Observe that the choice of coefficients $(r,1)$ in a scaled $R_2$ factor
is the general case for a super magic circulant, because scale factors will always be absorbed into the normalisation. Under an interchange of the real and complex parameters
in $M$, the row sums and norm square sums remain unaltered. This means that $(r,1)$ is equivalent to $(r^{-1},1)$, a kind of
S duality. For example, in
the tribimaximal case it doesn't matter if we use $\sqrt{2}$ or $\sqrt{2}^{-1}$. This is an inversion of parameters with respect to the unit circle, as in
Thus magic circulants, although very simple, provide an interesting parameter space in which to study mixing matrices. Note that the inverse of an $R_2$ factor is given by sending $i$ to $-i$, so we might think of the (charged lepton) identity mixing matrix as two inverse $R_2$ factors with parameters $a = r$ and $b = -r$. This gives three parameters in total
for lepton mixing.
is the general case for a super magic circulant, because scale factors will always be absorbed into the normalisation. Under an interchange of the real and complex parameters in $M$, the row sums and norm square sums remain unaltered. This means that $(r,1)$ is equivalent to $(r^{-1},1)$, a kind of S duality. For example, in the tribimaximal case it doesn't matter if we use $\sqrt{2}$ or $\sqrt{2}^{-1}$. This is an inversion of parameters with respect to the unit circle, as in 
Thus magic circulants, although very simple, provide an interesting parameter space in which to study mixing matrices. Note that the inverse of an $R_2$ factor is given by sending $i$ to $-i$, so we might think of the (charged lepton) identity mixing matrix as two inverse $R_2$ factors with parameters $a = r$ and $b = -r$. This gives three parameters in total for lepton mixing.
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