Vals and tuning space
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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]]. 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 Tenney-Euclidean norm, or TE norm, on a val v is given by ||v|| = sqrt(v2^2 + (v3/log2(3))^2 + ... + (vp/log2(p))^2) 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.
Original HTML content:
<html><head><title>Vals and Tuning Space</title></head><body>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 /> 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 Tenney-Euclidean norm, or TE norm, on a val v is given by<br /> <br /> ||v|| = sqrt(v2^2 + (v3/log2(3))^2 + ... + (vp/log2(p))^2) <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.</body></html>