3395edo: Difference between revisions

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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>2015-08-18 11:30:28 UTC</tt>.<br>
 
: The original revision id was <tt>556883155</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
3395edo is an extremely strong 17- and 19-limit system, and a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral, and gap edo]]. It has a lower 17-limit [[TE relative error]] than any edo until [[7033edo|7033]], and a lower 19-limit relative error than any edo until [[8269edo|8269]]. Besides, it provides the [[optimal patent val]] for the 13-limit rank-5 temperament tempering out [[6656/6655]], the jacobin comma, and for [[quartismic]], which also tempers out [[123201/123200]]. A basis for the 17-limit commas is {6656/6655, 12376/12375, 28561/28560, 31213/31212, 37180/37179, 937125/937024}, and for the 19-limit, {6656/6655, 12376/12375, 12636/12635, 13377/13376, 14365/14364, 23409/23408, 28561/28560}.
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>
=== Prime harmonics ===
<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 **3395** division divides the octave into 3395 equal steps of 0.35346 cents each. It is an extremely strong 17- and 19-limit system, and a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak, integral and gap edo]]. It has a lower 17-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] than any edo until [[7033edo|7033]], and a lower 19-limit relative error than any edo until [[8269edo|8269]]. A basis for the 17-limit commas is {6656/6655, 12376/12375, 28561/28560, 31213/31212, 37180/37179, 937125/937024}, and for the 19-limit, {6656/6655, 12376/12375, 12636/12635, 13377/13376, 14365/14364, 23409/23408, 28561/28560}.</pre></div>
{{Harmonics in equal|3395|columns=11}}
<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;3395edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;strong&gt;3395&lt;/strong&gt; division divides the octave into 3395 equal steps of 0.35346 cents each. It is an extremely strong 17- and 19-limit system, and a &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&gt;zeta peak, integral and gap edo&lt;/a&gt;. It has a lower 17-limit &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&gt;relative error&lt;/a&gt; than any edo until &lt;a class="wiki_link" href="/7033edo"&gt;7033&lt;/a&gt;, and a lower 19-limit relative error than any edo until &lt;a class="wiki_link" href="/8269edo"&gt;8269&lt;/a&gt;. A basis for the 17-limit commas is {6656/6655, 12376/12375, 28561/28560, 31213/31212, 37180/37179, 937125/937024}, and for the 19-limit, {6656/6655, 12376/12375, 12636/12635, 13377/13376, 14365/14364, 23409/23408, 28561/28560}.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
Since 3395 factors into {{factorization|3395}}, 3395edo has subset edos 5, 7, 35, 97, 485, and 679.
 
== Regular temperament properties ==
3395edo has a lower 17-limit [[TE relative error]] than any edo until [[7033edo|7033]], and a lower 19-limit relative error than any edo until [[8269edo|8269]].
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 2319\3395
| 819.676
| 55115776/34328125
| [[Genojacobin]]
|-
| 35
| 1409\3395<br />(51\3395)
| 498.027<br />(18.026)
| 4/3<br />(?)
| [[Bromine]]
|-
| 97
| 1409\3395<br />(9\3395)
| 498.027<br />(3.181)
| 4/3<br />(?)
| [[Berkelium]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
[[Category:Jacobin]]
[[Category:Quartismic]]

Latest revision as of 13:00, 10 April 2025

← 3394edo 3395edo 3396edo →
Prime factorization 5 × 7 × 97
Step size 0.353461 ¢ 
Fifth 1986\3395 (701.973 ¢)
Semitones (A1:m2) 322:255 (113.8 ¢ : 90.13 ¢)
Consistency limit 21
Distinct consistency limit 21

3395 equal divisions of the octave (abbreviated 3395edo or 3395ed2), also called 3395-tone equal temperament (3395tet) or 3395 equal temperament (3395et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3395 equal parts of about 0.353 ¢ each. Each step represents a frequency ratio of 21/3395, or the 3395th root of 2.

Theory

3395edo is an extremely strong 17- and 19-limit system, and a zeta peak, integral, and gap edo. It has a lower 17-limit TE relative error than any edo until 7033, and a lower 19-limit relative error than any edo until 8269. Besides, it provides the optimal patent val for the 13-limit rank-5 temperament tempering out 6656/6655, the jacobin comma, and for quartismic, which also tempers out 123201/123200. A basis for the 17-limit commas is {6656/6655, 12376/12375, 28561/28560, 31213/31212, 37180/37179, 937125/937024}, and for the 19-limit, {6656/6655, 12376/12375, 12636/12635, 13377/13376, 14365/14364, 23409/23408, 28561/28560}.

Prime harmonics

Approximation of prime harmonics in 3395edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.018 +0.019 +0.011 +0.081 +0.003 +0.022 +0.101 -0.174 +0.055 -0.175
Relative (%) +0.0 +5.2 +5.4 +3.0 +23.0 +0.7 +6.4 +28.6 -49.3 +15.5 -49.6
Steps
(reduced) 		3395
(0) 		5381
(1986) 		7883
(1093) 		9531
(2741) 		11745
(1560) 		12563
(2378) 		13877
(297) 		14422
(842) 		15357
(1777) 		16493
(2913) 		16819
(3239)

Subsets and supersets

Since 3395 factors into 5 × 7 × 97, 3395edo has subset edos 5, 7, 35, 97, 485, and 679.

Regular temperament properties

3395edo has a lower 17-limit TE relative error than any edo until 7033, and a lower 19-limit relative error than any edo until 8269.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 2319\3395 819.676 55115776/34328125 Genojacobin
35 1409\3395
(51\3395) 		498.027
(18.026) 		4/3
(?) 		Bromine
97 1409\3395
(9\3395) 		498.027
(3.181) 		4/3
(?) 		Berkelium

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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