954edo: 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>2011-07-14 19:33:35 UTC</tt>.<br>
 
: The original revision id was <tt>241410939</tt>.<br>
954edo is a very strong 17-limit system, [[consistency|distinctly consistent]] in the 17-limit, and is a [[zeta edo|zeta peak, integral and gap edo]]. The tuning of the primes to 17 are all flat, and the equal temperament [[tempering out|tempers out]] the [[ennealimma]], {{monzo| 1 -27 18 }}, in the 5-limit and [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament. In the 11-limit it tempers out [[3025/3024]], [[9801/9800]], 43923/43904, and 151263/151250 so that it supports hemiennealimmal. In the 13-limit it tempers out [[4225/4224]] and [[10648/10647]] and in the 17-limit 2431/2430 and [[2601/2600]]. It supports and gives the [[optimal patent val]] for the [[semihemiennealimmal]] temperament.
: 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>
=== Prime harmonics ===
<h4>Original Wikitext content:</h4>
{{Harmonics in equal|954|columns=13}}
<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 //954 equal division// divides the octave into 954 equal parts of 1.258 cents each. It is a very strong 17-limit system, uniquely [[consistent]] in the 17-limit, and is the fifteenth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]] and is also a zeta gap edo. The tuning of the primes to 17 are all flat, and it tempers out the ennealimma, |1 -27 18&gt;, in the 5-limit and 2401/2400 and 4375/4374 in the 7-limit, so that it supports [[Ragismic microtemperaments#Ennealimmal|ennealimmal temperament]]. In the 11-limit it tempers out 3025/3024, 9801/9800, 43923/43904, and 151263/151250 so that it supports hemiennealimmal. In the 13-limit it tempers out 4225/4224 and 10648/10647 and in the 17-limit 2431/2430 and 2601/2600. It supports and gives the optimal patent val for semihemiennealimmal temperament.</pre></div>
{{Harmonics in equal|954|start=14|columns=17|collapsed=1|title=Approximation of prime harmonics in 954edo (continued)}}
<h4>Original HTML content:</h4>
{{Harmonics in equal|954|start=31|columns=18|collapsed=1|title=Approximation of prime harmonics in 954edo (continued)}}
<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;954edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;954 equal division&lt;/em&gt; divides the octave into 954 equal parts of 1.258 cents each. It is a very strong 17-limit system, uniquely &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt; in the 17-limit, and is the fifteenth &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&gt;zeta integral edo&lt;/a&gt; and is also a zeta gap edo. The tuning of the primes to 17 are all flat, and it tempers out the ennealimma, |1 -27 18&amp;gt;, in the 5-limit and 2401/2400 and 4375/4374 in the 7-limit, so that it supports &lt;a class="wiki_link" href="/Ragismic%20microtemperaments#Ennealimmal"&gt;ennealimmal temperament&lt;/a&gt;. In the 11-limit it tempers out 3025/3024, 9801/9800, 43923/43904, and 151263/151250 so that it supports hemiennealimmal. In the 13-limit it tempers out 4225/4224 and 10648/10647 and in the 17-limit 2431/2430 and 2601/2600. It supports and gives the optimal patent val for semihemiennealimmal temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Subsets and supersets ===
Since 954 = {{factorization|954}}, 954edo has subset edos {{EDOs| 2, 3, 6, 9, 18, 53, 106, 159, 318, 477 }}.  
 
[[Category:Ennealimmal]]
[[Category:Hemiennealimmal]]
[[Category:Semihemiennealimmal]]

Latest revision as of 16:23, 20 February 2025

← 953edo 954edo 955edo →
Prime factorization 2 × 32 × 53
Step size 1.25786 ¢ 
Fifth 558\954 (701.887 ¢) (→ 31\53)
Semitones (A1:m2) 90:72 (113.2 ¢ : 90.57 ¢)
Consistency limit 17
Distinct consistency limit 17

954 equal divisions of the octave (abbreviated 954edo or 954ed2), also called 954-tone equal temperament (954tet) or 954 equal temperament (954et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 954 equal parts of about 1.26 ¢ each. Each step represents a frequency ratio of 21/954, or the 954th root of 2.

954edo is a very strong 17-limit system, distinctly consistent in the 17-limit, and is a zeta peak, integral and gap edo. The tuning of the primes to 17 are all flat, and the equal temperament tempers out the ennealimma, [1 -27 18⟩, in the 5-limit and 2401/2400 and 4375/4374 in the 7-limit, so that it supports the ennealimmal temperament. In the 11-limit it tempers out 3025/3024, 9801/9800, 43923/43904, and 151263/151250 so that it supports hemiennealimmal. In the 13-limit it tempers out 4225/4224 and 10648/10647 and in the 17-limit 2431/2430 and 2601/2600. It supports and gives the optimal patent val for the semihemiennealimmal temperament.

Prime harmonics

Approximation of prime harmonics in 954edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41
Error Absolute (¢) +0.000 -0.068 -0.150 -0.272 -0.375 -0.276 -0.553 +0.600 -0.601 +0.611 -0.381 +0.228 -0.132
Relative (%) +0.0 -5.4 -11.9 -21.7 -29.8 -21.9 -44.0 +47.7 -47.8 +48.6 -30.3 +18.1 -10.5
Steps
(reduced) 		954
(0) 		1512
(558) 		2215
(307) 		2678
(770) 		3300
(438) 		3530
(668) 		3899
(83) 		4053
(237) 		4315
(499) 		4635
(819) 		4726
(910) 		4970
(200) 		5111
(341)
Approximation of prime harmonics in 954edo (continued)
Harmonic 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113
Error Absolute (¢) +0.432 -0.098 -0.549 -0.052 +0.096 -0.062 +0.178 -0.117 +0.243 +0.267 +0.189 -0.399 +0.083 +0.099 -0.452 +0.191 -0.567
Relative (%) +34.3 -7.8 -43.6 -4.1 +7.7 -4.9 +14.1 -9.3 +19.3 +21.2 +15.0 -31.7 +6.6 +7.8 -36.0 +15.2 -45.1
Steps
(reduced) 		5177
(407) 		5299
(529) 		5464
(694) 		5612
(842) 		5658
(888) 		5787
(63) 		5867
(143) 		5905
(181) 		6014
(290) 		6082
(358) 		6178
(454) 		6296
(572) 		6352
(628) 		6379
(655) 		6431
(707) 		6457
(733) 		6506
(782)
Approximation of prime harmonics in 954edo (continued)
Harmonic 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223
Error Absolute (¢) -0.258 +0.144 +0.600 -0.591 -0.109 -0.551 -0.086 +0.394 -0.068 +0.459 +0.553 +0.185 +0.167 -0.257 -0.550 -0.428 +0.090 -0.074
Relative (%) -20.5 +11.4 +47.7 -47.0 -8.7 -43.8 -6.8 +31.3 -5.4 +36.5 +44.0 +14.7 +13.3 -20.4 -43.7 -34.0 +7.1 -5.9
Steps
(reduced) 		6667
(943) 		6710
(32) 		6772
(94) 		6791
(113) 		6887
(209) 		6905
(227) 		6959
(281) 		7011
(333) 		7044
(366) 		7093
(415) 		7140
(462) 		7155
(477) 		7229
(551) 		7243
(565) 		7271
(593) 		7285
(607) 		7366
(688) 		7442
(764)

Subsets and supersets

Since 954 = 2 × 32 × 53, 954edo has subset edos 2, 3, 6, 9, 18, 53, 106, 159, 318, 477.

Retrieved from "https://en.xen.wiki/index.php?title=954edo&oldid=181873"