1848edo - Revision history

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:33:10Z

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

← Older revision		Revision as of 13:33, 13 March 2026
Line 138:		Line 138:
	\| [[Iridium]]		\| [[Iridium]]
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki />* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==

Tristanbay: /* Subsets and supersets */mentioned 3696edo

2025-04-21T07:37:04Z

Subsets and supersets: mentioned 3696edo

← Older revision		Revision as of 07:37, 21 April 2025
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 }}.		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 ==		== Regular temperament properties ==

ArrowHead294 at 14:08, 20 February 2025

2025-02-20T14:08:40Z

← Older revision		Revision as of 14:08, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|1848~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 14:14, 16 January 2025

2025-01-16T14:14:38Z

← Older revision		Revision as of 14:14, 16 January 2025
Line 5:		Line 5:
	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]].		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]].		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 41:		Line 41:
	\| {{monzo\| -16 35 -17 }}, {{monzo\| 129 -14 -46 }}		\| {{monzo\| -16 35 -17 }}, {{monzo\| 129 -14 -46 }}
	\| {{mapping\| 1848 2929 4291 }}		\| {{mapping\| 1848 2929 4291 }}
	\| ~~−0~~.005705		\| −0.005705
	\| 0.011311		\| 0.011311
	\| 1.74		\| 1.74
Line 48:		Line 48:
	\| 250047/250000, {{monzo\| -4 17 1 -9 }}, {{monzo\| 43 -1 -13 -4 }}		\| 250047/250000, {{monzo\| -4 17 1 -9 }}, {{monzo\| 43 -1 -13 -4 }}
	\| {{mapping\| 1848 2929 4291 5188 }}		\| {{mapping\| 1848 2929 4291 5188 }}
	\| ~~−0~~.004748		\| −0.004748
	\| 0.009935		\| 0.009935
	\| 1.53		\| 1.53
Line 55:		Line 55:
	\| 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864		\| 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864
	\| {{mapping\| 1848 2929 4291 5188 6393 }}		\| {{mapping\| 1848 2929 4291 5188 6393 }}
	\| ~~−0~~.002686		\| −0.002686
	\| 0.009797		\| 0.009797
	\| 1.51		\| 1.51
Line 138:		Line 138:
	\| [[Iridium]]		\| [[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 ==		== Music ==

Eliora: /* Regular temperament properties */ put that plus in the right spot

2024-11-25T00:51:23Z

Regular temperament properties: put that plus in the right spot

← Older revision		Revision as of 00:51, 25 November 2024
Line 69:		Line 69:
	\| 5776/5775, 9801/9800, 10241/10240, 250047/250000, 233744896/233735625		\| 5776/5775, 9801/9800, 10241/10240, 250047/250000, 233744896/233735625
	\| {{mapping\| 1848 2929 4291 5188 6393 7850 }}		\| {{mapping\| 1848 2929 4291 5188 6393 7850 }}
	\| 0.002094		\| +0.002094
	\| 0.013936		\| 0.013936
	\| +2.15		\| 2.15
	\|}		\|}

ArrowHead294 at 12:08, 16 November 2024

2024-11-16T12:08:43Z

← Older revision		Revision as of 12:08, 16 November 2024
Line 20:		Line 20:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~{comma basis begin}}~~		{\| class="wikitable center-4 center-5 center-6"
			\|-
			! rowspan="2" \| [[Subgroup]]
			! rowspan="2" \| [[Comma list]]
			! rowspan="2" \| [[Mapping]]
			! rowspan="2" \| Optimal<br />8ve stretch (¢)
			! colspan="2" \| Tuning error
			\|-
			! [[TE error\|Absolute]] (¢)
			! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\| 2.3		\| 2.3
Line 63:		Line 72:
	\| 0.013936		\| 0.013936
	\| +2.15		\| +2.15
	~~{{comma basis end}~~}		\|}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{rank-2 ~~begin}}~~		{\| class="wikitable center-all left-5"
			\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
			\|-
			! Periods<br />per 8ve
			! Generator*
			! Cents*
			! Associated<br />ratio*
			! Temperaments
	\|-		\|-
	\| 1		\| 1
Line 121:		Line 137:
	\| 975/784<br />(?)		\| 975/784<br />(?)
	\| [[Iridium]]		\| [[Iridium]]
	~~{{rank-2 end}~~}		\|}
	~~{{orf}}~~		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	== Music ==		== Music ==

ArrowHead294 at 05:28, 16 November 2024

2024-11-16T05:28:51Z

← Older revision		Revision as of 05:28, 16 November 2024
Line 3:		Line 3:

	== 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]].		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 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]].
Line 20:		Line 20:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	\|-
	~~! rowspan="2" \| [[Subgroup]]~~
	~~! rowspan="2" \| [[Comma list\|Comma List]]~~
	~~! rowspan="2" \| [[Mapping]]~~
	~~! rowspan="2" \| Optimal<br />8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3		\| 2.3
Line 41:		Line 32:
	\| {{monzo\| -16 35 -17 }}, {{monzo\| 129 -14 -46 }}		\| {{monzo\| -16 35 -17 }}, {{monzo\| 129 -14 -46 }}
	\| {{mapping\| 1848 2929 4291 }}		\| {{mapping\| 1848 2929 4291 }}
	\| -0.005705		\| −0.005705
	\| 0.011311		\| 0.011311
	\| 1.74		\| 1.74
Line 48:		Line 39:
	\| 250047/250000, {{monzo\| -4 17 1 -9 }}, {{monzo\| 43 -1 -13 -4 }}		\| 250047/250000, {{monzo\| -4 17 1 -9 }}, {{monzo\| 43 -1 -13 -4 }}
	\| {{mapping\| 1848 2929 4291 5188 }}		\| {{mapping\| 1848 2929 4291 5188 }}
	\| -0.004748		\| −0.004748
	\| 0.009935		\| 0.009935
	\| 1.53		\| 1.53
Line 55:		Line 46:
	\| 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864		\| 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864
	\| {{mapping\| 1848 2929 4291 5188 6393 }}		\| {{mapping\| 1848 2929 4291 5188 6393 }}
	\| -0.002686		\| −0.002686
	\| 0.009797		\| 0.009797
	\| 1.51		\| 1.51
Line 65:		Line 56:
	\| 0.029378		\| 0.029378
	\| 4.52		\| 4.52
	\|-		\|- style="border-top: double;"
	\| style="border-top: double;" \| 2.3.5.7.11.19		\| 2.3.5.7.11.19
	~~\| style="border-top: double;"~~ \| 5776/5775, 9801/9800, 10241/10240, 250047/250000, 233744896/233735625		\| 5776/5775, 9801/9800, 10241/10240, 250047/250000, 233744896/233735625
	~~\| style="border-top: double;"~~ \| {{mapping\| 1848 2929 4291 5188 6393 7850 }}		\| {{mapping\| 1848 2929 4291 5188 6393 7850 }}
	~~\| style="border-top: double;"~~ \| 0.002094		\| 0.002094
	~~\| style="border-top: double;"~~ \| 0.013936		\| 0.013936
	~~\| style="border-top: double;"~~ \| +2.15		\| +2.15
	\|}		{{comma basis end}}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+ style="font-size: 105%;" \| Table of~~ rank-2 ~~temperaments by generator~~
	\|-
	~~! Periods<br />per 8ve~~
	! Generator*
	! Cents*
	! Associated<br />Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
Line 137:		Line 121:
	\| 975/784<br />(?)		\| 975/784<br />(?)
	\| [[Iridium]]		\| [[Iridium]]
	\|}		{{rank-2 end}}
	~~<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		{{orf}}

	== Music ==		== Music ==

Eliora: expand on 19 limit and relate the text

2024-11-14T16:14:31Z

expand on 19 limit and relate the text

← Older revision		Revision as of 16:14, 14 November 2024
Line 7:		Line 7:
	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 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]].

	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.		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~~/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. ~~Remarkably~~, ~~on the patent val 1848edo~~ tempers [[96/95]] ~~also~~ to [[66edo\|1\66]], ~~being~~ a ~~great 2.3.5.19 subgroup tuning~~.		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 ===
Line 71:		Line 71:
	\| style="border-top: double;" \| 0.002094		\| style="border-top: double;" \| 0.002094
	\| style="border-top: double;" \| 0.013936		\| style="border-top: double;" \| 0.013936
	\| style="border-top: double;" \| 2.15		\| style="border-top: double;" \| +2.15
	\|}		\|}

ArrowHead294: Formatting

2024-11-14T16:13:21Z

Formatting

← Older revision		Revision as of 16:13, 14 November 2024
Line 21:		Line 21:
	== 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\|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
	\|-		\|-
Line 75:		Line 76:
	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	\|-		\|-
	! Periods<br>per 8ve		! Periods<br />per 8ve
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br>Ratio*		! Associated<br />Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
Line 102:		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>({{monzo\|16 -13 2}})		\| 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>(8\1848)		\| 767\1848<br />(8\1848)
	\| 498.052<br>(5.195)		\| 498.052<br />(5.195)
	\| 4/3<br>(126/125)		\| 4/3<br />(126/125)
	\| [[Barium]]		\| [[Barium]]
	\|-		\|-
	\| 77		\| 77
	\| 581\1848<br>(42\1848)		\| 581\1848<br />(42\1848)
	\| 377.273<br>(27.273)		\| 377.273<br />(27.273)
	\| 975/784<br>(?)		\| 975/784<br />(?)
	\| [[Iridium]]		\| [[Iridium]]
	\|}		\|}
	<nowiki>*~~</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 it is distinct

	== Music ==		== Music ==

Eliora: /* Regular temperament properties */ this subgroup is valuable

2024-11-14T16:10:31Z

Regular temperament properties: this subgroup is valuable

← Older revision		Revision as of 16:10, 14 November 2024
Line 64:		Line 64:
	\| 0.029378		\| 0.029378
	\| 4.52		\| 4.52
			\|-
			\| style="border-top: double;" \| 2.3.5.7.11.19
			\| style="border-top: double;" \| 5776/5775, 9801/9800, 10241/10240, 250047/250000, 233744896/233735625
			\| style="border-top: double;" \| {{mapping\| 1848 2929 4291 5188 6393 7850 }}
			\| style="border-top: double;" \| 0.002094
			\| style="border-top: double;" \| 0.013936
			\| style="border-top: double;" \| 2.15
	\|}		\|}