Monarch - Revision history

Ground at 05:45, 20 August 2026

2026-08-20T05:45:21Z

← Older revision		Revision as of 05:45, 20 August 2026
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	Monarch is most associated with double-diatonic tunings (tunings with two diatonic generators) within a reasonable range to be called Meantone and Archy. There are also edos of a similar size with three diatonic generators: one deeptone (near 7edo), one pythagorean, and one hemiseven (near 5edo). These tunings are likely to temper out 81/80 d^3 h, equating 5 to three 3d/2 deeptone fifths and a 3h/2 hemiseven fifth. This comma is an unnamed SRTT variant of the syntonic comma. The smallest edo this works in is 47, but it's more notable in 82, 87, and so on. 49/48 m a^-2 is tempered out here as well, except it could also be written as 49/48 p h^-2 with p being the alteration of the pythagorean fifth. p may be so small that it may be ignored.		Monarch is most associated with double-diatonic tunings (tunings with two diatonic generators) within a reasonable range to be called Meantone and Archy. There are also edos of a similar size with three diatonic generators: one deeptone (near 7edo), one pythagorean, and one hemiseven (near 5edo). These tunings are likely to temper out 81/80 d^3 h, equating 5 to three 3d/2 deeptone fifths and a 3h/2 hemiseven fifth. This comma is an unnamed SRTT variant of the syntonic comma. The smallest edo this works in is 47, but it's more notable in 82, 87, and so on. 49/48 m a^-2 is tempered out here as well, except it could also be written as 49/48 p h^-2 with p being the alteration of the pythagorean fifth. p may be so small that it may be ignored.

	Triple-diatonic edos may be interpreted as triple-straddle-3, having three viable tunings of 3/1. This is less common than straddle-3 is the usual sense because it has one central 3 and two fringe ones, where the central 3 is too accurate to really straddle 3 at all. This can extend to triple-straddle-2, which works better than standard straddle-2 because all edos have a central 2 (a pure 2/1 in this case). If this were used, it would be more intuitive to triple-straddle both 2 and 3, both to pair versions of primes with error in the same direction and allow 3/1 or 3/2 to be the "anchor consonance" allowing 2/1 to be further off. Then it becomes possible to change the octave relative to the fifth to get scales otherwise not possible. With a just 3/2 generator, a period of a sharp octave resembles Meantone and a flat octave resembles Archy.		See also: [[Straddle primes#Straddle-2]]

	~~SRTT math, using n for the sharp octave that makes Meantone (letter after m) and b for the flat octave that makes Archy (letter after a):~~

	~~Just~~-~~3/2 tunings:~~

	~~(2n)^-2 (3/2))^4 = 5/4, n = sqrt(81/80), 2n = 1210.753¢<br>~~
	~~(2b)^-2 (3/2))^4 = 9/7, b = sqrt(63/64), 2b = 1186.368¢~~

	~~Just-4/3 tunings are also viable be replacing 3/2 in these equations with 2x / (4/3) = 3x/2, although doing this flips Meantone and Archy.~~

	~~These are two viable octaves that differ by 24¢, within the typical size range of an aberrisma. It may be incredibly jarring to use them together, but also interesting.~~

	This is distinct from something like (2n)^-2 (3/(2n))^4 = 5/4 or 81/(2n)^6 = 5/4, 2n = 1203.584¢. The result is an undertempered octave because the unaltered generator is assumed to be 3/1 (equivalent to 3/(2n)) rather than 3/2~~. This is still great for a more even distribution of error though.~~

	{{Navbox regtemp}}		{{Navbox regtemp}}
	{{Cat\|temperaments}}		{{Cat\|temperaments}}

Ground at 12:43, 18 August 2026

2026-08-18T12:43:57Z

← Older revision		Revision as of 12:43, 18 August 2026
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	Monarch is most associated with double-diatonic tunings (tunings with two diatonic generators) within a reasonable range to be called Meantone and Archy. There are also edos of a similar size with three diatonic generators: one deeptone (near 7edo), one pythagorean, and one hemiseven (near 5edo). These tunings are likely to temper out 81/80 d^3 h, equating 5 to three 3d/2 deeptone fifths and a 3h/2 hemiseven fifth. This comma is an unnamed SRTT variant of the syntonic comma. The smallest edo this works in is 47, but it's more notable in 82, 87, and so on. 49/48 m a^-2 is tempered out here as well, except it could also be written as 49/48 p h^-2 with p being the alteration of the pythagorean fifth. p may be so small that it may be ignored.		Monarch is most associated with double-diatonic tunings (tunings with two diatonic generators) within a reasonable range to be called Meantone and Archy. There are also edos of a similar size with three diatonic generators: one deeptone (near 7edo), one pythagorean, and one hemiseven (near 5edo). These tunings are likely to temper out 81/80 d^3 h, equating 5 to three 3d/2 deeptone fifths and a 3h/2 hemiseven fifth. This comma is an unnamed SRTT variant of the syntonic comma. The smallest edo this works in is 47, but it's more notable in 82, 87, and so on. 49/48 m a^-2 is tempered out here as well, except it could also be written as 49/48 p h^-2 with p being the alteration of the pythagorean fifth. p may be so small that it may be ignored.

	Triple-diatonic edos may be interpreted as triple-straddle-3, having three viable tunings of 3/1. This is less common than straddle-3 is the usual sense because it has one central 3 and two fringe ones, where the central 3 is too accurate to really straddle 3 at all. This can extend to triple-straddle-2, which works better than standard straddle-2 because all edos have a central 2. If this were used, it would be more intuitive to triple-straddle both 2 and 3. Then it becomes possible to change the octave relative to the fifth to get scales otherwise not possible. With a just 3/2 generator, a period of a sharp octave resembles Meantone and a flat octave resembles Archy.		Triple-diatonic edos may be interpreted as triple-straddle-3, having three viable tunings of 3/1. This is less common than straddle-3 is the usual sense because it has one central 3 and two fringe ones, where the central 3 is too accurate to really straddle 3 at all. This can extend to triple-straddle-2, which works better than standard straddle-2 because all edos have a central 2 (a pure 2/1 in this case). If this were used, it would be more intuitive to triple-straddle both 2 and 3, both to pair versions of primes with error in the same direction and allow 3/1 or 3/2 to be the "anchor consonance" allowing 2/1 to be further off. Then it becomes possible to change the octave relative to the fifth to get scales otherwise not possible. With a just 3/2 generator, a period of a sharp octave resembles Meantone and a flat octave resembles Archy.

	SRTT math, using n for the sharp octave that makes Meantone (letter after m) and b for the flat octave that makes Archy (letter after a):		SRTT math, using n for the sharp octave that makes Meantone (letter after m) and b for the flat octave that makes Archy (letter after a):

	~~If the intention is to get a just 4:5:6, do not do something like (2n)^~~-~~2 (3/(2n))^4 = 5/4 or 81/(2n)^6 = 5/4, 2n = 1203.584¢. The result is an undertempered octave because the unaltered generator is assumed to be 3/1 (equivalent to 3/(2n)) rather than~~ 3/2~~. This is still great for a more even distribution of error though.~~		Just-3/2 tunings:

	(2n)^-2 (3/2))^4 = 5/4, n = sqrt(81/80), 2n = 1210.753¢<br>		(2n)^-2 (3/2))^4 = 5/4, n = sqrt(81/80), 2n = 1210.753¢<br>
	(2b)^-2 (3/2))^4 = 9/7, b = sqrt(63/64), 2b = 1186.368¢		(2b)^-2 (3/2))^4 = 9/7, b = sqrt(63/64), 2b = 1186.368¢

			Just-4/3 tunings are also viable be replacing 3/2 in these equations with 2x / (4/3) = 3x/2, although doing this flips Meantone and Archy.

	These are two viable octaves that differ by 24¢, within the typical size range of an aberrisma. It may be incredibly jarring to use them together, but also interesting.		These are two viable octaves that differ by 24¢, within the typical size range of an aberrisma. It may be incredibly jarring to use them together, but also interesting.

			This is distinct from something like (2n)^-2 (3/(2n))^4 = 5/4 or 81/(2n)^6 = 5/4, 2n = 1203.584¢. The result is an undertempered octave because the unaltered generator is assumed to be 3/1 (equivalent to 3/(2n)) rather than 3/2. This is still great for a more even distribution of error though.

	{{Navbox regtemp}}		{{Navbox regtemp}}
	{{Cat\|temperaments}}		{{Cat\|temperaments}}

Ground at 03:53, 18 August 2026

2026-08-18T03:53:29Z

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	== Triple-straddle variants ==		== Triple-straddle variants ==

	Monarch is most associated with double-diatonic tunings (tunings with two diatonic generators) within a ~~specific~~ range. There are also edos of a similar size with three diatonic generators: one deeptone (near 7edo), one pythagorean, and one hemiseven (near 5edo). These tunings are likely to temper out 81/80 d^3 h, equating 5 to three 3d/2 deeptone fifths and a 3h/2 hemiseven fifth. The smallest edo this works in is 47, but it's more notable in 82, 87, and so on. 49/48 m a^-2 is tempered out here, except it could also be written as 49/48 p h^-2 with p being the alteration of the pythagorean fifth. p may be so small that it ~~can~~ be ignored.		Monarch is most associated with double-diatonic tunings (tunings with two diatonic generators) within a reasonable range to be called Meantone and Archy. There are also edos of a similar size with three diatonic generators: one deeptone (near 7edo), one pythagorean, and one hemiseven (near 5edo). These tunings are likely to temper out 81/80 d^3 h, equating 5 to three 3d/2 deeptone fifths and a 3h/2 hemiseven fifth. This comma is an unnamed SRTT variant of the syntonic comma. The smallest edo this works in is 47, but it's more notable in 82, 87, and so on. 49/48 m a^-2 is tempered out here as well, except it could also be written as 49/48 p h^-2 with p being the alteration of the pythagorean fifth. p may be so small that it may be ignored.

	Triple-diatonic edos may be interpreted as triple-straddle-3. ~~The same may be done with~~ 2, which works ~~well~~ because both 2 and 3 ~~would have one central mapping and two fringe ones~~. Then ~~instead of changing the fifth,~~ change the octave relative to the fifth. With a just 3/2 generator, a period of a sharp octave resembles Meantone and a flat octave resembles Archy.		Triple-diatonic edos may be interpreted as triple-straddle-3, having three viable tunings of 3/1. This is less common than straddle-3 is the usual sense because it has one central 3 and two fringe ones, where the central 3 is too accurate to really straddle 3 at all. This can extend to triple-straddle-2, which works better than standard straddle-2 because all edos have a central 2. If this were used, it would be more intuitive to triple-straddle both 2 and 3. Then it becomes possible to change the octave relative to the fifth to get scales otherwise not possible. With a just 3/2 generator, a period of a sharp octave resembles Meantone and a flat octave resembles Archy.

	~~Using~~ n for the sharp octave that makes Meantone (letter after m) and b for the flat octave that makes Archy (letter after a):		SRTT math, using n for the sharp octave that makes Meantone (letter after m) and b for the flat octave that makes Archy (letter after a):

	If the intention is to get a just 4:5:6, do not do something like (2n)^-2 (3/(2n))^4 = 5/4 or 81/(2n)^6 = 5/4, 2n = 1203.584¢. The result is undertempered because the unaltered generator is assumed to be 3/1 (equivalent to 3/(2n)) rather than 3/2. This is still great for a more even distribution of error though.		If the intention is to get a just 4:5:6, do not do something like (2n)^-2 (3/(2n))^4 = 5/4 or 81/(2n)^6 = 5/4, 2n = 1203.584¢. The result is an undertempered octave because the unaltered generator is assumed to be 3/1 (equivalent to 3/(2n)) rather than 3/2. This is still great for a more even distribution of error though.

	(2n)^-2 (3/2))^4 = 5/4, n = sqrt(81/80), 2n = 1210.753¢		(2n)^-2 (3/2))^4 = 5/4, n = sqrt(81/80), 2n = 1210.753¢<br>
	(2b)^-2 (3/2))^4 = 9/7, b = sqrt(63/64), 2b = 1186.368¢		(2b)^-2 (3/2))^4 = 9/7, b = sqrt(63/64), 2b = 1186.368¢

Ground at 03:45, 18 August 2026

2026-08-18T03:45:21Z

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 edo's Archy diatonic has a rather accurate 7/6 and a virtually just 17/11, making it stand out as a temperament I've been considering calling '''Lakefront'''. Instead of triads being 7/6*13/10 as in Oceanfront, they/re 7/6*22/17. So a lake is an ocean but smaller. I think it sounds better than any other diatonic scale between classic Archy and Oceanfront.
 }}
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Vector at 08:24, 14 August 2026

2026-08-14T08:24:09Z

← Older revision		Revision as of 08:24, 14 August 2026
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	'''Monarch''' is a rank-3 [[Straddle primes\|straddle]] temperament tempering out 81/80[-3] and 64/63[+3] in the 2.-3.+3.5.7 subgroup, or the Flat-3 Syntonic comma 81/80 m^4 and Sharp-3 Septimal comma 64/63 a^-2 in the 2.3m.3a.5.7 [[Straddle primes#SRTT\|SRTT]] subgroup. The name comes from the combination of Meantone and Archy, the two linearly independent temperaments that comprise it. Extensions and specific optimizations or erac tunings make use of combining the two tunings of 3.		'''Monarch''' is a rank-3 [[Straddle primes\|straddle]] temperament tempering out 81/80[-3] and 64/63[+3] in the 2.-3.+3.5.7 subgroup, or the Flat-3 Syntonic comma 81/80 m^4 and Sharp-3 Septimal comma 64/63 a^-2 in the 2.3m.3a.5.7 [[Straddle primes#SRTT\|SRTT]] subgroup. The name comes from the combination of Meantone and Archy, the two linearly independent temperaments that comprise it. Extensions and specific optimizations or erac tunings make use of combining the two tunings of 3.

	In other words, a meantone fifth and a superpyth fifth stack to make an in-tune 9/8, so a reasonable tuning is ~~696c~~ and 714c for the two fifths. The archy fifth is tuned sharp because a stack of two does not need to approximate 9/8.		In other words, a meantone fifth and a superpyth fifth stack to make an in-tune 9/8, so a reasonable tuning is 695c and 714c for the two fifths. The archy fifth is tuned sharp because a stack of two does not need to approximate 9/8; the meantone fifth is however tuned to about 19edo because a septimal interpretation of meantone is not necessary and to improve the tuning of 9/8 created by stacking both of them.

	{{UserTag\|g_\|Ground\|7766ff\|The name also implies that this is the king of all temperaments, and it very much is for me, the pinnacle of what I want out of an aberrismic diatonic straddle-3 tuning. It could also be called Archtone, making the same use of the "arch" affix.}}		{{UserTag\|g_\|Ground\|7766ff\|The name also implies that this is the king of all temperaments, and it very much is for me, the pinnacle of what I want out of an aberrismic diatonic straddle-3 tuning. It could also be called Archtone, making the same use of the "arch" affix.}}

Vector at 08:22, 14 August 2026

2026-08-14T08:22:25Z

← Older revision		Revision as of 08:22, 14 August 2026
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	'''Monarch''' is a rank-3 [[Straddle primes\|straddle]] temperament tempering out 81/80[-3] and 64/63[+3] in the 2.-3.+3.5.7 subgroup, or the Flat-3 Syntonic comma 81/80 m^4 and Sharp-3 Septimal comma 64/63 a^-2 in the 2.3m.3a.5.7 [[Straddle primes#SRTT\|SRTT]] subgroup. The name comes from the combination of Meantone and Archy, the two linearly independent temperaments that comprise it. Extensions and specific optimizations or erac tunings make use of combining the two tunings of 3.		'''Monarch''' is a rank-3 [[Straddle primes\|straddle]] temperament tempering out 81/80[-3] and 64/63[+3] in the 2.-3.+3.5.7 subgroup, or the Flat-3 Syntonic comma 81/80 m^4 and Sharp-3 Septimal comma 64/63 a^-2 in the 2.3m.3a.5.7 [[Straddle primes#SRTT\|SRTT]] subgroup. The name comes from the combination of Meantone and Archy, the two linearly independent temperaments that comprise it. Extensions and specific optimizations or erac tunings make use of combining the two tunings of 3.

			In other words, a meantone fifth and a superpyth fifth stack to make an in-tune 9/8, so a reasonable tuning is 696c and 714c for the two fifths. The archy fifth is tuned sharp because a stack of two does not need to approximate 9/8.

	{{UserTag\|g_\|Ground\|7766ff\|The name also implies that this is the king of all temperaments, and it very much is for me, the pinnacle of what I want out of an aberrismic diatonic straddle-3 tuning. It could also be called Archtone, making the same use of the "arch" affix.}}		{{UserTag\|g_\|Ground\|7766ff\|The name also implies that this is the king of all temperaments, and it very much is for me, the pinnacle of what I want out of an aberrismic diatonic straddle-3 tuning. It could also be called Archtone, making the same use of the "arch" affix.}}

Ground at 05:04, 13 August 2026

2026-08-13T05:04:29Z

← Older revision		Revision as of 05:04, 13 August 2026
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	Tensmajor 1/4-1/3-comma's biggest drawback is its more extreme Archy, missing a diatonic 9/7 and having a less usable diatonic scale. However, it is desirable for having a better sharp 7 and flat 11 superfourth, a slightly sharper Meantone fifth for Intergan, and a larger size difference between its two fifths which allows it to work in smaller edos. 86edo (2x43) is double a somewhat well-known RTT edo with an adequate Intergan generator, and which is only one step short of having 3 full octaves. 86edo also has near-optimal Mohajira. 91edo is slightly closer to optimal and better for Intergan, but it's a large "grime" number (not divisible by 2 or 3). 96edo (2x48, 3x32, 4x24) is further from optimal, but useful for its divisibility and near-optimal Porcupine. All three of these edos are Subcloud, a straddle-3-7 temperament with a generator of around 237¢. 74edo has the same optimization class, although 7/6 is very flat, so stacking a sharp 9/7 results in a Meantone fifth. It is a classic example of Supercloud, a straddle-3-7 temperament with a generator of around 243¢.		Tensmajor 1/4-1/3-comma's biggest drawback is its more extreme Archy, missing a diatonic 9/7 and having a less usable diatonic scale. However, it is desirable for having a better sharp 7 and flat 11 superfourth, a slightly sharper Meantone fifth for Intergan, and a larger size difference between its two fifths which allows it to work in smaller edos. 86edo (2x43) is double a somewhat well-known RTT edo with an adequate Intergan generator, and which is only one step short of having 3 full octaves. 86edo also has near-optimal Mohajira. 91edo is slightly closer to optimal and better for Intergan, but it's a large "grime" number (not divisible by 2 or 3). 96edo (2x48, 3x32, 4x24) is further from optimal, but useful for its divisibility and near-optimal Porcupine. All three of these edos are Subcloud, a straddle-3-7 temperament with a generator of around 237¢. 74edo has the same optimization class, although 7/6 is very flat, so stacking a sharp 9/7 results in a Meantone fifth. It is a classic example of Supercloud, a straddle-3-7 temperament with a generator of around 243¢.

	Tensmajor 1/4-1/2-comma continues the trend with an Archy fifth so sharp that the resulting diatonic scale is virtually unusable, but it's in the range to be a Hemiseven generator, two of them making a Slendric generator and three of them making some sort of semiquartal generator. The smaller edos to have this optimization class such as 62, 67, and 72 are all Interarch.		Tensmajor 1/4-1/2-comma continues the trend with an Archy fifth so sharp that the resulting diatonic scale is virtually unusable, but it's in the range to be a [[Hemiseven]] generator, two of them making a Slendric generator and three of them making some sort of semiquartal generator. The smaller edos to have this optimization class such as 62, 67, and 72 are all Interarch.

	None of these edos support Superpyth with the same sharp 5/4 as Meantone. The 1/4-comma Archy fifth is too flat and the 1/3-comma Archy fifth is too sharp. Edos in between the two such as 98 (2x49) straddle 9/7. 93edo is the only exception, which is possible because its 5 is virtually just whereas all the others use the sharp 5 of the tensmajor tuning.		None of these edos support Superpyth with the same sharp 5/4 as Meantone. The 1/4-comma Archy fifth is too flat and the 1/3-comma Archy fifth is too sharp. Edos in between the two such as 98 (2x49) straddle 9/7. 93edo is the only exception, which is possible because its 5 is virtually just whereas all the others use the sharp 5 of the tensmajor tuning.

Ground at 04:20, 10 August 2026

2026-08-10T04:20:07Z

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	{{UserTag\|g_\|Ground\|7766ff\|The name also implies that this is the king of all temperaments, and it very much is for me, the pinnacle of what I want out of an aberrismic diatonic straddle-3 tuning. It could also be called Archtone, making the same use of the "arch" affix.}}		{{UserTag\|g_\|Ground\|7766ff\|The name also implies that this is the king of all temperaments, and it very much is for me, the pinnacle of what I want out of an aberrismic diatonic straddle-3 tuning. It could also be called Archtone, making the same use of the "arch" affix.}}

	'''Interarch''' is a variant that splits the Meantone fourth 4/3[-3] in half, named after [[Intergan]], a useful [[Semiquartal\|semiquartal]] temperament generated by a mild Meantone semifourth often seen in optimizations of Monarch. Many new equivalences are possible in Interarch, but an Archy fifth plus a Meantone semifourth is almost universally a flat 7/4, which results in a more accurate 7/6 and 9/7 when combined with the Meantone fifth. This tempers out the Interarch Semaphore comma (7/4 / (3a/2 * sqrt(4/(3m))))^2 = 49/48 m a^-2. {{adv\|74edo does something analogous to Interarch, instead equating a meantone plus an Archy semifourth to 9/7. This tempers out ((9m^2/8 * sqrt(4/(3a)) / (9/7))^2 = 49/48 m^4 a^-1.}}		'''Interarch''' is a variant that splits the Meantone fourth 4/3[-3] in half, named after [[Intergan]], a useful [[Semiquartal\|semiquartal]] temperament generated by a mild Meantone semifourth often seen in optimizations of Monarch. Many new equivalences are possible in Interarch, but an Archy fifth plus a Meantone semifourth is almost universally a flat 7/4, which results in a more accurate 7/6 and 9/7 when combined with the Meantone fifth. This tempers out the Interarch Semaphore comma (7/4 / (3a/2 * sqrt(4/(3m))))^2 = 49/48 m a^-2. {{adv\|74edo does something analogous to Interarch, instead equating a meantone plus an Archy semifourth to 9/7. This tempers out <nowiki>((9m^2/8 * sqrt(4/(3a)) / (9/7))^2 = 49/48 m^4 a^-1.</nowiki>}}

	== Optimization ==		== Optimization ==

Ground at 04:17, 10 August 2026

2026-08-10T04:17:49Z

← Older revision		Revision as of 04:17, 10 August 2026
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	{{UserTag\|g_\|Ground\|7766ff\|The name also implies that this is the king of all temperaments, and it very much is for me, the pinnacle of what I want out of an aberrismic diatonic straddle-3 tuning. It could also be called Archtone, making the same use of the "arch" affix.}}		{{UserTag\|g_\|Ground\|7766ff\|The name also implies that this is the king of all temperaments, and it very much is for me, the pinnacle of what I want out of an aberrismic diatonic straddle-3 tuning. It could also be called Archtone, making the same use of the "arch" affix.}}

	'''Interarch''' is a variant that splits the Meantone fourth 4/3[-3] in half, named after [[Intergan]], a useful [[Semiquartal\|semiquartal]] temperament generated by a mild Meantone semifourth often seen in optimizations of Monarch. Many new equivalences are possible in Interarch, but an Archy fifth plus a Meantone semifourth is almost universally a flat 7/4, which results in a more accurate 7/6 and 9/7 when combined with the Meantone fifth. This tempers out the Interarch Semaphore comma (7/4 / (3a/2 * sqrt(4/(3m))))^2 = 49/48 ma^-2.		'''Interarch''' is a variant that splits the Meantone fourth 4/3[-3] in half, named after [[Intergan]], a useful [[Semiquartal\|semiquartal]] temperament generated by a mild Meantone semifourth often seen in optimizations of Monarch. Many new equivalences are possible in Interarch, but an Archy fifth plus a Meantone semifourth is almost universally a flat 7/4, which results in a more accurate 7/6 and 9/7 when combined with the Meantone fifth. This tempers out the Interarch Semaphore comma (7/4 / (3a/2 * sqrt(4/(3m))))^2 = 49/48 m a^-2. {{adv\|74edo does something analogous to Interarch, instead equating a meantone plus an Archy semifourth to 9/7. This tempers out ((9m^2/8 * sqrt(4/(3a)) / (9/7))^2 = 49/48 m^4 a^-1.}}

	== Optimization ==		== Optimization ==

Ground at 18:51, 27 July 2026

2026-07-27T18:51:50Z

← Older revision		Revision as of 18:51, 27 July 2026
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	Tensmajor 1/4-1/4-comma's fifths differ from 3/2 by a more similar amount, making them feel more equal in terms of usability. 105edo (3x35) is very close to optimal, although 110edo (2x55) has the better Intergan generator as well as Mohajira, splitting the Meantone fifth in half. 117edo (3x39) is also somewhat close, worth mentioning for being a small multiple of a diatonic edo with over 3 octaves of range in a 128-note piano roll, and getting Mohajira close to optimal. 93edo (3x31) is possibly the best option due to its small size and being a small multiple of one of the most popular edos, but its Meantone septal diasem with the accurate 9/7 uses the extremely flat 7 found in Supercloud.		Tensmajor 1/4-1/4-comma's fifths differ from 3/2 by a more similar amount, making them feel more equal in terms of usability. 105edo (3x35) is very close to optimal, although 110edo (2x55) has the better Intergan generator as well as Mohajira, splitting the Meantone fifth in half. 117edo (3x39) is also somewhat close, worth mentioning for being a small multiple of a diatonic edo with over 3 octaves of range in a 128-note piano roll, and getting Mohajira close to optimal. 93edo (3x31) is possibly the best option due to its small size and being a small multiple of one of the most popular edos, but its Meantone septal diasem with the accurate 9/7 uses the extremely flat 7 found in Supercloud.

	Tensmajor 1/4-1/3-comma's biggest drawback is its more extreme Archy, missing a diatonic 9/7 and having a less usable diatonic scale. However, it is desirable for having a better sharp 7 and flat 11 superfourth, a slightly sharper Meantone fifth for Intergan, and a larger size difference between its two fifths which allows it to work in smaller edos. 86edo (2x43) is double a somewhat well-known RTT edo with an adequate Intergan generator, and which is only one step short of having 3 full octaves. 86edo also has near-optimal Mohajira. 91edo is slightly closer to optimal and better for Intergan, but it's a large ~~prime~~ number. 96edo (2x48, 3x32, 4x24) is further from optimal, but useful for its divisibility and near-optimal Porcupine. All three of these edos are Subcloud, a straddle-3-7 temperament with a generator of around 237¢. 74edo has the same optimization class, although 7/6 is very flat, so stacking a sharp 9/7 results in a Meantone fifth. It is a classic example of Supercloud, a straddle-3-7 temperament with a generator of around 243¢.		Tensmajor 1/4-1/3-comma's biggest drawback is its more extreme Archy, missing a diatonic 9/7 and having a less usable diatonic scale. However, it is desirable for having a better sharp 7 and flat 11 superfourth, a slightly sharper Meantone fifth for Intergan, and a larger size difference between its two fifths which allows it to work in smaller edos. 86edo (2x43) is double a somewhat well-known RTT edo with an adequate Intergan generator, and which is only one step short of having 3 full octaves. 86edo also has near-optimal Mohajira. 91edo is slightly closer to optimal and better for Intergan, but it's a large "grime" number (not divisible by 2 or 3). 96edo (2x48, 3x32, 4x24) is further from optimal, but useful for its divisibility and near-optimal Porcupine. All three of these edos are Subcloud, a straddle-3-7 temperament with a generator of around 237¢. 74edo has the same optimization class, although 7/6 is very flat, so stacking a sharp 9/7 results in a Meantone fifth. It is a classic example of Supercloud, a straddle-3-7 temperament with a generator of around 243¢.

	Tensmajor 1/4-1/2-comma continues the trend with an Archy fifth so sharp that the resulting diatonic scale is virtually unusable, but it's in the range to be a Hemiseven generator, two of them making a Slendric generator and three of them making some sort of semiquartal generator. The smaller edos to have this optimization class such as 62, 67, and 72 are all Interarch.		Tensmajor 1/4-1/2-comma continues the trend with an Archy fifth so sharp that the resulting diatonic scale is virtually unusable, but it's in the range to be a Hemiseven generator, two of them making a Slendric generator and three of them making some sort of semiquartal generator. The smaller edos to have this optimization class such as 62, 67, and 72 are all Interarch.