Vals and tuning space
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[[toc|flat]]
=Abstract=
A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc. Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO.
A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation <a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit]].
For example, the 5-limit val <12 19 28| tells us that we're viewing 12 steps of 12-equal as representing 2/1, 19 steps of 12-equal as representing 3/1, and 28 steps of 12-equal as representing 5/1. This means that we view 3/2 as mapping to 19-12=7 steps of 12-equal, that we view 5/4 as mapping to 28-24=4 steps of 12-equal, and that we view 81/80 as mapping to 19*4 - 12*4 - 28 = 0 steps of 12-equal.
If you would like to assume the perspective that the 1000 cent interval in 12-equal is a very tempered 7/4, then that decision can be represented by using the 7-limit <12 19 28 34| val. If you would, for some reason, like to say that the 7/4 maps to 300 cents instead, as absurd as that may be, that would be reflected by the <12 19 28 27| val. It's not recommended to use vals like that, but the mathematics will allow you to do it if you want, in the same way that a brick will allow you to hit yourself in the face with it.
Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective one takes on an EDO. As such, they allow us to harness the very powerful realm of mathematics to manipulate our musical intuitions.
See also: [[patent val]].
=Definition=
A val "maps" just intonation to a certain number of steps; by putting vals together we can define the mapping of a [[Regular temperaments|regular temperament]] and thereby define the temperament. A val is written in the form <a1 a2 a3 ... ak|, where the numbers a1 a2 a3 ... are the number of steps the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any set of generators are mapped to, where a set of generators for a [[Just intonation subgroups|just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.
A //rank r// temperament has r generators, and thus is defined by r vals. In the usual coordinates for the [[Harmonic limit|p-limit]], the set of generators are the first k prime numbers and the set of vals for a p-limit temperament gives you the coordinates for each prime harmonic in the p-limit. For example, all 5-limit rank-1 temperaments will be defined by a val <a b c|, where a is the number of generators it takes to reach the 2nd harmonic (2/1), b is the number of generators to reach the 3rd harmonic (3/1), and c is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments are defined by two vals: [<a1 b1 c1|, <a2 b2 c2|] Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (a1, a2), meaning go up a1 of the first generator, and up a2 of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be located by (b1, b2) and (c1, c2) respectively.
As an example, consider meantone temperament, where 81/80 vanishes. Meantone can be considered a 5-limit rank-2 temperament, defined by the two-val mapping [<1 1 0|, <0 1 4|]. This tells us just about everything we need to know about how the 5-limit is mapped in meantone: since 2/1 is mapped (1, 0), that tells us that the first generator //is// a 2/1, and since 3/1 is mapped to (1,1), that tells us that the 2nd generator is a 3/2; then, since 5/1 is mapped to (0,4), aka four 3/2s up, that tells us that 81/64 (which is (3/2)^4) equals 5/1 (which is 80/64). Since 81/64 is equated with 80/64 here, that tells us that 81/80 is tempered out! Thus it is possible to derive from the mapping the approximate size of the two generators, the commas that are tempered out, and roughly the complexity of the temperament (the number of notes of the temperament we need to reach all the prime harmonics in the p-limit). This makes the val an extremely compact and useful bit of notation for describing regular temperaments, since we can discern almost everything we need to know about the temperament essentially at a glance. Whenever one of the generators of a temperament is a 2/1 the key information is carried by the other vals, assuming octave equivalence (i.e. 3/1=3/2=6/1 etc). Thus the essential character of 5-limit meantone is defined by a single val (the one for the 3/2 generator), written <0 1 4|.
==Defintion for mathematicians==
The p-limit [[Monzos and Interval Space|monzos]] M form a free abelian group, or Z-module, of finite rank pi(p), which is the number of primes up to and including p. The [[http://planetmath.org/encyclopedia/DualModule.html|dual Z-module]] M* is [[http://en.wikipedia.org/wiki/Group_isomorphism|isomorphic]] to M, but not in a canonical way. Hence it, the group (Z-module) of **vals**, is also a free abelian group of rank pi(p). Just as monzos are often written as [[http://mathworld.wolfram.com/Ket.html|kets]], vals are typically written as [[http://mathworld.wolfram.com/Bra.html|bras]].
=Vals and Monzos=
If V is a val and M is a monzo of the same rank, then the [[http://mathworld.wolfram.com/AngleBracket.html|angle bracket]] <V|M>, which can also be written V(M), is the result of applying the [[http://en.wikipedia.org/wiki/Group_homomorphism|homomorphism]] V to M. For example, if V = <12 19 28 34| and M = |-5 2 2 -1> then <V|M> equals 12*(-5) + 19*2 + 28*2 - 34 = 0
This tells us that in septimal 12 equal, represented by V, the interval 225/224, represented by M, is mapped to 0, which represents 1. Hence, 225/224 vanishes in septimal 12 equal; it is in the [[http://mathworld.wolfram.com/GroupKernel.html|kernel]] of V. One should note in particular that the coordinates of V represent where the successive primes 2, 3, 5 and 7 are mapped.
Norms may be placed on the monzos in various ways, turning them into [[http://mathworld.wolfram.com/PointLattice.html|lattices]] in a vector space. Given a vector space norm on a space of ket vectors, the [[http://mathworld.wolfram.com/DualNormedSpace.html|dual vector space norm]] on the space of bra vectors is defined as the least quantity ||V|| making
**|<V|M>| <= ||V|| ||M||**
to be always true. The dual of the [[http://mathworld.wolfram.com/L1-Norm.html|L1 norm]] is the [[http://mathworld.wolfram.com/L-Infinity-Norm.html|Linfty norm]], and the dual space of Tenney interval space is Tenney tuning space. In tuning space, the vals now define a lattice. Similarly, the dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is //Tenney-Euclidean tuning space//. The Euclidean norm on a val v is given by
[[math]]
\displaystyle
||v|| = \sqrt{\left({\frac{v_2}{log_2(2)}\right)^2 + \left({\frac{v_3}{log_2(3)}\right)^2 + \left(\frac{v_5}{log_2(5)}\right)^2 + ... + \left(\frac{v_p}{log_2(p)}\right)^2
%original was ||v|| = sqrt(v2^2 + (v3/log2(3))^2 + ... + (vp/log2(p))^2)
[[math]]
It useful to renormalize to the RMS (root mean square) instead, which requires dividing the above by sqrt(n), where n = pi(p) is the number of primes up to p. This is the TE, or Tenney-Euclidean, norm.
It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in weighted coordinates is <1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then <JIP|M>, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = <1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval.
=Example=
The rank-1 [[7-limit]] [[val]] corresponding to [[31edo]] is <31 49 72 87|. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes
[[math]]
\displaystyle
\left<31 \; \frac{49}{log_2(3)} \; \frac{72}{log_2(5)} \; \frac{87}{log_2(7)}\right|
%original was <31 49/log2(3) 72/log2(5) 87/log2(7)|
[[math]]
which is approximately <31.000 30.916 31.009 30.990|. The standard Euclidean norm would then be the square root of the sum of squares of this vector, which is approximately sqrt(3838.694), or 61.957. To use the RMS we divide that by sqrt(4)=2, giving 30.976 for the TE norm. Note that the TE norm for this val is approximately 31.Original HTML content:
<html><head><title>Vals and Tuning Space</title></head><body><!-- ws:start:WikiTextTocRule:12:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#Abstract">Abstract</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#Vals and Monzos">Vals and Monzos</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Example">Example</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: -->
<!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc0"><a name="Abstract"></a><!-- ws:end:WikiTextHeadingRule:2 -->Abstract</h1>
A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc. Sometimes there's more than one sensible way to map the intervals in an EDO back to JI, and hence in that situation there's more than one val you could apply to an EDO.<br />
<br />
A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation <a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some <a class="wiki_link" href="/harmonic%20limit">prime limit</a>.<br />
<br />
For example, the 5-limit val <12 19 28| tells us that we're viewing 12 steps of 12-equal as representing 2/1, 19 steps of 12-equal as representing 3/1, and 28 steps of 12-equal as representing 5/1. This means that we view 3/2 as mapping to 19-12=7 steps of 12-equal, that we view 5/4 as mapping to 28-24=4 steps of 12-equal, and that we view 81/80 as mapping to 19*4 - 12*4 - 28 = 0 steps of 12-equal.<br />
<br />
If you would like to assume the perspective that the 1000 cent interval in 12-equal is a very tempered 7/4, then that decision can be represented by using the 7-limit <12 19 28 34| val. If you would, for some reason, like to say that the 7/4 maps to 300 cents instead, as absurd as that may be, that would be reflected by the <12 19 28 27| val. It's not recommended to use vals like that, but the mathematics will allow you to do it if you want, in the same way that a brick will allow you to hit yourself in the face with it.<br />
<br />
Vals form the basis for all of regular temperament theory. They are important because they provide a way to mathematically formalize the chosen JI perspective one takes on an EDO. As such, they allow us to harness the very powerful realm of mathematics to manipulate our musical intuitions.<br />
<br />
See also: <a class="wiki_link" href="/patent%20val">patent val</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:4 -->Definition</h1>
A val "maps" just intonation to a certain number of steps; by putting vals together we can define the mapping of a <a class="wiki_link" href="/Regular%20temperaments">regular temperament</a> and thereby define the temperament. A val is written in the form <a1 a2 a3 ... ak|, where the numbers a1 a2 a3 ... are the number of steps the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any set of generators are mapped to, where a set of generators for a <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a> is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.<br />
<br />
A <em>rank r</em> temperament has r generators, and thus is defined by r vals. In the usual coordinates for the <a class="wiki_link" href="/Harmonic%20limit">p-limit</a>, the set of generators are the first k prime numbers and the set of vals for a p-limit temperament gives you the coordinates for each prime harmonic in the p-limit. For example, all 5-limit rank-1 temperaments will be defined by a val <a b c|, where a is the number of generators it takes to reach the 2nd harmonic (2/1), b is the number of generators to reach the 3rd harmonic (3/1), and c is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments are defined by two vals: [<a1 b1 c1|, <a2 b2 c2|] Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (a1, a2), meaning go up a1 of the first generator, and up a2 of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be located by (b1, b2) and (c1, c2) respectively.<br />
<br />
As an example, consider meantone temperament, where 81/80 vanishes. Meantone can be considered a 5-limit rank-2 temperament, defined by the two-val mapping [<1 1 0|, <0 1 4|]. This tells us just about everything we need to know about how the 5-limit is mapped in meantone: since 2/1 is mapped (1, 0), that tells us that the first generator <em>is</em> a 2/1, and since 3/1 is mapped to (1,1), that tells us that the 2nd generator is a 3/2; then, since 5/1 is mapped to (0,4), aka four 3/2s up, that tells us that 81/64 (which is (3/2)^4) equals 5/1 (which is 80/64). Since 81/64 is equated with 80/64 here, that tells us that 81/80 is tempered out! Thus it is possible to derive from the mapping the approximate size of the two generators, the commas that are tempered out, and roughly the complexity of the temperament (the number of notes of the temperament we need to reach all the prime harmonics in the p-limit). This makes the val an extremely compact and useful bit of notation for describing regular temperaments, since we can discern almost everything we need to know about the temperament essentially at a glance. Whenever one of the generators of a temperament is a 2/1 the key information is carried by the other vals, assuming octave equivalence (i.e. 3/1=3/2=6/1 etc). Thus the essential character of 5-limit meantone is defined by a single val (the one for the 3/2 generator), written <0 1 4|.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc2"><a name="Definition-Defintion for mathematicians"></a><!-- ws:end:WikiTextHeadingRule:6 -->Defintion for mathematicians</h2>
The p-limit <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a> M form a free abelian group, or Z-module, of finite rank pi(p), which is the number of primes up to and including p. The <a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/DualModule.html" rel="nofollow">dual Z-module</a> M* is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_isomorphism" rel="nofollow">isomorphic</a> to M, but not in a canonical way. Hence it, the group (Z-module) of <strong>vals</strong>, is also a free abelian group of rank pi(p). Just as monzos are often written as <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Ket.html" rel="nofollow">kets</a>, vals are typically written as <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Bra.html" rel="nofollow">bras</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc3"><a name="Vals and Monzos"></a><!-- ws:end:WikiTextHeadingRule:8 -->Vals and Monzos</h1>
If V is a val and M is a monzo of the same rank, then the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/AngleBracket.html" rel="nofollow">angle bracket</a> <V|M>, which can also be written V(M), is the result of applying the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_homomorphism" rel="nofollow">homomorphism</a> V to M. For example, if V = <12 19 28 34| and M = |-5 2 2 -1> then <V|M> equals 12*(-5) + 19*2 + 28*2 - 34 = 0<br />
<br />
This tells us that in septimal 12 equal, represented by V, the interval 225/224, represented by M, is mapped to 0, which represents 1. Hence, 225/224 vanishes in septimal 12 equal; it is in the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/GroupKernel.html" rel="nofollow">kernel</a> of V. One should note in particular that the coordinates of V represent where the successive primes 2, 3, 5 and 7 are mapped.<br />
<br />
Norms may be placed on the monzos in various ways, turning them into <a class="wiki_link_ext" href="http://mathworld.wolfram.com/PointLattice.html" rel="nofollow">lattices</a> in a vector space. Given a vector space norm on a space of ket vectors, the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/DualNormedSpace.html" rel="nofollow">dual vector space norm</a> on the space of bra vectors is defined as the least quantity ||V|| making<br />
<br />
<strong>|<V|M>| <= ||V|| ||M||</strong><br />
<br />
to be always true. The dual of the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow">L1 norm</a> is the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L-Infinity-Norm.html" rel="nofollow">Linfty norm</a>, and the dual space of Tenney interval space is Tenney tuning space. In tuning space, the vals now define a lattice. Similarly, the dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is <em>Tenney-Euclidean tuning space</em>. The Euclidean norm on a val v is given by<br />
<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]<br/>
\displaystyle<br />
||v|| = \sqrt{\left({\frac{v_2}{log_2(2)}\right)^2 + \left({\frac{v_3}{log_2(3)}\right)^2 + \left(\frac{v_5}{log_2(5)}\right)^2 + ... + \left(\frac{v_p}{log_2(p)}\right)^2<br />
%original was ||v|| = sqrt(v2^2 + (v3/log2(3))^2 + ... + (vp/log2(p))^2)<br/>[[math]]
--><script type="math/tex">\displaystyle
||v|| = \sqrt{\left({\frac{v_2}{log_2(2)}\right)^2 + \left({\frac{v_3}{log_2(3)}\right)^2 + \left(\frac{v_5}{log_2(5)}\right)^2 + ... + \left(\frac{v_p}{log_2(p)}\right)^2
%original was ||v|| = sqrt(v2^2 + (v3/log2(3))^2 + ... + (vp/log2(p))^2)</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
It useful to renormalize to the RMS (root mean square) instead, which requires dividing the above by sqrt(n), where n = pi(p) is the number of primes up to p. This is the TE, or Tenney-Euclidean, norm.<br />
<br />
It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in weighted coordinates is <1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then <JIP|M>, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = <1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc4"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:10 -->Example</h1>
The rank-1 <a class="wiki_link" href="/7-limit">7-limit</a> <a class="wiki_link" href="/val">val</a> corresponding to <a class="wiki_link" href="/31edo">31edo</a> is <31 49 72 87|. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes<br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]<br/>
\displaystyle<br />
\left<31 \; \frac{49}{log_2(3)} \; \frac{72}{log_2(5)} \; \frac{87}{log_2(7)}\right|<br />
%original was <31 49/log2(3) 72/log2(5) 87/log2(7)|<br/>[[math]]
--><script type="math/tex">\displaystyle
\left<31 \; \frac{49}{log_2(3)} \; \frac{72}{log_2(5)} \; \frac{87}{log_2(7)}\right|
%original was <31 49/log2(3) 72/log2(5) 87/log2(7)|</script><!-- ws:end:WikiTextMathRule:1 --><br />
<br />
which is approximately <31.000 30.916 31.009 30.990|. The standard Euclidean norm would then be the square root of the sum of squares of this vector, which is approximately sqrt(3838.694), or 61.957. To use the RMS we divide that by sqrt(4)=2, giving 30.976 for the TE norm. Note that the TE norm for this val is approximately 31.</body></html>