Mapping: Difference between revisions

(4 intermediate revisions by 2 users not shown)

Line 1:

{{interwiki

| en = Mapping

| ja = マッピング

}}

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 ==

Line 17:

Line 21:

<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 312/219 (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 312/219 (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 ===

Line 23:

Line 27:

=== 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 }}"<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 ===

Line 55:

Line 59:

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.

Line 63:

Line 70:

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:

{~~{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:

{~~{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.

Line 88:

Line 104:

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.

Line 96:

Line 115:

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.

Line 110:

Line 132:

* [[Reduced mapping]]

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

== Notes ==

[[Category:Regular temperament theory]]

@@ Line 1: / Line 1: @@
+{{interwiki
+| en = Mapping
+| ja = マッピング
+}}
 {{Beginner|Temperament mapping matrix}}
 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&mdash;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&mdash;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 ==
@@ Line 17: / Line 21: @@
 <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&#x266D; above that is 1, the A&#x266D; below it is &minus;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&#x266D; above that is 1, the A&#x266D; below it is &minus;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 ===
@@ Line 23: / Line 27: @@
 === 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 }}"<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 ===
@@ Line 55: / Line 59: @@
 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 }} &ndash; period
+{| class="right-all left-5 left-6"
+|-
-{{val| d e f }} &ndash; 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.
@@ Line 63: / Line 70: @@
 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 || _ || _ || ]]
+|}
 /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 || _ || ]]
+|}
 /1 is simpler&mdash;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.
@@ Line 88: / Line 104: @@
 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&mdash;it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.
@@ Line 96: / Line 115: @@
 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.
@@ Line 110: / Line 132: @@
 * [[Reduced mapping]]
 * [[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]]