612edo - Revision history

FloraC: "Skisma" allegedly comes from Gene

2026-04-04T13:25:40Z

"Skisma" allegedly comes from Gene

← Older revision		Revision as of 13:25, 4 April 2026
Line 5:		Line 5:
	612edo is an insanely strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].		612edo is an insanely strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].

	The 612edo step was proposed as the logarithmic [[interval size measure]] '''nil''' by James Paul White<ref>[https://dn721808.ca.archive.org/0/items/sim_music-a-monthly-magazine_november-1894-april-1895_7/sim_music-a-monthly-magazine_november-1894-april-1895_7.pdf White, James Paul. 1895. "Is Perfect Intonation Practicable?". ''Music a Monthly Magazine'' 7:446]</ref> (also '''skisma''' or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions.		The 612edo step was proposed as the logarithmic [[interval size measure]] '''nil''' by [[James Paul White]]<ref>[https://dn721808.ca.archive.org/0/items/sim_music-a-monthly-magazine_november-1894-april-1895_7/sim_music-a-monthly-magazine_november-1894-april-1895_7.pdf White, James Paul. 1895. "Is Perfect Intonation Practicable?". ''Music a Monthly Magazine'' 7:446]</ref>, and also '''skisma''' (symbol: '''sk''') by [[Gene Ward Smith]], since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 133:		Line 133:
	* [https://www.youtube.com/watch?v=_DrkrgkiaAY ''Theme and Variations in Hemiennealimmal''] (2023)		* [https://www.youtube.com/watch?v=_DrkrgkiaAY ''Theme and Variations in Hemiennealimmal''] (2023)

	== ~~Notes~~ ==		== References ==
	<references />		<references />

Overthink: /* Theory */ formatting

2026-04-03T22:04:53Z

Theory: formatting

← Older revision		Revision as of 22:04, 3 April 2026
Line 5:		Line 5:
	612edo is an insanely strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].		612edo is an insanely strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].

	The 612edo step was proposed as the logarithmic [[interval size measure]] '''nil''' by James Paul White<ref>[https://dn721808.ca.archive.org/0/items/sim_music-a-monthly-magazine_november-1894-april-1895_7/sim_music-a-monthly-magazine_november-1894-april-1895_7.pdf		The 612edo step was proposed as the logarithmic [[interval size measure]] '''nil''' by James Paul White<ref>[https://dn721808.ca.archive.org/0/items/sim_music-a-monthly-magazine_november-1894-april-1895_7/sim_music-a-monthly-magazine_november-1894-april-1895_7.pdf White, James Paul. 1895. "Is Perfect Intonation Practicable?". ''Music a Monthly Magazine'' 7:446]</ref> (also '''skisma''' or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions.
	White, James Paul. 1895. "Is Perfect Intonation Practicable?". ''Music a Monthly Magazine'' 7:446]</ref> (also '''skisma''' or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions.

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

X31eq: Fixed journal title

2026-04-03T18:42:41Z

Fixed journal title

← Older revision		Revision as of 18:42, 3 April 2026
Line 6:		Line 6:

	The 612edo step was proposed as the logarithmic [[interval size measure]] '''nil''' by James Paul White<ref>[https://dn721808.ca.archive.org/0/items/sim_music-a-monthly-magazine_november-1894-april-1895_7/sim_music-a-monthly-magazine_november-1894-april-1895_7.pdf		The 612edo step was proposed as the logarithmic [[interval size measure]] '''nil''' by James Paul White<ref>[https://dn721808.ca.archive.org/0/items/sim_music-a-monthly-magazine_november-1894-april-1895_7/sim_music-a-monthly-magazine_november-1894-april-1895_7.pdf
	White, James Paul. 1895. "Is Perfect Intonation ~~Practical~~?". ''Music a Monthly Magazine'' 7:446]</ref> (also '''skisma''' or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions.		White, James Paul. 1895. "Is Perfect Intonation Practicable?". ''Music a Monthly Magazine'' 7:446]</ref> (also '''skisma''' or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions.

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

X31eq: Citation for James Paul White

2026-04-03T18:30:19Z

Citation for James Paul White

← Older revision		Revision as of 18:30, 3 April 2026
Line 5:		Line 5:
	612edo is an insanely strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].		612edo is an insanely strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].

	The 612edo step ~~has been~~ proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions.		The 612edo step was proposed as the logarithmic [[interval size measure]] '''nil''' by James Paul White<ref>[https://dn721808.ca.archive.org/0/items/sim_music-a-monthly-magazine_november-1894-april-1895_7/sim_music-a-monthly-magazine_november-1894-april-1895_7.pdf
			White, James Paul. 1895. "Is Perfect Intonation Practical?". ''Music a Monthly Magazine'' 7:446]</ref> (also '''skisma''' or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions.

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

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"

2026-03-13T13:38:10Z

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 13:38, 13 March 2026
Line 127:		Line 127:
	\| [[Hemiennealimmal]] (11-limit)		\| [[Hemiennealimmal]] (11-limit)
	\|}		\|}
	<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

	== Music ==		== Music ==

Lucius Chiaraviglio: /* Theory */ Insert Intervals section under this to be consistent with other EDO pages having intervals

2025-11-23T23:12:00Z

Theory: Insert Intervals section under this to be consistent with other EDO pages having intervals

← Older revision		Revision as of 23:12, 23 November 2025
Line 5:		Line 5:
	612edo is an insanely strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].		612edo is an insanely strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].

	The 612edo step has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions~~. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]]~~.		The 612edo step has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|612}}		{{Harmonics in equal\|612}}

			== Intervals ==
			A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].

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

Eufalesio: I don't think "very" doesn't quite do justice to how spot-on 612edo is in the 5-limit

2025-10-27T12:27:43Z

I don't think "very" doesn't quite do justice to how spot-on 612edo is in the 5-limit

← Older revision		Revision as of 12:27, 27 October 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	612edo is ~~a very~~ strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].		612edo is an insanely strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].

	The 612edo step has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].		The 612edo step has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].

Lucius Chiaraviglio: /* Theory */ Fix missing word

2025-07-31T12:55:05Z

Theory: Fix missing word

← Older revision		Revision as of 12:55, 31 July 2025
Line 5:		Line 5:
	612edo is a very strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].		612edo is a very strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].

	The 612edo has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].		The 612edo step has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].

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

FloraC: No need to remind readers of what a regular temperament is everywhere

2025-04-11T11:19:00Z

No need to remind readers of what a regular temperament is everywhere

← Older revision		Revision as of 11:19, 11 April 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	612edo is a very strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}.		612edo is a very strong [[5-limit]] system, a fact noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3-4):223-48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>, {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}{{citation needed}} and {{w\|James Murray Barbour}}{{citation needed}}. As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]].

	As an equal temperament, it [[tempering out\|tempers out]] the {{monzo\| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo\| 1 -27 18 }} ([[ennealimma]]), {{monzo\| -52 -17 34 }} ([[septendecima]]), {{monzo\| -53 10 16 }} ([[kwazy comma]]), {{monzo\| 54 -37 2 }} ([[monzisma]]), {{monzo\| -107 47 14 }} (fortune comma), and {{monzo\| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]]. ~~(''See [[regular temperament]] for more about what all this means and how to use it.'')~~

	The 612edo has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].		The 612edo has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs\| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]].
Line 63:		Line 61:

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	~~''See [[regular temperament]] for more about what all this means and how to use it.''~~

	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator

BudjarnLambeth: /* Rank-2 temperaments */ (''See regular temperament for more about what all this means and how to use it.'')

2025-04-11T07:46:56Z

Rank-2 temperaments: (''See regular temperament for more about what all this means and how to use it.'')

← Older revision		Revision as of 07:46, 11 April 2025
Line 63:		Line 63:

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
			''See [[regular temperament]] for more about what all this means and how to use it.''

	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator