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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>2015-08-20 13:12:55 UTC</tt>.<br>
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| : The original revision id was <tt>557048419</tt>.<br>
| | While 78005edo is [[consistency|distinctly consistent]] through the [[21-odd-limit]], its notability stems from the fact that it is a very strong [[5-limit]] system, with lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any smaller edo, and lower 5-limit [[TE logflat badness]] than any smaller edo excepting [[4296edo|4296]]. The equal temperament [[tempering out|tempers out]] {{monzo| 232 -183 25 }}, {{monzo| 324 8 -145 }}, {{monzo| 92 191 -170 }}, {{monzo| 140 -374 195 }}, the selenia {{monzo| -433 -137 280 }}, and the quark {{monzo| -573 237 85 }}. |
| : The revision comment was: <tt></tt><br>
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| 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|78005}} |
| <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 78005 division divides the octave into 78005 equal parts of 0.01538 cents each. While it is distinctly consistent through the 21 limit, its notability stems from the fact that it is a very strong 5-limit division, with lower 5-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] than any smaller edo, and lower 5-limit [[Tenney-Euclidean metrics#Logflat TE badness| TE loglfat badness]] than any smaller edo excepting [[4296edo|4296]]. It tempers out |232 -183 25>, |324 8 -145>, |92 191 -170>, |140 -374 195>, the selenia, |-433 -137 280>, and the quark |-573 237 85>.</pre></div>
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| <h4>Original HTML content:</h4>
| | === Subsets and supersets === |
| <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"><html><head><title>78005edo</title></head><body>The 78005 division divides the octave into 78005 equal parts of 0.01538 cents each. While it is distinctly consistent through the 21 limit, its notability stems from the fact that it is a very strong 5-limit division, with lower 5-limit <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness">relative error</a> than any smaller edo, and lower 5-limit <a class="wiki_link" href="/Tenney-Euclidean%20metrics#Logflat TE badness"> TE loglfat badness</a> than any smaller edo excepting <a class="wiki_link" href="/4296edo">4296</a>. It tempers out |232 -183 25&gt;, |324 8 -145&gt;, |92 191 -170&gt;, |140 -374 195&gt;, the selenia, |-433 -137 280&gt;, and the quark |-573 237 85&gt;.</body></html></pre></div>
| | 78005edo contains [[15601edo]], from which the approximation of the 3rd harmonic is derived. |
| Prime factorization
|
5 × 15601
|
| Step size
|
0.0153836 ¢
|
| Fifth
|
45630\78005 (701.955 ¢) (→ 9126\15601)
|
| Semitones (A1:m2)
|
7390:5865 (113.7 ¢ : 90.22 ¢)
|
| Consistency limit
|
21
|
| Distinct consistency limit
|
21
|
78005 equal divisions of the octave (abbreviated 78005edo or 78005ed2), also called 78005-tone equal temperament (78005tet) or 78005 equal temperament (78005et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 78005 equal parts of about 0.0154 ¢ each. Each step represents a frequency ratio of 21/78005, or the 78005th root of 2.
While 78005edo is distinctly consistent through the 21-odd-limit, its notability stems from the fact that it is a very strong 5-limit system, with lower 5-limit relative error than any smaller edo, and lower 5-limit TE logflat badness than any smaller edo excepting 4296. The equal temperament tempers out [232 -183 25⟩, [324 8 -145⟩, [92 191 -170⟩, [140 -374 195⟩, the selenia [-433 -137 280⟩, and the quark [-573 237 85⟩.
Prime harmonics
Approximation of prime harmonics in 78005edo
| Harmonic
|
2
|
3
|
5
|
7
|
11
|
13
|
17
|
19
|
23
|
29
|
31
|
| Error
|
Absolute (¢)
|
+0.00000
|
+0.00000
|
-0.00002
|
+0.00430
|
+0.00056
|
+0.00307
|
+0.00709
|
+0.00637
|
-0.00693
|
+0.00296
|
-0.00128
|
| Relative (%)
|
+0.0
|
+0.0
|
-0.1
|
+27.9
|
+3.7
|
+20.0
|
+46.1
|
+41.4
|
-45.0
|
+19.2
|
-8.3
|
Steps
(reduced)
|
78005
(0)
|
123635
(45630)
|
181122
(25112)
|
218988
(62978)
|
269853
(35838)
|
288653
(54638)
|
318843
(6823)
|
331360
(19340)
|
352860
(40840)
|
378947
(66927)
|
386452
(74432)
|
Subsets and supersets
78005edo contains 15601edo, from which the approximation of the 3rd harmonic is derived.