POTE tuning

POTE tuning is the short form of Pure-Octaves Tenney-Euclidean tuning, a good choice for a standard tuning enforcing a just 2/1 octave.

The POTE tuning for a map matrix such as M = [⟨1 0 2 -1], ⟨0 5 1 12]] (the map for 7-limit magic, which consists of a linearly independent list of vals defining magic) can be found as follows:

1. Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log2(p) on the diagonal; in other words the diagonal is [1 1/log2(3) 1/log2(5) 1/log2(7)]. Another way to say this is that each val is "weighted" by dividing through by the logarithms, so that V = [⟨1 0 2/log2(5) -1/log2(7)], ⟨5/log2(3) 1/log2(5) 12/log2(7)]]
2. Find the matrix P = V^T(VV^T)^-1.
3. Find the TE = ⟨1 1 1 1]P.
4. Find the TE octave: O_TE = (TE*V)₁, that is, the first entry of TE*V.
5. Find the POTE = TE/O_TE; in other words TE scalar divided by O_TE.

If you carry out these operations, you should find

• V ~ [⟨1 0 0.861 -0.356], ⟨0 3.155 0.431 4.274]]

• TE ~ ⟨1.000902 0.317246]

• POTE ~ ⟨1 0.3169600]

The tuning of the POTE generator corresponding to the mapping M is therefore 0.31696 octaves, or 380.352 cents. Naturally, this only gives the single POTE generator in the rank two case, and only when the map M is in period-generator form, but the POTE tuning can still be found in this way for mappings defining higher rank temperaments. The method can be generalized to subgroup temperaments so long as the group contains 2 by POL2 tuning.