Meantone & superpythagorean

Meantone can be described as the 19&31 system; if we take
h19 = <19 30 44 53| and h31 = <31 49 72 87| then the meantone
2-division val is j2 = 5 h19 - 3 h31 = <2 3 4 4|. Similarly, the
meantone 5-division val is j5 = 2 h31 - 3 h19 = <5 8 12 15|.

Superpythagorean can be described as the 22&31 system; if we take
h22 = <22 35 51 62| and h27 = <27 43 63 76| then the superpythagorean
2-division val is k2 = 5 h22 - 4 h27 = <2 3 3 6|. Similarly, the
superpythagorean 5-division val is k5 = h27 - h22 = <5 8 12 14|.

If we take the matrix with columns [j5 j2 k5 k2] and invert, we obtain
four monzos: f1 = |18 -16 2 1>, t1 = |12 -9 1 0>, f2 = |-21 18 -2 -1>,
t2 = |-4 4 -1 0>. Any 7-limit interval u can now be expressed in the
form j5(u)f1 + j2(u)t1 + k5(u)f2 + k2(u)t2; where we may consider this
as an expression for monzos, but also as an expression for log u in
terms of logarithmic values for f1, t1, f2, t2. Taking the latter
point of view, we can write log u as (p1R1 + q1)t1 + (p2R2 + q2)t2,
where p1 = j5(u), q1 = j2(u), p2 = k5(u), q2 = k2(u), and where R1 =
f1/t1 and R2 = f2/t2. Numerically, R1 is 1.60325..., right in the
optimal range for a meantone Blackwood R, and R2 is 4.35857..., right
in the optimal range for a superpythagorean Blackwood R. The
corresponding values for the fifth are a meantone fifth of 696.039
cents 709.913. We might call these the meantone and superpythagorean
co-fifths.

We have that our interval u is expressed logarithmically by
p1f1 + q1t1 + p2f2 + q2t2 = t1(p1R1 + q1) + t2(p2R2 + q2) =
t1(5R1+2)[(p1R1+q1)/(5R1+2)] + t2(5R2+2)[(p2R2+q2)/(5R2+2)] =
(5f1+2t1)M + (5f2+2t2)S, where M = (p1R1+q1)/(5R1+2) is the meantone
value of u in the co-fifth tuning, and S = (p2R2+q2)/(5R2+2) is the
corresponding superpythagorean value of u in the co-fifth tuning. Any
7-limit interval is the above linear combination of the interval in
meantone, using the meantone co-fifth tuning, and the interval in
superpythagorean, using the superpythagorean co-fifth tuning.