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. The name comes from the combination of Meantone and Archy, the two linearly independent combined to make Monarch. Extensions and specific 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.

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. The Meantone diatonic major third 81/64[-3] is broadly a sharp 5/4, which has several benefits when used in Archy pental blackdye according to the above standards. It refines aberrismic melodic properties, preserves the tuning of 7/5, and happens to coincide with a +1+1 major triad. This method of using prime 5 to constrain the fifths will be used as the default in this article. The diatonic minor third cannot be tuned to 6/5 or sharper without a distinctly flat fifth.

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 two main ways for Meantone septal diasem:

• In 110edo, 9/7 is 81/64[+3], a straightforward Archy diatonic major third. This may be called 1/4-1/4-comma Monarch.
• In 86edo, 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 1/4-1/3-comma Monarch.
• In 67edo, 7/4 is tuned more accurately with 16/9[+3] and 9/7 is 9/4[-3] / (16/9[+3]), giving 1/4-1/2-comma Monarch.

SRTT targets

This math will use the 2.3m.3a.5.7 subgroup, using variables to represent alteration ratios as done in SRTT.

• All three of these will use the same 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.
• 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¢.

Equal tunings

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.

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 is also 1/4-1/3-comma, although 7/6 is very flat, so stacking a sharp 9/7 results in a Meantone fifth. 81edo tunes 9/7 significantly flat rather than sharp.

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 support 1/4-1/2-comma such as 62, 67, and 72, this flatter fifth is Meantone and the semiquartal is Intergan.

None of these tunings are Superpyth with the same 5/4 as Meantone. 1/4-1/4-comma's Archy fifth is too flat and 1/4-1/3-comma's Archy fifth is too sharp. Edos in between the two such as 98 (2x49) straddle 9/7.

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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.

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.