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1848edo: 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>2015-08-17 12:03:50 UTC</tt>.<br>
: The original revision id was <tt>556811273</tt>.<br>
: 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>
<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">The 1848 equal division divides the octave into 1848 equal parts of 0.64935 cents each. It is a super strong 11-limit division, having the lowest 11-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] than any division until [[6079edo|6079]]. It tempers out the 11-limit commas 9801/9800, 151263/151250, 1771561/1771470 and 3294225/3294172. It also tempers out the 7-limit landscape comma, 250047/250000. It is distinctly consistent through the 15-limit, and tempers out the 13-limit commas 4225/4224 and 6656/6655. In the 5-limit it is an atomic system, tempering out the atom, |161 -84 -12&gt;; and also the minortone comma, |-16 35 -17&gt;.


1848 is highly composite, factoring as 2^3 * 3 * 7 * 11. Among its divisors are [[12edo|12]], [[14edo|14]], [[22edo|22]], [[24edo|24]], [[28edo|28]], [[56edo|56]], [[77edo|77]], [[84edo|84]], [[88edo|88]], [[154edo|154]], and [[231edo|231]].
== Theory ==
                                </pre></div>
1848edo is an extremely strong 11-limit division, having the lowest 11-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until [[6079edo|6079]]. 
<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;1848edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 1848 equal division divides the octave into 1848 equal parts of 0.64935 cents each. It is a super strong 11-limit division, having the lowest 11-limit &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&gt;relative error&lt;/a&gt; than any division until &lt;a class="wiki_link" href="/6079edo"&gt;6079&lt;/a&gt;. It tempers out the 11-limit commas 9801/9800, 151263/151250, 1771561/1771470 and 3294225/3294172. It also tempers out the 7-limit landscape comma, 250047/250000. It is distinctly consistent through the 15-limit, and tempers out the 13-limit commas 4225/4224 and 6656/6655. In the 5-limit it is an atomic system, tempering out the atom, |161 -84 -12&amp;gt;; and also the minortone comma, |-16 35 -17&amp;gt;.&lt;br /&gt;
In the 5-limit it tempers out the minortone comma, {{monzo| -16 35 -17 }} and [[Kirnberger's atom]], {{monzo| 161 -84 -12 }} and thus tunes the [[atomic]] temperament, for which it also provides the [[optimal patent val]] in the 11-limit. In the 7-limit it tempers out the [[landscape comma]], 250047/250000, so it supports [[domain]] and [[akjayland]]. In the 11-limit it tempers out [[9801/9800]], 151263/151250, [[1771561/1771470]], 3294225/3294172, and the [[spoob]].
&lt;br /&gt;
 
1848 is highly composite, factoring as 2^3 * 3 * 7 * 11. Among its divisors are &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/14edo"&gt;14&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/24edo"&gt;24&lt;/a&gt;, &lt;a class="wiki_link" href="/28edo"&gt;28&lt;/a&gt;, &lt;a class="wiki_link" href="/56edo"&gt;56&lt;/a&gt;, &lt;a class="wiki_link" href="/77edo"&gt;77&lt;/a&gt;, &lt;a class="wiki_link" href="/84edo"&gt;84&lt;/a&gt;, &lt;a class="wiki_link" href="/88edo"&gt;88&lt;/a&gt;, &lt;a class="wiki_link" href="/154edo"&gt;154&lt;/a&gt;, and &lt;a class="wiki_link" href="/231edo"&gt;231&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
It is distinctly [[consistent]] through the [[15-odd-limit]] (though just barely), and tempers out the 13-limit commas [[4225/4224]] and [[6656/6655]]. Higher-limit prime harmonics represented by 1848edo with less than 10% error are 37, 61, and 83, of which 61 is accurate to 0.002 edosteps (and is inherited from [[231edo]]). The harmonics represented by less than 20% error are 19, 47, 59, 67, 89, and the 2.3.5.7.11.19 subgroup is the simplest and most natural choice for using 1848edo with higher limits. In the 2.3.5.7.11.19, it tempers out [[5776/5775]].
 
1848edo is unique in that it consistently tunes both [[81/80]] and [[64/63]] to an integer fraction of the octave, [[56edo|1\56]] and [[44edo|1\44]] respectively. As a corollary, it supports [[barium]] and [[ruthenium]] temperaments, which have periods 56 and 44 respectively. While every edo that is a multiple of 616 shares the property of directly mapping 81/80 and 64/63 to fractions of the octave, 1848edo is unique due to its strength in simple harmonics and it actually shows how 81/80 and 64/63 are produced. In 2.3.5.7.11.19, it also tempers [[96/95]] to [[66edo|1\66]], thus making it a valuable system where important raising or lowering commas are represented by intervals that fit evenly within the octave.  
 
=== Prime harmonics ===
{{Harmonics in equal|1848|columns=11}}
 
=== Subsets and supersets ===
Since 1848 factors into {{factorization|1848}}, 1848edo has subset edos {{EDOs| 2, 3, 4, 6, 7, 8, 11, 12, 14, 21, 22, 24, 28, 33, 42, 44, 56, 66, 77, 84, 88, 132, 154, 168, 231, 264, 308, 462, 616, 924 }}.
 
[[3696edo]], which divides the edostep into two, and [[5544edo]], which divides the edostep into three, provide decent corrections for the 13- and the 17-limit.
 
== 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| -2929 1848 }}
| {{mapping| 1848 2929 }}
| 0.002192
| 0.002192
| 0.34
|-
| 2.3.5
| {{monzo| -16 35 -17 }}, {{monzo| 129 -14 -46 }}
| {{mapping| 1848 2929 4291 }}
| −0.005705
| 0.011311
| 1.74
|-
| 2.3.5.7
| 250047/250000, {{monzo| -4 17 1 -9 }}, {{monzo| 43 -1 -13 -4 }}
| {{mapping| 1848 2929 4291 5188 }}
| −0.004748
| 0.009935
| 1.53
|-
| 2.3.5.7.11
| 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864
| {{mapping| 1848 2929 4291 5188 6393 }}
| −0.002686
| 0.009797
| 1.51
|-
| 2.3.5.7.11.13
| 4225/4224, 6656/6655, 9801/9800, 151263/151250, 1771561/1771470
| {{mapping| 1848 2929 4291 5188 6393 6838 }}
| +0.009828
| 0.029378
| 4.52
|- style="border-top: double;"
| 2.3.5.7.11.19
| 5776/5775, 9801/9800, 10241/10240, 250047/250000, 233744896/233735625
| {{mapping| 1848 2929 4291 5188 6393 7850 }}
| +0.002094
| 0.013936
| 2.15
|}
 
=== 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
| 281\1848
| 182.467
| 10/9
| [[Minortone]]
|-
| 1
| 523\1848
| 339.610
| {{monzo|36 -24 1}}
| [[Empress]]
|-
| 3
| 281\1848
| 182.467
| 10/9
| [[Minortonic_family#Domain|Domain]]
|-
| 12
| 767\1848<br />(3\1848)
| 498.052<br />(1.948)
| 4/3<br />(32805/32768)
| [[Atomic]]
|-
| 21
| 901\1848<br />(21\1848)
| 585.065<br />(13.636)
| 91875/65536<br />(126/125)
| [[Akjayland]]
|-
| 22
| 767\1848<br />(11\1848)
| 498.052<br />(7.143)
| 4/3<br />({{monzo|16 -13 2}})
| [[Major arcana]]
|-
| 44
| 767\1848<br />(11\1848)
| 498.052<br />(7.143)
| 4/3<br />(18375/18304)
| [[Ruthenium]]
|-
| 56
| 767\1848<br />(8\1848)
| 498.052<br />(5.195)
| 4/3<br />(126/125)
| [[Barium]]
|-
| 77
| 581\1848<br />(42\1848)
| 377.273<br />(27.273)
| 975/784<br />(?)
| [[Iridium]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=pDCBMziEPko ''Nocturne for Strings in Major Arcana and Minortone''] (2023)
* [https://www.youtube.com/watch?v=A-xeNdcudEY ''Frolicking in Spoob''] (2024)
 
[[Category:Akjayland]]
[[Category:Atomic]]
[[Category:Listen]]
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