306edo: Difference between revisions

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**Imported revision 244520207 - Original comment: **
 
→Subsets and supersets: Add superset 612edo
 
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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}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-05 14:34:57 UTC</tt>.<br>
 
: The original revision id was <tt>244520207</tt>.<br>
306edo provides a very accurate fifth, only 0.0058 cents stretched. In the 5-limit, the [[patent val]] [[tempering out|tempers out]] 78732/78125 ([[sensipent comma]]), whereas the alternative 306c val tempers out 32805/32768 ([[schisma]]). In the 7-limit the patent val tempers out [[6144/6125]], whereas 306c tempers out [[16875/16807]]. 306 is the denominator of 179\306, the continued fraction convergent after [[53edo|31\53]] and before [[665edo|389\665]] in the sequence of continued fraction approximations to to log<sub>2</sub>(3/2). On the 2*306 subgroup 2.3.25.7.55 it takes the same values as [[612edo]].
: 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>
306edo provides an excellent approximation of [[Well temperament#Historical well temperaments|Valotti temperament]] due to its representation of the [[Pythagorean comma]] as 6 steps.
<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 //306 equal division// divides the octave into 306 equal parts of  3.922 cents each, and thereby provides a very accurate fifth, only 0.006 cents sharp. In the 5-limit, the [[patent val]] tempers out 78732/78125, whereas the alternative 306c val tempers out 32805/32768. In the 7-limit the patent val tempers out 6144/6125, whereas 306c tempers out 16875/16807. 306 is the denominator of 179/306, the continued fraction convergent after 31/53 and before 389/665 in the sequence of continued fraction approximations to the log base 2 of 3/2. On the 2*306 subgroup 2.3.25.7.55 it takes the same values as [[612edo]].</pre></div>
=== Prime harmonics ===
<h4>Original HTML content:</h4>
{{Harmonics in equal|306|prec=3}}
<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;306edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;306 equal division&lt;/em&gt; divides the octave into 306 equal parts of  3.922 cents each, and thereby provides a very accurate fifth, only 0.006 cents sharp. In the 5-limit, the &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; tempers out 78732/78125, whereas the alternative 306c val tempers out 32805/32768. In the 7-limit the patent val tempers out 6144/6125, whereas 306c tempers out 16875/16807. 306 is the denominator of 179/306, the continued fraction convergent after 31/53 and before 389/665 in the sequence of continued fraction approximations to the log base 2 of 3/2. On the 2*306 subgroup 2.3.25.7.55 it takes the same values as &lt;a class="wiki_link" href="/612edo"&gt;612edo&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Subsets and supersets ===
Since 306 factors into {{factorization|306}}, 306edo has subset edos {{EDOs| 2, 3, 6, 9, 17, 18, 34, 51, 102, and 153 }}. [[612edo]], which doubles it, provides corrections for the 5th, 11th, and 29th harmonics.
 
[[Category:3-limit record edos|###]] <!-- 3-digit number -->

Latest revision as of 07:09, 29 November 2025

← 305edo 306edo 307edo →
Prime factorization 2 × 32 × 17
Step size 3.92157 ¢ 
Fifth 179\306 (701.961 ¢)
(convergent)
Semitones (A1:m2) 29:23 (113.7 ¢ : 90.2 ¢)
Consistency limit 5
Distinct consistency limit 5

306 equal divisions of the octave (abbreviated 306edo or 306ed2), also called 306-tone equal temperament (306tet) or 306 equal temperament (306et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 306 equal parts of about 3.92 ¢ each. Each step represents a frequency ratio of 21/306, or the 306th root of 2.

306edo provides a very accurate fifth, only 0.0058 cents stretched. In the 5-limit, the patent val tempers out 78732/78125 (sensipent comma), whereas the alternative 306c val tempers out 32805/32768 (schisma). In the 7-limit the patent val tempers out 6144/6125, whereas 306c tempers out 16875/16807. 306 is the denominator of 179\306, the continued fraction convergent after 31\53 and before 389\665 in the sequence of continued fraction approximations to to log2(3/2). On the 2*306 subgroup 2.3.25.7.55 it takes the same values as 612edo.

306edo provides an excellent approximation of Valotti temperament due to its representation of the Pythagorean comma as 6 steps.

Prime harmonics

Approximation of prime harmonics in 306edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.006 +1.922 -0.198 +1.623 -1.312 +0.927 +0.526 -0.823 +1.795 +0.062
Relative (%) +0.0 +0.1 +49.0 -5.1 +41.4 -33.5 +23.6 +13.4 -21.0 +45.8 +1.6
Steps
(reduced) 		306
(0) 		485
(179) 		711
(99) 		859
(247) 		1059
(141) 		1132
(214) 		1251
(27) 		1300
(76) 		1384
(160) 		1487
(263) 		1516
(292)

Subsets and supersets

Since 306 factors into 2 × 32 × 17, 306edo has subset edos 2, 3, 6, 9, 17, 18, 34, 51, 102, and 153. 612edo, which doubles it, provides corrections for the 5th, 11th, and 29th harmonics.

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