Tempered monzos and vals

Tmonzos (for tempered monzos) and tvals (for tempered vals) are like regular monzos and vals, except they work in a space of tempered intervals (for example, the intervals found in meantone) rather than in just intonation. A tval specifies a tuning or further temperament of the generators of a temperament. For example, the 31edo tval for meantone is ⟨31 49], assume the generators are ~2 and ~3. A tmonzo specifies a particular tempered interval in terms of stacking the temperament's generators. For example, the tmonzo form of the major third in meantone is [-6 4⟩. Taking the dot product of these (multiplying corresponding elements and adding up the results) yields the tuning of the major third in 31edo, 10\31.

Mathematically, a regular temperament is a homomorphism (a kind of function) from the space of just intervals to the space of tempered intervals generated by that temperament, where both these spaces are abelian groups. Technically, a regular temperament is an equivalence class of functions separated by unimodular transformations, which represent the same temperament. An element of the space of tempered intervals is called a tempered monzo, or for short tmonzo, and represents the number of steps of each generator required to reach a tempered interval, and an element of the dual module K* is called a tempered val, or for short tval.

Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they are linear functionals which map from tmonzos to a scalar representing a certain number of steps. Note that there is no restriction on which bases tmonzos can be written in, but one option is to use the basis corresponding to the mapping matrix for the temperament which is in normal val list form.

Example

As an example, consider the mapping matrix

$$ \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 4 \end{bmatrix} $$

This mapping represents meantone temperament. If we apply this mapping to the monzo [1 0 0⟩, representing 2/1, we get the tmonzo [1 0⟩ (one tempered 2/1). If we instead apply it to [-1 1 0⟩, we get the tmonzo [0 1⟩ (one tempered 3/2). That 2/1 and 3/2 map to [1 0⟩ and [0 1⟩ respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now apply this mapping to the monzo [-2 0 1⟩, representing 5/4, we get the tmonzo [-2 4⟩, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.

See also

• Mapped interval – a beginner-level introduction