121edo - Revision history

FloraC: Update links

2026-04-05T10:41:36Z

Update links

← Older revision		Revision as of 10:41, 5 April 2026
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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		\| 2.3
	\| {{~~monzo~~\| 192 -121 }}		\| {{Monzo\| 192 -121 }}
	\| {{~~mapping~~\| 121 192 }}		\| {{Mapping\| 121 192 }}
	\| −0.687		\| −0.687
	\| 0.687		\| 0.687
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	\| 2.3.5		\| 2.3.5
	\| 15625/15552, {{monzo\| 31 -21 1 }}		\| 15625/15552, {{monzo\| 31 -21 1 }}
	\| {{~~mapping~~\| 121 192 281 }}		\| {{Mapping\| 121 192 281 }}
	\| −0.524		\| −0.524
	\| 0.606		\| 0.606
Line 43:		Line 43:
	\| 2.3.5.7		\| 2.3.5.7
	\| 4000/3969, 6144/6125, 10976/10935		\| 4000/3969, 6144/6125, 10976/10935
	\| {{~~mapping~~\| 121 192 281 340 }}		\| {{Mapping\| 121 192 281 340 }}
	\| −0.667		\| −0.667
	\| 0.580		\| 0.580
Line 50:		Line 50:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 540/539, 896/891, 1375/1372, 4375/4356		\| 540/539, 896/891, 1375/1372, 4375/4356
	\| {{~~mapping~~\| 121 192 281 340 419 }}		\| {{Mapping\| 121 192 281 340 419 }}
	\| −0.768		\| −0.768
	\| 0.556		\| 0.556
Line 57:		Line 57:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 325/324, 352/351, 364/363, 540/539, 625/624		\| 325/324, 352/351, 364/363, 540/539, 625/624
	\| {{~~mapping~~\| 121 192 281 340 419 448 }}		\| {{Mapping\| 121 192 281 340 419 448 }}
	\| −0.750		\| −0.750
	\| 0.510		\| 0.510
Line 64:		Line 64:
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441		\| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
	\| {{~~mapping~~\| 121 192 281 340 419 448 495 }}		\| {{Mapping\| 121 192 281 340 419 448 495 }}
	\| −0.787		\| −0.787
	\| 0.480		\| 0.480
Line 71:		Line 71:
	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374		\| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
	\| {{~~mapping~~\| 121 192 281 340 419 448 495 514 }}		\| {{Mapping\| 121 192 281 340 419 448 495 514 }}
	\| −0.689		\| −0.689
	\| 0.519		\| 0.519
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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*
	! Temperament		! Temperament
	\|-		\|-
Line 170:		Line 170:
	\| 495.87		\| 495.87
	\| 4/3		\| 4/3
	\| [[~~Leapday]] / [[polypyth~~]]		\| [[Polypyth]]
	\|-		\|-
	\| 1		\| 1
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	\|-		\|-
	\| 11		\| 11
	\| 50\121<br />(5\121)		\| 50\121<br>(5\121)
	\| 495.87<br />(49.59)		\| 495.87<br>(49.59)
	\| 4/3<br />(36/35)		\| 4/3<br>(36/35)
	\| [[Hendecatonic]]		\| [[Hendecatonic (temperament)\|Hendecatonic]]
	\|}		\|}
	<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		<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

	== 13-limit detempering of 121et ==		== 13-limit detempering of 121et ==

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"

2026-03-13T13:33:16Z

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
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	\| [[Hendecatonic]]		\| [[Hendecatonic]]
	\|}		\|}
	<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

	== 13-limit detempering of 121et ==		== 13-limit detempering of 121et ==

Overthink: merge section with theory

2025-12-27T07:36:23Z

merge section with theory

← Older revision		Revision as of 07:36, 27 December 2025
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	Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.		Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

			Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-gothic]] tunings.

	=== Odd harmonics ===		=== Odd harmonics ===
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	[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]		[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

	~~== Miscellany ==~~
	~~Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.~~

	[[Category:Grendel]]		[[Category:Grendel]]
	[[Category:Quintupole]]		[[Category:Quintupole]]

Fredg999: 121 isn't consistent in the 21-odd-limit, hence the table shows odd harmonics as per the template doc

2025-06-24T15:38:42Z

121 isn't consistent in the 21-odd-limit, hence the table shows odd harmonics as per the template doc

← Older revision		Revision as of 15:38, 24 June 2025
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	Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.		Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

	=== ~~Prime~~ harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|121}}		{{Harmonics in equal\|121}}

ArrowHead294 at 16:57, 18 February 2025

2025-02-18T16:57:52Z

← Older revision		Revision as of 16:57, 18 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|121~~}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2025-01-17T15:19:20Z

Theory: cleanup

← Older revision		Revision as of 15:19, 17 January 2025
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	== Theory ==		== Theory ==
	121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempers out]] 15625/15552 ([[15625/15552\|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].		121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempering out\|tempers out]] 15625/15552 ([[15625/15552\|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

	Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.		Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.
Line 11:		Line 11:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 121 factors into ~~{{factorization\|121}}~~, 121edo contains [[11edo]] as its only nontrivial subset.		Since 121 factors into 11<sup>2</sup>, 121edo contains [[11edo]] as its only nontrivial subset.

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

ArrowHead294 at 15:03, 16 January 2025

2025-01-16T15:03:51Z

← Older revision		Revision as of 15:03, 16 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. ~~The equal temperament~~ [[~~tempering out\|~~tempers out]] 15625/15552 ([[15625/15552\|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].		121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempers out]] 15625/15552 ([[15625/15552\|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

	Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.		Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.
Line 28:		Line 28:
	\| {{monzo\| 192 -121 }}		\| {{monzo\| 192 -121 }}
	\| {{mapping\| 121 192 }}		\| {{mapping\| 121 192 }}
	\| ~~−0~~.687		\| −0.687
	\| 0.687		\| 0.687
	\| 6.93		\| 6.93
Line 35:		Line 35:
	\| 15625/15552, {{monzo\| 31 -21 1 }}		\| 15625/15552, {{monzo\| 31 -21 1 }}
	\| {{mapping\| 121 192 281 }}		\| {{mapping\| 121 192 281 }}
	\| ~~−0~~.524		\| −0.524
	\| 0.606		\| 0.606
	\| 6.11		\| 6.11
Line 42:		Line 42:
	\| 4000/3969, 6144/6125, 10976/10935		\| 4000/3969, 6144/6125, 10976/10935
	\| {{mapping\| 121 192 281 340 }}		\| {{mapping\| 121 192 281 340 }}
	\| ~~−0~~.667		\| −0.667
	\| 0.580		\| 0.580
	\| 5.85		\| 5.85
Line 49:		Line 49:
	\| 540/539, 896/891, 1375/1372, 4375/4356		\| 540/539, 896/891, 1375/1372, 4375/4356
	\| {{mapping\| 121 192 281 340 419 }}		\| {{mapping\| 121 192 281 340 419 }}
	\| ~~−0~~.768		\| −0.768
	\| 0.556		\| 0.556
	\| 5.61		\| 5.61
Line 56:		Line 56:
	\| 325/324, 352/351, 364/363, 540/539, 625/624		\| 325/324, 352/351, 364/363, 540/539, 625/624
	\| {{mapping\| 121 192 281 340 419 448 }}		\| {{mapping\| 121 192 281 340 419 448 }}
	\| ~~−0~~.750		\| −0.750
	\| 0.510		\| 0.510
	\| 5.14		\| 5.14
Line 63:		Line 63:
	\| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441		\| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
	\| {{mapping\| 121 192 281 340 419 448 495 }}		\| {{mapping\| 121 192 281 340 419 448 495 }}
	\| ~~−0~~.787		\| −0.787
	\| 0.480		\| 0.480
	\| 4.85		\| 4.85
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	\| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374		\| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
	\| {{mapping\| 121 192 281 340 419 448 495 514 }}		\| {{mapping\| 121 192 281 340 419 448 495 514 }}
	\| ~~−0~~.689		\| −0.689
	\| 0.519		\| 0.519
	\| 5.23		\| 5.23
Line 194:		Line 194:
	\| [[Hendecatonic]]		\| [[Hendecatonic]]
	\|}		\|}
	<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

	== 13-limit detempering of 121et ==		== 13-limit detempering of 121et ==

ArrowHead294 at 11:42, 16 November 2024

2024-11-16T11:42:33Z

← Older revision		Revision as of 11:42, 16 November 2024
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	== 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
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	\| 0.519		\| 0.519
	\| 5.23		\| 5.23
	~~{{comma basis end}~~}		\|}
	* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo\|111]] before being superseded by [[130edo\|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo\|140]].		* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo\|111]] before being superseded by [[130edo\|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo\|140]].

	=== 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*
			! Temperament
	\|-		\|-
	\| 1		\| 1
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	\| 4/3<br />(36/35)		\| 4/3<br />(36/35)
	\| [[Hendecatonic]]		\| [[Hendecatonic]]
	~~{{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

	== 13-limit detempering of 121et ==		== 13-limit detempering of 121et ==

ArrowHead294 at 06:08, 16 November 2024

2024-11-16T06:08:21Z

← Older revision		Revision as of 06:08, 16 November 2024
Line 14:		Line 14:

	== 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
	\| {{monzo\| 192 -121 }}		\| {{monzo\| 192 -121 }}
	\| {{mapping\| 121 192 }}		\| {{mapping\| 121 192 }}
	\| -0.687		\| −0.687
	\| 0.687		\| 0.687
	\| 6.93		\| 6.93
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	\| 15625/15552, {{monzo\| 31 -21 1 }}		\| 15625/15552, {{monzo\| 31 -21 1 }}
	\| {{mapping\| 121 192 281 }}		\| {{mapping\| 121 192 281 }}
	\| -0.524		\| −0.524
	\| 0.606		\| 0.606
	\| 6.11		\| 6.11
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	\| 4000/3969, 6144/6125, 10976/10935		\| 4000/3969, 6144/6125, 10976/10935
	\| {{mapping\| 121 192 281 340 }}		\| {{mapping\| 121 192 281 340 }}
	\| -0.667		\| −0.667
	\| 0.580		\| 0.580
	\| 5.85		\| 5.85
Line 48:		Line 40:
	\| 540/539, 896/891, 1375/1372, 4375/4356		\| 540/539, 896/891, 1375/1372, 4375/4356
	\| {{mapping\| 121 192 281 340 419 }}		\| {{mapping\| 121 192 281 340 419 }}
	\| -0.768		\| −0.768
	\| 0.556		\| 0.556
	\| 5.61		\| 5.61
Line 55:		Line 47:
	\| 325/324, 352/351, 364/363, 540/539, 625/624		\| 325/324, 352/351, 364/363, 540/539, 625/624
	\| {{mapping\| 121 192 281 340 419 448 }}		\| {{mapping\| 121 192 281 340 419 448 }}
	\| -0.750		\| −0.750
	\| 0.510		\| 0.510
	\| 5.14		\| 5.14
Line 62:		Line 54:
	\| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441		\| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
	\| {{mapping\| 121 192 281 340 419 448 495 }}		\| {{mapping\| 121 192 281 340 419 448 495 }}
	\| -0.787		\| −0.787
	\| 0.480		\| 0.480
	\| 4.85		\| 4.85
Line 69:		Line 61:
	\| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374		\| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
	\| {{mapping\| 121 192 281 340 419 448 495 514 }}		\| {{mapping\| 121 192 281 340 419 448 495 514 }}
	\| -0.689		\| −0.689
	\| 0.519		\| 0.519
	\| 5.23		\| 5.23
	\|}		{{comma basis end}}
	* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo\|111]] before being superseded by [[130edo\|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo\|140]].		* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo\|111]] before being superseded by [[130edo\|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo\|140]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+Table of~~ rank-2 ~~temperaments by generator~~
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
Line 187:		Line 173:
	\|-		\|-
	\| 11		\| 11
	\| 50\121<br>(5\121)		\| 50\121<br />(5\121)
	\| 495.87<br>(49.59)		\| 495.87<br />(49.59)
	\| 4/3<br>(36/35)		\| 4/3<br />(36/35)
	\| [[Hendecatonic]]		\| [[Hendecatonic]]
	\|}		{{rank-2 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~~		{{orf}}

	== 13-limit detempering of 121et ==		== 13-limit detempering of 121et ==
Line 200:		Line 186:

	== Miscellany ==		== Miscellany ==

	Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.		Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.

	[[Category:Grendel]]		[[Category:Grendel]]
	[[Category:Quintupole]]		[[Category:Quintupole]]

FloraC: /* Regular temperament properties */ 13- to 23-limit notability

2024-06-04T13:28:53Z

Regular temperament properties: 13- to 23-limit notability

← Older revision		Revision as of 13:28, 4 June 2024
Line 73:		Line 73:
	\| 5.23		\| 5.23
	\|}		\|}
			* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo\|111]] before being superseded by [[130edo\|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo\|140]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===