User:Frostburn/Lens RTT

Monzos can be seen to arise from the properties of the logarithm of a fixed base

[math]\log_b (x^u y^v z^w) = u\log_b x + v\log_b y + w\log_b z[/math].

Allowing us to treat the exponents as a linear vector [u, v, w⟩. (Using the primes x = 2, y = 3, z = 5, etc. as the basis.)

A similar situation happens with the logarithm of a fixed argument

[math]log_{x^u y^v z^w}(a) = u^{-1}\log_x a \oplus v^{-1}\log_y a \oplus w^{-1}\log_z a[/math],

where lens addition [math]\oplus[/math] is inspired by the thin lens equation f⁻¹ = u⁻¹ + v⁻¹

[math]f = u \oplus v[/math].

Corresponding lens subtraction [math]\ominus[/math] follows from the same equation

[math]u = f \ominus v[/math].

Now we can treat the exponents as a lens-linear vector L[u⁻¹, v⁻¹, w⁻¹⟩. (Again using the primes as the basis.)

For now the point is that the lens-vals corresponding to these lens-monzos better capture the fact that they correspond to maps of fractions of equal temperaments. E.g. L5edo [math]\oplus[/math] L7edo = L12edo

[math]\mathrm{L}\left \langle \frac{1}{5}, \frac{1}{8}, \frac{1}{12} \right ] \oplus \mathrm{L}\left \langle \frac{1}{7}, \frac{1}{11}, \frac{1}{16} \right ] = \mathrm{L}\left \langle \frac{1}{12}, \frac{1}{19}, \frac{1}{28} \right ][/math]