Intro to Mappings
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=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.
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).
==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).
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\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.
=Temperamental Rank=
**A temperament's "rank" denotes how many independent chains of** **generators exist within the temperament.** This is a mathematical term that's borrowed from the field of group theory. It can also be viewed as the "dimensionality" of the temperament.
For example:
# 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 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, etc.
A single val in isolation only maps JI onto temperaments that are rank 1. For us to deal with temperaments of rank > 1, we simply need to use more than one val. **In general, the number of vals that it requires to fully map a temperament is equal to the temperament's rank.**
=Example=
At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:
<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.
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
<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:
<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:
<1 1 0]
<0 1 4]
This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.
=Change of Basis=
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.
In the language of mathematics, you've simply **changed the basis** for your temperament, and the resulting temperamental spaces will be **isomorphic** to one another. This is just a fancy way of stating that they're the same temperament.
If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following list of vals:
<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.
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):
<1 0 -4]
<0 1 4]Original HTML content:
<html><head><title>Intro to Mappings</title></head><body><!-- ws:start:WikiTextHeadingRule:1:<h1> --><h1 id="toc0"><a name="Intro To Mappings"></a><!-- ws:end:WikiTextHeadingRule:1 -->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 "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.<br />
<br />
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 <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 />
<br />
<!-- ws:start:WikiTextHeadingRule:3:<h2> --><h2 id="toc1"><a name="Intro To Mappings-Equal temperament mappings"></a><!-- ws:end:WikiTextHeadingRule:3 -->Equal temperament mappings</h2>
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).<br />
<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 />
<!-- ws:start:WikiTextMathRule:0:
[[math]]<br/>
\left\{440\right\}<br/>[[math]]
--><script type="math/tex">\left\{440\right\}</script><!-- ws:end:WikiTextMathRule:0 --><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 />
<br />
<!-- ws:start:WikiTextHeadingRule:5:<h1> --><h1 id="toc2"><a name="Temperamental Rank"></a><!-- ws:end:WikiTextHeadingRule:5 -->Temperamental Rank</h1>
<strong>A temperament's "rank" 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 "dimensionality" of the temperament.<br />
<br />
For example:<br />
<ol><li>An equal temperament is rank 1, as it exists in its entirety as a stack of one single generator.</li><li>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</li><li>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).</li><li>7-limit JI would be rank 4, etc.</li></ol><br />
A single val in isolation only maps JI onto temperaments that are rank 1. For us to deal with temperaments of rank > 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 />
<!-- ws:start:WikiTextHeadingRule:7:<h1> --><h1 id="toc3"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:7 -->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 />
<a b c] - period<br />
<d e f] - generator<br />
<br />
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.<br />
<br />
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<br />
<br />
<1 _ _]<br />
<0 _ _]<br />
<br />
<br />
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:<br />
<br />
<1 1 _]<br />
<0 1 _]<br />
<br />
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:<br />
<br />
<1 1 0]<br />
<0 1 4]<br />
<br />
This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:9:<h1> --><h1 id="toc4"><a name="Change of Basis"></a><!-- ws:end:WikiTextHeadingRule:9 -->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 />
<br />
In the language of mathematics, you've simply <strong>changed the basis</strong> for your temperament, and the resulting temperamental spaces will be <strong>isomorphic</strong> to one another. This is just a fancy way of stating that they're the same temperament.<br />
<br />
If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following list of vals:<br />
<br />
<1 2 4]<br />
<0 -1 -4]<br />
<br />
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.<br />
<br />
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):<br />
<br />
<1 0 -4]<br />
<0 1 4]</body></html>