198edo - 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"

2026-03-13T13:31:47Z

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 13:31, 13 March 2026
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	\| [[Icosidillic]]		\| [[Icosidillic]]
	\|}		\|}
	<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

	[[Category:99edo]]		[[Category:99edo]]
	[[Category:Major minthmic]]		[[Category:Major minthmic]]
	[[Category:Namaka]]		[[Category:Namaka]]

Overthink: /* Rank-2 temperaments */ add quadrafifths

2026-02-11T23:06:28Z

Rank-2 temperaments: add quadrafifths

← Older revision		Revision as of 23:06, 11 February 2026
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	\| 13/12		\| 13/12
	\| [[Quasijerome]]		\| [[Quasijerome]]
			\|-
			\| 1
			\| 29\198
			\| 175.76
			\| 72/65
			\| [[Quadrafifths]]
	\|-		\|-
	\| 1		\| 1
	\| 65\198		\| 65\198
	\| 393.93		\| 393.94
	\| 49/39		\| 49/39
	\| [[Hitch]]		\| [[Hitch]]
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	\| 2		\| 2
	\| 31\198		\| 31\198
	\| 187.87		\| 187.88
	\| 39/35		\| 39/35
	\| [[Semiwitch]]		\| [[Semiwitch]]
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	\| 2		\| 2
	\| 74\198<br>(25\198)		\| 74\198<br>(25\198)
	\| 448.48<br>(151.51)		\| 448.48<br>(151.52)
	\| 35/27<br>(12/11)		\| 35/27<br>(12/11)
	\| [[Neusec]]		\| [[Neusec]]
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	\| 3		\| 3
	\| 41\198<br>(25\198)		\| 41\198<br>(25\198)
	\| 248.48<br>(151.51)		\| 248.48<br>(151.52)
	\| 15/13<br>(12/11)		\| 15/13<br>(12/11)
	\| [[Hemimist]]		\| [[Hemimist]]

Tristanbay: /* Intervals */ Whoops, forgot to add the main notation section itself

2025-10-14T07:28:03Z

Intervals: Whoops, forgot to add the main notation section itself

← Older revision		Revision as of 07:28, 14 October 2025
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	{{Main\| Table of 198edo intervals }}		{{Main\| Table of 198edo intervals }}

			== Notation ==
	=== Ups and downs notation ===		=== Ups and downs notation ===
	198edo can be notated using [[Kite's ups and downs notation\|ups and downs]] with quarter-tone accidentals:		198edo can be notated using [[Kite's ups and downs notation\|ups and downs]] with quarter-tone accidentals:

Tristanbay: Added notation section with ups and downs

2025-10-14T07:22:38Z

Added notation section with ups and downs

← Older revision		Revision as of 07:22, 14 October 2025
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	== Intervals ==		== Intervals ==
	{{Main\| Table of 198edo intervals }}		{{Main\| Table of 198edo intervals }}

			=== Ups and downs notation ===
			198edo can be notated using [[Kite's ups and downs notation\|ups and downs]] with quarter-tone accidentals:
			{{Ups and downs sharpness\|198\|true}}

	== Regular temperament properties ==		== Regular temperament properties ==

Fredg999: Markup

2025-07-28T18:54:56Z

Markup

← Older revision		Revision as of 18:54, 28 July 2025
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	Like 99, it [[tempering out\|tempers out]] [[2401/2400]], [[3136/3125]], [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[10976/10935]] in the 7-limit. In the 11-limit, [[3025/3024]], [[3388/3375]], [[9801/9800]], [[14641/14580]], and [[16384/16335]]; in the 13-limit, [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[2200/2197]] and [[6656/6655]].		Like 99, it [[tempering out\|tempers out]] [[2401/2400]], [[3136/3125]], [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[10976/10935]] in the 7-limit. In the 11-limit, [[3025/3024]], [[3388/3375]], [[9801/9800]], [[14641/14580]], and [[16384/16335]]; in the 13-limit, [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[2200/2197]] and [[6656/6655]].

	It provides the [[optimal patent val]] for the 13-limit rank-5 temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as [[hemimist]] and [[namaka]]. Besides [[major minthmic chords]], it enables [[essentially tempered ~~chords~~]] including [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] in the [[13-odd-limit]], in addition to [[island chords]] in the [[15-odd-limit]].		It provides the [[optimal patent val]] for the 13-limit rank-5 temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as [[hemimist]] and [[namaka]]. Besides [[major minthmic chords]], it enables [[essentially tempered chord]]s including [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] in the [[13-odd-limit]], in addition to [[island chords]] in the [[15-odd-limit]].

	Notably, it is the last edo to map [[64/63]] and [[81/80]] to the same step consistently.		Notably, it is the last edo to map [[64/63]] and [[81/80]] to the same step consistently.

FloraC: /* Theory */ +octave stretch

2025-04-19T12:04:05Z

Theory: +octave stretch

← Older revision		Revision as of 12:04, 19 April 2025
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	Like 99, it [[tempering out\|tempers out]] [[2401/2400]], [[3136/3125]], [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[10976/10935]] in the 7-limit. In the 11-limit, [[3025/3024]], [[3388/3375]], [[9801/9800]], [[14641/14580]], and [[16384/16335]]; in the 13-limit, [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[2200/2197]] and [[6656/6655]].		Like 99, it [[tempering out\|tempers out]] [[2401/2400]], [[3136/3125]], [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[10976/10935]] in the 7-limit. In the 11-limit, [[3025/3024]], [[3388/3375]], [[9801/9800]], [[14641/14580]], and [[16384/16335]]; in the 13-limit, [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[2200/2197]] and [[6656/6655]].

	It provides the [[optimal patent val]] for the 13-limit rank-5 temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as [[hemimist]] and [[namaka]]. Besides [[major minthmic chords]], it enables [[essentially tempered chords]] including [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[island chords]] in the 15-odd-limit.		It provides the [[optimal patent val]] for the 13-limit rank-5 temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as [[hemimist]] and [[namaka]]. Besides [[major minthmic chords]], it enables [[essentially tempered chords]] including [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] in the [[13-odd-limit]], in addition to [[island chords]] in the [[15-odd-limit]].

	Notably, it is the last edo to map [[64/63]] and [[81/80]] to the same step consistently.		Notably, it is the last edo to map [[64/63]] and [[81/80]] to the same step consistently.
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	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|198}}		{{Harmonics in equal\|198}}

			=== Octave stretch ===
			198edo can benefit from slightly [[stretched and compressed tuning\|compressing the octave]] if that is acceptable, using tunings such as [[512ed6]] or [[710ed12]]. This improves the approximated harmonics 3, 5, 7, 13, and 23; the 11 may become less accurate depending on the specific tuning. The 19 also gets worse on compression, so the compression should be very mild if the target is the no-17 23-limit.

	=== Subsets and supersets ===		=== Subsets and supersets ===

FloraC: /* Regular temperament properties */ +no-17 19-limit and no-17 23-limit

2025-03-23T14:49:32Z

Regular temperament properties: +no-17 19-limit and no-17 23-limit

← Older revision		Revision as of 14:49, 23 March 2025
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	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[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
	\|-		\|-
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 2401/2400, 3025/3024, 3136/3125, 4375/4374		\| 2401/2400, 3025/3024, 3136/3125, 4375/4374
	\| {{~~mapping~~\| 198 314 460 556 685 }}		\| {{Mapping\| 198 314 460 556 685 }}
	\| −0.344		\| −0.344
	\| 0.291		\| 0.291
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 352/351, 676/675, 847/845, 1716/1715, 3025/3024		\| 352/351, 676/675, 847/845, 1716/1715, 3025/3024
	\| {{~~mapping~~\| 198 314 460 556 685 733 }}		\| {{Mapping\| 198 314 460 556 685 733 }}
	\| −0.372		\| −0.372
	\| 0.273		\| 0.273
	\| 4.50		\| 4.50
			\|-
			\| 2.3.5.7.11.13.19
			\| 352/351, 361/360, 456/455, 676/675, 847/845, 1331/1330
			\| {{Mapping\| 198 314 460 556 685 733 841 }}
			\| −0.301
			\| 0.307
			\| 5.07
			\|-
			\| 2.3.5.7.11.13.19.23
			\| 352/351, 361/360, 456/455, 484/483, 576/575, 676/675, 847/845
			\| {{Mapping\| 198 314 460 556 685 733 841 896 }}
			\| −0.319
			\| 0.291
			\| 4.81
	\|}		\|}
	* 198et has a lower absolute error in the 13-limit than any previous equal temperaments, past [[190edo\|190]] and followed by [[224edo\|224]].		* 198et has a lower absolute error in the 13-limit than any previous equal temperaments, past [[190edo\|190]] and followed by [[224edo\|224]].
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	\|+ style="font-size: 105%;" \| 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
	\|-		\|-
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	\|-		\|-
	\| 2		\| 2
	\| 52\198<br />(47\198)		\| 52\198<br>(47\198)
	\| 315.15<br />(284.85)		\| 315.15<br>(284.85)
	\| 6/5<br />(33/28)		\| 6/5<br>(33/28)
	\| [[Semiparakleismic]]		\| [[Semiparakleismic]]
	\|-		\|-
	\| 2		\| 2
	\| 58\198<br />(41\198)		\| 58\198<br>(41\198)
	\| 351.52<br />(248.48)		\| 351.52<br>(248.48)
	\| 49/40<br />(15/13)		\| 49/40<br>(15/13)
	\| [[Semihemi]]		\| [[Semihemi]]
	\|-		\|-
	\| 2		\| 2
	\| 67\198<br />(32\198)		\| 67\198<br>(32\198)
	\| 406.06<br />(193.94)		\| 406.06<br>(193.94)
	\| 495/392<br />(28/25)		\| 495/392<br>(28/25)
	\| [[Semihemiwürschmidt]]		\| [[Semihemiwürschmidt]]
	\|-		\|-
	\| 2		\| 2
	\| 74\198<br />(25\198)		\| 74\198<br>(25\198)
	\| 448.48<br />(151.51)		\| 448.48<br>(151.51)
	\| 35/27<br />(12/11)		\| 35/27<br>(12/11)
	\| [[Neusec]]		\| [[Neusec]]
	\|-		\|-
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	\|-		\|-
	\| 3		\| 3
	\| 41\198<br />(25\198)		\| 41\198<br>(25\198)
	\| 248.48<br />(151.51)		\| 248.48<br>(151.51)
	\| 15/13<br />(12/11)		\| 15/13<br>(12/11)
	\| [[Hemimist]]		\| [[Hemimist]]
	\|-		\|-
	\| 6		\| 6
	\| 82\198<br />(16\198)		\| 82\198<br>(16\198)
	\| 496.97<br />(96.97)		\| 496.97<br>(96.97)
	\| 4/3<br />(200/189)		\| 4/3<br>(200/189)
	\| [[Semimist]]		\| [[Semimist]]
	\|-		\|-
	\| 18		\| 18
	\| 52\198<br />(3\198)		\| 52\198<br>(3\198)
	\| 315.15<br />(18.18)		\| 315.15<br>(18.18)
	\| 6/5<br />(99/98)		\| 6/5<br>(99/98)
	\| [[Hemiennealimmal]]		\| [[Hemiennealimmal]]
	\|-		\|-
	\| 22		\| 22
	\| 82\198<br />(1\198)		\| 82\198<br>(1\198)
	\| 496.97<br />(6.06)		\| 496.97<br>(6.06)
	\| 4/3<br />(385/384)		\| 4/3<br>(385/384)
	\| [[Icosidillic]]		\| [[Icosidillic]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	[[Category:99edo]]		[[Category:99edo]]
	[[Category:Major minthmic]]		[[Category:Major minthmic]]
	[[Category:Namaka]]		[[Category:Namaka]]

FloraC: /* Theory */ discuss the extensions beyond the 13-limit

2025-03-23T14:31:04Z

Theory: discuss the extensions beyond the 13-limit

← Older revision		Revision as of 14:31, 23 March 2025
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	== Theory ==		== Theory ==
	198edo ~~is [[consistency\|distinctly consistent]] through the [[15-odd-limit]] with [[harmonic]]s of 3 through 13 all tuned sharp. It~~ is [[enfactoring\|enfactored]] in the [[7-limit]], with the same tuning as [[99edo]], but makes for a good [[11-limit\|11-]] and [[13-limit]] system.		198edo is [[enfactoring\|enfactored]] in the [[7-limit]], with the same tuning as [[99edo]], but makes for a good [[11-limit\|11-]] and [[13-limit]] system. It is [[consistency\|distinctly consistent]] through the [[15-odd-limit]], and demonstrates a sharp tendency, with [[harmonic]]s 3 through 13 all tuned sharp.

	Like 99, it [[tempering out\|tempers out]] [[2401/2400]], [[3136/3125]], [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[10976/10935]] in the 7-limit. In the 11-limit, [[3025/3024]], [[3388/3375]], [[9801/9800]], [[14641/14580]], and [[16384/16335]]; in the 13-limit, [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[2200/2197]] and [[6656/6655]].		Like 99, it [[tempering out\|tempers out]] [[2401/2400]], [[3136/3125]], [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[10976/10935]] in the 7-limit. In the 11-limit, [[3025/3024]], [[3388/3375]], [[9801/9800]], [[14641/14580]], and [[16384/16335]]; in the 13-limit, [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[2200/2197]] and [[6656/6655]].
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	Notably, it is the last edo to map [[64/63]] and [[81/80]] to the same step consistently.		Notably, it is the last edo to map [[64/63]] and [[81/80]] to the same step consistently.

			Extending it beyond the 13-limit can be tricky, as the approximated [[17/1\|harmonic 17]] is almost 1/3-edostep flat of just, which does not blend well with the sharp tendency from the lower harmonics. The 198g val in turn gives you an alternative that is more than 2/3-edostep sharp. However, if we skip prime 17 altogether and treat 198edo as a no-17 [[23-limit]] system, it is almost consistent to the no-17 [[23-odd-limit]] with the sole exception of [[19/15]] and its [[octave complement]]. It tempers out [[361/360]] and [[456/455]] in the [[19-limit]], and [[484/483]] and [[576/575]] in the [[23-limit]]. Finally, the harmonics [[29/1\|29]] and [[31/1\|31]] are quite accurate, though the [[25/1\|25]] and [[27/1\|27]] are sharp enough to have incurred more inconsistencies.

	The 198b val [[support]]s a [[septimal meantone]] close to the [[CTE tuning]], although [[229edo]] is even closer, and besides, the 198be val supports an undecimal meantone almost identical to the [[POTE tuning]].		The 198b val [[support]]s a [[septimal meantone]] close to the [[CTE tuning]], although [[229edo]] is even closer, and besides, the 198be val supports an undecimal meantone almost identical to the [[POTE tuning]].
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 198 factors into {{~~factorisation~~\|~~198~~}}, 198edo has subset edos {{EDOs\| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}.		Since 198 factors into primes as {{nowrap\| 2 × 3<sup>2</sup> × 11 }}, 198edo has subset edos {{EDOs\| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}.

	A step of 198edo is exactly 50 [[purdal]]s or 62 [[prima]]s.		A step of 198edo is exactly 50 [[purdal]]s or 62 [[prima]]s.

ArrowHead294 at 14:12, 20 February 2025

2025-02-20T14:12:52Z

← Older revision		Revision as of 14:12, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|198~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 14:20, 16 January 2025

2025-01-16T14:20:41Z

← Older revision		Revision as of 14:20, 16 January 2025
Line 17:		Line 17:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 198 factors into {{~~factor~~\|198}}, 198edo has subset edos {{EDOs\| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}.		Since 198 factors into {{factorisation\|198}}, 198edo has subset edos {{EDOs\| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}.

	A step of 198edo is exactly 50 [[purdal]]s or 62 [[prima]]s.		A step of 198edo is exactly 50 [[purdal]]s or 62 [[prima]]s.