Equal-step tuning: Difference between revisions
Wikispaces>Andrew_Heathwaite **Imported revision 277294250 - Original comment: ** |
Wikispaces>keenanpepper **Imported revision 277413950 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-11-20 14:32:46 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>277413950</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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**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]]. | **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]]. | ||
==Simultaneous equal divisions== | |||
What do 12ed2, 19ed3, and 28ed5 all have in common? They're all approximately the same scale. This happens because 12ed2 is an accurate temperament (for its size) that contains relatively close approximations of 3/1 and 5/1. In contrast, 11ed2 does not correspond closely to any equal division of 3/1 or 5/1. | |||
The following plot shows equal divisions of 2/1, 3/1, 5/1, and 7/1, and points out some instances when three or more of them happen to be close together. Note that any equal division of 2/1 is automatically an equal division of 4/1; and if something is simultaneously a good equal division of both 2/1 and 3/1, then it's a good equal division of 6/1 as well. | |||
[[image:equal.png width="800" height="69"]] | |||
(Unlimited resolution version: [[http://xenharmonic.wikispaces.com/file/view/equal.svg|equal.svg]]) | |||
For the mathematically inclined, this kind of diagram is closely related to [[The Riemann Zeta Function and Tuning|the Riemann zeta function]]. | |||
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<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 /> | <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 /> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Equal-Simultaneous equal divisions"></a><!-- ws:end:WikiTextHeadingRule:2 -->Simultaneous equal divisions</h2> | |||
What do 12ed2, 19ed3, and 28ed5 all have in common? They're all approximately the same scale. This happens because 12ed2 is an accurate temperament (for its size) that contains relatively close approximations of 3/1 and 5/1. In contrast, 11ed2 does not correspond closely to any equal division of 3/1 or 5/1.<br /> | |||
<br /> | |||
The following plot shows equal divisions of 2/1, 3/1, 5/1, and 7/1, and points out some instances when three or more of them happen to be close together. Note that any equal division of 2/1 is automatically an equal division of 4/1; and if something is simultaneously a good equal division of both 2/1 and 3/1, then it's a good equal division of 6/1 as well.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextLocalImageRule:31:&lt;img src=&quot;/file/view/equal.png/277409084/800x69/equal.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 69px; width: 800px;&quot; /&gt; --><img src="/file/view/equal.png/277409084/800x69/equal.png" alt="equal.png" title="equal.png" style="height: 69px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:31 --><br /> | |||
<br /> | |||
(Unlimited resolution version: <a href="http://xenharmonic.wikispaces.com/file/view/equal.svg">equal.svg</a>)<br /> | |||
<br /> | |||
For the mathematically inclined, this kind of diagram is closely related to <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">the Riemann zeta function</a>.<br /> | |||
<br /> | <br /> | ||
<hr /> | <hr /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Scale gallery"></a><!-- ws:end:WikiTextHeadingRule:4 -->Scale gallery</h1> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Scale gallery-Equal divisions..."></a><!-- ws:end:WikiTextHeadingRule:6 -->Equal divisions...</h2> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Scale gallery-Equal divisions...-...of the Octave/Duple (2/1)"></a><!-- ws:end:WikiTextHeadingRule:8 --><a class="wiki_link" href="/edo">...of the Octave/Duple (2/1)</a></h3> | ||
by far the most widespread ones<br /> | by far the most widespread ones<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Scale gallery-Equal divisions...-...of the Tritave/Triple (3/1)"></a><!-- ws:end:WikiTextHeadingRule:10 --><a class="wiki_link" href="/edt">...of the Tritave/Triple (3/1)</a></h3> | ||
most famously the <a class="wiki_link" href="/BP">BP</a> scale, but lots of others, too<br /> | most famously the <a class="wiki_link" href="/BP">BP</a> scale, but lots of others, too<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- 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 --><a class="wiki_link" href="/edf">...of the Perfect Fifth (3/2)</a></h3> | ||
most famously Carlos Alpha, Beta and Gamma, but lots of others, too<br /> | most famously Carlos Alpha, Beta and Gamma, but lots of others, too<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Scale gallery-Equal divisions...-...of the Perfect Fourth (4/3)"></a><!-- ws:end:WikiTextHeadingRule:14 -->...of the Perfect Fourth (4/3)</h3> | ||
<a class="wiki_link" href="/Cube%20Root%20of%20P4">Cube Root of 4/3</a><br /> | <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 /> | 9 - '<a class="wiki_link" href="/Noleta">Noleta</a>' Scale<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Scale gallery-Equal divisions...-...of the Just Major 17th (5/1)"></a><!-- ws:end:WikiTextHeadingRule:16 --><a class="wiki_link" href="/ed5">...of the Just Major 17th (5/1)</a></h3> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Scale gallery-Equal divisions...-...of the Double Octave (4/1)"></a><!-- ws:end:WikiTextHeadingRule:18 --><a class="wiki_link" href="/ed4">...of the Double Octave (4/1)</a></h3> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Scale gallery-Equal divisions...-...of the Tridecimal Ultramajor Third (13/10)"></a><!-- ws:end:WikiTextHeadingRule:20 -->...of the Tridecimal Ultramajor Third (13/10)</h3> | ||
<a class="wiki_link" href="/Square%20root%20of%2013%20over%2010">Square root of 13 over 10</a><br /> | <a class="wiki_link" href="/Square%20root%20of%2013%20over%2010">Square root of 13 over 10</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Scale gallery-Equal multiplications"></a><!-- ws:end:WikiTextHeadingRule:22 -->Equal multiplications</h2> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Scale gallery-Equal multiplications-...of a given cents value"></a><!-- ws:end:WikiTextHeadingRule:24 -->...of a given cents value</h3> | ||
<a class="wiki_link" href="/88cET">88-cET</a><br /> | <a class="wiki_link" href="/88cET">88-cET</a><br /> | ||
<a class="wiki_link" href="/65cET">65cET</a><br /> | <a class="wiki_link" href="/65cET">65cET</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Scale gallery-Equal multiplications-.....of various whole tones"></a><!-- ws:end:WikiTextHeadingRule:26 -->.....of various whole tones</h3> | ||
<a class="wiki_link" href="/9%208ths%20equal%20temperament">9:8</a>, 10:9, 12:11, 13:12<br /> | <a class="wiki_link" href="/9%208ths%20equal%20temperament">9:8</a>, 10:9, 12:11, 13:12<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Scale gallery-Equal multiplications-See also:"></a><!-- ws:end:WikiTextHeadingRule:28 -->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>, <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">The Riemann Zeta Function and Tuning</a></body></html></pre></div> | <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>, <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">The Riemann Zeta Function and Tuning</a></body></html></pre></div> | ||