Mapped interval: Difference between revisions

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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).

Tmonzo represents mapped interval, but it is not popular as notation. 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 write 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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 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).
+Tmonzo represents mapped interval, but it is not popular as notation. 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 write 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 ==