Mapping - Revision history

Naren: Add harmonic template to See also.

2026-06-15T19:12:03Z

Add harmonic template to See also.

← Older revision		Revision as of 19:12, 15 June 2026
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	* [[Reduced mapping]]		* [[Reduced mapping]]
	* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Mappings]] – for a step-by-step textbook tutorial style introduction to this topic		* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Mappings]] – for a step-by-step textbook tutorial style introduction to this topic
			* [[Harmonic template]]

	== Notes ==		== Notes ==

Currywurst44: Clarify usage of mapping

2026-05-12T13:49:02Z

Clarify usage of mapping

← Older revision		Revision as of 13:49, 12 May 2026
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	A [[regular temperament]] is more than simply a set of pitches. It's a set of notes together with a ''consistent rule'' that maps any pitch of the relevant [[just intonation subgroup]] to a specific note from that set. (In fact, an abstract regular temperament is not a set of definite pitches at all! The pitches can vary, and the rule mapping JI pitches to notes is the thing that uniquely characterizes the temperament.) This consistent rule is known as the ''JI mapping'' or simply '''mapping'''. The mapping answers the question "how do I play this JI pitch as a note of this temperament?". The answer will be the "tempered version" of that JI pitch, which may be a very close approximation or a very distant approximation depending on the circumstances.		A [[regular temperament]] is more than simply a set of pitches. It's a set of notes together with a ''consistent rule'' that maps any pitch of the relevant [[just intonation subgroup]] to a specific note from that set. (In fact, an abstract regular temperament is not a set of definite pitches at all! The pitches can vary, and the rule mapping JI pitches to notes is the thing that uniquely characterizes the temperament.) This consistent rule is known as the ''JI mapping'' or simply '''mapping'''. The mapping answers the question "how do I play this JI pitch as a note of this temperament?". The answer will be the "tempered version" of that JI pitch, which may be a very close approximation or a very distant approximation depending on the circumstances.

	Naively, one might think that a simple rounding function might be suitable for a mapping: let the "tempered version" of each JI pitch simply be the tempered pitch that is closest to it. However, this (usually) does not result in a regular temperament at all! The reason is that, although this mapping assigns a tempered pitch to each JI pitch, it does not do so in a ''consistent'' way—some instances of the same JI interval are represented by different tempered intervals if they occur in different places. A regular temperament mapping always represents each JI interval by the ''same'' tempered interval, even if that tempered interval is not the closest tempered interval to the JI interval.		Naively, one might think that a simple rounding function might be suitable for a mapping: let the "tempered version" of each JI pitch simply be the tempered pitch that is closest to it. However, this (usually) does not result in a regular temperament at all! The reason is that, although this (non-linear) mapping assigns a tempered pitch to each JI pitch, it does not do so in a ''consistent'' way—some instances of the same JI interval are represented by different tempered intervals if they occur in different places. A regular temperament mapping always represents each JI interval by the ''same'' tempered interval, even if that tempered interval is not the closest tempered interval to the JI interval.

	== A note on mathematical terminology ==		== A note on mathematical terminology ==

FloraC: Turn the mappings into tables. This allows vertical alignment

2025-08-10T18:55:10Z

Turn the mappings into tables. This allows vertical alignment

← Older revision		Revision as of 18:55, 10 August 2025
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	At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:		At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:

	{~~{val~~\| a b c ~~}} –~~ period		{\| class="right-all left-5 left-6"
			\|-
	~~{{val~~\| d e f }~~} – generator~~		\| [⟨ \|\| ''a'' \|\| ''b'' \|\| ''c'' \|\| ] \|\| – period
			\|-
			\| ⟨ \|\| ''d'' \|\| ''e'' \|\| ''f'' \|\| ]] \|\| – generator
			\|}

	The top val is taken by convention to represent the generator chain which is the period, and the bottom one is taken to represent the one which is not.		The top val is taken by convention to represent the generator chain which is the period, and the bottom one is taken to represent the one which is not.
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	When mapping a prime in JI onto a rank 2 temperament, one must think about how many steps of each type of generator it takes to reach the final tempered prime interval. For an example, we'll look at meantone temperament, and we'll start by mapping 2/1. We'll assume that the period is 2/1, and the generator is 3/2. 2/1 maps to one period and zero generators, and hence we arrive at		When mapping a prime in JI onto a rank 2 temperament, one must think about how many steps of each type of generator it takes to reach the final tempered prime interval. For an example, we'll look at meantone temperament, and we'll start by mapping 2/1. We'll assume that the period is 2/1, and the generator is 3/2. 2/1 maps to one period and zero generators, and hence we arrive at

	{~~{val~~\| 1 _ _ }}		{\| class="right-all left-5"
			\|-
	~~{{val~~\| 0 _ _ }}		\| [⟨ \|\| 1 \|\| _ \|\| _ \|\| ]
			\|-
			\| ⟨ \|\| 0 \|\| _ \|\| _ \|\| ]]
			\|}

	3/1 is slightly more complicated – it requires one step along the 2/1 period chain, plus one step along the 3/2 generator chain, to get to 3/1. This is represented by the following mapping:		3/1 is slightly more complicated – it requires one step along the 2/1 period chain, plus one step along the 3/2 generator chain, to get to 3/1. This is represented by the following mapping:

	{~~{val~~\| 1 1 _ }}		{\| class="right-all left-5"
			\|-
	~~{{val~~\| 0 1 _ }}		\| [⟨ \|\| 1 \|\| 1 \|\| _ \|\| ]
			\|-
			\| ⟨ \|\| 0 \|\| 1 \|\| _ \|\| ]]
			\|}

	5/1 is simpler—we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:		5/1 is simpler—we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:

	{~~{val~~\| 1 1 0 }}		{\| class="right-all left-5"
			\|-
	~~{{val~~\| 0 1 4 }}		\| [⟨ \|\| 1 \|\| 1 \|\| 0 \|\| ]
			\|-
			\| ⟨ \|\| 0 \|\| 1 \|\| 4 \|\| ]]
			\|}

	This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.		This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.
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	If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following list of vals:		If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following list of vals:

	{~~{val~~\| 1 2 4 }}		{\| class="right-all left-5"
			\|-
	~~{{val~~\| 0 -1 -4 }}		\| [⟨ \|\| 1 \|\| 2 \|\| 4 \|\| ]
			\|-
			\| ⟨ \|\| 0 \|\| -1 \|\| -4 \|\| ]]
			\|}

	This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods ''minus'' a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's ''down'', plus four octaves—it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.		This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods ''minus'' a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's ''down'', plus four octaves—it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.
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	If you wanted your basis to be 2/1 and 3/1, you'd end up with the following list of vals (left as an exercise to the reader to derive):		If you wanted your basis to be 2/1 and 3/1, you'd end up with the following list of vals (left as an exercise to the reader to derive):

	{~~{val~~\| 1 0 -4 }}		{\| class="right-all left-5"
			\|-
	~~{{val~~\| 0 1 4 }}		\| [⟨ \|\| 1 \|\| 0 \|\| -4 \|\| ]
			\|-
			\| ⟨ \|\| 0 \|\| 1 \|\| 4 \|\| ]]
			\|}

	The [[normal lists #Normal val list\|normal val list]] is a normalized form among the variety of writing the mapping matrices, and it is what appears in temperament pages on this wiki.		The [[normal lists #Normal val list\|normal val list]] is a normalized form among the variety of writing the mapping matrices, and it is what appears in temperament pages on this wiki.

FloraC: Review

2025-08-10T18:36:53Z

Review

← Older revision		Revision as of 18:36, 10 August 2025
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	=== Notation ===		=== Notation ===
	In regular temperament theory there is a special notation for this kind of JI mapping. We notate the 3-limit 12edo temperament described above as "{{val\| 12 19 }}", because the first prime (2) is mapped to 12 steps, and the second prime (3) is mapped to 19 steps. This mathematical object is known as a "mapping matrix" and it summarizes all the information in the mapping in a very compact form. Since this is an equal temperament, the mapping matrix contains only one row, and since it's a 3-limit temperament, the mapping matrix contains two columns, representing the primes 2 and 3. Such mapping with one row ~~in [[Extended bra-ket notation\|bra notation]]~~ is called [[val]]~~. And~~ mapping matrix in multiple vals (described below) is called '''val list'''.		In regular temperament theory there is a special notation for this kind of JI mapping. We notate the 3-limit 12edo temperament described above as "{{val\| 12 19 }}"<ref group="note">Known as [[extended bra-ket notation]].</ref>, because the first prime (2) is mapped to 12 steps, and the second prime (3) is mapped to 19 steps. This mathematical object is known as a "mapping matrix" and it summarizes all the information in the mapping in a very compact form. Since this is an equal temperament, the mapping matrix contains only one row, and since it's a 3-limit temperament, the mapping matrix contains two columns, representing the primes 2 and 3. Such mapping with one row is called a [[val]], and a mapping matrix in multiple vals (described below) is also called a '''val list'''.

	=== Many 12edo temperaments ===		=== Many 12edo temperaments ===
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	* [[Reduced mapping]]		* [[Reduced mapping]]
	* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Mappings]] – for a step-by-step textbook tutorial style introduction to this topic		* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Mappings]] – for a step-by-step textbook tutorial style introduction to this topic

			== Notes ==
			<references group="note"/>

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

Dummy index: /* Equal temperament mappings */ +monzo, +val list

2025-08-09T05:08:43Z

Equal temperament mappings: +monzo, +val list

← Older revision		Revision as of 05:08, 9 August 2025
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	<math>\left\{440\cdot 2^a\cdot 3^b\,\middle\|\,a,b\in\mathbb Z\right\}</math>		<math>\left\{440\cdot 2^a\cdot 3^b\,\middle\|\,a,b\in\mathbb Z\right\}</math>

	Let's use integers to represent the 12edo notes, so that A440 is note 0, the B♭ above that is 1, the A♭ below it is −1, and so on. Then the mapping is simply expressed by saying that each factor of 2 counts for 12 steps, and each factor of 3 counts for 19 steps (because 3/1, or 1901.955… cents, is approximated as 1900 cents, or 19 steps of 12edo). (If you want a mathematical formula, that means that the above expression is mapped to {{nowrap\|12''a'' + 19''b''}}.) So, for example, 1/1 is mapped to note 0, which is exactly A440; 2/1 is mapped to note 12, the A one octave higher; 3/2 is mapped to note 7 (the E above A440); and 3<sup>12</sup>/2<sup>19</sup> (the Pythagorean comma) is mapped to 0, the same note as 1/1.		Let's use integers to represent the 12edo notes, so that A440 is note 0, the B♭ above that is 1, the A♭ below it is −1, and so on. Then the mapping is simply expressed by saying that each factor of 2 counts for 12 steps, and each factor of 3 counts for 19 steps (because 3/1, or 1901.955… cents, is approximated as 1900 cents, or 19 steps of 12edo). (If you want a mathematical formula, that means that the above expression is mapped to {{nowrap\|12''a'' + 19''b''}}.) So, for example, 1/1 is mapped to note 0, which is exactly A440; 2/1 is mapped to note 12, the A one octave higher; 3/2 is mapped to note 7 (the E above A440); and 3<sup>12</sup>/2<sup>19</sup> (the Pythagorean comma) is mapped to 0, the same note as 1/1. ''a'' and ''b'' are read from such prime factorization or [[monzo]].

	=== Contrast with rounding ===		=== Contrast with rounding ===
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	=== Notation ===		=== Notation ===
	In regular temperament theory there is a special notation for this kind of JI mapping. We notate the 3-limit 12edo temperament described above as "{{val\| 12 19 }}", because the first prime (2) is mapped to 12 steps, and the second prime (3) is mapped to 19 steps. This mathematical object is known as a "mapping matrix" and it summarizes all the information in the mapping in a very compact form. Since this is an equal temperament, the mapping matrix contains only one row, and since it's a 3-limit temperament, the mapping matrix contains two columns, representing the primes 2 and 3.		In regular temperament theory there is a special notation for this kind of JI mapping. We notate the 3-limit 12edo temperament described above as "{{val\| 12 19 }}", because the first prime (2) is mapped to 12 steps, and the second prime (3) is mapped to 19 steps. This mathematical object is known as a "mapping matrix" and it summarizes all the information in the mapping in a very compact form. Since this is an equal temperament, the mapping matrix contains only one row, and since it's a 3-limit temperament, the mapping matrix contains two columns, representing the primes 2 and 3. Such mapping with one row in [[Extended bra-ket notation\|bra notation]] is called [[val]]. And mapping matrix in multiple vals (described below) is called '''val list'''.

	=== Many 12edo temperaments ===		=== Many 12edo temperaments ===

Dummy index: add interwiki

2025-08-08T14:47:17Z

add interwiki

← Older revision		Revision as of 14:47, 8 August 2025
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			{{interwiki
			\| en = Mapping
			\| ja = マッピング
			}}
	{{Beginner\|Temperament mapping matrix}}		{{Beginner\|Temperament mapping matrix}}

Fredg999: Undo revision 187782 by VectorGraphics (talk) The term "matrix" is present throughout the page, so it is appropriate to introduce it at the beginning, alongside the rest of the note which is also useful to disambiguate from other meanings of the term "mapping".

2025-03-23T23:40:04Z

Undo revision 187782 by VectorGraphics (talk) The term "matrix" is present throughout the page, so it is appropriate to introduce it at the beginning, alongside the rest of the note which is also useful to disambiguate from other meanings of the term "mapping".

← Older revision		Revision as of 23:40, 23 March 2025
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	Naively, one might think that a simple rounding function might be suitable for a mapping: let the "tempered version" of each JI pitch simply be the tempered pitch that is closest to it. However, this (usually) does not result in a regular temperament at all! The reason is that, although this mapping assigns a tempered pitch to each JI pitch, it does not do so in a ''consistent'' way—some instances of the same JI interval are represented by different tempered intervals if they occur in different places. A regular temperament mapping always represents each JI interval by the ''same'' tempered interval, even if that tempered interval is not the closest tempered interval to the JI interval.		Naively, one might think that a simple rounding function might be suitable for a mapping: let the "tempered version" of each JI pitch simply be the tempered pitch that is closest to it. However, this (usually) does not result in a regular temperament at all! The reason is that, although this mapping assigns a tempered pitch to each JI pitch, it does not do so in a ''consistent'' way—some instances of the same JI interval are represented by different tempered intervals if they occur in different places. A regular temperament mapping always represents each JI interval by the ''same'' tempered interval, even if that tempered interval is not the closest tempered interval to the JI interval.

			== A note on mathematical terminology ==
			In mathematics generally, "mapping" is synonymous with "map" and "function". In RTT, "mapping" has the more specific meaning of a {{w\|Linear map\|''linear'' mapping}}, which is a function that can be represented by a matrix.

	== Equal temperament mappings ==		== Equal temperament mappings ==
	An equal temperament, also known as a rank-1 temperament (see below for a discussion of rank), is not merely a set of equally spaced pitches. An equal temperament consists of		An equal temperament, also known as a rank-1 temperament (see below for a discussion of rank), is not merely a set of equally spaced pitches. An equal temperament consists of
	# A JI subgroup that is being represented, such as "5-limit JI", and		# A JI subgroup that is being represented, such as "5-limit JI", and
	# A mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).		# A mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).

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	For example:		For example:
	# An equal temperament is rank 1, as it exists in its entirety as a stack of one single generator.		# An equal temperament is rank 1, as it exists in its entirety as a stack of one single generator.
	# Temperaments which consist of two generators, or more commonly a "period" and a generator, are rank-2. Meantone is a good example, as its separate chain of fifths and chain of octaves constitute two independent generator chains.		# Temperaments which consist of two generators, or more commonly a "period" and a generator, are rank-2. Meantone is a good example, as its separate chain of fifths and chain of octaves constitute two independent generator chains.
	# Temperaments which consist of three generators, or more commonly a period and two generators, are rank-3. 5-limit JI, while not being a "temperament" in the traditional sense, would nonetheless be considered rank 3, as its three generators are 2/1, 3/1, and 5/1 (or 2/1, 3/2, and 5/4 if you'd like).		# Temperaments which consist of three generators, or more commonly a period and two generators, are rank-3. 5-limit JI, while not being a "temperament" in the traditional sense, would nonetheless be considered rank 3, as its three generators are 2/1, 3/1, and 5/1 (or 2/1, 3/2, and 5/4 if you'd like).
	# 7-limit JI would be rank-4, 11-limit JI would be rank-5, 13-limit JI would be rank-6, etc.		# 7-limit JI would be rank-4, 11-limit JI would be rank-5, 13-limit JI would be rank-6, etc.
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	== Units ==		== Units ==
	It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].		It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].

	~~== A note on mathematical terminology ==~~
	In mathematics generally, "mapping" is synonymous with "map" and "function". In RTT, "mapping" has the more specific meaning of a {{w\|Linear map\|''linear'' mapping}}, which is a function that can be represented by a matrix.

	== See also ==		== See also ==

VectorGraphics: Move correspondence with mathematical terminology to the bottom, beginners don't need to know about matrices for this

2025-03-23T04:37:15Z

Move correspondence with mathematical terminology to the bottom, beginners don't need to know about matrices for this

← Older revision		Revision as of 04:37, 23 March 2025
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	Naively, one might think that a simple rounding function might be suitable for a mapping: let the "tempered version" of each JI pitch simply be the tempered pitch that is closest to it. However, this (usually) does not result in a regular temperament at all! The reason is that, although this mapping assigns a tempered pitch to each JI pitch, it does not do so in a ''consistent'' way—some instances of the same JI interval are represented by different tempered intervals if they occur in different places. A regular temperament mapping always represents each JI interval by the ''same'' tempered interval, even if that tempered interval is not the closest tempered interval to the JI interval.		Naively, one might think that a simple rounding function might be suitable for a mapping: let the "tempered version" of each JI pitch simply be the tempered pitch that is closest to it. However, this (usually) does not result in a regular temperament at all! The reason is that, although this mapping assigns a tempered pitch to each JI pitch, it does not do so in a ''consistent'' way—some instances of the same JI interval are represented by different tempered intervals if they occur in different places. A regular temperament mapping always represents each JI interval by the ''same'' tempered interval, even if that tempered interval is not the closest tempered interval to the JI interval.

	~~== A note on mathematical terminology ==~~
	In mathematics generally, "mapping" is synonymous with "map" and "function". In RTT, "mapping" has the more specific meaning of a {{w\|Linear map\|''linear'' mapping}}, which is a function that can be represented by a matrix.

	== Equal temperament mappings ==		== Equal temperament mappings ==
	An equal temperament, also known as a rank-1 temperament (see below for a discussion of rank), is not merely a set of equally spaced pitches. An equal temperament consists of		An equal temperament, also known as a rank-1 temperament (see below for a discussion of rank), is not merely a set of equally spaced pitches. An equal temperament consists of
	# A JI subgroup that is being represented, such as "5-limit JI", and		# A JI subgroup that is being represented, such as "5-limit JI", and
	# A mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).		# A mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).

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	For example:		For example:
	# An equal temperament is rank 1, as it exists in its entirety as a stack of one single generator.		# An equal temperament is rank 1, as it exists in its entirety as a stack of one single generator.
	# Temperaments which consist of two generators, or more commonly a "period" and a generator, are rank-2. Meantone is a good example, as its separate chain of fifths and chain of octaves constitute two independent generator chains.		# Temperaments which consist of two generators, or more commonly a "period" and a generator, are rank-2. Meantone is a good example, as its separate chain of fifths and chain of octaves constitute two independent generator chains.
	# Temperaments which consist of three generators, or more commonly a period and two generators, are rank-3. 5-limit JI, while not being a "temperament" in the traditional sense, would nonetheless be considered rank 3, as its three generators are 2/1, 3/1, and 5/1 (or 2/1, 3/2, and 5/4 if you'd like).		# Temperaments which consist of three generators, or more commonly a period and two generators, are rank-3. 5-limit JI, while not being a "temperament" in the traditional sense, would nonetheless be considered rank 3, as its three generators are 2/1, 3/1, and 5/1 (or 2/1, 3/2, and 5/4 if you'd like).
	# 7-limit JI would be rank-4, 11-limit JI would be rank-5, 13-limit JI would be rank-6, etc.		# 7-limit JI would be rank-4, 11-limit JI would be rank-5, 13-limit JI would be rank-6, etc.
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	== Units ==		== Units ==
	It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].		It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].

			== A note on mathematical terminology ==
			In mathematics generally, "mapping" is synonymous with "map" and "function". In RTT, "mapping" has the more specific meaning of a {{w\|Linear map\|''linear'' mapping}}, which is a function that can be represented by a matrix.

	== See also ==		== See also ==

FloraC: +link

2025-02-07T12:39:18Z

+link

← Older revision		Revision as of 12:39, 7 February 2025
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	{{val\| 0 1 4 }}		{{val\| 0 1 4 }}

	The [[~~Normal~~ lists #Normal val list\|normal val list]] is a normalized form among the variety of writing the mapping matrices, and it is what appears in temperament pages on this wiki.		The [[normal lists #Normal val list\|normal val list]] is a normalized form among the variety of writing the mapping matrices, and it is what appears in temperament pages on this wiki.

	For more information on these sorts of operations, see [[Generator size manipulation]].		For more information on these sorts of operations, see [[Generator size manipulation]].
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	== See also ==		== See also ==
	* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Mappings]] ~~–~~ for a step-by-step textbook tutorial style introduction to this topic		* [[Reduced mapping]]
			* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Mappings]] – for a step-by-step textbook tutorial style introduction to this topic

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

ArrowHead294 at 18:39, 31 October 2024

2024-10-31T18:39:22Z

← Older revision		Revision as of 18:39, 31 October 2024
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	<math>\left\{440\cdot 2^a\cdot 3^b\,\middle\|\,a,b\in\mathbb Z\right\}</math>		<math>\left\{440\cdot 2^a\cdot 3^b\,\middle\|\,a,b\in\mathbb Z\right\}</math>

	Let's use integers to represent the 12edo notes, so that A440 is note 0, the B♭ above that is 1, the A♭ below it is &~~minud~~;, and so on. Then the mapping is simply expressed by saying that each factor of 2 counts for 12 steps, and each factor of 3 counts for 19 steps (because 3/1, or 1901.955… cents, is approximated as 1900 cents, or 19 steps of 12edo). (If you want a mathematical formula, that means that the above expression is mapped to 12''a'' + 19''b''.) So, for example, 1/1 is mapped to note 0, which is exactly A440; 2/1 is mapped to note 12, the A one octave higher; 3/2 is mapped to note 7 (the E above A440); and 3<sup>12</sup>/2<sup>19</sup> (the Pythagorean comma) is mapped to 0, the same note as 1/1.		Let's use integers to represent the 12edo notes, so that A440 is note 0, the B♭ above that is 1, the A♭ below it is −1, and so on. Then the mapping is simply expressed by saying that each factor of 2 counts for 12 steps, and each factor of 3 counts for 19 steps (because 3/1, or 1901.955… cents, is approximated as 1900 cents, or 19 steps of 12edo). (If you want a mathematical formula, that means that the above expression is mapped to {{nowrap\|12''a'' + 19''b''}}.) So, for example, 1/1 is mapped to note 0, which is exactly A440; 2/1 is mapped to note 12, the A one octave higher; 3/2 is mapped to note 7 (the E above A440); and 3<sup>12</sup>/2<sup>19</sup> (the Pythagorean comma) is mapped to 0, the same note as 1/1.

	=== Contrast with rounding ===		=== Contrast with rounding ===
	Now, consider the pitch 3<sup>36</sup>/2<sup>57</sup>. In JI, this pitch is 70.38… cents above A440, so the closest 12edo note to it is B♭. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (B♭). Why is this? The pitch 3<sup>36</sup>/2<sup>57</sup>is three Pythagorean commas above A. If each Pythagorean comma is represented by 0 steps, then since 0 + 0 + 0 = 0 the pitch 3<sup>36</sup>/2<sup>57</sup> must be represented by A, even though in JI it's closer to B♭. Mapping it to B♭ would require one of the three Pythagorean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.		Now, consider the pitch 3<sup>36</sup>/2<sup>57</sup>. In JI, this pitch is 70.38… cents above A440, so the closest 12edo note to it is B♭. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (B♭). Why is this? The pitch 3<sup>36</sup>/2<sup>57</sup>is three Pythagorean commas above A. If each Pythagorean comma is represented by 0 steps, then since {{nowrap\|0 + 0 + 0 {{=}} 0}} the pitch 3<sup>36</sup>/2<sup>57</sup> must be represented by A, even though in JI it's closer to B♭. Mapping it to B♭ would require one of the three Pythagorean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.

	=== Notation ===		=== Notation ===
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	* A♯ = A440 + 4 octaves − 7 perfect fourths		* A♯ = A440 + 4 octaves − 7 perfect fourths

	In other words, every note can be represented as an ordered pair of integers (''x'', ''y'') where ''x'' is the number of octaves from A440 (positive is up, negative is down), and ''y'' is the number of perfect fourths.		In other words, every note can be represented as an ordered pair of integers {{nowrap\|(''x'', ''y'')}} where ''x'' is the number of octaves from A440 (positive is up, negative is down), and ''y'' is the number of perfect fourths.

	== Temperamental rank ==		== Temperamental rank ==
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	At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:		At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:

	{{val\| a b c }} – period		{{val\| a b c }} – period

	{{val\| d e f }} – generator		{{val\| d e f }} – generator

	The top val is taken by convention to represent the generator chain which is the period, and the bottom one is taken to represent the one which is not.		The top val is taken by convention to represent the generator chain which is the period, and the bottom one is taken to represent the one which is not.
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	{{val\| 0 1 _ }}		{{val\| 0 1 _ }}

	5/1 is simpler – we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:		5/1 is simpler—we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:

	{{val\| 1 1 0 }}		{{val\| 1 1 0 }}
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	{{val\| 0 -1 -4 }}		{{val\| 0 -1 -4 }}

	This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods ''minus'' a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's ''down'', plus four octaves – it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.		This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods ''minus'' a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's ''down'', plus four octaves—it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.

	If you wanted your basis to be 2/1 and 3/1, you'd end up with the following list of vals (left as an exercise to the reader to derive):		If you wanted your basis to be 2/1 and 3/1, you'd end up with the following list of vals (left as an exercise to the reader to derive):
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	== Units ==		== Units ==
	It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: units~~ analysis]].		It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].

	== See also ==		== See also ==
	* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT~~: mappings~~]], for a step-by-step textbook tutorial style introduction to this topic		* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Mappings]] – for a step-by-step textbook tutorial style introduction to this topic

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