Mapped interval

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⟩ = [-3 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 — two 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" 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.

See also

• Dave Keenan & Douglas Blumeyer's guide to RTT: tuning fundamentals#The RTT version: another take at explaining mapped intervals
• Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Mapped interval: a units analysis of mapped intervals (also the following section, for a rank-2 example)
• Tmonzos and tvals: a more mathematical explanation