68edo - Revision history

FloraC: /* Regular temperament properties */ + associated ratios; this fixes the alignment. Note hemiwuerschmidt

2026-10-03T09:43:18Z

Regular temperament properties: + associated ratios; this fixes the alignment. Note hemiwuerschmidt

← Older revision		Revision as of 09:43, 3 October 2026
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	\| 2.3.5.7		\| 2.3.5.7
	\| 245/243, 2048/2025, 2401/2400		\| 245/243, 2048/2025, 2401/2400
	\| {{~~mapping~~\| 68 108 158 191 }}		\| {{Mapping\| 68 108 158 191 }}
	\| −0.983		\| −0.983
	\| 0.915		\| 0.915
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 121/120, 176/175, 245/243, 1375/1372		\| 121/120, 176/175, 245/243, 1375/1372
	\| {{~~mapping~~\| 68 108 158 191 235 }}		\| {{Mapping\| 68 108 158 191 235 }}
	\| −0.541		\| −0.541
	\| 1.206		\| 1.206
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 121/120, 176/175, 196/195, 245/243, 275/273		\| 121/120, 176/175, 196/195, 245/243, 275/273
	\| {{~~mapping~~\| 68 108 158 191 235 252 }}		\| {{Mapping\| 68 108 158 191 235 252 }}
	\| −0.745		\| −0.745
	\| 1.191		\| 1.191
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 121/120, 136/135, 154/153, 176/175, 196/195, 275/273		\| 121/120, 136/135, 154/153, 176/175, 196/195, 275/273
	\| {{~~mapping~~\| 68 108 158 191 235 252 278 }}		\| {{Mapping\| 68 108 158 191 235 252 278 }}
	\| −0.671		\| −0.671
	\| 1.118		\| 1.118
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	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 121/120, 136/135, 154/153, 190/189, 176/175, 196/195, 275/273		\| 121/120, 136/135, 154/153, 190/189, 176/175, 196/195, 275/273
	\| {{~~mapping~~\| 68 108 158 191 235 252 278 289 }}		\| {{Mapping\| 68 108 158 191 235 252 278 289 }}
	\| −0.661		\| −0.661
	\| 1.046		\| 1.046
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	\|-		\|-
	! Periods<br>per 8ve		! Periods<br>per 8ve
	! Generator		! Generator*
	! Cents		! Cents*
			! Associated<br>ratio*
	! Temperament		! Temperament
	\|-		\|-
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	\| 5\68		\| 5\68
	\| 88.24		\| 88.24
			\| 20/19
	\| [[Octacot]]		\| [[Octacot]]
	\|-		\|-
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	\| 11\68		\| 11\68
	\| 194.12		\| 194.12
	\| [[~~Hemiwur~~]]		\| 28/25
			\| [[Hemiwürschmidt]] (68e) / hemiwur (68)
	\|-		\|-
	\| 2		\| 2
	\| 3\68		\| 3\68
	\| 52.94		\| 52.94
			\| 33/32
	\| [[Shrutar]]		\| [[Shrutar]]
	\|-		\|-
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	\| 6\68		\| 6\68
	\| 105.88		\| 105.88
			\| 17/16
	\| [[Bidia]]		\| [[Bidia]]
	\|}		\|}
			<nowiki/>* In [[normal forms #Minimal-generator form\|minimal-generator form]]

	== Scales ==		== Scales ==

BudjarnLambeth: /* Scales */

2026-10-03T02:52:48Z

Scales

← Older revision		Revision as of 02:52, 3 October 2026
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	* [[Deeptone]][7]: 10 10 9 10 10 10 9		* [[Deeptone]][7]: 10 10 9 10 10 10 9
	* [[Hemiwur]][31]: 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2		* [[Hemiwur]][31]: 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2
	* Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7		* Inverse half octave{{clarify}}: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7
	* Negative semitone: 14 14 -1 14 14 14 -1 (''E is sharper than F, and B is sharper than C'')		* Negative semitone: 14 14 -1 14 14 14 -1 (''E is sharper than F, and B is sharper than C'')
	* [[Octacot]][27]: 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3		* [[Octacot]][27]: 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3
	* [[Quartkeenlig]][23] (stretched [[23edo]]): 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2		* [[Quartkeenlig]][23] (stretched [[23edo]]): 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2
	* [[Shrutar]][22]: 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3		* [[Shrutar]][22]: 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3
	* [[Superpyth]] quarter octave: 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1		* [[Superpyth]] quarter octave{{clarify}}: 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1

	== Instruments ==		== Instruments ==

BudjarnLambeth: /* Scales */

2026-10-03T02:51:58Z

Scales

← Older revision		Revision as of 02:51, 3 October 2026
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	* [[Deeptone]][7]: 10 10 9 10 10 10 9		* [[Deeptone]][7]: 10 10 9 10 10 10 9
			* [[Hemiwur]][31]: 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2
	* Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7		* Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7
	* Negative semitone: 14 14 -1 14 14 14 -1 (''E is sharper than F, and B is sharper than C'')		* Negative semitone: 14 14 -1 14 14 14 -1 (''E is sharper than F, and B is sharper than C'')

BudjarnLambeth: /* Scales */

2026-10-03T02:50:30Z

Scales

← Older revision		Revision as of 02:50, 3 October 2026
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	* Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7		* Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7
	* Negative semitone: 14 14 -1 14 14 14 -1 (''E is sharper than F, and B is sharper than C'')		* Negative semitone: 14 14 -1 14 14 14 -1 (''E is sharper than F, and B is sharper than C'')
			* [[Octacot]][27]: 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3
	* [[Quartkeenlig]][23] (stretched [[23edo]]): 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2		* [[Quartkeenlig]][23] (stretched [[23edo]]): 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2
	* [[Shrutar]][22]: 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3		* [[Shrutar]][22]: 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3

BudjarnLambeth: /* Scales */ add 1

2026-10-03T02:46:20Z

Scales: add 1

← Older revision		Revision as of 02:46, 3 October 2026
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	: ''See also: [[List of MOS scales in 68edo]], [[34edo #Scales]], and [[17edo #Scales]].''		: ''See also: [[List of MOS scales in 68edo]], [[34edo #Scales]], and [[17edo #Scales]].''

	* ~~Negative semitone: 14 14 -1 14 14 14 -1 (E is sharper than F, and B is sharper than C)~~		* [[Deeptone]][7]: 10 10 9 10 10 10 9
	* Deeptone[7]: 10 10 9 10 10 10 9
	* Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7		* Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7
	* ~~Superpyth quarter octave~~: 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1		* Negative semitone: 14 14 -1 14 14 14 -1 (''E is sharper than F, and B is sharper than C'')
	* ~~Quartkeenlig~~[23] ~~(Stretched 23edo)~~: 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2		* [[Quartkeenlig]][23] (stretched [[23edo]]): 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2
			* [[Shrutar]][22]: 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3
			* [[Superpyth]] quarter octave: 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1

	== Instruments ==		== Instruments ==

BudjarnLambeth: /* Regular temperament properties */ begin rank 2 temp table

2026-10-03T02:32:53Z

Regular temperament properties: begin rank 2 temp table

@@ Line 402: / Line 402: @@
 | 1.046
 | 5.93
 |}

BudjarnLambeth: Todo:add rank 2 temperaments table

2026-10-02T14:03:58Z

Todo:add rank 2 temperaments table

← Older revision		Revision as of 14:03, 2 October 2026
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	[[Category:Shrutar]]		[[Category:Shrutar]]
	[[Category:Quartismic]]		[[Category:Quartismic]]
			[[Category:Todo:add rank 2 temperaments table]]

Fredg999: Update link

2026-07-08T04:03:51Z

Update link

← Older revision		Revision as of 04:03, 8 July 2026
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	68edo's step is half of the step size of [[34edo]], which does well in the 5-limit but not so well in the 7-limit, and one quarter the size of [[17edo]], which does well in the [[3-limit]], but not so well in the [[5-limit]]. The luck continues: 68 is a strong [[7-limit]] system, but does not do as well in the [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly since [[11/9]] is not mapped to its best approximation. However, 68edo maps many higher primes better than it does 11 (specifically 13 and 23 inherited from 17edo, 17 inherited from 34edo, and 19 and 31 new to 68edo), notably being [[consistent]] in the entire no-11s 25-[[odd limit]] add-31. It achieves this by having a consistent sharp tendency among all primes up to 31, save 11 and 29. Therefore, a slight octave compression, such as in [[158ed5]] or [[191ed7]], can improve upon the accuracy of 68edo's harmonic series.		68edo's step is half of the step size of [[34edo]], which does well in the 5-limit but not so well in the 7-limit, and one quarter the size of [[17edo]], which does well in the [[3-limit]], but not so well in the [[5-limit]]. The luck continues: 68 is a strong [[7-limit]] system, but does not do as well in the [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly since [[11/9]] is not mapped to its best approximation. However, 68edo maps many higher primes better than it does 11 (specifically 13 and 23 inherited from 17edo, 17 inherited from 34edo, and 19 and 31 new to 68edo), notably being [[consistent]] in the entire no-11s 25-[[odd limit]] add-31. It achieves this by having a consistent sharp tendency among all primes up to 31, save 11 and 29. Therefore, a slight octave compression, such as in [[158ed5]] or [[191ed7]], can improve upon the accuracy of 68edo's harmonic series.

	As a 7-limit system, 68et [[tempering out\|tempers out]] [[2048/2025]], [[245/243]], [[4000/3969]], [[15625/15552]], [[3136/3125]], [[6144/6125]], and [[2401/2400]]. It [[support]]s [[octacot]], [[shrutar]], [[hemiwürschmidt]], [[hemikleismic]], [[clyde]] and [[neptune]] temperaments, and supplies the [[optimal patent val]] for 11-limit [[hemikleismic]].		As a 7-limit system, 68et [[tempering out\|tempers out]] [[2048/2025]], [[245/243]], [[4000/3969]], [[15625/15552]], [[3136/3125]], [[6144/6125]], and [[2401/2400]]. It [[support]]s [[octacot]], [[shrutar]], [[hemiwürschmidt]], [[hemikleismic]], [[clyde]] and [[Neptune (temperament)\|neptune]] temperaments, and supplies the [[optimal patent val]] for 11-limit [[hemikleismic]].

	The 3rd degree of 68edo can be used as a generator for [[23edo and octave stretching\|stretched 23edo]], which also acts as the [[quartkeenlig]] temperament tempering out the quartisma, 385/384 and 6250/6237. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65{{c}}). It also works as a [[22L 1s]] MOS of the quartkeenlig temperament.		The 3rd degree of 68edo can be used as a generator for [[23edo and octave stretching\|stretched 23edo]], which also acts as the [[quartkeenlig]] temperament tempering out the quartisma, 385/384 and 6250/6237. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65{{c}}). It also works as a [[22L 1s]] MOS of the quartkeenlig temperament.

Fitzgerald Lee: /* Interval mappings */Added 68e interval mappings

2026-04-27T00:42:04Z

Interval mappings: Added 68e interval mappings

← Older revision		Revision as of 00:42, 27 April 2026
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	=== Interval mappings ===		=== Interval mappings ===
	{{Q-odd-limit intervals}}		{{Q-odd-limit intervals}}
			{{Q-odd-limit intervals\|68.1\|apx=val\|header=none\|tag=none\|title=15-odd-limit intervals by 68e val mapping}}

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

BudjarnLambeth: Remove the zeta peak table that had no explanation or context

2025-09-17T02:00:16Z

Remove the zeta peak table that had no explanation or context

@@ Line 354: / Line 354: @@
 === Interval mappings ===
 {{Q-odd-limit intervals}}
 == Regular temperament properties ==