All twelve modes belong to six order classes
On this page · Permalink · Source line 124
The primitive-twelve projector is one packet in a complete orthogonal decomposition. A frequency label \(k\) has root order \(d=12/\gcd(k,12)\). The possible orders are the divisors \(1,2,3,4,6,12\).
\[
P_d{}_{nm}=\frac{c_d(n-m)}{12},\quad
P_dP_e=\delta_{de}P_d,\quad
\sum_{d\mid12}P_d=I.
\]
These are character-order projectors. They select Fourier modes by root order. They are different from coefficient-residue selectors, which select formal-series coefficients by their index modulo an integer.