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:~~xenwolf~~|~~xenwolf~~]] and made on <tt>2011-06-~~24 04~~:31:18 UTC</tt>.

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-15 09:55:23 UTC</tt>.

: The original revision id was <tt>~~238542401~~</tt>.

: The original revision id was <tt>241474831</tt>.

: The revision comment was: <tt>~~better to start a stub~~</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.

<h4>Original Wikitext content:</h4>

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**Equal: a tuning in which every single step is the same interval; an equal-step scale.** See also [[edo]].

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~~.

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). Any interval, rational, Just, or irrational, may be used as the basis for an equal tuning, although divisions of the octave are most common. 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]].

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.

When a tuning is called "n-tone equal temperament" (abbreviated n-tET or n-ET), this usually means "n 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 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]].)

The less theory-laden term //EDO// (occasionally written ED2), meaning "equal divisions of the octave" (or "equal divisions of 2/1"), leaves comparison to JI out of the picture, aside from the octave itself. (There are other less standard terms, many in the [[http://www.tonalsoft.com/enc/encyclopedia.aspx|Tonalsoft Encyclopedia]].) More generally, the term //EDn// can be used, where n is any harmonic of the harmonic series. For example, the equal-tempered Bohlen-Pierce scale may also be referred to as 13-ED3, for 13 equal divisions of 3/1 (the 3rd harmonic).

**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. See also <a class="wiki_link" href="/edo">edo</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~~.

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). Any interval, rational, Just, or irrational, may be used as the basis for an equal tuning, although divisions of the octave are most common. 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>.

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.

When a tuning is called &quot;n-tone equal temperament&quot; (abbreviated n-tET or n-ET), this usually means &quot;n 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 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>.)

The less theory-laden term EDO (occasionally written ED2), meaning &quot;equal divisions of the octave&quot; (or &quot;equal divisions of 2/1&quot;), leaves comparison to JI out of the picture, aside from the octave itself. (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>.) More generally, the term EDn can be used, where n is any harmonic of the harmonic series. For example, the equal-tempered Bohlen-Pierce scale may also be referred to as 13-ED3, for 13 equal divisions of 3/1 (the 3rd harmonic).

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:xenwolf|xenwolf]] and made on <tt>2011-06-24 04:31:18 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-15 09:55:23 UTC</tt>.<br>
-: The original revision id was <tt>238542401</tt>.<br>
+: The original revision id was <tt>241474831</tt>.<br>
-: The revision comment was: <tt>better to start a stub</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>
 <h4>Original Wikitext content:</h4>
@@ Line 10: / Line 10: @@
 **Equal: a tuning in which every single step is the same interval; an equal-step scale.** See also [[edo]].
-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.
+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). Any interval, rational, Just, or irrational, may be used as the basis for an equal tuning, although divisions of the octave are most common. 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]].
-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.
+When a tuning is called "n-tone equal temperament" (abbreviated n-tET or n-ET), this usually means "n 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 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]].)
+The less theory-laden term //EDO// (occasionally written ED2), meaning "equal divisions of the octave" (or "equal divisions of 2/1"), leaves comparison to JI out of the picture, aside from the octave itself. (There are other less standard terms, many in the [[http://www.tonalsoft.com/enc/encyclopedia.aspx|Tonalsoft Encyclopedia]].) More generally, the term //EDn// can be used, where n is any harmonic of the harmonic series. For example, the equal-tempered Bohlen-Pierce scale may also be referred to as 13-ED3, for 13 equal divisions of 3/1 (the 3rd harmonic).
 **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 67: / Line 67: @@
 &lt;strong&gt;Equal: a tuning in which every single step is the same interval; an equal-step scale.&lt;/strong&gt; See also &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&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, 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;
+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). Any interval, rational, Just, or irrational, may be used as the basis for an equal tuning, although divisions of the octave are most common. 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;.&lt;br /&gt;
 &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;
+When a tuning is called &amp;quot;n-tone equal temperament&amp;quot; (abbreviated n-tET or n-ET), this usually means &amp;quot;n 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 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;
+The less theory-laden term &lt;em&gt;EDO&lt;/em&gt; (occasionally written ED2), meaning &amp;quot;equal divisions of the octave&amp;quot; (or &amp;quot;equal divisions of 2/1&amp;quot;), leaves comparison to JI out of the picture, aside from the octave itself. (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;.) More generally, the term &lt;em&gt;EDn&lt;/em&gt; can be used, where n is any harmonic of the harmonic series. For example, the equal-tempered Bohlen-Pierce scale may also be referred to as 13-ED3, for 13 equal divisions of 3/1 (the 3rd harmonic).&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;