Note also that the two dimensional Fourier matrix $F_2$ can be summed to a real $1$-circulant, in three ways.
Now
Carl has been talking a lot about taking exponentials of nice matrices. Let us look at the simplest Hermitian circulant, written in quark boson form,
It is fun to exponentiate this matrix, to obtain
up to whatever errors I may have made with the constants. Those $12$th roots just keep popping up everywhere. If the diagonal of $M$ was scaled by adding a factor $r \mathbf{1}$, as in a Koide mass matrix, this would affect the exponential by another factor of $\textrm{exp}(ir)$. In this way, the exponential of all Hermitian circulants is easily obtained. Note that for general phases $\omega$, the resulting matrix has only $3$ real parameters, coming from $r$ and $\omega$.
Now Carl has been talking a lot about taking exponentials of nice matrices. Let us look at the simplest Hermitian circulant, written in quark boson form, 
It is fun to exponentiate this matrix, to obtain 
up to whatever errors I may have made with the constants. Those $12$th roots just keep popping up everywhere. If the diagonal of $M$ was scaled by adding a factor $r \mathbf{1}$, as in a Koide mass matrix, this would affect the exponential by another factor of $\textrm{exp}(ir)$. In this way, the exponential of all Hermitian circulants is easily obtained. Note that for general phases $\omega$, the resulting matrix has only $3$ real parameters, coming from $r$ and $\omega$.
This business of rotating by i makes one wonder about the angle 2/9 of the mass matrices. Anyway, notice that $\textrm{sin}(2/9) = 0.2204$ while $\textrm{sin}(2/9 + \pi / 2) = 0.9754$. These are not the same as, but quite close to two entries in the CKM matrix.
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