Equal-step tuning
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=Equal= **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). 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 "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. If you are wondering how equal divisions of the octave can become associated with temperaments, [[EDOs to ETs|this page]] may help clarify. There are many reasons why one might choose to not consider JI approximations when dealing with equal tunings, and thus not treat equal tunings as temperaments. In such case, 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 (which is assumed to be Just). 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]]. ---- =Scale gallery= ==Equal divisions...== ===[[edo|...of the Octave/Duple (2/1)]]=== by far the most widespread ones ===[[edt|...of the Tritave/Triple (3/1)]]=== ===[[edf|...of the Perfect Fifth (3/2)]]=== ===...of the Perfect Fourth (4/3)=== [[Cube Root of P4|Cube Root of 4/3]] 9 - '[[Noleta]]' Scale ===[[ed5|...of the Just Major 17th (5/1)]]=== ===.....of various whole tones=== [[9 8ths equal temperament|9:8]], 10:9, 12:11, 13:12 ==Equal multiplications?== [[88cET|88-cET]], Alpha, Beta, Gamma ===See also:=== [[edo anatomy]], [[macrotonal edos]], [[quasi-equal]], [[Toctave]]
Original HTML content:
<html><head><title>Equal-step Tuning</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><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> See also <a class="wiki_link" href="/edo">edo</a>.<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). 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>.<br /> <br /> 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 <a class="wiki_link" href="/Regular%20Temperaments">temperament</a>—that is, of a harmony-centric, JI-approximation-based understanding of the scale. If you are wondering how equal divisions of the octave can become associated with temperaments, <a class="wiki_link" href="/EDOs%20to%20ETs">this page</a> may help clarify.<br /> <br /> There are many reasons why one might choose to not consider JI approximations when dealing with equal tunings, and thus not treat equal tunings as temperaments. In such case, the less theory-laden term <em>EDO</em>(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 (which is assumed to be Just). 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 <em>EDn</em> 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).<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:<h1> --><h1 id="toc1"><a name="Scale gallery"></a><!-- ws:end:WikiTextHeadingRule:2 -->Scale gallery</h1> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Scale gallery-Equal divisions..."></a><!-- ws:end:WikiTextHeadingRule:4 -->Equal divisions...</h2> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="Scale gallery-Equal divisions...-...of the Octave/Duple (2/1)"></a><!-- ws:end:WikiTextHeadingRule:6 --><a class="wiki_link" href="/edo">...of the Octave/Duple (2/1)</a></h3> by far the most widespread ones<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="Scale gallery-Equal divisions...-...of the Tritave/Triple (3/1)"></a><!-- ws:end:WikiTextHeadingRule:8 --><a class="wiki_link" href="/edt">...of the Tritave/Triple (3/1)</a></h3> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="Scale gallery-Equal divisions...-...of the Perfect Fifth (3/2)"></a><!-- ws:end:WikiTextHeadingRule:10 --><a class="wiki_link" href="/edf">...of the Perfect Fifth (3/2)</a></h3> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="Scale gallery-Equal divisions...-...of the Perfect Fourth (4/3)"></a><!-- ws:end:WikiTextHeadingRule:12 -->...of the Perfect Fourth (4/3)</h3> <a class="wiki_link" href="/Cube%20Root%20of%20P4">Cube Root of 4/3</a><br /> 9 - '<a class="wiki_link" href="/Noleta">Noleta</a>' Scale<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h3> --><h3 id="toc7"><a name="Scale gallery-Equal divisions...-...of the Just Major 17th (5/1)"></a><!-- ws:end:WikiTextHeadingRule:14 --><a class="wiki_link" href="/ed5">...of the Just Major 17th (5/1)</a></h3> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="Scale gallery-Equal divisions...-.....of various whole tones"></a><!-- ws:end:WikiTextHeadingRule:16 -->.....of various whole tones</h3> <br /> <a class="wiki_link" href="/9%208ths%20equal%20temperament">9:8</a>, 10:9, 12:11, 13:12<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Scale gallery-Equal multiplications?"></a><!-- ws:end:WikiTextHeadingRule:18 -->Equal multiplications?</h2> <a class="wiki_link" href="/88cET">88-cET</a>, Alpha, Beta, Gamma<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h3> --><h3 id="toc10"><a name="Scale gallery-Equal multiplications?-See also:"></a><!-- ws:end:WikiTextHeadingRule:20 -->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>, <a class="wiki_link" href="/Toctave">Toctave</a></body></html>