Equal-step tuning: 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:~~guest~~|~~guest~~]] and made on <tt>2010-02-~~22 12~~:56:59 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-15 18:11:44 UTC</tt>.

: The original revision id was <tt>~~122148663~~</tt>.

: The original revision id was <tt>142237599</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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**Equal: a tuning in which every single step is the same interval; an equal-step scale.**

The size of this single step is given explicitly (e.g. 88 cent equal temperament) or as a fraction of a larger interval (e.g. 13 equal tones per octave). (When a just interval is equally divided, none of the resulting intervals are just~~. See~~ [[roots]].)

The size of this single step is given explicitly (e.g. 88 cent equal temperament) or as a fraction of a larger interval (e.g. 13 equal tones per octave). When a just interval is equally divided, it is assumed none of the resulting intervals are just, because if the interval has a rational root it is seen as a division of that [[roots|root]]. Hence we do not talk of equal divisions of 4 or 16/9.

When a tuning is called "X tone equal temperament" (abbreviated -tET or -ET), this ~~usually~~ means "X divisions of 2/1, the octave," but it also implies a mindset of [[Regular Temperaments|temperament]]—that is, of a harmony-centric, JI-approximation-based understanding of the scale.

When a tuning is called "X tone equal temperament" (abbreviated -tET or -ET), this means "X divisions of 2/1, the octave, or some approximation thereof" but it also implies a mindset of [[Regular Temperaments|temperament]]—that is, of a harmony-centric, JI-approximation-based understanding of the scale.

The less ~~loaded~~ term //EDO//, meaning "equal divisions of the octave," ~~is helpful for leaving~~ comparison to JI ~~completely~~ out of the picture. (There are other less standard terms, many in the [[http://www.tonalsoft.com/enc/encyclopedia.aspx|Tonalsoft Encyclopedia]].)

The less theory-laden term //EDO//, meaning "equal divisions of the octave," leaves comparison to JI, aside from the octave itself, out of the picture. (There are other less standard terms, many in the [[http://www.tonalsoft.com/enc/encyclopedia.aspx|Tonalsoft Encyclopedia]].)

**As there are infinite intervals, there are infinite equal scales.** Barring technicalities there are large quantities of perceivably different equal scales. Seeing such a diverse menagerie at their disposal, some composers choose to combine multiple equal tunings [[ET surveys|sequentially]] or [[Polymicrotonality|simultaneously]].

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Equal: a tuning in which every single step is the same interval; an equal-step scale.

The size of this single step is given explicitly (e.g. 88 cent equal temperament) or as a fraction of a larger interval (e.g. 13 equal tones per octave). (When a just interval is equally divided, none of the resulting intervals are just~~. See~~ <a class="wiki_link" href="/roots">~~roots~~</a>.)

The size of this single step is given explicitly (e.g. 88 cent equal temperament) or as a fraction of a larger interval (e.g. 13 equal tones per octave). When a just interval is equally divided, it is assumed none of the resulting intervals are just, because if the interval has a rational root it is seen as a division of that <a class="wiki_link" href="/roots">root</a>. Hence we do not talk of equal divisions of 4 or 16/9.

When a tuning is called &quot;X tone equal temperament&quot; (abbreviated -tET or -ET), this ~~usually~~ means &quot;X divisions of 2/1, the octave,&quot; but it also implies a mindset of <a class="wiki_link" href="/Regular%20Temperaments">temperament</a>—that is, of a harmony-centric, JI-approximation-based understanding of the scale.

When a tuning is called &quot;X tone equal temperament&quot; (abbreviated -tET or -ET), this means &quot;X divisions of 2/1, the octave, or some approximation thereof&quot; but it also implies a mindset of <a class="wiki_link" href="/Regular%20Temperaments">temperament</a>—that is, of a harmony-centric, JI-approximation-based understanding of the scale.

The less ~~loaded~~ term EDO, meaning &quot;equal divisions of the octave,&quot; ~~is helpful for leaving~~ comparison to JI ~~completely~~ out of the picture. (There are other less standard terms, many in the <a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/encyclopedia.aspx" rel="nofollow">Tonalsoft Encyclopedia</a>.)

The less theory-laden term EDO, meaning &quot;equal divisions of the octave,&quot; leaves comparison to JI, aside from the octave itself, out of the picture. (There are other less standard terms, many in the <a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/encyclopedia.aspx" rel="nofollow">Tonalsoft Encyclopedia</a>.)

As there are infinite intervals, there are infinite equal scales. Barring technicalities there are large quantities of perceivably different equal scales. Seeing such a diverse menagerie at their disposal, some composers choose to combine multiple equal tunings <a class="wiki_link" href="/ET%20surveys">sequentially</a> or <a class="wiki_link" href="/Polymicrotonality">simultaneously</a>.

@@ 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:guest|guest]] and made on <tt>2010-02-22 12:56:59 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-15 18:11:44 UTC</tt>.<br>
-: The original revision id was <tt>122148663</tt>.<br>
+: The original revision id was <tt>142237599</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 10: / Line 10: @@
 **Equal: a tuning in which every single step is the same interval; an equal-step scale.**
-The size of this single step is given explicitly (e.g. 88 cent equal temperament) or as a fraction of a larger interval (e.g. 13 equal tones per octave). (When a just interval is equally divided, none of the resulting intervals are just. See [[roots]].)
+The size of this single step is given explicitly (e.g. 88 cent equal temperament) or as a fraction of a larger interval (e.g. 13 equal tones per octave). When a just interval is equally divided, it is assumed none of the resulting intervals are just, because if the interval has a rational root it is seen as a division of that [[roots|root]]. Hence we do not talk of equal divisions of 4 or 16/9.
-When a tuning is called "X tone equal temperament" (abbreviated -tET or -ET), this usually means "X divisions of 2/1, the octave," but it also implies a mindset of [[Regular Temperaments|temperament]]—that is, of a harmony-centric, JI-approximation-based understanding of the scale.
+When a tuning is called "X tone equal temperament" (abbreviated -tET or -ET), this means "X divisions of 2/1, the octave, or some approximation thereof" but it also implies a mindset of [[Regular Temperaments|temperament]]—that is, of a harmony-centric, JI-approximation-based understanding of the scale.
-The less loaded term //EDO//, meaning "equal divisions of the octave," is helpful for leaving comparison to JI completely out of the picture. (There are other less standard terms, many in the [[http://www.tonalsoft.com/enc/encyclopedia.aspx|Tonalsoft Encyclopedia]].)
+The less theory-laden term //EDO//, meaning "equal divisions of the octave," leaves comparison to JI, aside from the octave itself, out of the picture. (There are other less standard terms, many in the [[http://www.tonalsoft.com/enc/encyclopedia.aspx|Tonalsoft Encyclopedia]].)
 **As there are infinite intervals, there are infinite equal scales.** Barring technicalities there are large quantities of perceivably different equal scales. Seeing such a diverse menagerie at their disposal, some composers choose to combine multiple equal tunings [[ET surveys|sequentially]] or [[Polymicrotonality|simultaneously]].
@@ Line 62: / Line 62: @@
 &lt;strong&gt;Equal: a tuning in which every single step is the same interval; an equal-step scale.&lt;/strong&gt;&lt;br /&gt;
 &lt;br /&gt;
-The size of this single step is given explicitly (e.g. 88 cent equal temperament) or as a fraction of a larger interval (e.g. 13 equal tones per octave). (When a just interval is equally divided, none of the resulting intervals are just. See &lt;a class="wiki_link" href="/roots"&gt;roots&lt;/a&gt;.)&lt;br /&gt;
+The size of this single step is given explicitly (e.g. 88 cent equal temperament) or as a fraction of a larger interval (e.g. 13 equal tones per octave). When a just interval is equally divided, it is assumed none of the resulting intervals are just, because if the interval has a rational root it is seen as a division of that &lt;a class="wiki_link" href="/roots"&gt;root&lt;/a&gt;. Hence we do not talk of equal divisions of 4 or 16/9.&lt;br /&gt;
 &lt;br /&gt;
-When a tuning is called &amp;quot;X tone equal temperament&amp;quot; (abbreviated -tET or -ET), this usually means &amp;quot;X divisions of 2/1, the octave,&amp;quot; but it also implies a mindset of &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;temperament&lt;/a&gt;—that is, of a harmony-centric, JI-approximation-based understanding of the scale.&lt;br /&gt;
+When a tuning is called &amp;quot;X tone equal temperament&amp;quot; (abbreviated -tET or -ET), this means &amp;quot;X divisions of 2/1, the octave, or some approximation thereof&amp;quot; but it also implies a mindset of &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;temperament&lt;/a&gt;—that is, of a harmony-centric, JI-approximation-based understanding of the scale.&lt;br /&gt;
 &lt;br /&gt;
-The less loaded term &lt;em&gt;EDO&lt;/em&gt;, meaning &amp;quot;equal divisions of the octave,&amp;quot; is helpful for leaving comparison to JI completely out of the picture. (There are other less standard terms, many in the &lt;a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/encyclopedia.aspx" rel="nofollow"&gt;Tonalsoft Encyclopedia&lt;/a&gt;.)&lt;br /&gt;
+The less theory-laden term &lt;em&gt;EDO&lt;/em&gt;, meaning &amp;quot;equal divisions of the octave,&amp;quot; leaves comparison to JI, aside from the octave itself, out of the picture. (There are other less standard terms, many in the &lt;a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/encyclopedia.aspx" rel="nofollow"&gt;Tonalsoft Encyclopedia&lt;/a&gt;.)&lt;br /&gt;
 &lt;br /&gt;
 &lt;strong&gt;As there are infinite intervals, there are infinite equal scales.&lt;/strong&gt; Barring technicalities there are large quantities of perceivably different equal scales. Seeing such a diverse menagerie at their disposal, some composers choose to combine multiple equal tunings &lt;a class="wiki_link" href="/ET%20surveys"&gt;sequentially&lt;/a&gt; or &lt;a class="wiki_link" href="/Polymicrotonality"&gt;simultaneously&lt;/a&gt;.&lt;br /&gt;