Mapped interval - Revision history

VectorGraphics at 02:39, 31 August 2025

2025-08-31T02:39:07Z

← Older revision		Revision as of 02:39, 31 August 2025
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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 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]].		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 ==		== Terminology ==

Dummy index: add a notice for tmonzo

2025-08-30T16:22:54Z

add a notice for tmonzo

← Older revision		Revision as of 16:22, 30 August 2025
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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 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 ==		== Terminology ==

Dummy index: add interwiki

2025-08-30T15:46:12Z

add interwiki

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			{{interwiki
			\| en = Mapped interval
			\| ja = マップされた音程
			}}
	{{Beginner\|Tmonzos and tvals}}		{{Beginner\|Tmonzos and tvals}}
	A '''mapped interval''' is an [[interval]] that has been mapped by a [[mapping]] matrix for a [[regular temperament]].		A '''mapped interval''' is an [[interval]] that has been mapped by a [[mapping]] matrix for a [[regular temperament]].

Dummy index: fix calc

2025-08-29T15:27:24Z

fix calc

← Older revision		Revision as of 15:27, 29 August 2025
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	A '''mapped interval''' is an [[interval]] that has been mapped by a [[mapping]] matrix for a [[regular temperament]].		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.		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.

Fredg999: Merge from Tempered interval (mathematical theory), misc. edits

2025-03-24T01:50:04Z

Merge from Tempered interval (mathematical theory), misc. edits

← Older revision		Revision as of 01:50, 24 March 2025
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	* 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>.		* 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>.

	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.		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).

			== 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 ==		== See also ==

ArrowHead294: Update links

2024-11-02T17:58:28Z

Update links

← Older revision		Revision as of 17:58, 2 November 2024
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	Here are several mnemonics for the use of <math>\textbf{y}</math> as the symbol for mapped intervals:		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>.		* 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>.

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	== See also ==		== 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~~: 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]])
	* [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: units~~ analysis#Mapped interval]]: a units analysis of mapped intervals (also [[~~Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_units_analysis~~#Rank-~~2_mapped_interval~~\|the following section, for a rank-2 example]])

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Tuning]]		[[Category:Tuning]]

Fredg999: Beginner template

2023-07-08T19:02:03Z

Beginner template

← Older revision		Revision as of 19:02, 8 July 2023
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			{{Beginner\|Tmonzos and tvals}}
	A '''mapped interval''' is an [[interval]] that has been mapped by a [[mapping]] matrix for a [[regular temperament]].		A '''mapped interval''' is an [[interval]] that has been mapped by a [[mapping]] matrix for a [[regular temperament]].

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	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.		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==		== 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: 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_%26_Douglas_Blumeyer%27s_guide_to_RTT:_units_analysis#Rank-2_mapped_interval\|the following section, for a rank-2 example]])		* [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Mapped interval]]: a units analysis of mapped intervals (also [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_units_analysis#Rank-2_mapped_interval\|the following section, for a rank-2 example]])
	* [[Tmonzos and tvals]]: a more mathematical explanation

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Tuning]]		[[Category:Tuning]]

Cmloegcmluin: /* See also */ add another link

2022-12-16T15:50:16Z

Cmloegcmluin at 16:09, 15 December 2022

2022-12-15T16:09:16Z

← Older revision		Revision as of 16:09, 15 December 2022
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	* 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 — two 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>.		* 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>.

			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==		==See also==

Cmloegcmluin: include jargon terms

2022-12-15T00:26:25Z

include jargon terms

← Older revision		Revision as of 00:26, 15 December 2022
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	A '''mapped interval''' is an [[interval]] that has been mapped by a [[mapping]] matrix for a [[regular temperament]].		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\|-3 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.		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.

← Older revision		Revision as of 15:50, 16 December 2022
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	* [[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: 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_%26_Douglas_Blumeyer%27s_guide_to_RTT:_units_analysis#Rank-2_mapped_interval\|the following section, for a rank-2 example]])
	* [[Tmonzos and tvals]]: a more mathematical explanation		* [[Tmonzos and tvals]]: a more mathematical explanation

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Tuning]]		[[Category:Tuning]]