Intro to Mappings: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:clumma|clumma]] and made on <tt>2016-07-27 14:44:39 UTC</tt>.

: This revision was by author [[User:clumma|clumma]] and made on <tt>2016-07-27 14:47:13 UTC</tt>.

: The original revision id was <tt>~~588211341~~</tt>.

: The original revision id was <tt>588211841</tt>.

: The revision comment was: <tt></tt>

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A [[Regular Temperaments|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 subgroups|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.

~~Naïvely~~, 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).

=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 (1) a JI subgroup that is being represented, such as "5-limit JI", and (2) a mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).

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 **//1.//** a JI subgroup that is being represented, such as "5-limit JI", and **//2.//** a mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).

As an example, let's consider the familiar [[12edo]] considered as a 3-limit temperament. For concreteness, let's use A440 as the base note. In this case the JI subgroup is the [[3-limit]], that is, all pitches that form a JI interval with A440 whose prime factorization contains no primes other than 2 and 3. (For people familiar with mathematical notation, this can be written as

As an example, let's consider the familiar [[12edo]] considered as a 3-limit temperament. For concreteness, let's use A440 as the base note. In this case the JI subgroup is the [[3-limit]], that is, all pitches that form a JI interval with A440 whose prime factorization contains no primes other than 2 and 3. For people familiar with mathematical notation, this can be written as

[[math]]

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[[math]]

) Let's use integers to represent the 12edo notes, so that A440 is note 0, the Bb above that is 1, the Ab 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 12a+19b.) 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 Bb above that is 1, the Ab 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 12a+19b.) 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.

==Contrast with rounding==

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Intro to Mappings</title></head><body>A <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a> 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 <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a> 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 &quot;how do I play this JI pitch as a note of this temperament?&quot;. The answer will be the &quot;tempered version&quot; of that JI pitch, which may be a very close approximation or a very distant approximation depending on the circumstances.

~~Naïvely~~, one might think that a simple rounding function might be suitable for a mapping: let the &quot;tempered version&quot; 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 &quot;tempered version&quot; 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).

<h1 id="toc0"><a name="Equal temperament mappings"></a>Equal temperament mappings</h1>

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 (1) a JI subgroup that is being represented, such as &quot;5-limit JI&quot;, and (2) a mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).

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 1. a JI subgroup that is being represented, such as &quot;5-limit JI&quot;, and 2. a mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).

As an example, let's consider the familiar <a class="wiki_link" href="/12edo">12edo</a> considered as a 3-limit temperament. For concreteness, let's use A440 as the base note. In this case the JI subgroup is the <a class="wiki_link" href="/3-limit">3-limit</a>, that is, all pitches that form a JI interval with A440 whose prime factorization contains no primes other than 2 and 3. (For people familiar with mathematical notation, this can be written as

As an example, let's consider the familiar <a class="wiki_link" href="/12edo">12edo</a> considered as a 3-limit temperament. For concreteness, let's use A440 as the base note. In this case the JI subgroup is the <a class="wiki_link" href="/3-limit">3-limit</a>, that is, all pitches that form a JI interval with A440 whose prime factorization contains no primes other than 2 and 3. For people familiar with mathematical notation, this can be written as

<!-- ws:start:WikiTextMathRule:0:

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--><script type="math/tex">\left\{440\cdot 2^a\cdot 3^b\middle|a,b\in\mathbb Z\right\}</script>

) Let's use integers to represent the 12edo notes, so that A440 is note 0, the Bb above that is 1, the Ab 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 12a+19b.) 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 Bb above that is 1, the Ab 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 12a+19b.) 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.

<h2 id="toc1"><a name="Equal temperament mappings-Contrast with rounding"></a>Contrast with rounding</h2>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:clumma|clumma]] and made on <tt>2016-07-27 14:44:39 UTC</tt>.<br>
+: This revision was by author [[User:clumma|clumma]] and made on <tt>2016-07-27 14:47:13 UTC</tt>.<br>
-: The original revision id was <tt>588211341</tt>.<br>
+: The original revision id was <tt>588211841</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 8: / Line 8: @@
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A [[Regular Temperaments|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 subgroups|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.
-Naïvely, 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).
 =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 (1) a JI subgroup that is being represented, such as "5-limit JI", and (2) a mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).
+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 **//1.//** a JI subgroup that is being represented, such as "5-limit JI", and **//2.//** a mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).
-As an example, let's consider the familiar [[12edo]] considered as a 3-limit temperament. For concreteness, let's use A440 as the base note. In this case the JI subgroup is the [[3-limit]], that is, all pitches that form a JI interval with A440 whose prime factorization contains no primes other than 2 and 3. (For people familiar with mathematical notation, this can be written as
+As an example, let's consider the familiar [[12edo]] considered as a 3-limit temperament. For concreteness, let's use A440 as the base note. In this case the JI subgroup is the [[3-limit]], that is, all pitches that form a JI interval with A440 whose prime factorization contains no primes other than 2 and 3. For people familiar with mathematical notation, this can be written as
 [[math]]
@@ Line 20: / Line 20: @@
 [[math]]
-) Let's use integers to represent the 12edo notes, so that A440 is note 0, the Bb above that is 1, the Ab 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 12a+19b.) 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&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;/2&lt;span style="vertical-align: super;"&gt;19&lt;/span&gt; (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 Bb above that is 1, the Ab 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 12a+19b.) 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&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;/2&lt;span style="vertical-align: super;"&gt;19&lt;/span&gt; (the Pythagorean comma) is mapped to 0, the same note as 1/1.
 ==Contrast with rounding==
@@ Line 105: / Line 105: @@
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Intro to Mappings&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;regular temperament&lt;/a&gt; is more than simply a set of pitches. It's a set of notes together with a &lt;strong&gt;consistent rule&lt;/strong&gt; that maps any pitch of the relevant &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;just intonation subgroup&lt;/a&gt; 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 &lt;em&gt;JI mapping&lt;/em&gt; or simply &lt;em&gt;mapping&lt;/em&gt;. The mapping answers the question &amp;quot;how do I play this JI pitch as a note of this temperament?&amp;quot;. The answer will be the &amp;quot;tempered version&amp;quot; of that JI pitch, which may be a very close approximation or a very distant approximation depending on the circumstances.&lt;br /&gt;
 &lt;br /&gt;
-Naïvely, one might think that a simple rounding function might be suitable for a mapping: let the &amp;quot;tempered version&amp;quot; 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 &lt;strong&gt;consistent&lt;/strong&gt; 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 &lt;strong&gt;same&lt;/strong&gt; tempered interval (even if that tempered interval is not the closest tempered interval to the JI interval).&lt;br /&gt;
+Naively, one might think that a simple rounding function might be suitable for a mapping: let the &amp;quot;tempered version&amp;quot; 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 &lt;strong&gt;consistent&lt;/strong&gt; 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 &lt;strong&gt;same&lt;/strong&gt; tempered interval (even if that tempered interval is not the closest tempered interval to the JI interval).&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Equal temperament mappings"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;Equal temperament mappings&lt;/h1&gt;
 &lt;br /&gt;
-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 (1) a JI subgroup that is being represented, such as &amp;quot;5-limit JI&amp;quot;, and (2) a mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).&lt;br /&gt;
+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 &lt;strong&gt;&lt;em&gt;1.&lt;/em&gt;&lt;/strong&gt; a JI subgroup that is being represented, such as &amp;quot;5-limit JI&amp;quot;, and &lt;strong&gt;&lt;em&gt;2.&lt;/em&gt;&lt;/strong&gt; a mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).&lt;br /&gt;
 &lt;br /&gt;
-As an example, let's consider the familiar &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; considered as a 3-limit temperament. For concreteness, let's use A440 as the base note. In this case the JI subgroup is the &lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt;, that is, all pitches that form a JI interval with A440 whose prime factorization contains no primes other than 2 and 3. (For people familiar with mathematical notation, this can be written as&lt;br /&gt;
+As an example, let's consider the familiar &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; considered as a 3-limit temperament. For concreteness, let's use A440 as the base note. In this case the JI subgroup is the &lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt;, that is, all pitches that form a JI interval with A440 whose prime factorization contains no primes other than 2 and 3. For people familiar with mathematical notation, this can be written as&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:0:
@@ Line 118: / Line 118: @@
   --&gt;&lt;script type="math/tex"&gt;\left\{440\cdot 2^a\cdot 3^b\middle|a,b\in\mathbb Z\right\}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-) Let's use integers to represent the 12edo notes, so that A440 is note 0, the Bb above that is 1, the Ab 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 12a+19b.) 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&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;/2&lt;span style="vertical-align: super;"&gt;19&lt;/span&gt; (the Pythagorean comma) is mapped to 0, the same note as 1/1.&lt;br /&gt;
+Let's use integers to represent the 12edo notes, so that A440 is note 0, the Bb above that is 1, the Ab 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 12a+19b.) 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&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;/2&lt;span style="vertical-align: super;"&gt;19&lt;/span&gt; (the Pythagorean comma) is mapped to 0, the same note as 1/1.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Equal temperament mappings-Contrast with rounding"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Contrast with rounding&lt;/h2&gt;