1848edo: Difference between revisions

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== 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 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 is an atomic system, tempering out the [[atom]], {{monzo| 161 -84 -12 }} and also the minortone comma, {{monzo| -16 35 -17 }}. It also tempers out the 7-limit [[landscape comma]], 250047/250000. It is distinctly consistent through the 15-odd-limit, and tempers out the 13-limit commas [[4225/4224]] and [[6656/6655]].  


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


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}}.
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 }}.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|1848}}
{{Harmonics in equal|1848|columns=11}}


== 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]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal  
! rowspan="2" | Optimal<br>8ve Stretch (¢)
8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" |Tuning error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{Monzo|-2929, 1848}}
| {{monzo| -2929 1848 }}
|[{{val|1848 2929}}]
| [{{val| 1848 2929 }}]
|0.002192
| 0.002192
|0.002192
| 0.002192
|0.338
| 0.338
|-
|-
|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}}]
| [{{val| 1848 2929 4291 }}]
|<nowiki>-0.005705</nowiki>
| -0.005705
|0.011311
| 0.011311
|1.742
| 1.742
|-
|-
|2.3.5.7
| 2.3.5.7
|250047/250000, 645700815/645657712, {{Monzo|43, -1, -13, -4}}
| 250047/250000, 645700815/645657712, {{monzo| 43 -1 -13 -4 }}
|[{{val|1848 2929 4291 5188}}]
| [{{val| 1848 2929 4291 5188 }}]
| -0.004748
| -0.004748
|0.009935
| 0.009935
|1.530
| 1.530
|-
|-
|2.3.5.7.11
|2.3.5.7.11
|9801/9800, 250047/250000, 14348907/14348180, 67110351/67108864
| 9801/9800, 250047/250000, 14348907/14348180, 67110351/67108864
|[{{val|1848 2929 4291 5188 6393}}]
| [{{val|1848 2929 4291 5188 6393}}]
|<nowiki>-0.002686</nowiki>
| -0.002686
|0.009797
| 0.009797
|1.509
| 1.509
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|4225/4224, 9801/9800, 67392/67375, 250047/250000, 4429568/4428675
| 4225/4224, 9801/9800, 67392/67375, 250047/250000, 4429568/4428675
|[{{val|1848 2929 4291 5188 6393 6838}}]
| [{{val|1848 2929 4291 5188 6393 6838}}]
|0.009828
| 0.009828
|0.029378
| 0.029378
|4.524
| 4.524
|}
|}
[[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
! Periods<br>per 8ve
per Octave
! Generator<br>(Reduced)
!Generator
! Cents<br>(Reduced)
(Reduced)
! Associated<br>Ratio
!Cents
! Temperaments
(Reduced)
!Associated
Ratio
!Temperaments
|-
|-
|3
| 3
|281\1848
| 281\1848
|182.467
| 182.467
|10/9
| 10/9
|[[Domain]]
| [[Domain]]
|-
|-
|12
| 12
|3\1848
| 3\1848
|1.948
| 1.948
|32805/32768
| 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]]
|-
|-
|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>(?)
|[[Ruthenium]]
| [[Ruthenium]]
|-
|-
|56
| 56
|767\1848<br>(21\1848)
| 767\1848<br>(21\1848)
|498.052<br>(13.636)
| 498.052<br>(13.636)
|4/3<br>(126/125)
| 4/3<br>(126/125)
|[[Barium]]
| [[Barium]]
|}<!-- 4-digit number -->
|}
 
[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
[[Category:Akjayland]]
[[Category:Akjayland]]
[[Category:Atomic]]
[[Category:Atomic]]

Revision as of 19:32, 11 October 2022

← 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

Template:EDO intro

Theory

1848edo is a super strong 11-limit division, having the lowest 11-limit relative error than any division until 6079. It tempers out the 11-limit commas 9801/9800, 151263/151250, 1771561/1771470 and 3294225/3294172. In the 5-limit it is an atomic system, tempering out the atom, [161 -84 -12⟩ and also the minortone comma, [-16 35 -17⟩. It also tempers out the 7-limit landscape comma, 250047/250000. It is distinctly consistent through the 15-odd-limit, and tempers out the 13-limit commas 4225/4224 and 6656/6655.

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

1848 factors as 23 × 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 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.

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)

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-2929 1848⟩ [⟨1848 2929]] 0.002192 0.002192 0.338
2.3.5 [-16 35 -17⟩, [129 -14 -46⟩ [⟨1848 2929 4291]] -0.005705 0.011311 1.742
2.3.5.7 250047/250000, 645700815/645657712, [43 -1 -13 -4⟩ [⟨1848 2929 4291 5188]] -0.004748 0.009935 1.530
2.3.5.7.11 9801/9800, 250047/250000, 14348907/14348180, 67110351/67108864 [⟨1848 2929 4291 5188 6393]] -0.002686 0.009797 1.509
2.3.5.7.11.13 4225/4224, 9801/9800, 67392/67375, 250047/250000, 4429568/4428675 [⟨1848 2929 4291 5188 6393 6838]] 0.009828 0.029378 4.524

Rank-2 temperaments

Periods
per 8ve 		Generator
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
3 281\1848 182.467 10/9 Domain
12 3\1848 1.948 32805/32768 Atomic
21 901\1848
(21\1848) 		585.065
(13.636) 		91875/65536
(126/125) 		Akjayland
44 767\1848
(11\1848) 		498.052
(7.143) 		4/3
(?) 		Ruthenium
56 767\1848
(21\1848) 		498.052
(13.636) 		4/3
(126/125) 		Barium
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