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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1848}}
{{ED intro}}


== Theory ==
== Theory ==
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>; and also the minortone comma, |-16 35 -17>.
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]].


In the 7-limit, it supports [[domain]] and [[akjayland]].
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]]. 


1848 factors as 2^3 * 3 * 7 * 11. It is a superabundant number in the no-fives subgroup, that is if only numbers not divisible by 5 are counted. Its divisors are {{EDOs|1, 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}}.
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 ===
=== Prime harmonics ===
{{Harmonics in equal|1848}}
{{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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |Subgroup
! rowspan="2" |[[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal
8ve stretch (¢)
! colspan="2" |Tuning error
|-
|-
![[TE error|Absolute]] (¢)
! rowspan="2" | [[Subgroup]]
![[TE simple badness|Relative]] (%)
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|2.3
! [[TE error|Absolute]] (¢)
|{{Monzo|-2929, 1848}}
! [[TE simple badness|Relative]] (%)
|[{{val|1848 2929}}]
|0.002192
|0.002192
|0.338
|-
|-
|2.3.5
| 2.3
|{{Monzo|-16, 35, -17}}, {{Monzo|129, -14, -46}}
| {{monzo| -2929 1848 }}
|[{{val|1848 2929 4291}}]
| {{mapping| 1848 2929 }}
|<nowiki>-0.005705</nowiki>
| 0.002192
|0.011311
| 0.002192
|1.742
| 0.34
|-
|-
|2.3.5.7
| 2.3.5
|250047/250000, 645700815/645657712, {{Monzo|43, -1, -13, -4}}
| {{monzo| -16 35 -17 }}, {{monzo| 129 -14 -46 }}
|[{{val|1848 2929 4291 5188}}]
| {{mapping| 1848 2929 4291 }}
| -0.004748
| −0.005705
|0.009935
| 0.011311
|1.530
| 1.74
|-
|-
|2.3.5.7.11
| 2.3.5.7
|9801/9800, 250047/250000, 14348907/14348180, 67110351/67108864
| 250047/250000, {{monzo| -4 17 1 -9 }}, {{monzo| 43 -1 -13 -4 }}
|[{{val|1848 2929 4291 5188 6393}}]
| {{mapping| 1848 2929 4291 5188 }}
|<nowiki>-0.002686</nowiki>
| −0.004748
|0.009797
| 0.009935
|1.509
| 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
|}
|}
[[Category:Equal divisions of the octave|####]]


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
!Periods
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
per Octave
|-
!Generator
! Periods<br />per 8ve
(Reduced)
! Generator*
!Cents
! Cents*
(Reduced)
! Associated<br />ratio*
!Associated
! Temperaments
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]]
|-
|-
|3
| 12
|281\1848
| 767\1848<br />(3\1848)
|182.467
| 498.052<br />(1.948)
|10/9
| 4/3<br />(32805/32768)
|[[Domain]]
| [[Atomic]]
|-
|-
|12
| 21
|3\1848
| 901\1848<br />(21\1848)
|1.948
| 585.065<br />(13.636)
|32805/32768
| 91875/65536<br />(126/125)
|[[Atomic]]
| [[Akjayland]]
|-
|-
|21
| 22
|901\1848<br>(21\1848)
| 767\1848<br />(11\1848)
|585.065<br>(13.636)
| 498.052<br />(7.143)
|91875/65536<br>(126/125)
| 4/3<br />({{monzo|16 -13 2}})
|[[Akjayland]]
| [[Major arcana]]
|-
|-
|44
| 44
|767\1848<br>(11\1848)
| 767\1848<br />(11\1848)
|498.052<br>(7.143)
| 498.052<br />(7.143)
|4/3<br>(?)
| 4/3<br />(18375/18304)
|[[Ruthenium]]
| [[Ruthenium]]
|-
|-
|56
| 56
|767\1848<br>(21\1848)
| 767\1848<br />(8\1848)
|498.052<br>(13.636)
| 498.052<br />(5.195)
|4/3<br>(126/125)
| 4/3<br />(126/125)
|[[Barium]]
| [[Barium]]
|}<!-- 4-digit number -->
|-
| 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:Akjayland]]
[[Category:Atomic]]
[[Category:Atomic]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/1848edo"