Smonzos and svals

Given a just intonation subgroup, we can find a canonical form for its generators by means of the normal interval list which may be computed from any finite set of generators. In the case of the full p-limit group for any prime p, this consists of the primes from 2 to p in ascending order. This is precisely the ordered list used to define vals and monzos, and we may generalize the notation simply by using any normal interval list in place of the ascending primes to p. This generalization we may call the subgroup monzos and subgroup vals, or smonzos and svals for short.

Conversion

If m_G is a smonzo of the subgroup G, and if S is a subgroup basis matrix whose columns form a basis for the subgroup G, then the corresponding monzo m is given by

$$\vec m = S\vec m_G$$

Conversely, if m is a monzo for an interval in the subgroup, then we can take the pseudoinverse of S, S⁺, and the corresponding smonzo is given by

$$\vec m_G = S^+ \vec m$$

For example, consider the subgroup generated by the barbados triad, 1–13/10–3/2–2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix S with columns consisting of the monzos for 2, 3, and 13/10. Now the monzo for 676/675 is [2 -3 2 0 0 2⟩, and left-multiplying this by S⁺ gives the smonzo [2 -3 2⟩. We may check this is the correct smonzo from 2² 3^-3 (13/5)², which is 676/675 as desired.