Intro to Mappings: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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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">=Intro To Mappings=

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 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.

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).

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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]]

\left\{440\cdot 2^a\cdot 3^b\middle|a,b\in\mathbb Z\right\}

[[math]]

~~A val maps JI onto one chain~~ of ~~generators~~, or ~~relates~~ that ~~generator chain back~~ to JI. However, ~~many temperaments incorporate more than~~ one of ~~chain~~ of ~~generators~~. ~~The familiar meantone~~ temperament is an ~~example~~, as it ~~requires~~ two: the ~~fifth~~ and the ~~octave~~. ~~However~~, ~~vals only relate~~ a ~~single generator chain~~ to JI. ~~If we want to evolve out~~ of the ~~realm~~ of ~~isolated EDOs and consider higher~~-~~dimensional~~ temperaments ~~in their full glory~~, we'~~re going~~ to ~~have to raise~~ the ~~bar on what vals~~ can do for ~~us. Luckily~~, the ~~mathematics to do so~~ is ~~simple enough~~.

) 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<span style="vertical-align: super;">12</span>/2<span style="vertical-align: super;">19</span> (the Pythagorean comma) is mapped to 0, the same note as 1/1.

===Contrast with rounding===

Now, consider the pitch 3<span style="vertical-align: super;">36</span>/2<span style="vertical-align: super;">57</span>. In JI, this pitch is 70.38... cents above A440, so the closest 12edo note to it is Bb. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (Bb). Why is this? The pitch 3<span style="vertical-align: super;">36</span>/2<span style="vertical-align: super;">57</span>is three Pythagorean commas above A. If each Pythagorean comma is represented by 0 steps, then since 0+0+0=0 the pitch 3<span style="vertical-align: super;">36</span>/2<span style="vertical-align: super;">57 </span>must be represented by A, even though in JI it's closer to Bb. Mapping it to Bb would require one of the three Pythaogrean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.

===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 "<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.

===Many 12edo temperaments===

Now, let's consider 12edo, not as a 3-limit temperament, but as a [[5-limit]] temperament. This temperament maps all the 3-limit JI intervals in the same way as above, but in addition also maps the rest of the 5-limit JI intervals. Its mapping matrix is <12 19 28|. It's important to keep in mind that this is, technically speaking, a **different** regular temperament than <12 19|, even though they would both be referred to as "12-tone equal temperament" in common parlance.

Furthermore, consider 12edo as an 11-limit temperament. What is its mapping matrix? It actually depends whether you consider 11/8 a "very sharp D" or a "very flat D#". This choice results in two different mappings, <12 19 28 34 40| and <12 19 28 34 41|. The latter has a more accurate 11/8, but the former has more accurate versions of other intervals, including 12/11. In the language of regular temperament theory, these are simply two different 11-limit temperaments that both happen to have 12 steps per octave. Phrases like "11-limit 12edo" are thus ambiguous because they don't specify the mapping, and therefore don't refer to a specific temperament.

(Strictly speaking, "5-limit 12edo" or even "3-limit 12edo" are also ambiguous, because <12 19 27|, for example, is a valid temperament even though it's much less accurate than <12 19 28|. In this temperament 5/4 would be represented as 3 steps of 12edo, or 300 cents. For practical purposes, of course, the ambiguity doesn't appear until higher limits.)

==Linear temperament mappings==

Now let's consider a temperament that does not consist of a single chain of equally spaced notes. For example, consider conventional music notation **without** enharmonic equivalence. Every note of this system can be expressed as some combination of octaves and perfect fourths, for example

E5 = A440 + 1 octave - 1 perfect fourths

Bb4 = A440 - 3 octaves + 5 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.

=Temperamental Rank=

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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><h1 id="toc0"><a name="Intro To Mappings"></a>Intro To Mappings</h1>

<br />

A <a class="wiki_link" href="/~~regular~~%~~20temperament~~">regular temperament</a> is more than simply a set of pitches. It's a set of notes together with a <strong>consistent rule</strong> 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 <em>JI mapping</em> or simply <em>mapping</em>. 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.<br />

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 <strong>consistent rule</strong> 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 <em>JI mapping</em> or simply <em>mapping</em>. 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.<br />

<br />

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 <strong>consistent</strong> 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 <strong>same</strong> tempered interval (even if that tempered interval is not the closest tempered interval to the JI interval).<br />

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<br />

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<br />

<br />

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

[[math]]&lt;br/&gt;

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

<br />

~~A val maps JI onto one chain~~ of ~~generators~~, or ~~relates~~ that ~~generator chain back~~ to JI. However, ~~many temperaments incorporate more than~~ one of ~~chain~~ of ~~generators~~. ~~The familiar meantone~~ temperament is an ~~example~~, as it ~~requires~~ two: the ~~fifth and~~ the ~~octave~~. ~~However~~, ~~vals only relate~~ a ~~single generator chain~~ to JI. ~~If we want to evolve out~~ of the ~~realm~~ of ~~isolated EDOs and consider higher~~-~~dimensional~~ temperaments ~~in their full glory~~, we'~~re going~~ to ~~have to raise~~ the ~~bar on what vals~~ can do for ~~us. Luckily~~, the ~~mathematics to do so~~ is ~~simple enough~~.<br />

) 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<span style="vertical-align: super;">12</span>/2<span style="vertical-align: super;">19</span> (the Pythagorean comma) is mapped to 0, the same note as 1/1.<br />

<br />

<h3 id="toc2"><a name="Intro To Mappings-Equal temperament mappings-Contrast with rounding"></a>Contrast with rounding</h3>

Now, consider the pitch 3<span style="vertical-align: super;">36</span>/2<span style="vertical-align: super;">57</span>. In JI, this pitch is 70.38... cents above A440, so the closest 12edo note to it is Bb. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (Bb). Why is this? The pitch 3<span style="vertical-align: super;">36</span>/2<span style="vertical-align: super;">57</span>is three Pythagorean commas above A. If each Pythagorean comma is represented by 0 steps, then since 0+0+0=0 the pitch 3<span style="vertical-align: super;">36</span>/2<span style="vertical-align: super;">57 </span>must be represented by A, even though in JI it's closer to Bb. Mapping it to Bb would require one of the three Pythaogrean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.<br />

<br />

<h3 id="toc3"><a name="Intro To Mappings-Equal temperament mappings-Notation"></a>Notation</h3>

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 &quot;&lt;12 19|&quot;, 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 &quot;mapping matrix&quot; 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.<br />

<br />

<h3 id="toc4"><a name="Intro To Mappings-Equal temperament mappings-Many 12edo temperaments"></a>Many 12edo temperaments</h3>

Now, let's consider 12edo, not as a 3-limit temperament, but as a <a class="wiki_link" href="/5-limit">5-limit</a> temperament. This temperament maps all the 3-limit JI intervals in the same way as above, but in addition also maps the rest of the 5-limit JI intervals. Its mapping matrix is &lt;12 19 28|. It's important to keep in mind that this is, technically speaking, a <strong>different</strong> regular temperament than &lt;12 19|, even though they would both be referred to as &quot;12-tone equal temperament&quot; in common parlance.<br />

<br />

Furthermore, consider 12edo as an 11-limit temperament. What is its mapping matrix? It actually depends whether you consider 11/8 a &quot;very sharp D&quot; or a &quot;very flat D#&quot;. This choice results in two different mappings, &lt;12 19 28 34 40| and &lt;12 19 28 34 41|. The latter has a more accurate 11/8, but the former has more accurate versions of other intervals, including 12/11. In the language of regular temperament theory, these are simply two different 11-limit temperaments that both happen to have 12 steps per octave. Phrases like &quot;11-limit 12edo&quot; are thus ambiguous because they don't specify the mapping, and therefore don't refer to a specific temperament.<br />

<br />

(Strictly speaking, &quot;5-limit 12edo&quot; or even &quot;3-limit 12edo&quot; are also ambiguous, because &lt;12 19 27|, for example, is a valid temperament even though it's much less accurate than &lt;12 19 28|. In this temperament 5/4 would be represented as 3 steps of 12edo, or 300 cents. For practical purposes, of course, the ambiguity doesn't appear until higher limits.)<br />

<br />

<h2 id="toc5"><a name="Intro To Mappings-Linear temperament mappings"></a>Linear temperament mappings</h2>

Now let's consider a temperament that does not consist of a single chain of equally spaced notes. For example, consider conventional music notation <strong>without</strong> enharmonic equivalence. Every note of this system can be expressed as some combination of octaves and perfect fourths, for example<br />

E5 = A440 + 1 octave - 1 perfect fourths<br />

Bb4 = A440 - 3 octaves + 5 perfect fourths<br />

A# = A440 + 4 octaves - 7 perfect fourths<br />

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.<br />

<br />

<h1 id="~~toc2~~"><a name="Temperamental Rank"></a>Temperamental Rank</h1>

<h1 id="toc6"><a name="Temperamental Rank"></a>Temperamental Rank</h1>

<strong>A temperament's &quot;rank&quot; denotes how many independent chains of</strong> <strong>generators exist within the temperament.</strong> This is a mathematical term that's borrowed from the field of group theory. It can also be viewed as the &quot;dimensionality&quot; of the temperament.<br />

<br />

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A single val in isolation only maps JI onto temperaments that are rank 1. For us to deal with temperaments of rank &gt; 1, we simply need to use more than one val. <strong>In general, the number of vals that it requires to fully map a temperament is equal to the temperament's rank.</strong><br />

<br />

<h1 id="~~toc3~~"><a name="Example"></a>Example</h1>

<h1 id="toc7"><a name="Example"></a>Example</h1>

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

<br />

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This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.<br />

<br />

<h1 id="~~toc4~~"><a name="Change of Basis"></a>Change of Basis</h1>

<h1 id="toc8"><a name="Change of Basis"></a>Change of Basis</h1>

<br />

In the above example, we wrote out the meantone mapping matrix from the perspective of the two generators 2/1 and 3/2. What if we instead wanted to treat the generators as being 2/1 and 4/3? Or, what if we wanted to write it out from the perspective that the generators are 2/1 and 3/1? All of these will lead to different val lists, but will still represent the same temperament.<br />

@@ 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:keenanpepper|keenanpepper]] and made on <tt>2013-07-26 21:28:06 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2013-07-26 23:06:57 UTC</tt>.<br>
-: The original revision id was <tt>442486156</tt>.<br>
+: The original revision id was <tt>442491528</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">=Intro To Mappings=
-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 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.
+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).
@@ Line 16: / Line 16: @@
 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]]
 \left\{440\cdot 2^a\cdot 3^b\middle|a,b\in\mathbb Z\right\}
 [[math]]
-A val maps JI onto one chain of generators, or relates that generator chain back to JI. However, many temperaments incorporate more than one of chain of generators. The familiar meantone temperament is an example, as it requires two: the fifth and the octave. However, vals only relate a single generator chain to JI. If we want to evolve out of the realm of isolated EDOs and consider higher-dimensional temperaments in their full glory, we're going to have to raise the bar on what vals can do for us. Luckily, the mathematics to do so is simple enough.
+) 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===
+Now, consider the pitch 3&lt;span style="vertical-align: super;"&gt;36&lt;/span&gt;/2&lt;span style="vertical-align: super;"&gt;57&lt;/span&gt;. In JI, this pitch is 70.38... cents above A440, so the closest 12edo note to it is Bb. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (Bb). Why is this? The pitch 3&lt;span style="vertical-align: super;"&gt;36&lt;/span&gt;/2&lt;span style="vertical-align: super;"&gt;57&lt;/span&gt;is three Pythagorean commas above A. If each Pythagorean comma is represented by 0 steps, then since 0+0+0=0 the pitch 3&lt;span style="vertical-align: super;"&gt;36&lt;/span&gt;/2&lt;span style="vertical-align: super;"&gt;57 &lt;/span&gt;must be represented by A, even though in JI it's closer to Bb. Mapping it to Bb would require one of the three Pythaogrean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.
+===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 "&lt;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.
+===Many 12edo temperaments===
+Now, let's consider 12edo, not as a 3-limit temperament, but as a [[5-limit]] temperament. This temperament maps all the 3-limit JI intervals in the same way as above, but in addition also maps the rest of the 5-limit JI intervals. Its mapping matrix is &lt;12 19 28|. It's important to keep in mind that this is, technically speaking, a **different** regular temperament than &lt;12 19|, even though they would both be referred to as "12-tone equal temperament" in common parlance.
+Furthermore, consider 12edo as an 11-limit temperament. What is its mapping matrix? It actually depends whether you consider 11/8 a "very sharp D" or a "very flat D#". This choice results in two different mappings, &lt;12 19 28 34 40| and &lt;12 19 28 34 41|. The latter has a more accurate 11/8, but the former has more accurate versions of other intervals, including 12/11. In the language of regular temperament theory, these are simply two different 11-limit temperaments that both happen to have 12 steps per octave. Phrases like "11-limit 12edo" are thus ambiguous because they don't specify the mapping, and therefore don't refer to a specific temperament.
+(Strictly speaking, "5-limit 12edo" or even "3-limit 12edo" are also ambiguous, because &lt;12 19 27|, for example, is a valid temperament even though it's much less accurate than &lt;12 19 28|. In this temperament 5/4 would be represented as 3 steps of 12edo, or 300 cents. For practical purposes, of course, the ambiguity doesn't appear until higher limits.)
+==Linear temperament mappings==
+Now let's consider a temperament that does not consist of a single chain of equally spaced notes. For example, consider conventional music notation **without** enharmonic equivalence. Every note of this system can be expressed as some combination of octaves and perfect fourths, for example
+E5 = A440 + 1 octave - 1 perfect fourths
+Bb4 = A440 - 3 octaves + 5 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.
 =Temperamental Rank=
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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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Intro to Mappings&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Intro To Mappings"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;Intro To Mappings&lt;/h1&gt;
   &lt;br /&gt;
-A &lt;a class="wiki_link" href="/regular%20temperament"&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;
+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;
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 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;
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 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;
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   --&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;
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-A val maps JI onto one chain of generators, or relates that generator chain back to JI. However, many temperaments incorporate more than one of chain of generators. The familiar meantone temperament is an example, as it requires two: the fifth and the octave. However, vals only relate a single generator chain to JI. If we want to evolve out of the realm of isolated EDOs and consider higher-dimensional temperaments in their full glory, we're going to have to raise the bar on what vals can do for us. Luckily, the mathematics to do so is simple enough.&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:5:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="Intro To Mappings-Equal temperament mappings-Contrast with rounding"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Contrast with rounding&lt;/h3&gt;
+ Now, consider the pitch 3&lt;span style="vertical-align: super;"&gt;36&lt;/span&gt;/2&lt;span style="vertical-align: super;"&gt;57&lt;/span&gt;. In JI, this pitch is 70.38... cents above A440, so the closest 12edo note to it is Bb. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (Bb). Why is this? The pitch 3&lt;span style="vertical-align: super;"&gt;36&lt;/span&gt;/2&lt;span style="vertical-align: super;"&gt;57&lt;/span&gt;is three Pythagorean commas above A. If each Pythagorean comma is represented by 0 steps, then since 0+0+0=0 the pitch 3&lt;span style="vertical-align: super;"&gt;36&lt;/span&gt;/2&lt;span style="vertical-align: super;"&gt;57 &lt;/span&gt;must be represented by A, even though in JI it's closer to Bb. Mapping it to Bb would require one of the three Pythaogrean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="Intro To Mappings-Equal temperament mappings-Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Notation&lt;/h3&gt;
+ 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 &amp;quot;&amp;lt;12 19|&amp;quot;, 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 &amp;quot;mapping matrix&amp;quot; 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.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="Intro To Mappings-Equal temperament mappings-Many 12edo temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Many 12edo temperaments&lt;/h3&gt;
+ Now, let's consider 12edo, not as a 3-limit temperament, but as a &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; temperament. This temperament maps all the 3-limit JI intervals in the same way as above, but in addition also maps the rest of the 5-limit JI intervals. Its mapping matrix is &amp;lt;12 19 28|. It's important to keep in mind that this is, technically speaking, a &lt;strong&gt;different&lt;/strong&gt; regular temperament than &amp;lt;12 19|, even though they would both be referred to as &amp;quot;12-tone equal temperament&amp;quot; in common parlance.&lt;br /&gt;
+&lt;br /&gt;
+Furthermore, consider 12edo as an 11-limit temperament. What is its mapping matrix? It actually depends whether you consider 11/8 a &amp;quot;very sharp D&amp;quot; or a &amp;quot;very flat D#&amp;quot;. This choice results in two different mappings, &amp;lt;12 19 28 34 40| and &amp;lt;12 19 28 34 41|. The latter has a more accurate 11/8, but the former has more accurate versions of other intervals, including 12/11. In the language of regular temperament theory, these are simply two different 11-limit temperaments that both happen to have 12 steps per octave. Phrases like &amp;quot;11-limit 12edo&amp;quot; are thus ambiguous because they don't specify the mapping, and therefore don't refer to a specific temperament.&lt;br /&gt;
+&lt;br /&gt;
+(Strictly speaking, &amp;quot;5-limit 12edo&amp;quot; or even &amp;quot;3-limit 12edo&amp;quot; are also ambiguous, because &amp;lt;12 19 27|, for example, is a valid temperament even though it's much less accurate than &amp;lt;12 19 28|. In this temperament 5/4 would be represented as 3 steps of 12edo, or 300 cents. For practical purposes, of course, the ambiguity doesn't appear until higher limits.)&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Intro To Mappings-Linear temperament mappings"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;Linear temperament mappings&lt;/h2&gt;
+ Now let's consider a temperament that does not consist of a single chain of equally spaced notes. For example, consider conventional music notation &lt;strong&gt;without&lt;/strong&gt; enharmonic equivalence. Every note of this system can be expressed as some combination of octaves and perfect fourths, for example&lt;br /&gt;
+E5 = A440 + 1 octave - 1 perfect fourths&lt;br /&gt;
+Bb4 = A440 - 3 octaves + 5 perfect fourths&lt;br /&gt;
+A# = A440 + 4 octaves - 7 perfect fourths&lt;br /&gt;
+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.&lt;br /&gt;
+&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Temperamental Rank"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Temperamental Rank&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Temperamental Rank"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;Temperamental Rank&lt;/h1&gt;
   &lt;strong&gt;A temperament's &amp;quot;rank&amp;quot; denotes how many independent chains of&lt;/strong&gt; &lt;strong&gt;generators exist within the temperament.&lt;/strong&gt; This is a mathematical term that's borrowed from the field of group theory. It can also be viewed as the &amp;quot;dimensionality&amp;quot; of the temperament.&lt;br /&gt;
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 A single val in isolation only maps JI onto temperaments that are rank 1. For us to deal with temperaments of rank &amp;gt; 1, we simply need to use more than one val. &lt;strong&gt;In general, the number of vals that it requires to fully map a temperament is equal to the temperament's rank.&lt;/strong&gt;&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Example&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:15:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:15 --&gt;Example&lt;/h1&gt;
   At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:&lt;br /&gt;
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 This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Change of Basis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Change of Basis&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:17:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Change of Basis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&gt;Change of Basis&lt;/h1&gt;
   &lt;br /&gt;
 In the above example, we wrote out the meantone mapping matrix from the perspective of the two generators 2/1 and 3/2. What if we instead wanted to treat the generators as being 2/1 and 4/3? Or, what if we wanted to write it out from the perspective that the generators are 2/1 and 3/1? All of these will lead to different val lists, but will still represent the same temperament.&lt;br /&gt;