Tempered monzos and vals
Tempered monzos and 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 tempered val (for short, "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 tempered monzo (for short, "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.
Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos, in that they map from tmonzos to a scalar representing a certain number of steps. A similar concept is a generator tuning map, which maps from tmonzos to tunings in cents (or another logarithmic measure).
Note that there is no restriction on which bases tmonzos can be written in (i.e. what intervals should be considered the generators of the temperament), 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.
Now, let's use a tval ⟨31 18] to figure out what the tuning of the tempered 5/4 is in 31edo. By applying this tval to the tmonzo [-2 4⟩, we get -62 + 72 = 10 edosteps.
See also
- Mapped interval – a beginner-level introduction