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Six 13-limit nonoctave temperaments

🔗Gene Ward Smith <gwsmith@svpal.org>

4/15/2004 5:22:08 PM

Nonoctave temperaments of dimension n are essentially identical to
octave temperaments of dimension n+1; in particular a linear nonoctave
temperament is essentially the same beast as a planar temperament such
that the kernel has only commas which are ratios of odd integers. I
took four commas 245/243, 275/273, 847/845, 1575/1573 from Lehmer's
N/(N-2) list, and added a comma from his N/(N-4) list, namely
4459/4455, and then took this in sets of three, finding the
corresponding planar temperaments (equivalent to nonoctave linears.)
Here is what I got, listing (planar) wedgie, mapping, and TOP tuning
for six temperaments.

[21, 9, 19, 13, -24, -18, -30, 14, 2, -16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[[1, 0, 0, 0, 0, 0], [0, 1, 9, 5, 9, 7], [0, 0, 21, 9, 19, 13]]
[1200.0, 1902.277228, 2785.467655, 3367.802970, 4150.708360, 4441.876018]

[6, -3, -2, 13, -15, -16, 5, 3, 30, 33, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[[1, 0, 0, 0, 0, 0], [0, 1, 5, 0, 1, 10], [0, 0, 6, -3, -2, 13]]
[1200.0, 1903.771872, 2783.652046, 3367.603656, 4148.840976, 4444.769540]

[0, 0, 13, 13, 0, 19, 19, 23, 23, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[[1, 0, 0, 0, 0, 0], [0, 13, 19, 23, 0, 2], [0, 0, 0, 0, 1, 1]]
[1200.0, 1904.187463, 2783.043215, 3368.947050, 4146.445241, 4439.397158]

[9, 15, 10, 13, 6, -5, -2, -15, -12, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[[1, 0, 0, 0, 0, 0], [0, 1, 4, 6, 5, 6], [0, 0, 9, 15, 10, 13]]
[1200.0, 1902.661549, 2784.022381, 3371.596268, 4150.392396, 4444.179340]

[12, -6, 9, 0, -30, -13, -28, 29, 14, -21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[[1, 0, 0, 0, 0, 0], [0, 3, 1, 7, 4, 7], [0, 0, 4, -2, 3, 0]]
[1200.0, 1903.372996, 2784.236389, 3366.314293, 4150.164703, 4441.203655]