1848edo: Difference between revisions

(3 intermediate revisions by 2 users not shown)

Line 1:

== Theory ==

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 5-limit it tempers out the minortone comma, {{monzo| -16 35 -17 }} and ~~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. 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]].

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]].

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]].

Line 17:

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 }}.

[[5544edo]], which divides the edostep into three, ~~provides a good correction~~ for 13- and the 17-limit.

[[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 ==

Line 41:

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

| {{mapping| 1848 2929 4291 }}

| ~~−0~~.005705

| −0.005705

| 0.011311

| 1.74

Line 48:

| 250047/250000, {{monzo| -4 17 1 -9 }}, {{monzo| 43 -1 -13 -4 }}

| {{mapping| 1848 2929 4291 5188 }}

| ~~−0~~.004748

| −0.004748

| 0.009935

| 1.53

Line 55:

| 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864

| {{mapping| 1848 2929 4291 5188 6393 }}

| ~~−0~~.002686

| −0.002686

| 0.009797

| 1.51

Line 69:

| 5776/5775, 9801/9800, 10241/10240, 250047/250000, 233744896/233735625

| {{mapping| 1848 2929 4291 5188 6393 7850 }}

| 0.002094

| +0.002094

| 0.013936

| +2.15

| 2.15

|}

Line 138:

| [[Iridium]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if ~~it is~~ distinct

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|1848}}
+{{ED intro}}
 == Theory ==
 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]].
-In the 5-limit it tempers out the minortone comma, {{monzo| -16 35 -17 }} and 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. 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]].
+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]].
 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]].
@@ Line 17: / Line 17: @@
 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 }}.
-[[5544edo]], which divides the edostep into three, provides a good correction for 13- and the 17-limit.
+[[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 ==
@@ Line 41: / Line 41: @@
 | {{monzo| -16 35 -17 }}, {{monzo| 129 -14 -46 }}
 | {{mapping| 1848 2929 4291 }}
-| &minus;0.005705
+| −0.005705
 | 0.011311
 | 1.74
@@ Line 48: / Line 48: @@
 | 250047/250000, {{monzo| -4 17 1 -9 }}, {{monzo| 43 -1 -13 -4 }}
 | {{mapping| 1848 2929 4291 5188 }}
-| &minus;0.004748
+| −0.004748
 | 0.009935
 | 1.53
@@ Line 55: / Line 55: @@
 | 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864
 | {{mapping| 1848 2929 4291 5188 6393 }}
-| &minus;0.002686
+| −0.002686
 | 0.009797
 | 1.51
@@ Line 69: / Line 69: @@
 | 5776/5775, 9801/9800, 10241/10240, 250047/250000, 233744896/233735625
 | {{mapping| 1848 2929 4291 5188 6393 7850 }}
-| 0.002094
+| +0.002094
 | 0.013936
-| +2.15
+| 2.15
 |}
@@ Line 138: / Line 138: @@
 | [[Iridium]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Music ==