Equal-step tuning
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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/Duple (2/1)=== These are, by far, the most widespread ones. See a separate [[edo|dedicated page.]] ===...of the Tritave/Triple (3/1)=== [[7edt]] 11 (Euler Temperament) [[12edt]] [[BP|13 (Bohlen-Pierce)]] [[17edt]] [[15edt]] [[19ED3|19 (Bernhard Stopper)]] 39 Triple Bohlen-Pierce (Erlich) ===...of the Perfect Fifth (3/2)=== [[4edf]] [[6edf]] [[8edf]] ([[88cET]]) [[Carlos Alpha|9 (Carlos Alpha)]] [[Carlos Beta|11 (Carlos Beta)]] [[Carlos Gamma|20 (Carlos Gamma)]] ===...of the Perfect Fourth (4/3)=== 9 - 'Noleta' Scale ===...of the Just Major 17th (5/1)=== 25 (Stockhausen, McLaren) ===.....of various whole tones=== 9:8, 10:9, 12:11, 13:12 ==Equal multiplications?== [[88cET|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:<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><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 "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 <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 "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 <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:<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 -->...of the Octave/Duple (2/1)</h3> These are, by far, the most widespread ones. See a separate <a class="wiki_link" href="/edo">dedicated page.</a><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 -->...of the Tritave/Triple (3/1)</h3> <a class="wiki_link" href="/7edt">7edt</a><br /> 11 (Euler Temperament)<br /> <a class="wiki_link" href="/12edt">12edt</a><br /> <a class="wiki_link" href="/BP">13 (Bohlen-Pierce)</a><br /> <a class="wiki_link" href="/17edt">17edt</a><br /> <a class="wiki_link" href="/15edt">15edt</a><br /> <a class="wiki_link" href="/19ED3">19 (Bernhard Stopper)</a><br /> 39 Triple Bohlen-Pierce (Erlich)<br /> <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 -->...of the Perfect Fifth (3/2)</h3> <a class="wiki_link" href="/4edf">4edf</a><br /> <a class="wiki_link" href="/6edf">6edf</a><br /> <a class="wiki_link" href="/8edf">8edf</a> (<a class="wiki_link" href="/88cET">88cET</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: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> 9 - 'Noleta' Scale<br /> <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 -->...of the Just Major 17th (5/1)</h3> 25 (Stockhausen, McLaren)<br /> <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 /> 9:8, 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></body></html>