Monarch
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 temperaments that comprise it. Extensions and specific optimizations or erac tunings make use of combining the two tunings of 3.
Optimization
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
This math will use the 2.3m.3a.5.7 subgroup, using variables to represent alteration ratios as done in 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.
- 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
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 1/4-1/2-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.
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.
Edos using 1/3-comma Meantone instead include 64, 76, and 81.
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.
