Mapped interval

This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily.
The corresponding expert page for this topic is Tmonzos and tvals.

A mapped interval is an interval that has been mapped by a mapping matrix for a regular temperament.

For example, if we begin with an unmapped, JI interval [math]\displaystyle{ \frac{10}{9} }[/math] with prime-count vector (or "monzo") [math]\displaystyle{ \textbf{i} = }[/math] [1 -2 1⟩, the mapped interval ~[math]\displaystyle{ \frac{10}{9} }[/math] under meantone temperament [⟨1 1 0] ⟨0 1 4]} would have generator-count vector (or "tmonzo") [math]\displaystyle{ \textbf{y} = }[/math] [⟨1 1 0] ⟨0 1 4]}[1 -2 1⟩ = [-1 2}.

Note that we've notated the mapped interval with a tilde, ~[math]\displaystyle{ \frac{10}{9} }[/math], to indicate that its size is now approximate.

Here are several mnemonics for the use of [math]\displaystyle{ \textbf{y} }[/math] as the symbol for mapped intervals:

The letter "y" is linguistically similar to the letter "i", the obvious letter for (just) intervals.
Visually, a "Y" also looks like a diagram showing—from the top—wo just intervals getting mapped to the same size.
A 'y' also looks like a 'g', which is fitting because [math]\displaystyle{ \mathbf{y} }[/math] is a generator-count vector, associated with the generator tuning map [math]\displaystyle{ 𝒈 }[/math], in the sense that intervals are associated with (tempered-prime) tuning maps [math]\displaystyle{ 𝒕 }[/math], or in other words, [math]\displaystyle{ 𝒕\textbf{i} = 𝒈\textbf{y} }[/math].

A mapped interval therefore refers not to any particular JI interval, but to an equivalence class of JI intervals separated by any combination of the commas that are tempered out by a given temperament. Thus, a regular temperament can be seen as a sort of vector generalization of the "modulus" in modular arithmetic - that is, 100/81, 5/4, 81/64, 6561/5120, etc., belong to the same equivalence class "modulo" 81/80 (and in fact, due to the fundamental theorem of arithmetic, this can be seen as an actual modulus in logarithmic space, though this uses constructions involving real numbers rather than integers).

A tmonzo represents a mapped interval, but it is not very popular as a notation scheme. To establish this as notation for intervals, it is not enough to simply have the temperament clarified in context; it is also necessary to explicitly state the generator basis of the temperament (often effectively by writing out the mapping matrix). In the above example, [-1 2} actually has generators that are an octave and a perfect fifth, namely ~2 and ~3/2. This basis is explicitly stated by writing ~2.~3/2 [-1 2} or P8.P5 [-1 2}. If the generator basis is an octave and a perfect twelfth, the same pitch would be written as ~2.~3 [-3 2} or P8.P12 [-3 2}. For transformations between these, refer to generator form manipulation.

Terminology

A "mapped interval" could also be called a "tempered interval"; however, "tempered interval" is more ambiguous: "tempered interval" could also refer to a size resulting from mapping an interval by a tuning map for a temperament (in the same sense that "interval" is used to refer to a "(just) interval's (size)", or it could even refer to a projected interval such as the [0 0 1/4⟩ generator of quarter-comma meantone. Only "mapped interval" unambiguously refers to an interval that has been transformed only by the mapping matrix for a temperament.

Mapped interval

Terminology

See also