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248edo: Difference between revisions

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**Imported revision 338045074 - Original comment: **
m Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"
 
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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>2012-05-21 16:21:36 UTC</tt>.<br>
 
: The original revision id was <tt>338045074</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
248edo shares the mapping of [[harmonic]]s [[5/1|5]] and [[7/1|7]] with [[31edo]]. It has a decent 13-limit interpretation despite not being [[consistent]]. The equal temperament [[tempering out|tempers out]] [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]]. 248edo, additionally, has the interesting property of its mapping for all prime harmonics 3 to 23 being a multiple of 3, and therefore derived from [[131edt]]. Similarly, using the lower-error 248[[Wart notation|h]] val, the mappings of all its [[2.5.7_subgroup|no-3]] harmonics up to [[23-limit|28]] are multiples of 2 and derived from [[124edo]].
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>
It [[support]]s the [[bischismic]] temperament, providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the [[essence]] temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.  
<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 //248 equal division// divides the octave into 248 equal parts of 4.389 cents each. It tempers out 32805/32768 in the 5-limit; 3136/3125 and 420175/419904 in the 7-limit; 441/440 and 8019/8000 in the 11-limit; 729/728, 847/845, 1001/1000, 1575/1573 and 2200/2197 in the 13-limit. It supports [[Schismatic family#Bischismic|bischismic temperament]], providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7 and 13-limits. It also provides the optimal patent val for [[Quince clan#Essence|essence temperament]]. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.</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;248edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;248 equal division&lt;/em&gt; divides the octave into 248 equal parts of 4.389 cents each. It tempers out 32805/32768 in the 5-limit; 3136/3125 and 420175/419904 in the 7-limit; 441/440 and 8019/8000 in the 11-limit; 729/728, 847/845, 1001/1000, 1575/1573 and 2200/2197 in the 13-limit. It supports &lt;a class="wiki_link" href="/Schismatic%20family#Bischismic"&gt;bischismic temperament&lt;/a&gt;, providing the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for 11-limit bischismic, and excellent tunings in the 7 and 13-limits. It also provides the optimal patent val for &lt;a class="wiki_link" href="/Quince%20clan#Essence"&gt;essence temperament&lt;/a&gt;. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|248|}}
 
=== Subsets and supersets ===
Since 248 factors into {{factorization|248}}, 248edo has subset edos {{EDOs| 2, 4, 8, 31, 62, and 124 }}.
 
== 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| 287 -181 }}
| {{mapping| 248 393 }}
| +0.108
| 0.108
| 2.23
|-
| 2.3.5
| 32805/32768, {{monzo| 12 32 -27 }}
| {{mapping| 248 393 576 }}
| -0.041
| 0.228
| 4.70
|-
| 2.3.5.7
| 3136/3125, 32805/32768, 420175/419904
| {{mapping| 248 393 576 696 }}
| +0.066
| 0.270
| 5.58
|-
| 2.3.5.7.11
| 441/440, 3136/3125, 8019/8000, 41503/41472
| {{mapping| 248 393 576 696 858 }}
| +0.036
| 0.249
| 5.15
|-
| 2.3.5.7.11.13
| 441/440, 729/728, 847/845, 1001/1000, 3136/3125
| {{mapping| 248 393 576 696 858 918 }}
| +0.079
| 0.275
| 5.69
|}
 
=== 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
| 5\248
| 24.19
| 686/675
| [[Sengagen]]
|-
| 1
| 103\248
| 498.39
| 4/3
| [[Helmholtz (temperament)|Helmholtz]]
|-
| 2
| 77\248<br />(47\248)
| 372.58<br />(227.42)
| 26/21<br />(154/135)
| [[Essence]]
|-
| 2
| 103\248
| 498.39
| 4/3
| [[Bischismic]]
|-
| 8
| 117\248<br />(7\248)
| 566.13<br />(33.87)
| 104/75<br />(49/48)
| [[Octowerck]]
|-
| 31
| 103\248<br />(1\248)
| 498.39<br />(4.84)
| 4/3<br />(385/384)
| [[Birds]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
[[Category:Bischismic]]
[[Category:Essence]]
Retrieved from "https://en.xen.wiki/w/248edo"