298edo - 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"

2026-03-13T13:32:43Z

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 13:32, 13 March 2026
Line 91:		Line 91:
	\| [[Bison]]		\| [[Bison]]
	\|}		\|}
	<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

	== Scales ==		== Scales ==

Eliora: /* Theory */ +barton

2025-08-14T05:25:28Z

Theory: +barton

← Older revision		Revision as of 05:25, 14 August 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	298edo is [[enfactoring\|enfactored]] in the [[5-limit]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo—it retains the mapping for [[harmonic]]s [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[17/1\|17]] but differs on the mapping for [[7/4\|7]], [[11/8\|11]], [[13/8\|13]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed {{nowrap\|77 & 298}} temperament with [[13/8]] as its generator.		298edo is [[enfactoring\|enfactored]] in the [[5-limit]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo—it retains the mapping for [[harmonic]]s [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[17/1\|17]] but differs on the mapping for [[7/4\|7]], [[11/8\|11]], [[13/8\|13]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]] and is a strong tuning for [[barton]]. In the full 13-limit, 298edo supports an unnamed {{nowrap\|77 & 298}} temperament with [[13/8]] as its generator.

	Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val\|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the {{nowrap\|31 & 298d}} variant and the {{nowrap\|118 & 298d}} variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val—for example, it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].		Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val\|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the {{nowrap\|31 & 298d}} variant and the {{nowrap\|118 & 298d}} variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val—for example, it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].

ArrowHead294 at 22:57, 20 February 2025

2025-02-20T22:57:01Z

← Older revision		Revision as of 22:57, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|298~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 19:42, 15 January 2025

2025-01-15T19:42:29Z

← Older revision		Revision as of 19:42, 15 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	298edo is [[enfactoring\|enfactored]] in the [[5-limit]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to ~~149edo—it~~ retains the mapping for [[harmonic]]s [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[17/1\|17]] but differs on the mapping for [[7/4\|7]], [[11/8\|11]], [[13/8\|13]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed 77 & 298 temperament with [[13/8]] as its generator.		298edo is [[enfactoring\|enfactored]] in the [[5-limit]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo—it retains the mapping for [[harmonic]]s [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[17/1\|17]] but differs on the mapping for [[7/4\|7]], [[11/8\|11]], [[13/8\|13]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed {{nowrap\|77 & 298}} temperament with [[13/8]] as its generator.

	Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val\|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent ~~val—for~~ example, it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].		Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val\|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the {{nowrap\|31 & 298d}} variant and the {{nowrap\|118 & 298d}} variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val—for example, it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].

	The 298cd val, {{val\| 298 472 '''691''' '''836''' 1031 }} supports [[miracle]].		The 298cd val, {{val\| 298 472 '''691''' '''836''' 1031 }} supports [[miracle]].
Line 50:		Line 50:
	\| 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925		\| 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925
	\| {{mapping\| 298 472 692 837 1031 1103 }} (298)		\| {{mapping\| 298 472 692 837 1031 1103 }} (298)
	\| ~~−0~~.0478		\| −0.0478
	\| 0.4271		\| 0.4271
	\| 10.6		\| 10.6
Line 57:		Line 57:
	\| 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925		\| 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925
	\| {{mapping\| 298 472 692 837 1031 1103 1218 }} (298)		\| {{mapping\| 298 472 692 837 1031 1103 1218 }} (298)
	\| ~~−0~~.0320		\| −0.0320
	\| 0.3974		\| 0.3974
	\| 9.87		\| 9.87
Line 91:		Line 91:
	\| [[Bison]]		\| [[Bison]]
	\|}		\|}
	<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

	== Scales ==		== Scales ==
	The [[concoctic]] scale for 298edo is a scale produced by a generator of 105 steps (paraconcoctic), and the associated rank-2 temperament is 105 & 298.		The [[concoctic]] scale for 298edo is a scale produced by a generator of 105 steps (paraconcoctic), and the associated rank-2 temperament is {{nowrap\|105 & 298}}.

	[[Category:Bison]]		[[Category:Bison]]
	[[Category:Emka]]		[[Category:Emka]]

ArrowHead294: Partial undo

2024-11-16T12:16:09Z

Partial undo

← Older revision		Revision as of 12:16, 16 November 2024
Line 15:		Line 15:

	== 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]] (%)
	\|-		\|-
	\| 2.3.5.7		\| 2.3.5.7
Line 51:		Line 60:
	\| 0.3974		\| 0.3974
	\| 9.87		\| 9.87
	~~{{comma basis end}~~}		\|}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	Note: 5-limit temperaments supported by 149et are not listed.		Note: 5-limit temperaments supported by 149et are not listed.
	{{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 74:		Line 90:
	\| 35/32		\| 35/32
	\| [[Bison]]		\| [[Bison]]
	~~{{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

	== Scales ==		== Scales ==

ArrowHead294 at 05:10, 16 November 2024

2024-11-16T05:10:15Z

← Older revision		Revision as of 05:10, 16 November 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	298edo is [[enfactoring\|enfactored]] in the [[5-limit]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo – it retains the mapping for [[harmonic]]s [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[17/1\|17]] but differs on the mapping for [[7/4\|7]], [[11/8\|11]], [[13/8\|13]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed 77 & 298 temperament with [[13/8]] as its generator.		298edo is [[enfactoring\|enfactored]] in the [[5-limit]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo—it retains the mapping for [[harmonic]]s [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[17/1\|17]] but differs on the mapping for [[7/4\|7]], [[11/8\|11]], [[13/8\|13]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed 77 & 298 temperament with [[13/8]] as its generator.

	Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val\|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val – for example it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].		Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val\|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val—for example, it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].

	The 298cd val, {{val\| 298 472 '''691''' '''836''' 1031 }} supports [[miracle]].		The 298cd val, {{val\| 298 472 '''691''' '''836''' 1031 }} supports [[miracle]].
Line 15:		Line 15:

	== 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]] (%)~~
	\|-		\|-
	\| 2.3.5.7		\| 2.3.5.7
Line 49:		Line 41:
	\| 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925		\| 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925
	\| {{mapping\| 298 472 692 837 1031 1103 }} (298)		\| {{mapping\| 298 472 692 837 1031 1103 }} (298)
	\| -0.0478		\| −0.0478
	\| 0.4271		\| 0.4271
	\| 10.6		\| 10.6
Line 56:		Line 48:
	\| 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925		\| 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925
	\| {{mapping\| 298 472 692 837 1031 1103 1218 }} (298)		\| {{mapping\| 298 472 692 837 1031 1103 1218 }} (298)
	\| -0.0320		\| −0.0320
	\| 0.3974		\| 0.3974
	\| 9.87		\| 9.87
	\|}		{{comma basis end}}
	~~<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	Note: 5-limit temperaments supported by 149et are not listed.		Note: 5-limit temperaments supported by 149et are not listed.
	{~~\| class="wikitable center~~-~~all left-5"~~		{{rank-2 begin}}
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
Line 88:		Line 74:
	\| 35/32		\| 35/32
	\| [[Bison]]		\| [[Bison]]
	\|}		{{rank-2 end}}
			{{orf}}

	== Scales ==		== Scales ==

FloraC: /* Theory */

2024-03-04T08:26:18Z

Theory

← Older revision		Revision as of 08:26, 4 March 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	298edo is [[enfactoring\|enfactored]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo – it retains the mapping for [[harmonic]]s [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[17/1\|17]] but differs on the mapping for [[7/4\|7]], [[11/8\|11]], [[13/8\|13]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed 77 & 298 temperament with [[13/8]] as its generator.		298edo is [[enfactoring\|enfactored]] in the [[5-limit]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo – it retains the mapping for [[harmonic]]s [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[17/1\|17]] but differs on the mapping for [[7/4\|7]], [[11/8\|11]], [[13/8\|13]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed 77 & 298 temperament with [[13/8]] as its generator.

	Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val\|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val – for example it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].		Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val\|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val – for example it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].

FloraC: /* Regular temperament properties */

2024-03-04T08:18:29Z

Regular temperament properties

← Older revision		Revision as of 08:18, 4 March 2024
Line 48:		Line 48:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925		\| 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925
	\| {{mapping\| 298 472 692 837 1031 1103 }}		\| {{mapping\| 298 472 692 837 1031 1103 }} (298)
	\| -0.0478		\| -0.0478
	\| 0.4271		\| 0.4271
Line 55:		Line 55:
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925		\| 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925
	\| {{mapping\| 298 472 692 837 1031 1103 1218 }}		\| {{mapping\| 298 472 692 837 1031 1103 1218 }} (298)
	\| 0.~~3974~~		\| -0.0320
	\| 0.3974		\| 0.3974
	\| 9.87		\| 9.87
Line 75:		Line 75:
	\| 455.033		\| 455.033
	\| 13/10		\| 13/10
	\| [[Petrtri]] ~~(2.11/5.13/5)~~		\| [[Petrtri]]
	\|-		\|-
	\| 1		\| 1

FloraC: Cleanup

2024-03-04T08:16:53Z

Cleanup

← Older revision		Revision as of 08:16, 4 March 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	298edo is [[consistent]] in the 5-odd-limit~~, where it is [[enfactoring\|enfactored~~]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo - it retains the mapping for 2.3.5.17 but differs on the mapping for ~~harmonics~~ [[7/4\|7]], [[11/8\|11]], [[13/8\|13]]. ~~298edo~~ tempers out the [[rastma]], splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]].		298edo is [[enfactoring\|enfactored]] and only [[consistent]] in the [[5-odd-limit]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo – it retains the mapping for [[harmonic]]s [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[17/1\|17]] but differs on the mapping for [[7/4\|7]], [[11/8\|11]], [[13/8\|13]]. Using the [[patent val]], the equal temperament [[tempering out\|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 [[subgroup]], 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed 77 & 298 temperament with [[13/8]] as its generator.
	~~The patent val supports~~ the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 subgroup, 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed 77 & 298 temperament with [[13/8]] as its generator.

	Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val\|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val - for example it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].		Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val\|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val – for example it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].

	The 298cd val, {{val\|298 472 '''691''' '''836''' 1031}} supports [[miracle]].		The 298cd val, {{val\| 298 472 '''691''' '''836''' 1031 }} supports [[miracle]].

	In higher limits, 298edo has excellent representation of the 2.5.11.17.23.43.53.59 subgroup, with all the harmonics having errors of less than 10 r¢. A comma basis for the 2.5.11.17.23.43.53.59 subgroup is {1376/1375, 3128/3127, 4301/4300, 25075/25069, 38743/38720, 58351/58300, 973360/972961}.		In higher limits, 298edo has excellent representation of the 2.5.11.17.23.43.53.59 subgroup, with all the harmonics having errors of less than 10 r¢. A comma basis for the 2.5.11.17.23.43.53.59 subgroup is {1376/1375, 3128/3127, 4301/4300, 25075/25069, 38743/38720, 58351/58300, 973360/972961}.
	=== ~~Prime~~ harmonics ===
			=== Odd harmonics ===
	{{Harmonics in equal\|298}}		{{Harmonics in equal\|298}}

	== 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]] (%)
	\|-		\|-
	\|2.3.5.7		\| 2.3.5.7
	\|6144/6125, ~~321489~~/~~320000~~, 3796875/3764768		\| 6144/6125, 78732/78125, 3796875/3764768
	\|[{{~~val~~\|298 472 692 837}}]		\| {{mapping\| 298 472 692 837 }} (298)
	\|0.0275		\| +0.0275
	\|0.5022		\| 0.5022
	\|?		\| 12.5
	\|-		\|-
	\|2.3.5.7.11		\| 2.3.5.7.11
	\|243/242, 1375/1372, 6144/6125, 72171/71680		\| 243/242, 1375/1372, 6144/6125, 72171/71680
	\|[{{~~val~~\|298 472 692 837 1031}}]		\| {{mapping\| 298 472 692 837 1031 }} (298)
	\|0.0012		\| +0.0012
	\|0.4523		\| 0.4523
	\|?		\| 11.2
	\|-		\|-
	\|2.3.5.7.11		\| 2.3.5.7.11
	\|243/242, 1029/1024, 3136/3125, 9801/9800		\| 243/242, 1029/1024, 3136/3125, 9801/9800
	\|[{{~~val~~\|298 472 692 836 1031}}] (298d)		\| {{mapping\| 298 472 692 836 1031 }} (298d)
	\|0.2882		\| +0.2882
	\|0.4439		\| 0.4439
	\|?		\| 11.0
	\|-		\|-
	\|2.3.5.7.11.13		\| 2.3.5.7.11.13
	\|243/242, 351/350, 1375/1372, 4096/4095, 16038/15925		\| 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925
	\|[{{~~val~~\|298 472 692 837 1031 1103}}]		\| {{mapping\| 298 472 692 837 1031 1103 }}
	\|		\| -0.0478
	\|		\| 0.4271
	\|?		\| 10.6
	\|-		\|-
	\|2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\|243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925		\| 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925
	\|[{{~~val~~\|298 472 692 837 1031 1103 1218}}]		\| {{mapping\| 298 472 692 837 1031 1103 1218 }}
	\|		\| 0.3974
	\|		\| 0.3974
	\| ?		\| 9.87
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	Note: 5-limit temperaments ~~represented~~ by ~~149edo~~ are not ~~included~~.		Note: 5-limit temperaments supported by 149et are not listed.
	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	! Periods<br>per ~~Octave~~		! Periods<br>per 8ve
	! Generator~~<br>(Reduced)~~		! Generator*
	! Cents~~<br>(Reduced)~~		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
	\|1		\| 1
	\|113\298		\| 113\298
	\|455.033		\| 455.033
	\|13/10		\| 13/10
	\|[[Petrtri]] (2.11/5.13/5)		\| [[Petrtri]] (2.11/5.13/5)
	\|-		\|-
	\| 1		\| 1
Line 93:		Line 93:
	The [[concoctic]] scale for 298edo is a scale produced by a generator of 105 steps (paraconcoctic), and the associated rank-2 temperament is 105 & 298.		The [[concoctic]] scale for 298edo is a scale produced by a generator of 105 steps (paraconcoctic), and the associated rank-2 temperament is 105 & 298.

	~~[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->~~
	[[Category:Bison]]		[[Category:Bison]]
	[[Category:Emka ~~family~~]]		[[Category:Emka]]

Eliora: /* Theory */ maybe paragraphs like that are better, the page looks too broad

2024-02-03T21:47:17Z

Theory: maybe paragraphs like that are better, the page looks too broad

← Older revision		Revision as of 21:47, 3 February 2024
Line 4:		Line 4:
	== Theory ==		== Theory ==
	298edo is [[consistent]] in the 5-odd-limit, where it is [[enfactoring\|enfactored]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo - it retains the mapping for 2.3.5.17 but differs on the mapping for harmonics [[7/4\|7]], [[11/8\|11]], [[13/8\|13]]. 298edo tempers out the [[rastma]], splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]].		298edo is [[consistent]] in the 5-odd-limit, where it is [[enfactoring\|enfactored]], with the same tuning as [[149edo]]. Since 149edo is notable for being the smallest edo distinctly consistent in the [[17-odd-limit]], 298edo is related to 149edo - it retains the mapping for 2.3.5.17 but differs on the mapping for harmonics [[7/4\|7]], [[11/8\|11]], [[13/8\|13]]. 298edo tempers out the [[rastma]], splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]].

	The patent val supports the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 subgroup, 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed 77 & 298 temperament with [[13/8]] as its generator.		The patent val supports the [[bison]] temperament and the rank-3 temperament [[hemimage]]. In the 2.5.11.13 subgroup, 298edo supports [[emka]]. In the full 13-limit, 298edo supports an unnamed 77 & 298 temperament with [[13/8]] as its generator.

	Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step.		Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, {{val\|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val - for example it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].

	The 298d val, {{val\|298 472 692 '''836''' 1031}}, which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. Some of the commas it tempers out make for much more interesting temperaments than the patent val - for example it still tempers out 243/242, but now it adds [[1029/1024]], [[3136/3125]], and [[9801/9800]].

	The 298cd val, {{val\|298 472 '''691''' '''836''' 1031}} supports [[miracle]].		The 298cd val, {{val\|298 472 '''691''' '''836''' 1031}} supports [[miracle]].