Mapped interval: Difference between revisions

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

{{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]] <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]] <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.

Here are several mnemonics for the use of <math>\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.

* 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.

* 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>\mathbf{y}</math> is a generator-count vector, associated with the generator tuning map <math>𝒈</math>, in the sense that intervals are associated with (tempered-prime) tuning maps <math>𝒕</math>, or in other words, <math>𝒕\textbf{i} = 𝒈\textbf{y}</math>.

~~==See also==~~

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).

* [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: tuning~~ fundamentals#The RTT version]]: another take at explaining mapped intervals

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]].

* [[~~Tmonzos and tvals~~]]: a ~~more mathematical explanation~~

== Terminology ==

A "mapped interval" could also be called a "tempered interval"; however, "tempered interval" is more ambiguous: "tempered interval" could also refer to a [[span|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 {{ket|0 0 1/4}} generator of quarter-comma meantone. Only "''mapp''ed interval" unambiguously refers to an interval that has been transformed only by the ''mapp''ing 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 [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis#Rank-2 mapped interval|the following section, for a rank-2 example]])

[[Category:Regular temperament theory]]

[[Category:Tuning]]

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-A '''mapped interval''' is an interval that has been mapped by a [[mapping]] matrix for a [[regular temperament]].
+{{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]] <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]] <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.
 Here are several mnemonics for the use of <math>\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.
+* 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.
+* Visually, a "Y" also looks like a diagram showing&mdash;from the top&mdash;wo just intervals getting mapped to the same size.
 * A 'y' also looks like a 'g', which is fitting because <math>\mathbf{y}</math> is a generator-count vector, associated with the generator tuning map <math>𝒈</math>, in the sense that intervals are associated with (tempered-prime) tuning maps <math>𝒕</math>, or in other words, <math>𝒕\textbf{i} = 𝒈\textbf{y}</math>.
-==See also==
+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).
-* [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning fundamentals#The RTT version]]: another take at explaining mapped intervals
+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]].
-* [[Tmonzos and tvals]]: a more mathematical explanation
+== Terminology ==
+A "mapped interval" could also be called a "tempered interval"; however, "tempered interval" is more ambiguous: "tempered interval" could also refer to a [[span|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 {{ket|0 0 1/4}} generator of quarter-comma meantone. Only "''mapp''ed interval" unambiguously refers to an interval that has been transformed only by the ''mapp''ing 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 [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis#Rank-2 mapped interval|the following section, for a rank-2 example]])
 [[Category:Regular temperament theory]]
 [[Category:Tuning]]