36edo
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2010-08-10 15:48:29 UTC.
- The original revision id was 155970161.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
36edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33.333... cents.
36 is a highly composite number, factoring into 2x2x3x3. Since 36 is divisible by 12, it contains the overly-familiar [[12edo]] as a subset. It divides 12edo's 100-cent half step into three microtonal step of approximately 33 cents, which could be called "sixth tones." 36edo also contains [[18edo]] ("third tones") and [[9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo]] whole tone scale, [[4edo]] full-diminished seventh chord, and the [[3edo]] augmented triad, all of which are present in 12edo.
That 36edo contains 12edo as a subset makes in compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, [[http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/|De-quinin']]). Three 12edo instruments could play the entire gamut.
=As a harmonic temperament=
For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 & 7. As a 3 & 7 tuning, 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [[http://en.wikipedia.org/wiki/Septimal_diesis|Slendro diesis]] of around 36 cents, and as 64:63, the so-called [[http://en.wikipedia.org/wiki/Septimal_comma|septimal comma]] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [[http://en.wikipedia.org/wiki/Septimal_third-tone|Septimal third-tone]] (which = 49:48 x 64:63).
==3-limit (Pythagorean) approximations (same as 12edo):==
3/2 = 701.955... cents; 21 degrees of 36edo = 700 cents.
4/3 = 498.045... cents; 15 degrees of 36edo = 500 cents.
9/8 = 203.910... cents; 6 degrees of 36edo = 200 cents.
16/9 = 996.090... cents; 30 degrees of 36edo = 1000 cents.
27/16 = 905.865... cents; 27 degrees of 36edo = 900 cents.
32/27 = 294.135... cents; 9 degrees of 36edo = 300 cents.
81/64 = 407.820... cents; 12 degrees of 36edo = 400 cents.
128/81 = 792.180... cents; 24 degrees of 36edo = 800 cents.
==7-limit approximations:==
===7 only:===
7/4 = 968.826... cents; 29 degrees of 36edo = 966.666... cents.
8/7 = 231.174... cents; 7 degrees of 36edo = 233.333... cents.
49/32 = 737.652... cents; 22 degrees of 36edo = 733.333... cents.
64/49 = 462.348... cents; 14 degrees of 36edo = 466.666... cents.
===7 & 3:===
7/6 = 266.871... cents; 8 degrees of 36edo = 266.666... cents.
12/7 = 933.129... cents; 28 degrees of 36 = 933.333... cents.
9/7 = 435.084... cents; 13 degrees of 36edo = 433.333... cents.
14/9 = 764.916... cents; 23 degrees of 36edo = 766.666... cents.
28/27 = 62.961... cents; 2 degrees of 36edo = 66.666... cents.
27/14 = 1137.039... cents; 34 degrees of 36edo = 1133.333... cents.
21/16 = 470.781... cents; 14 degrees of 36edo = 466.666... cents.
32/21 = 729.219... cents; 22 degrees of 36edo = 733.333... cents.
49/48 = 35.697... cents; 1 degree of 36edo = 33.333... cents.
96/49 = 1164.303... cents; 35 degrees of 36edo = 1166.666... cents.
49/36 = 533.742... cents; 16 degrees of 36edo = 533.333... cents.
72/49 = 666.258... cents; 20 degrees of 36edo = 666.666... cents.
64/63 = 27.264... cents; 1 degree of 36edo = 33.333... cents.
63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.Original HTML content:
<html><head><title>36edo</title></head><body>36edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33.333... cents.<br /> <br /> 36 is a highly composite number, factoring into 2x2x3x3. Since 36 is divisible by 12, it contains the overly-familiar <a class="wiki_link" href="/12edo">12edo</a> as a subset. It divides 12edo's 100-cent half step into three microtonal step of approximately 33 cents, which could be called "sixth tones." 36edo also contains <a class="wiki_link" href="/18edo">18edo</a> ("third tones") and <a class="wiki_link" href="/9edo">9edo</a> ("two-thirds tones") as subsets, not to mention the <a class="wiki_link" href="/6edo">6edo</a> whole tone scale, <a class="wiki_link" href="/4edo">4edo</a> full-diminished seventh chord, and the <a class="wiki_link" href="/3edo">3edo</a> augmented triad, all of which are present in 12edo.<br /> <br /> That 36edo contains 12edo as a subset makes in compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, <a class="wiki_link_ext" href="http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/" rel="nofollow">De-quinin'</a>). Three 12edo instruments could play the entire gamut.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="As a harmonic temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->As a harmonic temperament</h1> <br /> For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 & 7. As a 3 & 7 tuning, 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_diesis" rel="nofollow">Slendro diesis</a> of around 36 cents, and as 64:63, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_comma" rel="nofollow">septimal comma</a> of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_third-tone" rel="nofollow">Septimal third-tone</a> (which = 49:48 x 64:63).<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="As a harmonic temperament-3-limit (Pythagorean) approximations (same as 12edo):"></a><!-- ws:end:WikiTextHeadingRule:2 -->3-limit (Pythagorean) approximations (same as 12edo):</h2> <br /> 3/2 = 701.955... cents; 21 degrees of 36edo = 700 cents.<br /> 4/3 = 498.045... cents; 15 degrees of 36edo = 500 cents.<br /> 9/8 = 203.910... cents; 6 degrees of 36edo = 200 cents.<br /> 16/9 = 996.090... cents; 30 degrees of 36edo = 1000 cents.<br /> 27/16 = 905.865... cents; 27 degrees of 36edo = 900 cents.<br /> 32/27 = 294.135... cents; 9 degrees of 36edo = 300 cents.<br /> 81/64 = 407.820... cents; 12 degrees of 36edo = 400 cents.<br /> 128/81 = 792.180... cents; 24 degrees of 36edo = 800 cents.<br /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="As a harmonic temperament-7-limit approximations:"></a><!-- ws:end:WikiTextHeadingRule:4 -->7-limit approximations:</h2> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="As a harmonic temperament-7-limit approximations:-7 only:"></a><!-- ws:end:WikiTextHeadingRule:6 -->7 only:</h3> 7/4 = 968.826... cents; 29 degrees of 36edo = 966.666... cents.<br /> 8/7 = 231.174... cents; 7 degrees of 36edo = 233.333... cents.<br /> 49/32 = 737.652... cents; 22 degrees of 36edo = 733.333... cents.<br /> 64/49 = 462.348... cents; 14 degrees of 36edo = 466.666... cents.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="As a harmonic temperament-7-limit approximations:-7 & 3:"></a><!-- ws:end:WikiTextHeadingRule:8 -->7 & 3:</h3> 7/6 = 266.871... cents; 8 degrees of 36edo = 266.666... cents.<br /> 12/7 = 933.129... cents; 28 degrees of 36 = 933.333... cents.<br /> 9/7 = 435.084... cents; 13 degrees of 36edo = 433.333... cents.<br /> 14/9 = 764.916... cents; 23 degrees of 36edo = 766.666... cents.<br /> 28/27 = 62.961... cents; 2 degrees of 36edo = 66.666... cents.<br /> 27/14 = 1137.039... cents; 34 degrees of 36edo = 1133.333... cents.<br /> 21/16 = 470.781... cents; 14 degrees of 36edo = 466.666... cents.<br /> 32/21 = 729.219... cents; 22 degrees of 36edo = 733.333... cents.<br /> 49/48 = 35.697... cents; 1 degree of 36edo = 33.333... cents.<br /> 96/49 = 1164.303... cents; 35 degrees of 36edo = 1166.666... cents.<br /> 49/36 = 533.742... cents; 16 degrees of 36edo = 533.333... cents.<br /> 72/49 = 666.258... cents; 20 degrees of 36edo = 666.666... cents.<br /> 64/63 = 27.264... cents; 1 degree of 36edo = 33.333... cents.<br /> 63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.</body></html>