Equal-step tuning: Difference between revisions

Revision as of 18:11, 15 May 2010

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=Equal= 

**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, 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 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 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]].

----
== == 
=Scale gallery= 

==Equal divisions...== 

===...of the Octave (2/1)=== 
(wildly popular; [[edo|dedicated page]])
|| //[[1edo]]// || //[[2edo]]// || //[[3edo]]// || //[[4edo]]// || [[5edo]] || //[[6edo]]// || [[7edo]] || [[8edo]] || [[9edo]] || [[10edo]] || [[11edo]] || //[[12edo]]// ||
|| [[13edo]] || [[14edo]] || [[15edo]] || [[16edo]] || [[17edo]] || [[18edo]] || [[19edo]] || [[20edo]] || [[21edo]] || [[22edo]] || [[23edo]] || [[24edo]] ||
|| [[25edo]] || [[26edo]] || [[27edo]] || [[28edo]] || [[29edo]] || [[30edo]] || [[31edo]] || [[32edo]] || [[33edo]] || [[34edo]] || [[35edo]] || [[36edo]] ||
|| [[37edo]] || [[38edo]] || [[39edo]] || [[40edo]] || [[41edo]] || [[42edo]] || [[43edo]] || [[44edo]] || [[45edo]] || [[46edo]] || [[47edo]] || [[48edo]] ||
|| [[49edo]] || [[50edo]] || [[51edo]] || [[52edo]] || [[53edo]] || [[54edo]] || [[55edo]] || [[56edo]] || [[57edo]] || [[58edo]] || [[59edo]] || [[60edo]] ||
and so on to less popular areas... [[72edo]] [[76edo]] [[88edo]] [[96edo]] [[120edo]]

The equal temperaments formatted in //italic// build subsets of //[[12edo]]// and can be taken as "[[macrotonal tuning]]s".

===...of the Tritave (3/1)=== 
7
[[12edt|12]]
[[BP|13 (Bohlen-Pierce)]]
[[19ED3|19 (Bernhard Stopper)]]
39 Triple Bohlen-Pierce
===...of the Perfect Fifth (3/2)=== 
[[4edf|4]]
[[6edf|6]]
[[88cET|8 (88-cET)]]
[[Carlos Alpha|9 (Carlos Alpha)]]
[[Carlos Beta|11 (Carlos Beta)]]
[[Carlos Gamma|20 (Carlos Gamma)]]

===...of the Just Major 17th (5/1)=== 
25 (Stockhausen, McLaren)

==Equal multiplications?== 
88-cET, Alpha, Beta, Gamma

===See also:=== 
[[edo anatomy]], [[macrotonal edos]], [[quasi-equal]]

Original HTML content:

<html><head><title>Equal-step Tuning</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Equal"></a><!-- ws:end:WikiTextHeadingRule:0 -->Equal</h1>
 <br />
<strong>Equal: a tuning in which every single step is the same interval; an equal-step scale.</strong><br />
<br />
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.<br />
<br />
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.<br />
<br />
The less theory-laden term <em>EDO</em>, 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>.)<br />
<br />
<strong>As there are infinite intervals, there are infinite equal scales.</strong> 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>.<br />
<br />
<hr />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><!-- ws:end:WikiTextHeadingRule:2 --> </h2>
 <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Scale gallery"></a><!-- ws:end:WikiTextHeadingRule:4 -->Scale gallery</h1>
 <br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Scale gallery-Equal divisions..."></a><!-- ws:end:WikiTextHeadingRule:6 -->Equal divisions...</h2>
 <br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Scale gallery-Equal divisions...-...of the Octave (2/1)"></a><!-- ws:end:WikiTextHeadingRule:8 -->...of the Octave (2/1)</h3>
 (wildly popular; <a class="wiki_link" href="/edo">dedicated page</a>)<br />


<table class="wiki_table">
    <tr>
        <td><em><a class="wiki_link" href="/1edo">1edo</a></em><br />
</td>
        <td><em><a class="wiki_link" href="/2edo">2edo</a></em><br />
</td>
        <td><em><a class="wiki_link" href="/3edo">3edo</a></em><br />
</td>
        <td><em><a class="wiki_link" href="/4edo">4edo</a></em><br />
</td>
        <td><a class="wiki_link" href="/5edo">5edo</a><br />
</td>
        <td><em><a class="wiki_link" href="/6edo">6edo</a></em><br />
</td>
        <td><a class="wiki_link" href="/7edo">7edo</a><br />
</td>
        <td><a class="wiki_link" href="/8edo">8edo</a><br />
</td>
        <td><a class="wiki_link" href="/9edo">9edo</a><br />
</td>
        <td><a class="wiki_link" href="/10edo">10edo</a><br />
</td>
        <td><a class="wiki_link" href="/11edo">11edo</a><br />
</td>
        <td><em><a class="wiki_link" href="/12edo">12edo</a></em><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/13edo">13edo</a><br />
</td>
        <td><a class="wiki_link" href="/14edo">14edo</a><br />
</td>
        <td><a class="wiki_link" href="/15edo">15edo</a><br />
</td>
        <td><a class="wiki_link" href="/16edo">16edo</a><br />
</td>
        <td><a class="wiki_link" href="/17edo">17edo</a><br />
</td>
        <td><a class="wiki_link" href="/18edo">18edo</a><br />
</td>
        <td><a class="wiki_link" href="/19edo">19edo</a><br />
</td>
        <td><a class="wiki_link" href="/20edo">20edo</a><br />
</td>
        <td><a class="wiki_link" href="/21edo">21edo</a><br />
</td>
        <td><a class="wiki_link" href="/22edo">22edo</a><br />
</td>
        <td><a class="wiki_link" href="/23edo">23edo</a><br />
</td>
        <td><a class="wiki_link" href="/24edo">24edo</a><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/25edo">25edo</a><br />
</td>
        <td><a class="wiki_link" href="/26edo">26edo</a><br />
</td>
        <td><a class="wiki_link" href="/27edo">27edo</a><br />
</td>
        <td><a class="wiki_link" href="/28edo">28edo</a><br />
</td>
        <td><a class="wiki_link" href="/29edo">29edo</a><br />
</td>
        <td><a class="wiki_link" href="/30edo">30edo</a><br />
</td>
        <td><a class="wiki_link" href="/31edo">31edo</a><br />
</td>
        <td><a class="wiki_link" href="/32edo">32edo</a><br />
</td>
        <td><a class="wiki_link" href="/33edo">33edo</a><br />
</td>
        <td><a class="wiki_link" href="/34edo">34edo</a><br />
</td>
        <td><a class="wiki_link" href="/35edo">35edo</a><br />
</td>
        <td><a class="wiki_link" href="/36edo">36edo</a><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/37edo">37edo</a><br />
</td>
        <td><a class="wiki_link" href="/38edo">38edo</a><br />
</td>
        <td><a class="wiki_link" href="/39edo">39edo</a><br />
</td>
        <td><a class="wiki_link" href="/40edo">40edo</a><br />
</td>
        <td><a class="wiki_link" href="/41edo">41edo</a><br />
</td>
        <td><a class="wiki_link" href="/42edo">42edo</a><br />
</td>
        <td><a class="wiki_link" href="/43edo">43edo</a><br />
</td>
        <td><a class="wiki_link" href="/44edo">44edo</a><br />
</td>
        <td><a class="wiki_link" href="/45edo">45edo</a><br />
</td>
        <td><a class="wiki_link" href="/46edo">46edo</a><br />
</td>
        <td><a class="wiki_link" href="/47edo">47edo</a><br />
</td>
        <td><a class="wiki_link" href="/48edo">48edo</a><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/49edo">49edo</a><br />
</td>
        <td><a class="wiki_link" href="/50edo">50edo</a><br />
</td>
        <td><a class="wiki_link" href="/51edo">51edo</a><br />
</td>
        <td><a class="wiki_link" href="/52edo">52edo</a><br />
</td>
        <td><a class="wiki_link" href="/53edo">53edo</a><br />
</td>
        <td><a class="wiki_link" href="/54edo">54edo</a><br />
</td>
        <td><a class="wiki_link" href="/55edo">55edo</a><br />
</td>
        <td><a class="wiki_link" href="/56edo">56edo</a><br />
</td>
        <td><a class="wiki_link" href="/57edo">57edo</a><br />
</td>
        <td><a class="wiki_link" href="/58edo">58edo</a><br />
</td>
        <td><a class="wiki_link" href="/59edo">59edo</a><br />
</td>
        <td><a class="wiki_link" href="/60edo">60edo</a><br />
</td>
    </tr>
</table>

and so on to less popular areas... <a class="wiki_link" href="/72edo">72edo</a> <a class="wiki_link" href="/76edo">76edo</a> <a class="wiki_link" href="/88edo">88edo</a> <a class="wiki_link" href="/96edo">96edo</a> <a class="wiki_link" href="/120edo">120edo</a><br />
<br />
The equal temperaments formatted in <em>italic</em> build subsets of <em><a class="wiki_link" href="/12edo">12edo</a></em> and can be taken as &quot;<a class="wiki_link" href="/macrotonal%20tuning">macrotonal tuning</a>s&quot;.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Scale gallery-Equal divisions...-...of the Tritave (3/1)"></a><!-- ws:end:WikiTextHeadingRule:10 -->...of the Tritave (3/1)</h3>
 7<br />
<a class="wiki_link" href="/12edt">12</a><br />
<a class="wiki_link" href="/BP">13 (Bohlen-Pierce)</a><br />
<a class="wiki_link" href="/19ED3">19 (Bernhard Stopper)</a><br />
39 Triple Bohlen-Pierce<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Scale gallery-Equal divisions...-...of the Perfect Fifth (3/2)"></a><!-- ws:end:WikiTextHeadingRule:12 -->...of the Perfect Fifth (3/2)</h3>
 <a class="wiki_link" href="/4edf">4</a><br />
<a class="wiki_link" href="/6edf">6</a><br />
<a class="wiki_link" href="/88cET">8 (88-cET)</a><br />
<a class="wiki_link" href="/Carlos%20Alpha">9 (Carlos Alpha)</a><br />
<a class="wiki_link" href="/Carlos%20Beta">11 (Carlos Beta)</a><br />
<a class="wiki_link" href="/Carlos%20Gamma">20 (Carlos Gamma)</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Scale gallery-Equal divisions...-...of the Just Major 17th (5/1)"></a><!-- ws:end:WikiTextHeadingRule:14 -->...of the Just Major 17th (5/1)</h3>
 25 (Stockhausen, McLaren)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Scale gallery-Equal multiplications?"></a><!-- ws:end:WikiTextHeadingRule:16 -->Equal multiplications?</h2>
 88-cET, Alpha, Beta, Gamma<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Scale gallery-Equal multiplications?-See also:"></a><!-- ws:end:WikiTextHeadingRule:18 -->See also:</h3>
 <a class="wiki_link" href="/edo%20anatomy">edo anatomy</a>, <a class="wiki_link" href="/macrotonal%20edos">macrotonal edos</a>, <a class="wiki_link" href="/quasi-equal">quasi-equal</a></body></html>

@@ 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;

Equal-step tuning: Difference between revisions

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IMPORTED REVISION FROM WIKISPACES

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