Yahoo Tuning Groups Ultimate Backup tuning-math Survey VI

<225/224, 4000/3969>

Minkowski reduction: <335/224, 3125/3087>

Ets: 12, 29, 41, 53, 94

Map:

[ 1 2]
[ 2 3]
[-1 6]
[-3 8]

Reduced map:

[ 0 1]
[-1 2]
[ 8 -1]
[14 -3]

Generators: a = .4148831986 (~4/3) = 38.99902067 / 94; b = 1

Septimal schismic system; effectively this is 53+41

Errors and 94-et:

3: .185 .173
5: -3.44 -3.33
7: 1.21 1.39

<4375/4374, 225/224> Minkowski reduced

Ets: 19, 53, 72

Map:

[ 3 1]
[ 6 0]
[ 8 1]
[13 -3]

Adjusted map:

[ 0 1]
[ 6 0]
[ 5 1]
[22 -3]

Generators: a = .2639365655 (~6/5) = 19.00343272 / 72; b = 1

This is pretty much the 53+19 system of the 72 et, but a bit better
in tune; in particular the long string of generators to 7 helps to
get it in better tune.

Errors and 72-et:

3: -1.61 -1.96
5: -2.69 -2.98
7: -0.90 -2.16

Because the 7's are so inacessible, it's tempting to treat this as a
purely 5-limit system, in which case it's closer to the 53 et, with a
generator of 14.00435/53, and triads which are very good; this
version is the kleismic temperament. Another possibility would be a
planar temperament.

<1728/1715, 81/80>

Minkowski reduction: <1029/1024, 81/80>

Ets: 5, 26, 31, 57

Map (no adjustment)

[ 0 1]
[ 3 1]
[ 12 0]
[-1 3]

Generators: a = .1934362896 (~8/7) = 5.996524978 / 31; b = 1

Errors and 31-et

3: -5.58 -5.18
5: -0.83 0.78
7: -0.95 -1.08

Pretty much the 26+5 system of the 31-et.

<64/63, 686/685> Minkowski reduced

Ets: 10, 27

Map:

[1 -1]
[1 -3]
[5 4]
[4 0]

Adjusted map:

[ 0 1]
[ 2 1]
[-9 5]
[-4 4]

Generators: a = .2969233588 (~11/9) = 8.016930688 / 27; b = 1

Errors and 27-et

3: 10.66 9.15
5: 6.91 13.68
7: 5.94 8.95

This is one version of a neutral third temperament, though Graham
does not even mention the 27-et in connection to neutral thirds, so
it's hardly canonical.

Survey VI

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com