68edo - Revision history

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

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 === Interval mappings ===
 {{Q-odd-limit intervals}}
 == Regular temperament properties ==

Lériendil: /* Subsets and supersets */

2025-09-14T07:58:06Z

Subsets and supersets

← Older revision		Revision as of 07:58, 14 September 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 68 factors into {{factorization\|68}}, 68edo has subset edos {{EDOs\| 2, 4, 17, and 34 }}.		Since 68 factors into {{factorization\|68}}, 68edo has subset edos {{EDOs\| 2, 4, 17, and 34 }}. Important supersets include [[612edo]], a 5-limit record-holder, and [[680edo]].

	== Intervals ==		== Intervals ==

Lériendil: /* Theory */

2025-09-12T15:46:03Z

Theory

← Older revision		Revision as of 15:46, 12 September 2025
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	== Theory ==		== Theory ==
	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. 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]] temperaments, and supplies the [[optimal patent val]] for 11-limit [[hemikleismic]].

Lucius Chiaraviglio: /* Music */ Add Bryan Deister's ''microtonal improvisation in 68edo'' (2025)

2025-07-29T21:01:15Z

Music: Add Bryan Deister's ''microtonal improvisation in 68edo'' (2025)

← Older revision		Revision as of 21:01, 29 July 2025
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	== Music ==		== Music ==
			; [[Bryan Deister]]
			* [https://www.youtube.com/shorts/CO1AslAu9E0 ''microtonal improvisation in 68edo''] (2025)

	; [[The Mercury Tree]]		; [[The Mercury Tree]]
	* [https://m.youtube.com/watch?v=ZRFumaIM02E Grown Apart] from ''Self Similar'' (2023)		* [https://m.youtube.com/watch?v=ZRFumaIM02E Grown Apart] from ''Self Similar'' (2023)

Yourmusic Productions: Add lumatone mapping link.

2025-07-17T06:35:27Z

Add lumatone mapping link.

← Older revision		Revision as of 06:35, 17 July 2025
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	* 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: 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
	* 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

			== Instruments ==
			* [[Lumatone mapping for 68edo]]

	== Music ==		== Music ==

Lériendil: /* Theory */redundancy

2025-06-23T13:58:36Z

Theory: redundancy

← Older revision		Revision as of 13:58, 23 June 2025
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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. 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. 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]]~~. It is a sharp-tending system, with the 3rd, 5th and 7th harmonics all sharp~~.		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]].

	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.

Lériendil: mentioned potential octave compressions

2025-06-23T12:44:31Z

mentioned potential octave compressions

← Older revision		Revision as of 12:44, 23 June 2025
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	== Theory ==		== Theory ==
	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. 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.		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. 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]]. It is a sharp-tending system, with the 3rd, 5th and 7th harmonics all sharp.		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]]. It is a sharp-tending system, with the 3rd, 5th and 7th harmonics all sharp.

ArrowHead294: /* Scales */

2025-06-18T12:15:02Z

Scales

← Older revision		Revision as of 12:15, 18 June 2025
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	== Scales ==		== Scales ==
	: ''See also: [[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)		* Negative semitone: 14 14 -1 14 14 14 -1 (E is sharper than F, and B is sharper than C)

Lériendil at 18:20, 16 June 2025

2025-06-16T18:20:04Z

← Older revision		Revision as of 18:20, 16 June 2025
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	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.

	The 5th degree of 68edo can be used as a generator for [[88cET]], while ~~one step~~ almost perfectly ~~approximates~~ [[50/49]], allowing for the tempering out of 2/(50/49)<sup>34</sup>.		The 5th degree of 68edo can be used as a generator for [[88cET]], while two steps almost perfectly approximate [[50/49]], allowing for the tempering out of 2/(50/49)<sup>34</sup>.

	=== Prime harmonics ===		=== Prime harmonics ===