Equal-step tuning

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This revision was by author xenwolf and made on 2010-05-22 05:58:25 UTC.

The original revision id was 143894133.

The revision comment was: I included edos up to into the table. I stopped doing so with the first row that would be entirely empty.

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Original Wikitext content:

=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]] ||
|| || || || || [[65edo]] || || || || || || || [[72edo]] ||
|| || || || [[76edo]] ||  ||  || || || || || || ||
|| || || [[87edo]] || [[88edo]] || || || || || || [[94edo]] || || [[96edo]] ||
|| || || [[99edo]] || || || || || || || || || ||
|| || || [[111edo]] || || || || || || || [[118edo]] || || [[120edo]] ||
|| || || || || || || || || || [[130edo]] || || ||
|| || || || || || || || [[140edo]] || || [[142edo]] || || ||
and so on. Some larger systems include [[171edo]] [[200edo]] [[224edo]] __[[253edo]]__ [[270edo]] [[311edo]] [[441edo]] [[494edo]] [[612edo]] __[[665edo]]__

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>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/65edo">65edo</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/72edo">72edo</a><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/76edo">76edo</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/87edo">87edo</a><br />
</td>
        <td><a class="wiki_link" href="/88edo">88edo</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/94edo">94edo</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/96edo">96edo</a><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/99edo">99edo</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/111edo">111edo</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/118edo">118edo</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/120edo">120edo</a><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/130edo">130edo</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/140edo">140edo</a><br />
</td>
        <td><br />
</td>
        <td><a class="wiki_link" href="/142edo">142edo</a><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
</table>

and so on. Some larger systems include <a class="wiki_link" href="/171edo">171edo</a> <a class="wiki_link" href="/200edo">200edo</a> <a class="wiki_link" href="/224edo">224edo</a> <u><a class="wiki_link" href="/253edo">253edo</a></u> <a class="wiki_link" href="/270edo">270edo</a> <a class="wiki_link" href="/311edo">311edo</a> <a class="wiki_link" href="/441edo">441edo</a> <a class="wiki_link" href="/494edo">494edo</a> <a class="wiki_link" href="/612edo">612edo</a> <u><a class="wiki_link" href="/665edo">665edo</a></u><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>