Bearings

My new type of transformation was discovered in the course of looking at the position of the minor triad 7/6-7/5-7/4 when reduced to the 5-limit lattice by means of a 7-limit comma in which 7 appears only to a power of +-1, such as 64/63, 126/125, 225/224 or 4375/4374. In this case we can describe the position by a bearing (here in degrees on one or another side of one of the six 5-limit consonances) and distance. One way to produce scales suitable for these planar temperaments would be to produce 5-limit scales running in the general direction of the bearings below. Hence, 64/63 works well with chains of fifths, 126/125 with chains of minor thirds, 225/224 with chains of 16/15s (secors), and 4375/4374 with chains of minor thirds.

I obtained the following:

Bearing for 64/63

7/4 ~ 16/9 distance = 2, bearing 4/3

7/6 ~ 32/27 distance = 3, bearing 4/3

7/5 ~ 64/45 distance sqrt(7) = 2.64575, bearing 19.10661 4/3 by 8/5

Triad distance sqrt(57)/3 = 2.51661, bearing 6.58678 4/3 by 8/5

Bearing for 126/125

7/4 ~ 125/72 distance sqrt(7) = 2.64575, bearing 19.10661 5/3 by 5/4

7/6 ~ 125/108 distance 3, bearing 5/3

7/5 ~ 25/18 distance 2, bearing 5/3

Triad distance sqrt(57)/3 = 2.51661, bearing 6.58678 5/3 by 5/4

Bearing for 225/224

7/4 ~ 225/128 distance 2sqrt(3) = 3.46410, bearing 30 3/2 by 5/4

7/6 ~ 75/64 distance sqrt(7) = 2.64575, bearing 19.10661 5/4 by 3/2

7/5 ~ 45/32 distance sqrt(7), bearing 19.10661 3/2 by 5/4

Triad distance 5sqrt(3)/3 = 2.88675, bearing 30 3/2 by 5/4
Straight at 15/8 = 2/secor

Bearing for 4375/4374

7/4~2187/1250 distance sqrt(37) = 6.08276, bearing 25.28500 6/5 by 3/2

7/6~729/625 distance 2sqrt(7) = 5.29150, bearing 19.10661 6/5 by 3/2

7/5~4374/3125 distance sqrt(39) = 6.24500, bearing 16.10211 6/5 by 3/2

Triad distance sqrt(309)/3 = 5.85947 bearing 20.17357 6/5 by 3/2