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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}} It is also known as the '''Triple Bohlen–Pierce scale''' ('''Triple BP'''), since it divides each step of the equal-tempered [[Bohlen–Pierce]] scale ([[13edt]]) into three equal parts.
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-11-19 20:02:43 UTC</tt>.<br>
: The original revision id was <tt>277286364</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>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 39 equal division of 3, the tritave, divides it into 39 equal parts of 48.678 cents each, a [[nonoctave]] tuning corresponding to 24.606 edo. It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]], and like [[26edt]] and [[52edt]] it is a multiple of [[13edt]] and so contains the [[Bohlen-Pierce]] scale. It is contorted in the 7-limit, tempering out the same BP commas 245/243 and 3125/3087 as 13edt. In the 11-limit it tempers out 1331/1323 and in the 13-limit 275/273, 847/845 and 1575/1573. It is related to the 49f&amp;172f temperament tempering out 245/243, 275/273, 847/845 and 1575/1573, which has map [&lt;1 0 0 0 0 0|, &lt;0 39 57 69 85 91|]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth [[The Riemann Zeta Function and Tuning#Removing%20primes|no-twos zeta peak edt]].


==Intervals of 39EDT==
39edt can be described as approximately 24.606[[edo]]. This implies that each step of 39edt can be approximated by 5 steps of [[123edo]]. 39edt contains within it a close approximation of [[4ed11/5]]: every seventh step of 39edt equates to a step of 4ed11/5.


||~ Degrees of 39EDT ||~ Cents Value ||~ ¢ Octave-Reduced ||~ Degrees of [[BP]] ||~ Comments ||
== Theory ==
|| 0 || 0.000 ||  || 0 || 1/1 ||
It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]; in fact it has a better no-twos 13-[[odd limit]] relative error than any other edt up to [[914edt]]. Like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being [[contorted]] in the no-twos 7-limit, tempering out the same BP commas, [[245/243]] and [[3125/3087]], as 13edt. In the [[11-limit]] it tempers out [[1331/1323]] and in the [[13-limit]] [[275/273]], [[1575/1573]], and [[847/845]]. An efficient traversal is therefore given by [[Mintra]] temperament, which in the 13-limit tempers out 275/273 and 1331/1323 alongside 245/243, and is generated by the interval of [[11/7]], which serves as a [[macrodiatonic]] "superpyth" fourth and splits the [[BPS]] generator of [[9/7]], up a tritave, in three.
|| 1 || 48.768 ||  ||  || 39th root of 3 ||
|| 2 || 97.536 ||  ||  ||  ||
|| 3 || 146.304 ||  || 1 || 13th root of 3 ||
|| 4 || 195.072 ||  ||  ||  ||
|| 5 || 243.840 ||  ||  ||  ||
|| 6 || 292.608 ||  || 2 ||  ||
|| 7 || 341.377 ||  ||  ||  ||
|| 8 || 390.145 ||  ||  ||  ||
|| 9 || 438.913 ||  || 3 ||  ||
|| 10 || 487.681 ||  ||  ||  ||
|| 11 || 536.449 ||  ||  ||  ||
|| 12 || 585.217 ||  || 4 ||  ||
|| 13 || 633.985 ||  ||  ||  ||
|| 14 || 682.753 ||  ||  ||  ||
|| 15 || 731.521 ||  || 5 ||  ||
|| 16 || 780.289 ||  ||  ||  ||
|| 17 || 829.057 ||  ||  ||  ||
|| 18 || 877.825 ||  || 6 ||  ||
|| 19 || 926.593 ||  ||  ||  ||
|| 20 || 975.362 ||  ||  ||  ||
|| 21 || 1024.130 ||  || 7 ||  ||
|| 22 || 1072.898 ||  ||  ||  ||
|| 23 || 1121.666 ||  ||  ||  ||
|| 24 || 1170.434 ||  || 8 ||  ||
|| 25 || 1219.202 || 19.202 ||  ||  ||
|| 26 || 1267.970 || 67.970 ||  ||  ||
|| 27 || 1316.738 || 116.738 || 9 ||  ||
|| 28 || 1365.506 || 165.506 ||  ||  ||
|| 29 || 1414.274 || 214.274 ||  ||  ||
|| 30 || 1463.042 || 263.042 || 10 ||  ||
|| 31 || 1511.810 || 311.810 ||  ||  ||
|| 32 || 1560.578 || 360.578 ||  ||  ||
|| 33 || 1609.347 || 409.347 || 11 ||  ||
|| 34 || 1658.115 || 458.115 ||  ||  ||
|| 35 || 1706.883 || 506.883 ||  ||  ||
|| 36 || 1755.651 || 555.651 || 12 ||  ||
|| 37 || 1804.419 || 604.419 ||  ||  ||
|| 38 || 1853.187 || 653.187 ||  ||  ||
|| 39 || 1901.955 || 701.955 || 13 || 3/1 (tritave) ||
|| 40 || 1950.723 || 750.723 ||  ||  ||
|| 41 || 1999.491 || 799.491 ||  ||  ||
|| 42 || 2048.259 || 848.259 || 14 ||  ||
|| 43 || 2097.027 || 897.027 ||  ||  ||
|| 44 || 2145.795 || 945.795 ||  ||  ||
|| 45 || 2194.563 || 994.563 || 15 ||  ||
|| 46 || 2243.332 || 1043.332 ||  ||  ||
|| 47 || 2292.100 || 1092.100 ||  ||  ||
|| 48 || 2340.868 || 1140.868 || 16 ||  ||
|| 49 || 2389.636 || 1189.636 ||  ||  ||
|| 50 || 2438.404 || 38.404 ||  ||  ||
|| 51 || 2487.172 || 87.172 || 17 ||  ||
|| 52 || 2535.940 || 135.940 ||  ||  ||
|| 53 || 2584.708 || 184.708 ||  ||  ||
|| 54 || 2633.476 || 233.476 || 18 ||  ||
|| 55 || 2682.244 || 282.244 ||  ||  ||
|| 56 || 2731.012 || 331.012 ||  ||  ||
|| 57 || 2779.780 || 379.780 || 19 ||  ||
|| 58 || 2828.548 || 428.548 ||  ||  ||
|| 59 || 2877.317 || 477.317 ||  ||  ||
|| 60 || 2926.085 || 526.085 || 20 ||  ||
|| 61 || 2974.853 || 574.853 ||  ||  ||
|| 62 || 3023.621 || 623.621 ||  ||  ||
|| 63 || 3072.389 || 672.389 || 21 ||  ||
|| 64 || 3121.157 || 721.157 ||  ||  ||
|| 65 || 3169.925 || 769.925 ||  ||  ||
|| 66 || 3218.693 || 818.693 || 22 ||  ||
|| 67 || 3267.461 || 867.461 ||  ||  ||
|| 68 || 3316.229 || 916.229 ||  ||  ||
|| 69 || 3364.997 || 964.997 || 23 ||  ||
|| 70 || 3413.765 || 1013.765 ||  ||  ||
|| 71 || 3462.533 || 1062.533 ||  ||  ||
|| 72 || 3511.302 || 1111.302 || 24 ||  ||
|| 73 || 3560.070 || 1160.070 ||  ||  ||
|| 74 || 3608.838 || 8.838 ||  ||  ||
|| 75 || 3657.606 || 57.606 || 25 ||  ||
|| 76 || 3706.374 || 106.374 ||  ||  ||
|| 77 || 3755.142 || 155.142 ||  ||  ||
|| 78 || 3803.910 || 203.910 || 26 || 9/1 ||</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;39edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 39 equal division of 3, the tritave, divides it into 39 equal parts of 48.678 cents each, a &lt;a class="wiki_link" href="/nonoctave"&gt;nonoctave&lt;/a&gt; tuning corresponding to 24.606 edo. It is a strong no-twos 13-limit system, a fact first noted by &lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt;, and like &lt;a class="wiki_link" href="/26edt"&gt;26edt&lt;/a&gt; and &lt;a class="wiki_link" href="/52edt"&gt;52edt&lt;/a&gt; it is a multiple of &lt;a class="wiki_link" href="/13edt"&gt;13edt&lt;/a&gt; and so contains the &lt;a class="wiki_link" href="/Bohlen-Pierce"&gt;Bohlen-Pierce&lt;/a&gt; scale. It is contorted in the 7-limit, tempering out the same BP commas 245/243 and 3125/3087 as 13edt. In the 11-limit it tempers out 1331/1323 and in the 13-limit 275/273, 847/845 and 1575/1573. It is related to the 49f&amp;amp;172f temperament tempering out 245/243, 275/273, 847/845 and 1575/1573, which has map [&amp;lt;1 0 0 0 0 0|, &amp;lt;0 39 57 69 85 91|]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Removing%20primes"&gt;no-twos zeta peak edt&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Intervals of 39EDT"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Intervals of 39EDT&lt;/h2&gt;
&lt;br /&gt;


If octaves are inserted, 39edt is related to the {{nowrap|49f &amp; 172f}} temperament in the full 13-limit, known as [[Sensamagic clan#Triboh|triboh]], tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [{{val|1 0 0 0 0 0}}, {{val|0 39 57 69 85 91}}]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth [[The Riemann zeta function and tuning#Removing primes|no-twos zeta peak edt]].


&lt;table class="wiki_table"&gt;
When treated as an octave-repeating tuning with the sharp octave of 25 steps (about 1219 cents), and the other primes chosen by their best octave-reduced mappings, it functions as a tuning of [[mavila]] temperament, analogous to [[25edo]]'s mavila.  
    &lt;tr&gt;
        &lt;th&gt;Degrees of 39EDT&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Cents Value&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;¢ Octave-Reduced&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Degrees of &lt;a class="wiki_link" href="/BP"&gt;BP&lt;/a&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Comments&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;48.768&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;39th root of 3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;97.536&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;146.304&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13th root of 3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;195.072&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;243.840&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;292.608&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;341.377&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;390.145&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;438.913&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;487.681&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;536.449&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;585.217&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;633.985&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;682.753&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;731.521&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;780.289&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;829.057&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;877.825&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;19&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;926.593&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;975.362&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1024.130&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1072.898&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;23&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1121.666&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;24&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1170.434&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1219.202&lt;br /&gt;
&lt;/td&gt;
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        &lt;td&gt;49&lt;br /&gt;
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        &lt;td&gt;54&lt;br /&gt;
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        &lt;td&gt;55&lt;br /&gt;
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        &lt;td&gt;57&lt;br /&gt;
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        &lt;td&gt;58&lt;br /&gt;
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        &lt;td&gt;59&lt;br /&gt;
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        &lt;td&gt;60&lt;br /&gt;
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&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;71&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3462.533&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1062.533&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;72&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3511.302&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1111.302&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;24&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;73&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3560.070&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1160.070&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;74&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3608.838&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8.838&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;75&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3657.606&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;57.606&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;76&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3706.374&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;106.374&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;77&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3755.142&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;155.142&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;78&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3803.910&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;203.910&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;26&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
Mavila is one of the few places where octave-stretching makes sense, due to how flat the fifth and often the major third are; this fifth of 683 cents is much more recognizable as a perfect fifth of 3/2 than the 672-cent tuning with just octaves.
{{Harmonics in equal|39|3|1|intervals=prime|columns=12}}
 
== Intervals ==
All intervals shown are within the 91-[[odd limit#Nonoctave equaves|throdd limit]] and are consistently represented.
 
{| class="wikitable center-all right-2 right-3"
|-
! Steps
! [[Cent]]s
! [[Hekt]]s
! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree
! Corresponding 3.5.7.11.13 subgroup<br />intervals
! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs,<br />{{nowrap|J {{=}} 1/1}})
! Mintaka[7]<br />(E macro-Phrygian)
|-
| 0
| 0
| 0
| P1
| [[1/1]]
| J
| E
|-
| 1
| 48.8
| 33.3
| SP1
| [[77/75]] (+3.2¢); [[65/63]] (&minus;5.3¢)
| ^J
| ^E, vF
|-
| 2
| 97.5
| 66.7
| sA1/sm2
| [[35/33]] (&minus;4.3¢); [[81/77]] (+9.9¢)
| vK
| F
|-
| 3
| 146.3
| 100
| A1/m2
| [[99/91]] (+0.4¢); [[49/45]] (&minus;1.1¢); [[27/25]] (+13.1¢)
| K
| ^F, vGb, Dx
|-
| 4
| 195.1
| 133.3
| SA1/Sm2
| [[55/49]] (&minus;4.9¢); [[91/81]] (&minus;6.5¢); [[39/35]] (+7.7¢)
| ^K
| Gb, vE#
|-
| 5
| 243.8
| 166.7
| sM2/sd3
| [[15/13]] (&minus;3.9¢); [[63/55]] (+8.7¢)
| vK#, vLb
| ^Gb, E#
|-
| 6
| 292.6
| 200
| M2/d3
| [[77/65]] (&minus;0.7¢); [[13/11]] (+3.4¢); [[25/21]] (&minus;9.2¢)
| K#, Lb
| vF#, ^E#
|-
| 7
| 341.4
| 233.3
| SM2/Sd3
| [[11/9]] (&minus;6.0¢); [[91/75]] (+6.6¢)
| ^K#, ^Lb
| F#
|-
| 8
| 390.1
| 266.7
| sA2/sP3/sd4
| [[49/39]] (&minus;5.0¢); [[81/65]] (+9.2¢)
| vL
| vG, ^F#
|-
| 9
| 438.9
| 300
| A2/P3/d4
| [[9/7]] (+3.8¢); [[35/27]] (&minus;10.3¢)
| L
| G
|-
| 10
| 487.7
| 333.3
| SA2/SP3/Sd4
| [[65/49]] (&minus;1.5¢); [[33/25]] (+7.0¢)
| ^L
| ^G, vAb
|-
| 11
| 536.4
| 366.7
| sA3/sm4/sd5
| [[15/11]] (&minus;0.5¢)
| vM
| Ab
|-
| 12
| 585.2
| 400
| A3/m4/d5
| [[7/5]] (+2.7¢)
| M
| ^Ab, Fx
|-
| 13
| 634.0
| 433.3
| SA3/Sm4/Sd5
| [[13/9]] (&minus;2.6¢)
| ^M
| vG#
|-
| 14
| 682.7
| 466.7
| sM4/sm5
| [[135/91]] (+0.07¢); [[49/33]] (&minus;1.6¢); [[81/55]] (+12.6¢)
| vM#, vNb
| G#
|-
| 15
| 731.5
| 500
| M4/m5
| [[75/49]] (&minus;5.4¢); [[117/77]] (+7.2¢)
| M#, Nb
| vA, ^G#
|-
| 16
| 780.3
| 533.3
| SM4/Sm5
| [[11/7]] (&minus;2.2¢); [[39/25]] (+10.4¢)
| ^M#, ^Nb
| A
|-
| 17
| 829.0
| 566.7
| sA4/sM5
| [[21/13]] (&minus;1.2¢)
| vN
| ^A, vBb
|-
| 18
| 877.8
| 600
| A4/M5
| [[91/55]] (+6.1¢); [[5/3]] (&minus;6.5¢); [[81/49]] (+7.7¢)
| N
| Bb
|-
| 19
| 926.6
| 633.3
| SA4/SM5
| [[77/45]] (&minus;3.3¢)
| ^N
| ^Bb, vCb, Gx
|-
| 20
| 975.3
| 666.7
| sA5/sm6/sd7
| [[135/77]] (+3.3¢)
| vO
| vA#, Cb
|-
| 21
| 1024.1
| 700
| A5/m6/d7
| [[165/91]] (&minus;6.1¢); [[9/5]] (+6.5¢); [[49/27]] (&minus;7.7¢)
| O
| A#, ^Cb
|-
| 22
| 1072.9
| 733.3
| SA5/Sm6/Sd7
| [[13/7]] (+1.2¢)
| ^O
| vB, ^A#
|-
| 23
| 1121.6
| 766.7
| sM6/sm7
| [[21/11]] (+2.2¢); [[25/13]] (&minus;10.4¢)
| vO#, vPb
| B
|-
| 24
| 1170.4
| 800
| M6/m7
| [[49/25]] (+5.4¢); [[77/39]] (&minus;7.2¢)
| O#, Pb
| ^B, vC
|-
| 25
| 1219.2
| 833.3
| SM6/Sm7
| [[91/45]] (+0.07¢); [[99/49]] (+1.6¢); [[55/27]] (&minus;12.6¢)
| ^O#, ^Pb
| C
|-
| 26
| 1267.9
| 866.7
| sA6/sM7/sd8
| [[27/13]] (+2.6¢)
| vP
| ^C, vDb
|-
| 27
| 1316.7
| 900
| A6/M7/d8
| [[15/7]] (&minus;2.7¢)
| P
| Db, vB#
|-
| 28
| 1365.5
| 933.3
| SA6/SM7/Sd8
| [[11/5]] (+0.5¢)
| ^P
| ^Db, B#
|-
| 29
| 1414.2
| 966.7
| sP8/sd9
| [[147/65]] (+1.5¢); [[25/11]] (&minus;7.0¢)
| vQ
| vC#, ^B#
|-
| 30
| 1463.0
| 1000
| P8/d9
| [[7/3]] (&minus;3.8¢); [[81/35]] (+10.3¢)
| Q
| C#
|-
| 31
| 1511.8
| 1033.3
| SP8/Sd9
| [[117/49]] (+5.0¢); [[65/27]] (&minus;9.2¢)
| ^Q
| vD, ^C#
|-
| 32
| 1560.5
| 1066.7
| sA8/sm9
| [[27/11]] (+6.0¢); [[225/91]] (+6.6¢)
| vQ#, vRb
| D
|-
| 33
| 1609.3
| 1100
| A8/m9
| [[195/77]] (&minus;0.7¢); [[33/13]] (&minus;3.4¢); [[63/25]] (+9.2¢)
| Q#, Rb
| ^D, vEb
|-
| 34
| 1658.1
| 1133.3
| SA8/Sm9
| [[13/5]] (+3.9¢); [[55/21]] (&minus;8.7¢)
| ^Q#, ^Rb
| Eb
|-
| 35
| 1706.9
| 1166.7
| sM9/sd10
| [[147/55]] (+4.9¢); [[243/91]] (+6.5¢); [[35/13]] (&minus;7.7¢)
| vR
| ^Eb, vFb, Cx
|-
| 36
| 1755.7
| 1200
| M9/d10
| [[91/33]] (+0.4¢); [[135/49]] (+1.1¢); [[25/9]] (&minus;13.1¢)
| R
| vD#, Fb
|-
| 37
| 1804.5
| 1233.3
| SM9/Sd10
| [[99/35]] (+4.3¢); [[77/27]] (&minus;9.9¢)
| ^R
| D#, ^Fb
|-
| 38
| 1853.2
| 1266.7
| sA9/sP10
| [[225/77]] (&minus;3.2¢); [[189/65]] (+5.3¢)
| vJ
| vE, ^D#
|-
| 39
| 1902.0
| 1300
| A9/P10
| [[3/1]]
| J
| E
|}
 
== Approximation to JI ==
 
=== No-2 zeta peak ===
{| class="wikitable"
|+
!Steps
per octave
!Steps
per tritave
!Step size
(cents)
!Height
!Tritave size
(cents)
!Tritave stretch
(cents)
|-
|24.573831630
|38.948601633
|48.832433543
|4.665720
|1904.464908194
|2.509907328
|}
 
Every 7 steps of the [[172edo|172f]] val is an excellent approximation of the ninth no-2 zeta peak in the 15-limit.
 
== Music ==
; [[Francium]]
* [https://www.youtube.com/watch?v=jstg4_B0jfY ''Strange Juice''] (2025)
;[https://www.youtube.com/@PhanomiumMusic Phanomium]
* ''[https://www.youtube.com/watch?v=GX79ZX1Z8C8 Polygonal]'' (2025)

Latest revision as of 13:30, 30 June 2025

← 38edt 39edt 40edt →
Prime factorization 3 × 13
Step size 48.7681 ¢ 
Octave 25\39edt (1219.2 ¢)
Consistency limit 3
Distinct consistency limit 3

39 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 39edt or 39ed3), is a nonoctave tuning system that divides the interval of 3/1 into 39 equal parts of about 48.8 ¢ each. Each step represents a frequency ratio of 31/39, or the 39th root of 3. It is also known as the Triple Bohlen–Pierce scale (Triple BP), since it divides each step of the equal-tempered Bohlen–Pierce scale (13edt) into three equal parts.

39edt can be described as approximately 24.606edo. This implies that each step of 39edt can be approximated by 5 steps of 123edo. 39edt contains within it a close approximation of 4ed11/5: every seventh step of 39edt equates to a step of 4ed11/5.

Theory

It is a strong no-twos 13-limit system, a fact first noted by Paul Erlich; in fact it has a better no-twos 13-odd limit relative error than any other edt up to 914edt. Like 26edt and 52edt, it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being contorted in the no-twos 7-limit, tempering out the same BP commas, 245/243 and 3125/3087, as 13edt. In the 11-limit it tempers out 1331/1323 and in the 13-limit 275/273, 1575/1573, and 847/845. An efficient traversal is therefore given by Mintra temperament, which in the 13-limit tempers out 275/273 and 1331/1323 alongside 245/243, and is generated by the interval of 11/7, which serves as a macrodiatonic "superpyth" fourth and splits the BPS generator of 9/7, up a tritave, in three.

If octaves are inserted, 39edt is related to the 49f & 172f temperament in the full 13-limit, known as triboh, tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [⟨1 0 0 0 0 0], ⟨0 39 57 69 85 91]]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth no-twos zeta peak edt.

When treated as an octave-repeating tuning with the sharp octave of 25 steps (about 1219 cents), and the other primes chosen by their best octave-reduced mappings, it functions as a tuning of mavila temperament, analogous to 25edo's mavila.

Mavila is one of the few places where octave-stretching makes sense, due to how flat the fifth and often the major third are; this fifth of 683 cents is much more recognizable as a perfect fifth of 3/2 than the 672-cent tuning with just octaves.

Approximation of prime harmonics in 39edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +19.2 +0.0 -6.5 -3.8 -6.0 -2.6 +20.6 +23.1 -15.0 +22.6 +4.7 -9.0
Relative (%) +39.4 +0.0 -13.4 -7.9 -12.4 -5.4 +42.3 +47.4 -30.8 +46.3 +9.6 -18.5
Steps
(reduced) 		25
(25) 		39
(0) 		57
(18) 		69
(30) 		85
(7) 		91
(13) 		101
(23) 		105
(27) 		111
(33) 		120
(3) 		122
(5) 		128
(11)

Intervals

All intervals shown are within the 91-throdd limit and are consistently represented.

Steps Cents Hekts Enneatonic
degree 		Corresponding 3.5.7.11.13 subgroup
intervals 		Lambda
(sLsLsLsLs,
J = 1/1) 		Mintaka[7]
(E macro-Phrygian)
0 0 0 P1 1/1 J E
1 48.8 33.3 SP1 77/75 (+3.2¢); 65/63 (−5.3¢) ^J ^E, vF
2 97.5 66.7 sA1/sm2 35/33 (−4.3¢); 81/77 (+9.9¢) vK F
3 146.3 100 A1/m2 99/91 (+0.4¢); 49/45 (−1.1¢); 27/25 (+13.1¢) K ^F, vGb, Dx
4 195.1 133.3 SA1/Sm2 55/49 (−4.9¢); 91/81 (−6.5¢); 39/35 (+7.7¢) ^K Gb, vE#
5 243.8 166.7 sM2/sd3 15/13 (−3.9¢); 63/55 (+8.7¢) vK#, vLb ^Gb, E#
6 292.6 200 M2/d3 77/65 (−0.7¢); 13/11 (+3.4¢); 25/21 (−9.2¢) K#, Lb vF#, ^E#
7 341.4 233.3 SM2/Sd3 11/9 (−6.0¢); 91/75 (+6.6¢) ^K#, ^Lb F#
8 390.1 266.7 sA2/sP3/sd4 49/39 (−5.0¢); 81/65 (+9.2¢) vL vG, ^F#
9 438.9 300 A2/P3/d4 9/7 (+3.8¢); 35/27 (−10.3¢) L G
10 487.7 333.3 SA2/SP3/Sd4 65/49 (−1.5¢); 33/25 (+7.0¢) ^L ^G, vAb
11 536.4 366.7 sA3/sm4/sd5 15/11 (−0.5¢) vM Ab
12 585.2 400 A3/m4/d5 7/5 (+2.7¢) M ^Ab, Fx
13 634.0 433.3 SA3/Sm4/Sd5 13/9 (−2.6¢) ^M vG#
14 682.7 466.7 sM4/sm5 135/91 (+0.07¢); 49/33 (−1.6¢); 81/55 (+12.6¢) vM#, vNb G#
15 731.5 500 M4/m5 75/49 (−5.4¢); 117/77 (+7.2¢) M#, Nb vA, ^G#
16 780.3 533.3 SM4/Sm5 11/7 (−2.2¢); 39/25 (+10.4¢) ^M#, ^Nb A
17 829.0 566.7 sA4/sM5 21/13 (−1.2¢) vN ^A, vBb
18 877.8 600 A4/M5 91/55 (+6.1¢); 5/3 (−6.5¢); 81/49 (+7.7¢) N Bb
19 926.6 633.3 SA4/SM5 77/45 (−3.3¢) ^N ^Bb, vCb, Gx
20 975.3 666.7 sA5/sm6/sd7 135/77 (+3.3¢) vO vA#, Cb
21 1024.1 700 A5/m6/d7 165/91 (−6.1¢); 9/5 (+6.5¢); 49/27 (−7.7¢) O A#, ^Cb
22 1072.9 733.3 SA5/Sm6/Sd7 13/7 (+1.2¢) ^O vB, ^A#
23 1121.6 766.7 sM6/sm7 21/11 (+2.2¢); 25/13 (−10.4¢) vO#, vPb B
24 1170.4 800 M6/m7 49/25 (+5.4¢); 77/39 (−7.2¢) O#, Pb ^B, vC
25 1219.2 833.3 SM6/Sm7 91/45 (+0.07¢); 99/49 (+1.6¢); 55/27 (−12.6¢) ^O#, ^Pb C
26 1267.9 866.7 sA6/sM7/sd8 27/13 (+2.6¢) vP ^C, vDb
27 1316.7 900 A6/M7/d8 15/7 (−2.7¢) P Db, vB#
28 1365.5 933.3 SA6/SM7/Sd8 11/5 (+0.5¢) ^P ^Db, B#
29 1414.2 966.7 sP8/sd9 147/65 (+1.5¢); 25/11 (−7.0¢) vQ vC#, ^B#
30 1463.0 1000 P8/d9 7/3 (−3.8¢); 81/35 (+10.3¢) Q C#
31 1511.8 1033.3 SP8/Sd9 117/49 (+5.0¢); 65/27 (−9.2¢) ^Q vD, ^C#
32 1560.5 1066.7 sA8/sm9 27/11 (+6.0¢); 225/91 (+6.6¢) vQ#, vRb D
33 1609.3 1100 A8/m9 195/77 (−0.7¢); 33/13 (−3.4¢); 63/25 (+9.2¢) Q#, Rb ^D, vEb
34 1658.1 1133.3 SA8/Sm9 13/5 (+3.9¢); 55/21 (−8.7¢) ^Q#, ^Rb Eb
35 1706.9 1166.7 sM9/sd10 147/55 (+4.9¢); 243/91 (+6.5¢); 35/13 (−7.7¢) vR ^Eb, vFb, Cx
36 1755.7 1200 M9/d10 91/33 (+0.4¢); 135/49 (+1.1¢); 25/9 (−13.1¢) R vD#, Fb
37 1804.5 1233.3 SM9/Sd10 99/35 (+4.3¢); 77/27 (−9.9¢) ^R D#, ^Fb
38 1853.2 1266.7 sA9/sP10 225/77 (−3.2¢); 189/65 (+5.3¢) vJ vE, ^D#
39 1902.0 1300 A9/P10 3/1 J E

Approximation to JI

No-2 zeta peak

Steps

per octave

Steps

per tritave

Step size

(cents)

Height Tritave size

(cents)

Tritave stretch

(cents)

24.573831630 38.948601633 48.832433543 4.665720 1904.464908194 2.509907328

Every 7 steps of the 172f val is an excellent approximation of the ninth no-2 zeta peak in the 15-limit.

Music

Francium
Phanomium
Retrieved from "https://en.xen.wiki/index.php?title=39edt&oldid=203919"