Monarch - Revision history

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.

Ground at 00:06, 27 July 2026

2026-07-27T00:06:51Z

← Older revision		Revision as of 00:06, 27 July 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. 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 ma^-2.

	== Optimization ==		== Optimization ==

Ground at 00:00, 27 July 2026

2026-07-27T00:00:42Z

← Older revision		Revision as of 00:00, 27 July 2026
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	{{proposed}}		{{proposed}}

	'''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]] 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.

	{{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 23:47, 26 July 2026

2026-07-26T23:47:54Z

← Older revision		Revision as of 23:47, 26 July 2026
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	{{proposed}}		{{proposed}}

	'''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. 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]] 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.

	{{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. 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.		'''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. 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.

	== Optimization ==		== Optimization ==
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	* In edos such as 86, 7/6 is tuned more accurately with 32/27[+3] due to the sharper 3. 9/7 is obtained by 3/2[-3] / (32/27[+3]), subtracting the Archy minor third from the Meantone fifth. This may be called '''tensmajor 1/4-1/3-comma''' Monarch.		* In edos such as 86, 7/6 is tuned more accurately with 32/27[+3] due to the sharper 3. 9/7 is obtained by 3/2[-3] / (32/27[+3]), subtracting the Archy minor third from the Meantone fifth. This may be called '''tensmajor 1/4-1/3-comma''' Monarch.
	* In edos such as 67, 7/4 is tuned more accurately with 16/9[+3] and 9/7 is 9/4[-3] / (16/9[+3]), giving '''tensmajor 1/4-1/2-comma''' Monarch.		* In edos such as 67, 7/4 is tuned more accurately with 16/9[+3] and 9/7 is 9/4[-3] / (16/9[+3]), giving '''tensmajor 1/4-1/2-comma''' Monarch.

	=== SRTT targets ===		=== SRTT targets ===

	This math will use the 2.3m.3a.5.7 subgroup, using variables to represent alteration ratios as done in [[Straddle primes\|SRTT]]. 3m/2 is the Meantone fifth and 3a/2 is the Archy fifth. Monarch tempers out the Flat-3 Syntonic comma 81/80 m^4 and Sharp-3 Septimal comma 64/63 a^-2, and Interarch additionally tempers out the Monarch Semaphore comma (7/4 / (3a/2 * sqrt(4/(3m))))^2 = 49/48 ma^-2.		All three of these will use the same tensmajor DR math explained above, equating the 1/4-comma Meantone ~5/4 to the ~5/4 used in the +1+1 chord with the Archy fifth: (3m)^4/64 = (3a/2-1)/2+1.
	* All three of these will use the same tensmajor DR math explained above, equating the 1/4-comma Meantone ~5/4 to the ~5/4 used in the +1+1 chord with the Archy fifth: (3m)^4/64 = (3a/2-1)/2+1.
	* In 1/4-1/4-comma: and (3a)^4/64 = 9/7. This system simplifies to {48a+32 = 81m^4, 63a^4 = 64}. The solution gives fifths of 697.602¢ and 708.771¢.		* In 1/4-1/4-comma: and (3a)^4/64 = 9/7. This system simplifies to {48a+32 = 81m^4, 63a^4 = 64}. The solution gives fifths of 697.602¢ and 708.771¢.
	* In 1/4-1/3-comma: and 3m/2 / (32/(3a)^3) = 9/7. This system simplifies to {48a+32 = 81m^4, 63ma^3 = 64}. The solution gives fifths of 698.135¢ and 712.317¢.		* In 1/4-1/3-comma: and 3m/2 / (32/(3a)^3) = 9/7. This system simplifies to {48a+32 = 81m^4, 63ma^3 = 64}. The solution gives fifths of 698.135¢ and 712.317¢.
	* In 1/4-1/2-comma: and (3m/2)^2 / (16/(3a)^2) = 9/7. This system simplifies to {48a+32 = 81m^4, 63m^2a^2 = 64}. The solution gives fifths of 699.062¢ and 718.480¢.		* In 1/4-1/2-comma: and (3m/2)^2 / (16/(3a)^2) = 9/7. This system simplifies to {48a+32 = 81m^4, 63m^2a^2 = 64}. The solution gives fifths of 699.062¢ and 718.480¢.
	* It's also possible to constrain the generators with both tensmajor and tractsubminor. Take 1/4-1/3-comma for example, using the aforementioned (3m)^4/64 = (3a/2-1)/2+1 and 32/(3a)^3 = (3m/2-1)/3+1 with fifths of 698.027¢ and 711.604¢, almost exactly 86edo.		It's also possible to constrain the generators with both tensmajor and tractsubminor. Take 1/4-1/3-comma for example, using the aforementioned (3m)^4/64 = (3a/2-1)/2+1 and 32/(3a)^3 = (3m/2-1)/3+1 with fifths of 698.027¢ and 711.604¢, almost exactly 86edo.

	== Equal tunings ==		== Equal tunings ==

Ground at 23:42, 26 July 2026

2026-07-26T23:42:19Z

← Older revision		Revision as of 23:42, 26 July 2026
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	=== SRTT targets ===		=== SRTT targets ===

	This math will use the 2.3m.3a.5.7 subgroup, using variables to represent alteration ratios as done in [[Straddle primes\|SRTT]]. 3m/2 is the Meantone fifth and 3a/2 is the Archy fifth. Monarch tempers out the Flat-3 Syntonic comma 81/80 m^4 and Sharp-3 Septimal comma 64/63 a^-2, and Interarch additionally tempers out the ~~Interarch~~ comma (7/4 / (3a/2 * sqrt(4/(3m))))^2 = 49/48 ma^-2.		This math will use the 2.3m.3a.5.7 subgroup, using variables to represent alteration ratios as done in [[Straddle primes\|SRTT]]. 3m/2 is the Meantone fifth and 3a/2 is the Archy fifth. Monarch tempers out the Flat-3 Syntonic comma 81/80 m^4 and Sharp-3 Septimal comma 64/63 a^-2, and Interarch additionally tempers out the Monarch Semaphore comma (7/4 / (3a/2 * sqrt(4/(3m))))^2 = 49/48 ma^-2.
	* All three of these will use the same tensmajor DR math explained above, equating the 1/4-comma Meantone ~5/4 to the ~5/4 used in the +1+1 chord with the Archy fifth: (3m)^4/64 = (3a/2-1)/2+1.		* All three of these will use the same tensmajor DR math explained above, equating the 1/4-comma Meantone ~5/4 to the ~5/4 used in the +1+1 chord with the Archy fifth: (3m)^4/64 = (3a/2-1)/2+1.
	* In 1/4-1/4-comma: and (3a)^4/64 = 9/7. This system simplifies to {48a+32 = 81m^4, 63a^4 = 64}. The solution gives fifths of 697.602¢ and 708.771¢.		* In 1/4-1/4-comma: and (3a)^4/64 = 9/7. This system simplifies to {48a+32 = 81m^4, 63a^4 = 64}. The solution gives fifths of 697.602¢ and 708.771¢.

Ground at 23:41, 26 July 2026

2026-07-26T23:41:45Z

Ground at 18:27, 26 July 2026

2026-07-26T18:27:02Z

← Older revision		Revision as of 18:27, 26 July 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.}}

			'''Semonarch''' is a variant that splits the Meantone fourth 4/3[-3] in half, named after [[Semiquartal\|semiquartal]]. It typically has a good [[Intergan]] generator. Many equivalences and extensions are possible, 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.

	== Optimization ==		== Optimization ==

	{{UserTag\|g_\|Ground\|7766ff\|I had been using 110edo as my standard for both Meantone septal diasem and Archy pental blackdye, with 110d 18:6:4 (50 patent 9:3:2) diasem and 110bc 16:9:4 blackdye. This exact tuning system is somewhat impractical to use, but the two scales share the same 5/4 and 9/7, which led to the temperament now called Monarch. I believe that tuning major thirds accurately rather than minor thirds is generally more important to a pure sound. 110edo also happens to contain a very good [[Intergan]] generator, which splits the Meantone fourth in half, and makes for a good superset of Monarch.}}		{{UserTag\|g_\|Ground\|7766ff\|I had been using 110edo as my standard for both Meantone septal diasem and Archy pental blackdye, with 110d 18:6:4 (50 patent 9:3:2) diasem and 110bc 16:9:4 blackdye. This exact tuning system is somewhat impractical to use, but the two scales share the same 5/4 and 9/7, which led to the temperament now called Monarch. I believe that tuning major thirds accurately rather than minor thirds is generally more important to a pure sound. 110edo also happens to contain a very good Intergan generator, which splits the Meantone fourth in half, and makes for a good superset of Monarch.}}

	There are multiple ways to optimize Monarch. This article will discuss two Meantone targets and three Archy targets for a total of six combinations:		There are multiple ways to optimize Monarch. This article will discuss two Meantone targets and three Archy targets for a total of six combinations:

Ground at 03:48, 26 July 2026

2026-07-26T03:48:38Z

← Older revision		Revision as of 03:48, 26 July 2026
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	86edo'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 7/613/10 as in Oceanfront, they/re 7/622/17. So a lake is an ocean but smaller. I think it sounds better than any other diatonic scale between classic Archy and Oceanfront.		86edo'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 7/613/10 as in Oceanfront, they/re 7/622/17. So a lake is an ocean but smaller. I think it sounds better than any other diatonic scale between classic Archy and Oceanfront.
	}}		}}

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

Ground at 01:00, 26 July 2026

2026-07-26T01:00:56Z

← Older revision		Revision as of 01:00, 26 July 2026
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	This math will use the 2.3m.3a.5.7 subgroup, using variables to represent alteration ratios as done in [[Straddle primes\|SRTT]]. 3m/2 is the Meantone fifth and 3a/2 is the Archy fifth.		This math will use the 2.3m.3a.5.7 subgroup, using variables to represent alteration ratios as done in [[Straddle primes\|SRTT]]. 3m/2 is the Meantone fifth and 3a/2 is the Archy fifth.
	* All three of these will use the same tensmajor DR math explained above, equating the Meantone ~5/4 to the ~5/4 used in the +1+1 chord with the Archy fifth: (3m)^4/64 = (3a/2-1)/2+1.		* All three of these will use the same tensmajor DR math explained above, equating the 1/4-comma Meantone ~5/4 to the ~5/4 used in the +1+1 chord with the Archy fifth: (3m)^4/64 = (3a/2-1)/2+1.
	* In 1/4-1/4-comma: and (3a)^4/64 = 9/7. This system simplifies to {48a+32 = 81m^4, 63a^4 = 64}. The solution gives fifths of 697.602¢ and 708.771¢.		* In 1/4-1/4-comma: and (3a)^4/64 = 9/7. This system simplifies to {48a+32 = 81m^4, 63a^4 = 64}. The solution gives fifths of 697.602¢ and 708.771¢.
	* In 1/4-1/3-comma: and 3m/2 / (32/(3a)^3) = 9/7. This system simplifies to {48a+32 = 81m^4, 63ma^3 = 64}. The solution gives fifths of 698.135¢ and 712.317¢.		* In 1/4-1/3-comma: and 3m/2 / (32/(3a)^3) = 9/7. This system simplifies to {48a+32 = 81m^4, 63ma^3 = 64}. The solution gives fifths of 698.135¢ and 712.317¢.
	* In 1/4-1/2-comma: and (3m/2)^2 / (16/(3a)^2) = 9/7. This system simplifies to {48a+32 = 81m^4, 63m^2a^2 = 64}. The solution gives fifths of 699.062¢ and 718.480¢.		* In 1/4-1/2-comma: and (3m/2)^2 / (16/(3a)^2) = 9/7. This system simplifies to {48a+32 = 81m^4, 63m^2a^2 = 64}. The solution gives fifths of 699.062¢ and 718.480¢.
			* It's also possible to constrain the generators with both tensmajor and tractsubminor. Take 1/4-1/3-comma for example, using the aforementioned (3m)^4/64 = (3a/2-1)/2+1 and 32/(3a)^3 = (3m/2-1)/3+1 with fifths of 698.027¢ and 711.604¢, almost exactly 86edo.

	== Equal tunings ==		== Equal tunings ==

Ground at 22:34, 25 July 2026

2026-07-25T22:34:14Z

← Older revision		Revision as of 22:34, 25 July 2026
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	Edos using 1/3-comma Meantone instead include 64, 76, and 81.		Edos using 1/3-comma Meantone instead include 64, 76, and 81.

	{{UserTag\|g_\|Ground\|7766ff\|Optimizing Intergan Monarch is the culmination of my search for large edos that are good for my particular perspective and workflow. I've investigated all of these edos as candidates in the past, and the SRTT math finally explains why, including why I settled on 86edo after so much deliberation.		{{UserTag\|g_\|Ground\|7766ff\|Optimizing Intergan Monarch is the culmination of my search for large edos that are good for my particular perspective and workflow. I've investigated all of these edos as candidates in the past, and the SRTT math finally explains why, including why I settled on 86edo after so much deliberation. It balances every aspect of the temperament with practical considerations.

	86edo'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 7/613/10 as in Oceanfront, they/re 7/622/17. So a lake is an ocean but smaller. I think it sounds better than any other diatonic scale between classic Archy and Oceanfront.		86edo'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 7/613/10 as in Oceanfront, they/re 7/622/17. So a lake is an ocean but smaller. I think it sounds better than any other diatonic scale between classic Archy and Oceanfront.
	}}		}}

← Older revision		Revision as of 23:41, 26 July 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.}}

	'''~~Semonarch~~''' is a variant that splits the Meantone fourth 4/3[-3] in half, named after [[~~Semiquartal\|semiquartal~~]]~~. It typically has~~ a ~~good~~ [[~~Intergan~~]] ~~generator~~. Many equivalences ~~and extensions~~ are possible, 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.		'''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. 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.

	== Optimization ==		== Optimization ==
Line 30:		Line 30:
	=== SRTT targets ===		=== SRTT targets ===

	This math will use the 2.3m.3a.5.7 subgroup, using variables to represent alteration ratios as done in [[Straddle primes\|SRTT]]. 3m/2 is the Meantone fifth and 3a/2 is the Archy fifth.		This math will use the 2.3m.3a.5.7 subgroup, using variables to represent alteration ratios as done in [[Straddle primes\|SRTT]]. 3m/2 is the Meantone fifth and 3a/2 is the Archy fifth. Monarch tempers out the Flat-3 Syntonic comma 81/80 m^4 and Sharp-3 Septimal comma 64/63 a^-2, and Interarch additionally tempers out the Interarch comma (7/4 / (3a/2 * sqrt(4/(3m))))^2 = 49/48 ma^-2.
	* All three of these will use the same tensmajor DR math explained above, equating the 1/4-comma Meantone ~5/4 to the ~5/4 used in the +1+1 chord with the Archy fifth: (3m)^4/64 = (3a/2-1)/2+1.		* All three of these will use the same tensmajor DR math explained above, equating the 1/4-comma Meantone ~5/4 to the ~5/4 used in the +1+1 chord with the Archy fifth: (3m)^4/64 = (3a/2-1)/2+1.
	* In 1/4-1/4-comma: and (3a)^4/64 = 9/7. This system simplifies to {48a+32 = 81m^4, 63a^4 = 64}. The solution gives fifths of 697.602¢ and 708.771¢.		* In 1/4-1/4-comma: and (3a)^4/64 = 9/7. This system simplifies to {48a+32 = 81m^4, 63a^4 = 64}. The solution gives fifths of 697.602¢ and 708.771¢.
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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 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 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 1/4-1/2-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, three of them making some sort of semiquartal generator~~, and six of them making a flatter fifth~~. ~~In the~~ smaller edos to have this optimization class such as 62, 67, and 72~~, this flatter fifth is Meantone and the semiquartal is Intergan~~.		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.		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.

	Edos using 1/3-comma Meantone instead include 64, 76, and 81.		Edos using 1/3-comma Meantone instead include 64, 76, and 81.

	{{UserTag\|g_\|Ground\|7766ff\|Optimizing ~~Intergan Monarch~~ is the culmination of my search for large edos that are good for my particular perspective and workflow. I've investigated all of these edos as candidates in the past, and the SRTT math finally explains why, including why I settled on 86edo after so much deliberation. It balances every aspect of the temperament with practical considerations.		{{UserTag\|g_\|Ground\|7766ff\|Optimizing Interarch is the culmination of my search for large edos that are good for my particular perspective and workflow. I've investigated all of these edos as candidates in the past, and the SRTT math finally explains why, including why I settled on 86edo after so much deliberation. It balances every aspect of the temperament with practical considerations.

	86edo'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 7/613/10 as in Oceanfront, they/re 7/622/17. So a lake is an ocean but smaller. I think it sounds better than any other diatonic scale between classic Archy and Oceanfront.		86edo'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 7/613/10 as in Oceanfront, they/re 7/622/17. So a lake is an ocean but smaller. I think it sounds better than any other diatonic scale between classic Archy and Oceanfront.