View source for 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 {{monzo | 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 [[Wikipedia:Thin lens|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{{monzo | 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>