Subgroup monzos and vals

A subgroup monzo is a monzo whose elements refer to powers of the basis elements of a JI subgroup (specified rationals such as 2, 3, 35, and 13/11 which are combined to form the set of intervals in a subgroup), rather than strictly to ordered primes. For cases where the basis elements are all prime numbers, a subgroup monzo can be seen as "abbreviating" entries from a standard monzo: for example, the monzo [0 0 -1 0 0 1⟩ (13/5) may be abbreviated to the subgroup monzo 5.13 [-1 1⟩. However, subgroup monzos can refer to intervals in subgroups whose basis elements are not primes, for example 2.3.13/5 [1 -1 1⟩ for 26/15. In that case, conversion to and from a subgroup monzo is a little more complicated, and is covered in the next section.

A subgroup val is like a standard val, but the entries are the mappings of subgroup basis elements rather than strictly of primes (and can be derived from a standard val by applying the val to the basis intervals). Since converting to a subgroup val loses information, there is no clear way to convert a subgroup val back to a standard val.

For example, the subgroup val 2.3.7 ⟨5 8 14] tells you that prime 2 is mapped to 5 steps, prime 3 is mapped to 8 steps, and prime 7 is mapped to 14 steps, without specifying a mapping for 5.

For short, a subgroup monzo may be referred to as an smonzo, and a subgroup val may be referred to as an sval.

Conversion

If m_G is an 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.

Here is a tutorial that goes through the steps if you need to calculate them by hand.

Tutorial

Say we want to convert the subgroup monzo 2.3.13/5 [1 -1 1⟩ to a standard monzo, look at the monzo for each basis element of the subgroup monzo:

2 = [1 0 0 0 0 0⟩

3 = [0 1 0 0 0 0⟩

13/5 = [0 0 -1 0 0 1⟩

Now, we take each of these monzos and multiply each of the entries by the corresponding entry in the subgroup monzo.

[1 0 0 0 0 0⟩ × 1 = [1 0 0 0 0 0⟩

[0 1 0 0 0 0⟩ × (-1) = [0 -1 0 0 0 0⟩

[0 0 -1 0 0 1⟩ × 1 = [0 0 -1 0 0 1⟩

Now, we add all the entries together vertically, to get:

26/15 = [1 -1 -1 0 0 1⟩.

To go back from [1 -1 -1 0 0 1⟩ to our subgroup monzo, the most reasonable option is to repeatedly subtract and add the monzos for our subgroup basis elements until we reach the unison [0 0 0 0 0 0⟩.

[1 -1 -1 0 0 1⟩ - [0 0 -1 0 0 1⟩ + [0 1 0 0 0 0⟩ - [1 0 0 0 0 0⟩ = [0 0 0 0 0 0⟩

If we keep track of how many times we subtract each basis monzo (negative for adding it), it is 1 for 2, -1 for 3, and 1 for 13/5. Thus, we re-derive our subgroup monzo [1 -1 1⟩.

In elementary math

An elementary way to convert between standard monzos and subgroup monzos is to evaluate the monzo as a rational number, then start factoring out either primes or subgroup basis elements. To convert 2.3.13/5 [1 -1 1⟩ to a standard monzo, we first look at the interval itself, which is 26/15. You can see that there is 2 × 13 on the top, and then 3 × 5 on the bottom, and so the correct monzo is [1 -1 -1 0 0 1⟩.

To convert back, knowing the subgroup 2.3.13/5, we evaluate this monzo as a fraction (26/15). Recognizing the factor of 13/5 is a little tricky, but you can rearrange the expression as (26/5)/3 to make it more visible. Once you factor it out you have 13/5 × 2 on top, and 3 on the bottom, which can be used to write the subgroup monzo [1 -1 1⟩.