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. The name implies that this is a temperament of greater scope or importance. It may 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. The diatonic minor third cannot be tuned to 6/5 or sharper without an excessively 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, which sacrifices 9/7 to improve sharp 7 and flat 11.

SRTT targets

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

• In both 1/4-1/4-comma and 1/4-1/3-comma, the DR math is the same, equating the Meantone major third to the Archy +1+1 major third: (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¢.

Tunings

105edo (3x35) is very close to the ideal for 1/4-1/4-comma, 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.

Despite the drawback of a more extreme Archy lacking the good 9/7, 1/4-1/3-comma is desirable for having a larger size difference between its two fifths, allowing 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¢.

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.