1848edo: Difference between revisions

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== 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 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, so it supports [[domain]] and [[akjayland]]. It is distinctly [[consistent]] through the 15-odd-limit, and tempers out the 13-limit commas [[4225/4224]] and [[6656/6655]].

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 provides the [[optimal patent val]] ~~for~~ 11-limit ~~atomic~~.

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 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 ~~{{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]].

~~=== Fractional-octave temperaments ===~~

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.

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.

=== Prime harmonics ===

=== 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~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

Line 28:

Line 33:

| 2.3

| {{monzo| -2929 1848 }}

| [{{~~val~~| 1848 2929 }}]

| {{mapping| 1848 2929 }}

| 0.002192

Line 35:

Line 40:

| 2.3.5

| {{monzo| -16 35 -17 }}, {{monzo| 129 -14 -46 }}

| [{{~~val~~| 1848 2929 4291 }}]

| {{mapping| 1848 2929 4291 }}

| -0.005705

| −0.005705

| 0.011311

| 1.74

Line 42:

Line 47:

| 2.3.5.7

| 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

| 1.53

Line 49:

Line 54:

| 2.3.5.7.11

| 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

| 1.51

Line 56:

Line 61:

| 2.3.5.7.11.13

| 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

| 0.029378

| 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"

! Periods per 8ve

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Generator~~ (Reduced)~~

|-

! Cents~~ (Reduced)~~

! Periods per 8ve

! Associated ~~Ratio~~

! Generator*

! Cents*

! Associated ratio*

! 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

Line 83:

Line 103:

|-

| 12

| 767\1848 (3\1848)

| 767\1848 (3\1848)

| 498.052 (1.948)

| 498.052 (1.948)

| 4/3 (32805/32768)

| 4/3 (32805/32768)

| [[Atomic]]

|-

| 21

| 901\1848 (21\1848)

| 901\1848 (21\1848)

| 585.065 (13.636)

| 585.065 (13.636)

| 91875/65536 (126/125)

| 91875/65536 (126/125)

| [[Akjayland]]

|-

| 22

| 767\1848 (11\1848)

| 498.052 (7.143)

| 4/3 ({{monzo|16 -13 2}})

| [[Major arcana]]

|-

| 44

| 767\1848 (11\1848)

| 767\1848 (11\1848)

| 498.052 (7.143)

| 498.052 (7.143)

| 4/3 (18375/18304)

| 4/3 (18375/18304)

| [[Ruthenium]]

|-

| 56

| 767\1848 (21\1848)

| 767\1848 (8\1848)

| 498.052 (13.~~636~~)

| 498.052 (5.195)

| 4/3 (126/125)

| 4/3 (126/125)

| [[Barium]]

|-

| 77

| 581\1848 (42\1848)

| 377.273 (27.273)

| 975/784 (?)

| [[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|####]] ~~

[[Category:Akjayland]]

[[Category:Atomic]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|1848}}
+{{ED intro}}
 == Theory ==
-edo 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, so it supports [[domain]] and [[akjayland]]. It is distinctly [[consistent]] through the 15-odd-limit, and tempers out the 13-limit commas [[4225/4224]] and [[6656/6655]].
+edo 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 provides the [[optimal patent val]] for 11-limit atomic.
+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]].
-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 }}.
+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]].
-=== Fractional-octave temperaments ===
+edo 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.
-edo 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.
 === 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|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 28: / Line 33: @@
 | 2.3
 | {{monzo| -2929 1848 }}
-| [{{val| 1848 2929 }}]
+| {{mapping| 1848 2929 }}
 | 0.002192
 | 0.002192
@@ Line 35: / Line 40: @@
 | 2.3.5
 | {{monzo| -16 35 -17 }}, {{monzo| 129 -14 -46 }}
-| [{{val| 1848 2929 4291 }}]
+| {{mapping| 1848 2929 4291 }}
-| -0.005705
+| −0.005705
 | 0.011311
 | 1.74
@@ Line 42: / Line 47: @@
 | 2.3.5.7
 | 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
 | 1.53
@@ Line 49: / Line 54: @@
 | 2.3.5.7.11
 | 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
 | 1.51
@@ Line 56: / Line 61: @@
 | 2.3.5.7.11.13
 | 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
 | 0.029378
 | 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"
-! 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
 |-
-|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
@@ Line 83: / Line 103: @@
 |-
 | 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]]
 |-
 | 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]]
+|-
+| 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)
+| 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]]
 |-
 | 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]]
+|-
+| 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:Atomic]]
+[[Category:Listen]]