Xenharmonic Wiki

347edo: Difference between revisions

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Wikispaces>genewardsmith
**Imported revision 241111101 - Original comment: **
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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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>2011-07-12 23:50:44 UTC</tt>.<br>
 
: The original revision id was <tt>241111101</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
The equal temperament [[tempering out|tempers out]] {{monzo| 32 -7 -9 }} ([[escapade comma]]) and {{monzo| 54 -37 2 }} ([[monzisma]]), [[3136/3125]], 420175/419904, and 5250987/5242880 in the 7-limit. It provides an excellent tuning for [[sengagen]], the {{nowrap|99 &amp; 248}} temperament tempering out both 3136/3125 and 420175/419904, and the rank-3 [[hemimean]] temperament tempering out 3136/3125.  
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>
Extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. It makes more sense as a 2.3.5.7.13 [[subgroup]] temperament, where it tempers out [[676/675]] and [[4096/4095]], or as a 2.3.5.7.13.19 subgroup temperament, where it tempers out [[1521/1520]] and [[1729/1728]].  
<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 //347 equal temperament// divides the octave into 347 equal parts of 3.458 cents each. It tempers out 3136/3125 and 420175/419904 in the 7-limit, and provides an excellent tuning for the temperament tempering out both, the 49&amp;50 temperament, and the planar hemimean temperament tempering out 3136/3125.</pre></div>
 
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;347edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;347 equal temperament&lt;/em&gt; divides the octave into 347 equal parts of 3.458 cents each. It tempers out 3136/3125 and 420175/419904 in the 7-limit, and provides an excellent tuning for the temperament tempering out both, the 49&amp;amp;50 temperament, and the planar hemimean temperament tempering out 3136/3125.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|347}}
 
=== Subsets and supersets ===
347edo is the 69th [[prime edo]].
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 550 -347 }}
| {{mapping| 347 550 }}
| -0.0197
| 0.0197
| 0.57
|-
| 2.3.5
| {{monzo| 32 -7 -9 }}, {{monzo| -22 30 -11 }}
| {{mapping| 347 550 806 }}
| -0.1576
| 0.1956
| 5.66
|-
| 2.3.5.7
| 3136/3125, 420175/419904, 5250987/5242880
| {{mapping| 347 550 806 974 }}
| -0.0713
| 0.2259
| 6.53
|}
 
=== 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
| 7\347
| 24.21
| 686/675
| [[Sengagen]]
|-
| 1
| 16\347
| 55.33
| 16875/16384
| [[Escapade]]
|-
| 1
| 69\347
| 238.62
| 147/128
| [[Tokko]]
|-
| 1
| 72\347
| 248.99
| {{monzo| -26 18 -1 }}
| [[Monzismic]]
|-
| 1
| 146\347
| 504.90
| 104976/78125
| [[Countermeantone]]
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
[[Category:Hemimean]]
[[Category:Sengagen]]
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