margin-left: 2em;"><a href="#EDO FAQ-Size of an EDO">Size of an EDO</a></div>

</div>

<div style="margin-left: 2em;"><a href="#EDO FAQ-What's the difference between EDOs and Equal Temperaments?">What's the difference between EDOs and Equal Temperaments?</a></div>

The abbreviation <strong>EDO</strong> stands for <strong><a class="wiki_link" href="/Equal">Equal</a> Divisions of the <a class="wiki_link" href="/Octave">Octave</a></strong> (not to be confused with the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Edo_period" rel="nofollow">Edo period</a> in Japanese history).<br />

<div style="margin-left: 1em;"><a href="#Individual pages for EDOs">Individual pages for EDOs</a></div>

<div style="margin-left: 1em;"><a href="#Articles">Articles</a></div>

</div>

The abbreviation <strong>EDO</strong> stands for <strong><a class="wiki_link" href="/Equal">Equal</a> Divisions of the <a class="wiki_link" href="/Octave">Octave</a></strong> (not to be confused with the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Edo_period" rel="nofollow">Edo period</a> in Japanese history).<br />

<br />

<h1 id="toc0"><a name="EDO FAQ"></a>EDO FAQ</h1>

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<br />

EDOs can be further subdivided and classified according to the size of the fifth, such as with <a class="wiki_link" href="/Margo%20Schulter">Margo Schulter</a>'s <a class="wiki_link" href="/gentle%20region">gentle region</a> or the distinction between negative, positive, doubly negative and doubly positive of <a class="wiki_link" href="/RHM%20Bosanquet">RHM Bosanquet</a>.<br />

<br />

<h2 id="toc4"><a name="EDO FAQ-Non-tuning properties"></a>Non-tuning properties</h2>

<br />

You will quickly find that the <em>factorization</em> of the total number of notes in each EDO has consequences for its structure and the way it relates to other EDOs. For example, 6 = 2 x 3, so 6-edo contains all of the intervals in both 2-edo and 3-edo. On the other hand, 7 is a prime number, so no 7-edo intervals are redundant with those of smaller EDOs. See <a class="wiki_link" href="/prime%20numbers#prime numbers in EDOs">prime numbers in EDOs</a> for more details.<br />

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The <a class="wiki_link" href="/MOSScales">Moments of Symmetry</a> paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.<br />

<br />

Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). ~~Most obviously~~, ~~an interval~~ that ~~they both approximate will also be approximated~~. ~~(Since there is only one sum~~, ~~you could call~~ this &~~quot~~;~~Chinese Parenting~~&~~quot~~; ~~har har har)~~. ~~Taking this further~~, ~~any MOS~~ they ~~both support~~, ~~may be found in~~ the ~~child~~ - this is ~~because~~ of the ~~nature~~ of ~~the Moment~~ of ~~SymmetryrtemmyS fo tnemoM as~~ a X(L)+~~Y(s~~)~~. The breakthrough here would~~ be ~~figuring out mathematically how multiple-limit harmony interacts rather than~~ the ~~dual~~-~~limit perspective of MOSes, but that is more important and less immediately obvious~~.<br />

<h2 id="toc5"><a name="EDO FAQ-Adding EDOs"></a>Adding EDOs</h2>

<br />

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 <em>vals</em>, 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. <br />

<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 13) can have a much smaller error due to the cancellation.<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.<br />

<br />

~~Dissimilar pairings have less mathematically well defined (yet) results. FFFF pointed out that 15+16~~=~~31, which triggered my insight. The tip~~ of ~~the iceberg in this example is that 15edo,~~ a ~~multiple of 5, tunes the fifth sharp 20 cents and 16 edo tunes the fifth flat 25 cents. More importantly would be the higher limit implications, and the route~~ of tempering between them (and thus, I suspect, the mood) are hybrid too, though certainly not chimeric. While many pairs add up to the same number, it seems that in each case that heritage could be seen as relevant.<br />

<h2 id="toc6"><a name="EDO FAQ-Size of an EDO"></a>Size of an EDO</h2>

<br />

When an edo divides the octave into fewer than 12 divisions (so that each step exceeds 100 cents), you might call it a <a class="wiki_link" href="/macrotonal%20edos">macrotonal edo</a>. Of these, 1, 2, 3, 4 and 6 divide 12 and so are already available to anyone wishing to explore them. The 5, 7 and 9 edos have arguably been used in various kinds of musical traditions in different parts of the world.<br />

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All of these tools are also applicable to equal divisions of other (<a class="wiki_link" href="/nonoctave">nonoctave</a>) intervals as well.<br />

<br />

<h2 id="~~toc4~~"><a name="EDO FAQ-What's the difference between EDOs and Equal Temperaments?"></a>What's the difference between EDOs and Equal Temperaments?</h2>

<h2 id="toc7"><a name="EDO FAQ-What's the difference between EDOs and Equal Temperaments?"></a>What's the difference between EDOs and Equal Temperaments?</h2>

See <a class="wiki_link" href="/edovset">here.</a><br />

<br />

<h1 id="~~toc5~~"><a name="Individual pages for EDOs"></a>Individual pages for EDOs</h1>

<h1 id="toc8"><a name="Individual pages for EDOs"></a>Individual pages for EDOs</h1>

<br />

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and so on. Some larger systems include <a class="wiki_link" href="/415edo">415edo</a>, <a class="wiki_link" href="/422edo">422edo</a>, <a class="wiki_link" href="/441edo">441edo</a>, <a class="wiki_link" href="/446edo">446edo</a>, <a class="wiki_link" href="/458edo">458edo</a>, <a class="wiki_link" href="/460edo">460edo</a>, <a class="wiki_link" href="/473edo">473edo</a>, <a class="wiki_link" href="/491edo">491edo</a>, <a class="wiki_link" href="/494edo">494edo</a>, <a class="wiki_link" href="/534edo">534edo</a>, <a class="wiki_link" href="/539edo">539edo</a>, <a class="wiki_link" href="/578edo">578edo</a>, <a class="wiki_link" href="/581edo">581edo</a>, <a class="wiki_link" href="/587edo">587edo</a>, <a class="wiki_link" href="/612edo">612edo</a>, <a class="wiki_link" href="/640edo">640edo</a>, <a class="wiki_link" href="/643edo">643edo</a>, <a class="wiki_link" href="/665edo">665edo</a>, <a class="wiki_link" href="/695edo">695edo</a>, <a class="wiki_link" href="/703edo">703edo</a>, <a class="wiki_link" href="/730edo">730edo</a>, <a class="wiki_link" href="/742edo">742edo</a>, <a class="wiki_link" href="/749edo">749edo</a>, <a class="wiki_link" href="/764edo">764edo</a>, <a class="wiki_link" href="/771edo">771edo</a>, <a class="wiki_link" href="/814edo">814edo</a>, <a class="wiki_link" href="/873edo">873edo</a>, <a class="wiki_link" href="/935edo">935edo</a>, <a class="wiki_link" href="/940edo">940edo</a>, <a class="wiki_link" href="/954edo">954edo</a>, <a class="wiki_link" href="/971edo">971edo</a>, <a class="wiki_link" href="/1178edo">1178edo</a>, <a class="wiki_link" href="/1200edo">1200edo</a>, <a class="wiki_link" href="/1236edo">1236edo</a>, <a class="wiki_link" href="/1506edo">1506edo</a>, <a class="wiki_link" href="/1578edo">1578edo</a>, <a class="wiki_link" href="/1817edo">1817edo</a>, <a class="wiki_link" href="/1848edo">1848edo</a>, <a class="wiki_link" href="/1889edo">1889edo</a>, <a class="wiki_link" href="/2000edo">2000edo</a>, <a class="wiki_link" href="/2190edo">2190edo</a>, <a class="wiki_link" href="/2460edo">2460edo</a>, <a class="wiki_link" href="/2684edo">2684edo</a>, <a class="wiki_link" href="/2966edo">2966edo</a>, <a class="wiki_link" href="/3125edo">3125edo</a>, <a class="wiki_link" href="/3395edo">3395edo</a>, <a class="wiki_link" href="/3558edo">3558edo</a>, <a class="wiki_link" href="/3600edo">3600edo</a>, <a class="wiki_link" href="/4296edo">4296edo</a>, <a class="wiki_link" href="/4573edo">4573edo</a>, <a class="wiki_link" href="/5585edo">5585edo</a>, <a class="wiki_link" href="/6079edo">6079edo</a>, <a class="wiki_link" href="/6691edo">6691edo</a>, <a class="wiki_link" href="/8269edo">8269edo</a>, <a class="wiki_link" href="/8539edo">8539edo</a>, <a class="wiki_link" href="/9033edo">9033edo</a>, <a class="wiki_link" href="/15601edo">15601edo</a>, <a class="wiki_link" href="/46032edo">46032edo</a><br />

<br />

<h1 id="~~toc6~~"><a name="Articles"></a>Articles</h1>

<h1 id="toc9"><a name="Articles"></a>Articles</h1>

<ul><li><a class="wiki_link_ext" href="http://www.webcitation.org/5xZz8RtQB" rel="nofollow">Teen Tunes</a> by <a class="wiki_link" href="/Ivor%20Darreg">Ivor Darreg</a></li></ul></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>2011-08-13 22:50:26 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-14 00:21:35 UTC</tt>.<br>
-: The original revision id was <tt>245821255</tt>.<br>
+: The original revision id was <tt>245828887</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 29: / Line 29: @@
 EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of [[RHM Bosanquet]].
+==Non-tuning properties==
 You will quickly find that the //factorization// of the total number of notes in each EDO has consequences for its structure and the way it relates to other EDOs. For example, 6 = 2 x 3, so 6-edo contains all of the intervals in both 2-edo and 3-edo. On the other hand, 7 is a prime number, so no 7-edo intervals are redundant with those of smaller EDOs. See [[prime numbers#prime numbers in EDOs|prime numbers in EDOs]] for more details.
@@ Line 34: / Line 36: @@
 The [[MOSScales|Moments of Symmetry]] paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.
-Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). Most obviously, an interval that they both approximate will also be approximated. (Since there is only one sum, you could call this "Chinese Parenting" har har har). Taking this further, any MOS they both support, may be found in the child - this is because of the nature of the Moment of SymmetryrtemmyS fo tnemoM as a X(L)+Y(s). The breakthrough here would be figuring out mathematically how multiple-limit harmony interacts rather than the dual-limit perspective of MOSes, but that is more important and less immediately obvious.
+==Adding EDOs==
+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.
+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.
-Dissimilar pairings have less mathematically well defined (yet) results. FFFF pointed out that 15+16=31, which triggered my insight. The tip of the iceberg in this example is that 15edo, a multiple of 5, tunes the fifth sharp 20 cents and 16 edo tunes the fifth flat 25 cents. More importantly would be the higher limit implications, and the route of tempering between them (and thus, I suspect, the mood) are hybrid too, though certainly not chimeric. While many pairs add up to the same number, it seems that in each case that heritage could be seen as relevant.
+==Size of an EDO==
 When an edo divides the octave into fewer than 12 divisions (so that each step exceeds 100 cents), you might call it a [[macrotonal edos|macrotonal edo]]. Of these, 1, 2, 3, 4 and 6 divide 12 and so are already available to anyone wishing to explore them. The 5, 7 and 9 edos have arguably been used in various kinds of musical traditions in different parts of the world.
@@ Line 88: / Line 96: @@
 * [[http://www.webcitation.org/5xZz8RtQB|Teen Tunes]] by [[Ivor Darreg]]</pre></div>
 <h4>Original HTML content:</h4>
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-&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#EDO FAQ-Why would I want to use an EDO?"&gt;Why would I want to use an EDO?&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#EDO FAQ-Why would I want to use an EDO?"&gt;Why would I want to use an EDO?&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#EDO FAQ-How do I explore so many?"&gt;How do I explore so many?&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#EDO FAQ-How do I explore so many?"&gt;How do I explore so many?&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#EDO FAQ-What's the difference between EDOs and Equal Temperaments?"&gt;What's the difference between EDOs and Equal Temperaments?&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#EDO FAQ-Non-tuning properties"&gt;Non-tuning properties&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Individual pages for EDOs"&gt;Individual pages for EDOs&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#EDO FAQ-Adding EDOs"&gt;Adding EDOs&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Articles"&gt;Articles&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#EDO FAQ-Size of an EDO"&gt;Size of an EDO&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#EDO FAQ-What's the difference between EDOs and Equal Temperaments?"&gt;What's the difference between EDOs and Equal Temperaments?&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:22 --&gt;The abbreviation &lt;strong&gt;EDO&lt;/strong&gt; stands for &lt;strong&gt;&lt;a class="wiki_link" href="/Equal"&gt;Equal&lt;/a&gt; Divisions of the &lt;a class="wiki_link" href="/Octave"&gt;Octave&lt;/a&gt;&lt;/strong&gt; (not to be confused with the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Edo_period" rel="nofollow"&gt;Edo period&lt;/a&gt; in Japanese history).&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Individual pages for EDOs"&gt;Individual pages for EDOs&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Articles"&gt;Articles&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:31 --&gt;The abbreviation &lt;strong&gt;EDO&lt;/strong&gt; stands for &lt;strong&gt;&lt;a class="wiki_link" href="/Equal"&gt;Equal&lt;/a&gt; Divisions of the &lt;a class="wiki_link" href="/Octave"&gt;Octave&lt;/a&gt;&lt;/strong&gt; (not to be confused with the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Edo_period" rel="nofollow"&gt;Edo period&lt;/a&gt; in Japanese history).&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="EDO FAQ"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;EDO FAQ&lt;/h1&gt;
@@ Line 118: / Line 129: @@
 &lt;br /&gt;
 EDOs can be further subdivided and classified according to the size of the fifth, such as with &lt;a class="wiki_link" href="/Margo%20Schulter"&gt;Margo Schulter&lt;/a&gt;'s &lt;a class="wiki_link" href="/gentle%20region"&gt;gentle region&lt;/a&gt; or the distinction between negative, positive, doubly negative and doubly positive of &lt;a class="wiki_link" href="/RHM%20Bosanquet"&gt;RHM Bosanquet&lt;/a&gt;.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="EDO FAQ-Non-tuning properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Non-tuning properties&lt;/h2&gt;
 &lt;br /&gt;
 You will quickly find that the &lt;em&gt;factorization&lt;/em&gt; of the total number of notes in each EDO has consequences for its structure and the way it relates to other EDOs. For example, 6 = 2 x 3, so 6-edo contains all of the intervals in both 2-edo and 3-edo. On the other hand, 7 is a prime number, so no 7-edo intervals are redundant with those of smaller EDOs. See &lt;a class="wiki_link" href="/prime%20numbers#prime numbers in EDOs"&gt;prime numbers in EDOs&lt;/a&gt; for more details.&lt;br /&gt;
@@ Line 123: / Line 136: @@
 The &lt;a class="wiki_link" href="/MOSScales"&gt;Moments of Symmetry&lt;/a&gt; paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.&lt;br /&gt;
 &lt;br /&gt;
-Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). Most obviously, an interval that they both approximate will also be approximated. (Since there is only one sum, you could call this &amp;quot;Chinese Parenting&amp;quot; har har har). Taking this further, any MOS they both support, may be found in the child - this is because of the nature of the Moment of SymmetryrtemmyS fo tnemoM as a X(L)+Y(s). The breakthrough here would be figuring out mathematically how multiple-limit harmony interacts rather than the dual-limit perspective of MOSes, but that is more important and less immediately obvious.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="EDO FAQ-Adding EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Adding EDOs&lt;/h2&gt;
+&lt;br /&gt;
+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;
+&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;
 &lt;br /&gt;
-Dissimilar pairings have less mathematically well defined (yet) results. FFFF pointed out that 15+16=31, which triggered my insight. The tip of the iceberg in this example is that 15edo, a multiple of 5, tunes the fifth sharp 20 cents and 16 edo tunes the fifth flat 25 cents. More importantly would be the higher limit implications, and the route of tempering between them (and thus, I suspect, the mood) are hybrid too, though certainly not chimeric. While many pairs add up to the same number, it seems that in each case that heritage could be seen as relevant.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="EDO FAQ-Size of an EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Size of an EDO&lt;/h2&gt;
 &lt;br /&gt;
 When an edo divides the octave into fewer than 12 divisions (so that each step exceeds 100 cents), you might call it a &lt;a class="wiki_link" href="/macrotonal%20edos"&gt;macrotonal edo&lt;/a&gt;. Of these, 1, 2, 3, 4 and 6 divide 12 and so are already available to anyone wishing to explore them. The 5, 7 and 9 edos have arguably been used in various kinds of musical traditions in different parts of the world.&lt;br /&gt;
@@ Line 133: / Line 152: @@
 All of these tools are also applicable to equal divisions of other (&lt;a class="wiki_link" href="/nonoctave"&gt;nonoctave&lt;/a&gt;) intervals as well.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="EDO FAQ-What's the difference between EDOs and Equal Temperaments?"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;What's the difference between EDOs and Equal Temperaments?&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="EDO FAQ-What's the difference between EDOs and Equal Temperaments?"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;What's the difference between EDOs and Equal Temperaments?&lt;/h2&gt;
   See &lt;a class="wiki_link" href="/edovset"&gt;here.&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Individual pages for EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Individual pages for EDOs&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Individual pages for EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Individual pages for EDOs&lt;/h1&gt;
   &lt;br /&gt;
@@ Line 1,029: / Line 1,048: @@
 and so on. Some larger systems include &lt;a class="wiki_link" href="/415edo"&gt;415edo&lt;/a&gt;, &lt;a class="wiki_link" href="/422edo"&gt;422edo&lt;/a&gt;, &lt;a class="wiki_link" href="/441edo"&gt;441edo&lt;/a&gt;, &lt;a class="wiki_link" href="/446edo"&gt;446edo&lt;/a&gt;, &lt;a class="wiki_link" href="/458edo"&gt;458edo&lt;/a&gt;, &lt;a class="wiki_link" href="/460edo"&gt;460edo&lt;/a&gt;, &lt;a class="wiki_link" href="/473edo"&gt;473edo&lt;/a&gt;, &lt;a class="wiki_link" href="/491edo"&gt;491edo&lt;/a&gt;, &lt;a class="wiki_link" href="/494edo"&gt;494edo&lt;/a&gt;, &lt;a class="wiki_link" href="/534edo"&gt;534edo&lt;/a&gt;, &lt;a class="wiki_link" href="/539edo"&gt;539edo&lt;/a&gt;, &lt;a class="wiki_link" href="/578edo"&gt;578edo&lt;/a&gt;, &lt;a class="wiki_link" href="/581edo"&gt;581edo&lt;/a&gt;, &lt;a class="wiki_link" href="/587edo"&gt;587edo&lt;/a&gt;, &lt;a class="wiki_link" href="/612edo"&gt;612edo&lt;/a&gt;, &lt;a class="wiki_link" href="/640edo"&gt;640edo&lt;/a&gt;, &lt;a class="wiki_link" href="/643edo"&gt;643edo&lt;/a&gt;, &lt;a class="wiki_link" href="/665edo"&gt;665edo&lt;/a&gt;, &lt;a class="wiki_link" href="/695edo"&gt;695edo&lt;/a&gt;, &lt;a class="wiki_link" href="/703edo"&gt;703edo&lt;/a&gt;, &lt;a class="wiki_link" href="/730edo"&gt;730edo&lt;/a&gt;, &lt;a class="wiki_link" href="/742edo"&gt;742edo&lt;/a&gt;, &lt;a class="wiki_link" href="/749edo"&gt;749edo&lt;/a&gt;, &lt;a class="wiki_link" href="/764edo"&gt;764edo&lt;/a&gt;, &lt;a class="wiki_link" href="/771edo"&gt;771edo&lt;/a&gt;, &lt;a class="wiki_link" href="/814edo"&gt;814edo&lt;/a&gt;, &lt;a class="wiki_link" href="/873edo"&gt;873edo&lt;/a&gt;, &lt;a class="wiki_link" href="/935edo"&gt;935edo&lt;/a&gt;, &lt;a class="wiki_link" href="/940edo"&gt;940edo&lt;/a&gt;, &lt;a class="wiki_link" href="/954edo"&gt;954edo&lt;/a&gt;, &lt;a class="wiki_link" href="/971edo"&gt;971edo&lt;/a&gt;, &lt;a class="wiki_link" href="/1178edo"&gt;1178edo&lt;/a&gt;, &lt;a class="wiki_link" href="/1200edo"&gt;1200edo&lt;/a&gt;, &lt;a class="wiki_link" href="/1236edo"&gt;1236edo&lt;/a&gt;, &lt;a class="wiki_link" href="/1506edo"&gt;1506edo&lt;/a&gt;, &lt;a class="wiki_link" href="/1578edo"&gt;1578edo&lt;/a&gt;, &lt;a class="wiki_link" href="/1817edo"&gt;1817edo&lt;/a&gt;, &lt;a class="wiki_link" href="/1848edo"&gt;1848edo&lt;/a&gt;, &lt;a class="wiki_link" href="/1889edo"&gt;1889edo&lt;/a&gt;, &lt;a class="wiki_link" href="/2000edo"&gt;2000edo&lt;/a&gt;, &lt;a class="wiki_link" href="/2190edo"&gt;2190edo&lt;/a&gt;, &lt;a class="wiki_link" href="/2460edo"&gt;2460edo&lt;/a&gt;, &lt;a class="wiki_link" href="/2684edo"&gt;2684edo&lt;/a&gt;, &lt;a class="wiki_link" href="/2966edo"&gt;2966edo&lt;/a&gt;, &lt;a class="wiki_link" href="/3125edo"&gt;3125edo&lt;/a&gt;, &lt;a class="wiki_link" href="/3395edo"&gt;3395edo&lt;/a&gt;, &lt;a class="wiki_link" href="/3558edo"&gt;3558edo&lt;/a&gt;, &lt;a class="wiki_link" href="/3600edo"&gt;3600edo&lt;/a&gt;, &lt;a class="wiki_link" href="/4296edo"&gt;4296edo&lt;/a&gt;, &lt;a class="wiki_link" href="/4573edo"&gt;4573edo&lt;/a&gt;, &lt;a class="wiki_link" href="/5585edo"&gt;5585edo&lt;/a&gt;, &lt;a class="wiki_link" href="/6079edo"&gt;6079edo&lt;/a&gt;, &lt;a class="wiki_link" href="/6691edo"&gt;6691edo&lt;/a&gt;, &lt;a class="wiki_link" href="/8269edo"&gt;8269edo&lt;/a&gt;, &lt;a class="wiki_link" href="/8539edo"&gt;8539edo&lt;/a&gt;, &lt;a class="wiki_link" href="/9033edo"&gt;9033edo&lt;/a&gt;, &lt;a class="wiki_link" href="/15601edo"&gt;15601edo&lt;/a&gt;, &lt;a class="wiki_link" href="/46032edo"&gt;46032edo&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Articles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Articles&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Articles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Articles&lt;/h1&gt;
   &lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xZz8RtQB" rel="nofollow"&gt;Teen Tunes&lt;/a&gt; by &lt;a class="wiki_link" href="/Ivor%20Darreg"&gt;Ivor Darreg&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>