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


== Theory ==
== Theory ==
1848edo 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. In the 5-limit it tempers out the the [[atom]], {{monzo| 161 -84 -12 }} and thus tunes the [[atomic]] temperament, for which it also provides the [[optimal patent val]] in the 11-limit. and also the minortone comma, {{monzo| -16 35 -17 }}. It also tempers out the 7-limit [[landscape comma]], 250047/250000, so it supports [[domain]] and [[akjayland]].   
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]].   


It is distinctly [[consistent]] through the 15-odd-limit, 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.  
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]].


1848edo is unique in that it consistently tunes both [[81/80]] and [[64/63]] to an integer fraction of the octave, 1/56th and 1/44th 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.  
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]].


=== Subsets and supersets ===
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.  
1848 factors as 2<sup>3</sup> × 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 }}.
 
[[5544edo]], which divides the edostep into three, provides a good correction for 13- and the 17-limit.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|1848|columns=11}}
{{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" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 30: Line 33:
| 2.3
| 2.3
| {{monzo| -2929 1848 }}
| {{monzo| -2929 1848 }}
| [{{val| 1848 2929 }}]
| {{mapping| 1848 2929 }}
| 0.002192
| 0.002192
| 0.002192
| 0.002192
Line 37: Line 40:
| 2.3.5
| 2.3.5
| {{monzo| -16 35 -17 }}, {{monzo| 129 -14 -46 }}
| {{monzo| -16 35 -17 }}, {{monzo| 129 -14 -46 }}
| [{{val| 1848 2929 4291 }}]
| {{mapping| 1848 2929 4291 }}
| -0.005705
| −0.005705
| 0.011311
| 0.011311
| 1.74
| 1.74
Line 44: Line 47:
| 2.3.5.7
| 2.3.5.7
| 250047/250000, {{monzo| -4 17 1 -9 }}, {{monzo| 43 -1 -13 -4 }}
| 250047/250000, {{monzo| -4 17 1 -9 }}, {{monzo| 43 -1 -13 -4 }}
| [{{val| 1848 2929 4291 5188 }}]
| {{mapping| 1848 2929 4291 5188 }}
| -0.004748
| −0.004748
| 0.009935
| 0.009935
| 1.53
| 1.53
Line 51: Line 54:
| 2.3.5.7.11
| 2.3.5.7.11
| 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864
| 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864
| [{{val| 1848 2929 4291 5188 6393 }}]
| {{mapping| 1848 2929 4291 5188 6393 }}
| -0.002686
| −0.002686
| 0.009797
| 0.009797
| 1.51
| 1.51
Line 58: Line 61:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 4225/4224, 6656/6655, 9801/9800, 151263/151250, 1771561/1771470
| 4225/4224, 6656/6655, 9801/9800, 151263/151250, 1771561/1771470
| [{{val| 1848 2929 4291 5188 6393 6838 }}]
| {{mapping| 1848 2929 4291 5188 6393 6838 }}
| +0.009828
| +0.009828
| 0.029378
| 0.029378
| 4.52
| 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 ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per 8ve
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Generator<br>(Reduced)
|-
! Cents<br>(Reduced)
! Periods<br />per 8ve
! Associated<br>Ratio
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|281\1848
| 281\1848
|182.467
| 182.467
|10/9
| 10/9
|[[Minortone]]
| [[Minortone]]
|-
| 1
| 523\1848
| 339.610
| {{monzo|36 -24 1}}
| [[Empress]]
|-
|-
| 3
| 3
Line 85: Line 103:
|-
|-
| 12
| 12
| 767\1848<br>(3\1848)
| 767\1848<br />(3\1848)
| 498.052<br>(1.948)
| 498.052<br />(1.948)
| 4/3<br>(32805/32768)
| 4/3<br />(32805/32768)
| [[Atomic]]
| [[Atomic]]
|-
|-
| 21
| 21
| 901\1848<br>(21\1848)
| 901\1848<br />(21\1848)
| 585.065<br>(13.636)
| 585.065<br />(13.636)
| 91875/65536<br>(126/125)
| 91875/65536<br />(126/125)
| [[Akjayland]]
| [[Akjayland]]
|-
|-
|22
| 22
|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 />({{monzo|16 -13 2}})
|[[Major Arcana]]
| [[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>(18375/18304)
| 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]]
|-
| 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:Equal divisions of the octave|####]] <!-- 4-digit number -->
[[Category:Akjayland]]
[[Category:Akjayland]]
[[Category:Atomic]]
[[Category:Atomic]]
[[Category:Listen]]

Latest revision as of 07:37, 21 April 2025

← 1847edo 1848edo 1849edo →
Prime factorization 23 × 3 × 7 × 11
Step size 0.649351 ¢ 
Fifth 1081\1848 (701.948 ¢)
Semitones (A1:m2) 175:139 (113.6 ¢ : 90.26 ¢)
Consistency limit 15
Distinct consistency limit 15

1848 equal divisions of the octave (abbreviated 1848edo or 1848ed2), also called 1848-tone equal temperament (1848tet) or 1848 equal temperament (1848et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1848 equal parts of about 0.649 ¢ each. Each step represents a frequency ratio of 21/1848, or the 1848th root of 2.

Theory

1848edo is an extremely strong 11-limit division, having the lowest 11-limit relative error than any division until 6079.

In the 5-limit it tempers out the minortone comma, [-16 35 -17⟩ and Kirnberger's atom, [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.

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, 1\56 and 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 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

Approximation of prime harmonics in 1848edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.007 +0.050 +0.005 -0.019 -0.268 +0.239 -0.110 +0.297 +0.293 -0.230
Relative (%) +0.0 -1.1 +7.7 +0.8 -3.0 -41.3 +36.9 -17.0 +45.8 +45.1 -35.5
Steps
(reduced) 		1848
(0) 		2929
(1081) 		4291
(595) 		5188
(1492) 		6393
(849) 		6838
(1294) 		7554
(162) 		7850
(458) 		8360
(968) 		8978
(1586) 		9155
(1763)

Subsets and supersets

Since 1848 factors into 23 × 3 × 7 × 11, 1848edo has subset 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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-2929 1848⟩ [⟨1848 2929]] 0.002192 0.002192 0.34
2.3.5 [-16 35 -17⟩, [129 -14 -46⟩ [⟨1848 2929 4291]] −0.005705 0.011311 1.74
2.3.5.7 250047/250000, [-4 17 1 -9⟩, [43 -1 -13 -4⟩ [⟨1848 2929 4291 5188]] −0.004748 0.009935 1.53
2.3.5.7.11 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864 [⟨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 [⟨1848 2929 4291 5188 6393 6838]] +0.009828 0.029378 4.52
2.3.5.7.11.19 5776/5775, 9801/9800, 10241/10240, 250047/250000, 233744896/233735625 [⟨1848 2929 4291 5188 6393 7850]] +0.002094 0.013936 2.15

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 281\1848 182.467 10/9 Minortone
1 523\1848 339.610 [36 -24 1⟩ Empress
3 281\1848 182.467 10/9 Domain
12 767\1848
(3\1848) 		498.052
(1.948) 		4/3
(32805/32768) 		Atomic
21 901\1848
(21\1848) 		585.065
(13.636) 		91875/65536
(126/125) 		Akjayland
22 767\1848
(11\1848) 		498.052
(7.143) 		4/3
([16 -13 2⟩) 		Major arcana
44 767\1848
(11\1848) 		498.052
(7.143) 		4/3
(18375/18304) 		Ruthenium
56 767\1848
(8\1848) 		498.052
(5.195) 		4/3
(126/125) 		Barium
77 581\1848
(42\1848) 		377.273
(27.273) 		975/784
(?) 		Iridium

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Eliora
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