Mapped interval: Difference between revisions

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{{interwiki

| en = Mapped interval

| ja = マップされた音程

}}

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>\frac{10}{9}</math> with [[prime-count vector]] (or "[[monzo]]") <math>\textbf{i} =</math> {{ket|1 -2 1}}, the mapped interval ~<math>\frac{10}{9}</math> under [[meantone temperament]] {{rket|{{bra|1 1 0}} {{bra|0 1 4}}}} would have [[generator-count vector]] (or "[[tmonzo]]") <math>\textbf{y} =</math> {{rket|{{bra|1 1 0}} {{bra|0 1 4}}}}{{ket|1 -2 1}} = {{rket|-3 2}}.

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

Note that we've notated the mapped interval with a tilde, <span style="background-color: yellow">~</span><math>\frac{10}{9}</math>, to indicate that its size is now approximate.

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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 [[comma]]s that are [[tempered out]] by a given temperament. Thus, a regular temperament can be seen as a sort of [[monzo|vector]] generalization of the "modulus" in {{w|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 [[Prime number|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, {{rket|-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 {{rket|-1 2}} or P8.P5 {{rket|-1 2}}. If the generator basis is an octave and a perfect twelfth, the same pitch would be written as ~2.~3 {{rket|-3 2}} or P8.P12 {{rket|-3 2}}. For transformations between these, refer to [[generator form manipulation]].

== Terminology ==

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+{{interwiki
+| en = Mapped interval
+| ja = マップされた音程
+}}
 {{Beginner|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>\frac{10}{9}</math> with [[prime-count vector]] (or "[[monzo]]") <math>\textbf{i} =</math> {{ket|1 -2 1}}, the mapped interval ~<math>\frac{10}{9}</math> under [[meantone temperament]] {{rket|{{bra|1 1 0}} {{bra|0 1 4}}}} would have [[generator-count vector]] (or "[[tmonzo]]") <math>\textbf{y} =</math> {{rket|{{bra|1 1 0}} {{bra|0 1 4}}}}{{ket|1 -2 1}} = {{rket|-3 2}}.
+For example, if we begin with an unmapped, [[JI]] interval <math>\frac{10}{9}</math> with [[prime-count vector]] (or "[[monzo]]") <math>\textbf{i} =</math> {{ket|1 -2 1}}, the mapped interval ~<math>\frac{10}{9}</math> under [[meantone temperament]] {{rket|{{bra|1 1 0}} {{bra|0 1 4}}}} would have [[generator-count vector]] (or "[[tmonzo]]") <math>\textbf{y} =</math> {{rket|{{bra|1 1 0}} {{bra|0 1 4}}}}{{ket|1 -2 1}} = {{rket|-1 2}}.
 Note that we've notated the mapped interval with a tilde, <span style="background-color: yellow">~</span><math>\frac{10}{9}</math>, to indicate that its size is now approximate.
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 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 [[comma]]s that are [[tempered out]] by a given temperament. Thus, a regular temperament can be seen as a sort of [[monzo|vector]] generalization of the "modulus" in {{w|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 [[Prime number|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, {{rket|-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 {{rket|-1 2}} or P8.P5 {{rket|-1 2}}. If the generator basis is an octave and a perfect twelfth, the same pitch would be written as ~2.~3 {{rket|-3 2}} or P8.P12 {{rket|-3 2}}. For transformations between these, refer to [[generator form manipulation]].
 == Terminology ==