Monarch

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Monarch is a rank-3 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 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.

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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 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

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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:

• 1/4-comma Meantone: 81/64[-3] = 5/4
• 1/3-comma Meantone: 32/27[-3] = 6/5
• 1/4-comma Archy: 81/64[+3] = 9/7
• 1/3-comma Archy: 32/27[+3] = 7/6
• 1/2-comma Archy: 16/9[+3] = 7/4

The intervals on the right side of the equation may also be altered from JI using a DR chord. There are two main options for this:

• Tensmajor: 5 is altered such that 4:5:6[+3] in Archy pental blackdye is +1+1.
• Tractsubminor: 7 is altered such that 6:7:9[-3] in Meantone septal diasem is +1+2.

The format for naming optimization classes is [alteration] [Meantone fraction]-[Archy fraction]-comma.

According to the standards seen in 110edo, the Meantone major third is broadly a sharp 5/4, which has several benefits when used in Archy pental blackdye. It refines aberrismic melodic properties, preserves the tuning of 7/5, and happens to coincide with a +1+1 major triad. Tensmajor will thus be used as the default in this article. On the other hand, the narrower capture zone of 9/7 and multiple non-2 primes in its factorization imply that it should be tuned more accurately. This may be done in about three main ways for Meantone septal diasem:

• In edos such as 110, 9/7 is 81/64[+3], a straightforward Archy diatonic major third. This may be called tensmajor 1/4-1/4-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.

SRTT targets

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 1/4-1/3-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¢.

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

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 "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.

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.

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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 7/6*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.