Bird's eye view of temperaments by accuracy - Revision history

Overthink: /* Orwell */ link

2026-06-30T16:15:15Z

Orwell: link

← Older revision		Revision as of 16:15, 30 June 2026
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	[[#Generator tunings\|Generator tunings]]: 5\22, 7\31, 12\53, 19\84		[[#Generator tunings\|Generator tunings]]: 5\22, 7\31, 12\53, 19\84

	Orwell is a tempermant generated by a slightly sharp subminor third representing [[7/6]], seven of which reach [[6/1]]. Its name stems from George Orwell's book 1984, as a good generator is [[84edo\|19\84]]. Three subminor thirds reach 8/5, the [[5th]] subharmonic. This temperament is supported by notable edos [[22edo\|22]], [[31edo\|31]], and [[53edo\|53]], as well as the less known though still notable [[84edo]]. This temperament represents the 7-limit with good accuracy and relatively low complexity. However, the mapping for 11 is very simple, so this temperament really comes into its own in the 11-limit (see [[#Undecimal Orwell]]). Important commas tempered out by orwell include [[225/224]], [[1728/1715]], [[2430/2401]], [[6144/6125]], [[65625/65536]], and [[2109375/2097152]].		Orwell is a tempermant generated by a slightly sharp subminor third representing [[7/6]], seven of which reach [[6/1]]. Its name stems from George Orwell's book 1984, as a good generator is [[84edo\|19\84]]. Three subminor thirds reach [[8/5]], the 5th subharmonic. This temperament is supported by notable edos [[22edo\|22]], [[31edo\|31]], and [[53edo\|53]], as well as the less known though still notable [[84edo]]. This temperament represents the 7-limit with good accuracy and relatively low complexity. However, the mapping for 11 is very simple, so this temperament really comes into its own in the 11-limit (see [[#Undecimal Orwell]]). Important commas tempered out by orwell include [[225/224]], [[1728/1715]], [[2430/2401]], [[6144/6125]], [[65625/65536]], and [[2109375/2097152]].

	=== 11-limit focus ===		=== 11-limit focus ===

Godtone: /* Gary */ sort odds

2026-06-06T23:03:37Z

Gary: sort odds

← Older revision		Revision as of 23:03, 6 June 2026
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	Note counts:		Note counts:

	* 41 for {3, 9~~, 7~~, 11, 21, 27, 33, 77, 81, 99} ([[12L 29s]])		* 41 for {3, 7, 9, 11, 21, 27, 33, 77, 81, 99} ([[12L 29s]])

	Generator tunings: 24\41, 55\94, 79\135, 498\851		Generator tunings: 24\41, 55\94, 79\135, 498\851

Godtone: /* ~17-limit focus */ missing curly brace

2026-06-06T23:03:06Z

~17-limit focus: missing curly brace

← Older revision		Revision as of 23:03, 6 June 2026
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	* 82: see [[#Gary]]		* 82: see [[#Gary]]
	* 176 for {3, 5, 7, 9, 11, 13, 15, 21, 25		* 176 for {3, 5, 7, 9, 11, 13, 15, 21, 25}

	Generator tunings: (55\94, 1\2), (79\176, 1\2), (158\270, 1\2)		Generator tunings: (55\94, 1\2), (79\176, 1\2), (158\270, 1\2)

Godtone: /* Sensipent */ i dont intend on documenting 17-limit sensor as being too inaccurate for its complexity

2026-05-31T23:43:49Z

Sensipent: i dont intend on documenting 17-limit sensor as being too inaccurate for its complexity

← Older revision		Revision as of 23:43, 31 May 2026
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	* the most accurate is [[#Sendai]] which finds primes 23 and 29		* the most accurate is [[#Sendai]] which finds primes 23 and 29
	* the second most accurate is [[#Sensible]], which finds primes 11, 17 and 23		* the second most accurate is [[#Sensible]], which finds primes 11, 17 and 23
	* the simplest but least accurate is [[#Sensor]] (commonly just called "sensi"), which interprets it as a full 17-limit temperament.		* the simplest but least accurate is [[Sensipent family#Sensor\|Sensor]] (commonly just called "sensi"), which interprets it as a full 17-limit temperament, for which the best tuning is [[46edo]].

	==== [[Würschmidt]] ====		==== [[Würschmidt]] ====

Godtone: /* Garibaldi */ just noticed tetracot is conceived fairly naturally by this explanation

2026-05-08T00:56:52Z

Garibaldi: just noticed tetracot is conceived fairly naturally by this explanation

← Older revision		Revision as of 00:56, 8 May 2026
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	Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about prime 11 or prime 13:		Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about prime 11 or prime 13:
	* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] (as in [[cassandra]]) or a comma below [[~]][[5/4]] (as in [[andromeda]]). However, there is significant damage to 15/13 and 13/10.		* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] (as in [[cassandra]]) or a comma below [[~]][[5/4]] (as in [[andromeda]]), corresponding to [[tetracot]] in 2.3.5.11 (by tempering out (S9/S11 = [[243/242]],) S10 = [[100/99]] and S10/(S9/S11) = [[2200/2187]] respectively) which splits the halved fifth into two small major seconds of [[~]][[11/10]][[~]][[10/9]] around 175.6 cents. However, there is significant damage to 15/13 and 13/10.
	* For primes 5 and 13, [[53edo]] is better, as it finds [[interseptimal interval]]s distinctly from adjacent [[septimal]] intervals so that [[~]][[15/13]] is half of a practically-just [[4/3]] (tempering out [[676/675\|S13/S15]]) and is (resultantly) found as a comma above [[~]][[8/7]] or a comma below [[~]][[7/6]], which reflects to (3/2)/(15/13) = [[~]][[13/10]] being made the midpoint of [[~]][[21/16]] and [[~]][[9/7]] respectively. It also makes [[~]][[16/13]] a comma below [[~]][[5/4]] (by tempering out ((5/4)/(16/13))/(81/80) = 325/324). This corresponds to a number of temperaments; the most relevant of which for [[#Schismic]] is the very accurate extension to prime 13 called [[Schismatic family#Tridecaschismic (2.3.5.13)\|tridecaschismic]], corresponding to reaching 13/4 through (9/8)<sup>10</sup> (tempering out the [[tridecapyth comma]]) and also corresponding to tempering out [[325/324]] = S25*S26 = S10/S12 as mentioned. However, there is significant damage to 14/11. (Also, 53edo's fifth is flatter so better tuned for schismic/for the 5-limit, as implicitly aforementioned.)		* For primes 5 and 13, [[53edo]] is better, as it finds [[interseptimal interval]]s distinctly from adjacent [[septimal]] intervals so that [[~]][[15/13]] is half of a practically-just [[4/3]] (tempering out [[676/675\|S13/S15]]) and is (resultantly) found as a comma above [[~]][[8/7]] or a comma below [[~]][[7/6]], which reflects to (3/2)/(15/13) = [[~]][[13/10]] being made the midpoint of [[~]][[21/16]] and [[~]][[9/7]] respectively. It also makes [[~]][[16/13]] a comma below [[~]][[5/4]] (by tempering out ((5/4)/(16/13))/(81/80) = 325/324). This corresponds to a number of temperaments; the most relevant of which for [[#Schismic]] is the very accurate extension to prime 13 called [[Schismatic family#Tridecaschismic (2.3.5.13)\|tridecaschismic]], corresponding to reaching 13/4 through (9/8)<sup>10</sup> (tempering out the [[tridecapyth comma]]) and also corresponding to tempering out [[325/324]] = S25*S26 = S10/S12 as mentioned. However, there is significant damage to 14/11. (Also, 53edo's fifth is flatter so better tuned for schismic/for the 5-limit, as implicitly aforementioned.)

Godtone: /* Garibaldi */ merge into one paragraph as the ratio appears on its own at some scales which feels weird/unstylistic

2026-05-08T00:44:33Z

Garibaldi: merge into one paragraph as the ratio appears on its own at some scales which feels weird/unstylistic

← Older revision		Revision as of 00:44, 8 May 2026
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	[[#Generator tunings\|Generator tunings]]: 24\31, 31\53, 55\94		[[#Generator tunings\|Generator tunings]]: 24\31, 31\53, 55\94

	Garibaldi is arguably the best way to bestow prime 7 upon [[#Schismic]] effectively, at the cost of some accuracy. It uses a slightly sharper fifth that tunes the 5-limit worse, making it no longer a microtemperament. This is done by interpreting (9/8)<sup>3</sup> as [[~]][[10/7]] by tempering out S8/S9 = [[5120/5103]] so that 8/7 and 10/9 are equidistant from 9/8, corresponding to equating S8 = [[64/63]] and S9 = [[81/80]] respectively. This results in a conveniently general tempered comma-sized interval that also represents the [[Pythagorean comma]], which is equal to (9/8)<sup>6</sup> / (2/1).		Garibaldi is arguably the best way to bestow prime 7 upon [[#Schismic]] effectively, at the cost of some accuracy. It uses a slightly sharper fifth that tunes the 5-limit worse, making it no longer a microtemperament. This is done by interpreting (9/8)<sup>3</sup> as [[~]][[10/7]] by tempering out S8/S9 = [[5120/5103]] so that 8/7 and 10/9 are equidistant from 9/8, corresponding to equating S8 = [[64/63]] and S9 = [[81/80]] respectively. This results in a conveniently general tempered comma-sized interval that also represents the [[Pythagorean comma]], which is equal to (9/8)<sup>6</sup> / (2/1). [[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors.

	[[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors.

	Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about prime 11 or prime 13:		Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about prime 11 or prime 13:

Godtone: /* Garibaldi */ it does seem very subjective as far as i can tell as to which is a better 7-limit system

2026-05-07T13:28:11Z

Garibaldi: it does seem very subjective as far as i can tell as to which is a better 7-limit system

← Older revision		Revision as of 13:28, 7 May 2026
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	[[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors.		[[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors.

	~~The choice~~ of 41edo and 53edo ~~for~~ the 7-limit ~~is hard to determine~~, so a better way of choosing is ~~choosing between primes 7,~~ 11 ~~and primes 5,~~ 13.		Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about prime 11 or prime 13:
	* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] (as in [[cassandra]]) or a comma below [[~]][[5/4]] (as in [[andromeda]]). However, there is significant damage to 15/13 and 13/10.		* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] (as in [[cassandra]]) or a comma below [[~]][[5/4]] (as in [[andromeda]]). However, there is significant damage to 15/13 and 13/10.
	* For primes 5 and 13, [[53edo]] is better, as it finds [[interseptimal interval]]s distinctly from adjacent [[septimal]] intervals so that [[~]][[15/13]] is half of a practically-just [[4/3]] (tempering out [[676/675\|S13/S15]]) and is (resultantly) found as a comma above [[~]][[8/7]] or a comma below [[~]][[7/6]], which reflects to (3/2)/(15/13) = [[~]][[13/10]] being made the midpoint of [[~]][[21/16]] and [[~]][[9/7]] respectively. It also makes [[~]][[16/13]] a comma below [[~]][[5/4]] (by tempering out ((5/4)/(16/13))/(81/80) = 325/324). This corresponds to a number of temperaments; the most relevant of which for [[#Schismic]] is the very accurate extension to prime 13 called [[Schismatic family#Tridecaschismic (2.3.5.13)\|tridecaschismic]], corresponding to reaching 13/4 through (9/8)<sup>10</sup> (tempering out the [[tridecapyth comma]]) and also corresponding to tempering out [[325/324]] = S25*S26 = S10/S12 as mentioned. However, there is significant damage to 14/11.		* For primes 5 and 13, [[53edo]] is better, as it finds [[interseptimal interval]]s distinctly from adjacent [[septimal]] intervals so that [[~]][[15/13]] is half of a practically-just [[4/3]] (tempering out [[676/675\|S13/S15]]) and is (resultantly) found as a comma above [[~]][[8/7]] or a comma below [[~]][[7/6]], which reflects to (3/2)/(15/13) = [[~]][[13/10]] being made the midpoint of [[~]][[21/16]] and [[~]][[9/7]] respectively. It also makes [[~]][[16/13]] a comma below [[~]][[5/4]] (by tempering out ((5/4)/(16/13))/(81/80) = 325/324). This corresponds to a number of temperaments; the most relevant of which for [[#Schismic]] is the very accurate extension to prime 13 called [[Schismatic family#Tridecaschismic (2.3.5.13)\|tridecaschismic]], corresponding to reaching 13/4 through (9/8)<sup>10</sup> (tempering out the [[tridecapyth comma]]) and also corresponding to tempering out [[325/324]] = S25*S26 = S10/S12 as mentioned. However, there is significant damage to 14/11. (Also, 53edo's fifth is flatter so better tuned for schismic/for the 5-limit, as implicitly aforementioned.)

	Both support [[cassandra]], a 13-limit extension which finds [[~]][[16/13]] as a comma below [[~]][[5/4]] and equates (3/2)/(16/13) = [[39/32]] with [[11/9]]. (This means that in 41edo, we have a single neutral third at the cost of damage to prime 13, while in 53edo we have two neutral thirds at the cost of damage to prime 11, hence 41 + 53 = [[94edo]] is a lot more characteristic of cassandra's tuning.)		Both support [[cassandra]], a 13-limit extension which finds [[~]][[16/13]] as a comma below [[~]][[5/4]] and equates (3/2)/(16/13) = [[39/32]] with [[11/9]]. (This means that in 41edo, we have a single neutral third at the cost of damage to prime 13, while in 53edo we have two neutral thirds at the cost of damage to prime 11, hence 41 + 53 = [[94edo]] is a lot more characteristic of cassandra's tuning.)

Godtone: /* Garibaldi */ address a bunch of things discussed on my user page where it was brought up. i'll do a diff again to see if i missed anything, also cuz theres some repeat info i want to address

2026-05-07T12:53:44Z

Garibaldi: address a bunch of things discussed on my user page where it was brought up. i'll do a diff again to see if i missed anything, also cuz theres some repeat info i want to address

← Older revision		Revision as of 12:53, 7 May 2026
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	[[#Generator tunings\|Generator tunings]]: 24\31, 31\53, 55\94		[[#Generator tunings\|Generator tunings]]: 24\31, 31\53, 55\94

	Garibaldi is arguably the ~~simplest~~ way to bestow prime 7 upon [[#Schismic]] effectively, at the cost of some accuracy. It uses a slightly sharper fifth that tunes the 5-limit worse, making it no longer a microtemperament. This is done by interpreting (9/8)<sup>3</sup> as [[~]][[10/7]] by tempering out [[5120/5103]] so that 8/7 and 10/9 are equidistant from 9/8~~. The distance is a tempered [[Pythagorean comma]] that also represents~~ [[64/63]] and [[81/80]]~~, which necessarily~~ results in [[~~225/224~~]] ~~tempered out as 5120~~/~~5103 * 32805~~/~~32768 = 225~~/~~224~~.		Garibaldi is arguably the best way to bestow prime 7 upon [[#Schismic]] effectively, at the cost of some accuracy. It uses a slightly sharper fifth that tunes the 5-limit worse, making it no longer a microtemperament. This is done by interpreting (9/8)<sup>3</sup> as [[~]][[10/7]] by tempering out S8/S9 = [[5120/5103]] so that 8/7 and 10/9 are equidistant from 9/8, corresponding to equating S8 = [[64/63]] and S9 = [[81/80]] respectively. This results in a conveniently general tempered comma-sized interval that also represents the [[Pythagorean comma]], which is equal to (9/8)<sup>6</sup> / (2/1).

	[[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a barely inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors.		[[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors.

	The choice of 41edo and 53edo for the 7-limit is hard to determine, so a better way of choosing is choosing between primes 7,11 and primes 5,13. ~~Both support~~ [[~~cassandra~~]], ~~a 13-limit extension which tempers out~~ [[~~352/351~~]] ~~and~~ [[~~325~~/~~324~~]]~~, so that~~ [[~]][[16/13]] is a comma below [[~]][[5/4]] ~~and~~ [[~~39/32~~]] is ~~equated with [[11~~/~~9]]~~.		The choice of 41edo and 53edo for the 7-limit is hard to determine, so a better way of choosing is choosing between primes 7, 11 and primes 5, 13.
	* For ~~prime 11~~, [[~~41edo~~]] is better, as it finds [[~]][[11/9]] as a ~~hemififth~~ and as a comma above [[~]][[6/5]] ~~(as in~~ [[~~cassandra~~]]~~) or~~ a comma below [[~]][[5/4]] (~~as in~~ [[~~andromeda~~]]~~). Primes 5 and 13 are worse. This~~ is ~~reflected in~~ the [[~~tetracot comma~~]] ~~also being tempered out~~, ~~which necessarily tunes 16/~~13 ~~and 5~~/4 ~~flatter, and~~ [[~]][[16/13]] ~~also being a hemififth~~.		* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] (as in [[cassandra]]) or a comma below [[~]][[5/4]] (as in [[andromeda]]). However, there is significant damage to 15/13 and 13/10.
			* For primes 5 and 13, [[53edo]] is better, as it finds [[interseptimal interval]]s distinctly from adjacent [[septimal]] intervals so that [[~]][[15/13]] is half of a practically-just [[4/3]] (tempering out [[676/675\|S13/S15]]) and is (resultantly) found as a comma above [[~]][[8/7]] or a comma below [[~]][[7/6]], which reflects to (3/2)/(15/13) = [[~]][[13/10]] being made the midpoint of [[~]][[21/16]] and [[~]][[9/7]] respectively. It also makes [[~]][[16/13]] a comma below [[~]][[5/4]] (by tempering out ((5/4)/(16/13))/(81/80) = 325/324). This corresponds to a number of temperaments; the most relevant of which for [[#Schismic]] is the very accurate extension to prime 13 called [[Schismatic family#Tridecaschismic (2.3.5.13)\|tridecaschismic]], corresponding to reaching 13/4 through (9/8)<sup>10</sup> (tempering out the [[tridecapyth comma]]) and also corresponding to tempering out [[325/324]] = S25*S26 = S10/S12 as mentioned. However, there is significant damage to 14/11.

	* For primes 5 and 13, [[~~53edo~~]] ~~is better~~, ~~as it has~~ a ~~close to just fifth that benefits [[Schismatic family#Tridecaschismic (2.3.5.13)\|tridecaschismic]], finding very well tuned ~5/4 and ~16/~~13~~. It also has half~~-~~fourths corresponding to~~ [[~~15/13~~]]~~, tempering out~~ [[~~676~~/~~675~~]]~~, telling apart~~ [[~~Interseptimal interval\|Interseptimal intervals~~]] ~~from adjacent~~ [[~~septimal~~]] ~~intervals; 15~~/~~13 is here the midpoint between 8~~/~~7 and 7~~/~~6, making it part of~~ [[~~The Archipelago~~]]~~. Primes 7 and~~ 11 ~~are worse~~. This ~~is reflected~~ in ~~14/11 being equated with 9/7~~, ~~and 11/9 being equated with~~ a ~~flat 39/32~~, ~~inflating~~ the [[~~rastma~~]] to a ~~whole step~~.		Both support [[cassandra]], a 13-limit extension which finds [[~]][[16/13]] as a comma below [[~]][[5/4]] and equates (3/2)/(16/13) = [[39/32]] with [[11/9]]. (This means that in 41edo, we have a single neutral third at the cost of damage to prime 13, while in 53edo we have two neutral thirds at the cost of damage to prime 11, hence 41 + 53 = [[94edo]] is a lot more characteristic of cassandra's tuning.)

	=== 11-limit focus ===		=== 11-limit focus ===

Eufalesio: Partial reversal on garibaldi: let's not beat around the bush it IS the simplest good 7-limit schismic extension; gracecordial is more complex and schism is not good; reduced bloat and acknowledged interseptimals of 53edo, relationship between primes 5,13 and 7,11 I consider to be very important in cassandra

2026-05-07T09:30:24Z

Partial reversal on garibaldi: let's not beat around the bush it IS the simplest good 7-limit schismic extension; gracecordial is more complex and schism is not good; reduced bloat and acknowledged interseptimals of 53edo, relationship between primes 5,13 and 7,11 I consider to be very important in cassandra

← Older revision		Revision as of 09:30, 7 May 2026
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	[[#Generator tunings\|Generator tunings]]: (24\41,) 31\53, 69\118, 100\171, 131\224		[[#Generator tunings\|Generator tunings]]: (24\41,) 31\53, 69\118, 100\171, 131\224

	Schismic is ~~a very~~ accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])<sup>4</sup>, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, as it is practically a relabeling of the [[3-limit]]. 41edo arguably qualifies as the coarsest equal temperament to support schismic well enough, but it is way too tempered for a 5-limit tuning, as it also it is [[magic]].		Schismic is an extremely accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])<sup>4</sup>, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, as it is practically a relabeling of the [[3-limit]]. 41edo arguably qualifies as the coarsest equal temperament to support schismic well enough, but it is way too tempered for a 5-limit tuning, as it also it is [[magic]].

	In schismic, (9/8)<sup>6</sup> overshoots the octave by [[~]][[81/80]] so that the syntonic comma and the [[Pythagorean comma]] are equated.		In schismic, (9/8)<sup>6</sup> overshoots the octave by [[~]][[81/80]] so that the syntonic comma and the [[Pythagorean comma]] are equated.
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	[[#Generator tunings\|Generator tunings]]: 24\31, 31\53, 55\94		[[#Generator tunings\|Generator tunings]]: 24\31, 31\53, 55\94

	Garibaldi is ~~a very natural and very efficient (for its accuracy)~~ way ~~of bestowing~~ prime 7 upon [[#Schismic]], at the cost of accuracy ~~as needing~~ a slightly sharper fifth tunes the 5-limit worse ~~so that~~ it is no longer a microtemperament. This is done by interpreting ([[9/8]])<sup>3</sup> as [[~]][[10/7]] by tempering out [[5120/5103~~\|S8/S9~~]] so that 8/7 and 10/9 are equidistant from 9/8~~, with the step being~~ a ~~conveniently general~~ tempered comma~~-sized interval~~ that ~~simultaneously not only~~ represents ~~not only~~ [[64/63]] ~~= S8~~ and [[81/80]] ~~= S9 but also the~~ [[~~Pythagorean comma~~]] (as ~~per schismic), equal to (9~~/~~8)<sup>6<~~/~~sup>~~ / (2/1). [[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors.		Garibaldi is arguably the simplest way to bestow prime 7 upon [[#Schismic]] effectively, at the cost of some accuracy. It uses a slightly sharper fifth that tunes the 5-limit worse, making it no longer a microtemperament. This is done by interpreting (9/8)<sup>3</sup> as [[~]][[10/7]] by tempering out [[5120/5103]] so that 8/7 and 10/9 are equidistant from 9/8. The distance is a tempered [[Pythagorean comma]] that also represents [[64/63]] and [[81/80]], which necessarily results in [[225/224]] tempered out as 5120/5103 * 32805/32768 = 225/224.

	~~Which~~ of 41edo and 53edo ~~do better in~~ the 7-limit ~~depends on how you measure them and who you ask; therefore~~, a better way of choosing is ~~based on whether you care more about prime~~ 11 ~~or prime~~ 13:		[[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a barely inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors.
	* For prime 11, [[~~41edo~~]] ~~is better~~, ~~as it finds~~ [[~]][[11/9]] ~~as half of the fifth and as a comma above~~ [[~]][[6/5]] or a comma below [[~]][[5/4]]~~. This corresponds to being~~ [[~~cassandra~~]] + [[~~andromeda~~]] ~~'''(respectively)'''~~.
	* For prime 13, [[~~53edo~~]] is better, as it finds ~~[[interseptimal interval]]s distinctly from adjacent [[septimal]] intervals so that~~ [[~]][[15/13]] ~~is half of~~ a ~~practically-just~~ [[~~4/3~~]] ~~(tempering out~~ [[~~676~~/~~675\|S13/S15~~]]~~) and is~~ (~~resultantly) found~~ as ~~a comma above~~ [[~~~]][[8/7~~]] or a comma below [[~]][[7/6]] (~~which reflects to~~ [[~]][[13~~/10]] being made~~ the ~~midpoint of~~ [[~]]~~[[9~~/~~7]]~~ and ~~[[~]][[21~~/~~16]]). It also makes~~ [[~]][[16/13]] a ~~comma below [[~]][[~~5~~/4]] (by tempering out 325/324). This corresponds to a number of temperaments; the most relevant of which for~~ [[~~#Schismic~~]] is ~~the very accurate extension~~ to ~~prime 13 called~~ [[Schismatic family#Tridecaschismic (2.3.5.13)\|tridecaschismic]], corresponding to ~~reaching~~ 13/~~4 through (9~~/8~~)<sup>10<~~/~~sup> (tempering out the~~ [[~~tridecapyth comma~~]]) and ~~also corresponding to tempering out~~ [[~~325/324~~]] ~~= S25*S26 = S10/S12~~.		The choice of 41edo and 53edo for the 7-limit is hard to determine, so a better way of choosing is choosing between primes 7,11 and primes 5,13. Both support [[cassandra]], a 13-limit extension which tempers out [[352/351]] and [[325/324]], so that [[~]][[16/13]] is a comma below [[~]][[5/4]] and [[39/32]] is equated with [[11/9]].
			* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as a hemififth and as a comma above [[~]][[6/5]] (as in [[cassandra]]) or a comma below [[~]][[5/4]] (as in [[andromeda]]). Primes 5 and 13 are worse. This is reflected in the [[tetracot comma]] also being tempered out, which necessarily tunes 16/13 and 5/4 flatter, and [[~]][[16/13]] also being a hemififth.

			* For primes 5 and 13, [[53edo]] is better, as it has a close to just fifth that benefits [[Schismatic family#Tridecaschismic (2.3.5.13)\|tridecaschismic]], finding very well tuned ~5/4 and ~16/13. It also has half-fourths corresponding to [[15/13]], tempering out [[676/675]], telling apart [[Interseptimal interval\|Interseptimal intervals]] from adjacent [[septimal]] intervals; 15/13 is here the midpoint between 8/7 and 7/6, making it part of [[The Archipelago]]. Primes 7 and 11 are worse. This is reflected in 14/11 being equated with 9/7, and 11/9 being equated with a flat 39/32, inflating the [[rastma]] to a whole step.

	=== 11-limit focus ===		=== 11-limit focus ===

Eufalesio: Clarified some things

2026-05-06T22:02:45Z

Clarified some things

← Older revision		Revision as of 22:02, 6 May 2026
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	Gariwizmic has a 1/2-octave period representing [[99/70]], two of them being ~2/1 and tempering out [[9801/9800]]. It is generated by a [[#Gary]] fifth. As its name implies, it also tempers out the [[wizma]].		Gariwizmic has a 1/2-octave period representing [[99/70]], two of them being ~2/1 and tempering out [[9801/9800]]. It is generated by a [[#Gary]] fifth. As its name implies, it also tempers out the [[wizma]].

	Tempering out the kalisma allows the pythagorean comma to be split into two [[2835/2816\|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243\|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (-27 fifths - 1 period).		Tempering out the kalisma allows the pythagorean comma to be split into two [[2835/2816\|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243\|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (-27 fifths - 1 period). This structure is practically identical to that of [[cassaschismic]], only that the minicomma is not an independent generator and is instead found in the deep in the diploid chain of fifths.

	It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It is very complex mapping-wise despite its great accuracy, but it can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 (and 19) which is still reachable within the rank-2 structure (It isn't a true generator; were it independent, the temperament would be [[cassaschismic]]).		It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It is very complex mapping-wise despite its great accuracy, but it can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 (and 19) which is still reachable within the rank-2 structure. It isn't a true generator; were it independent, the temperament would be [[cassaschismic]].

	It naturally extends into the 2.3.5.7.11.13.19 subgroup by adding [[1216/1215]] to the comma list, finding the major third to be [[19/15]] and 19/16 to be the minor third + minicomma, thus also working as [[361/360]]. Interestingly, gariwizmic tempers out the smallest superparticular of the 19- and 23-limit: the [[tredekisma]].		It naturally extends into the 2.3.5.7.11.13.19 subgroup by adding [[1216/1215]] to the comma list, finding the major third to be [[19/15]] and 19/16 to be the minor third + minicomma, thus also working as [[361/360]]. Interestingly, gariwizmic tempers out the smallest superparticular of the 19- and 23-limit: the [[tredekisma]].