Vals and tuning space
This is an expert page. It is written to allow experienced readers to learn more about the advanced elements of the topic.
The corresponding beginner page for this topic is Val.
Definition
A val "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a 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 along the chain that the first k primes are mapped to. This can be generalized so that a1 a2 a3 … represent the number of steps any JI basis is mapped to, whereas a JI basis for a 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 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, or equal 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), sometimes written as [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 reached 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 to [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 3/2's 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 readily find where all of the primes are mapped along the temperament's chain of generators 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].
Definition for mathematicians
The p-limit monzos M form a free abelian group, or ℤ-module, of finite rank π (p), which is the number of primes up to and including p. The dual ℤ-module M* is 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 π (p). Just as monzos are often written as kets, vals are typically written as bras. Vals are homomorphisms from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist Yves Hellegouarch seems to have been the first to write about them, under the name "degrees".
Vals and monzos
If V is a val and M is a monzo of the same rank, then the angle bracket, written ⟨V|M⟩ (or occasionally V(M)), is the result of applying the 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 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.
By embedding the monzos into a suitable vector space, norms may be placed on the monzos in various ways, turning them into lattices in a vector space. Given a vector space norm on a space of ket vectors, the dual vector space norm on the space of bra vectors is defined as the least quantity ||V|| making
[math]\displaystyle \lvert \langle V|M \rangle \rvert \le \lVert V \rVert \lVert M \rVert [/math]
to be always true. The dual of the L1 norm is the L-infinity norm, and the dual space of Tenney interval space is Tenney tuning space. The embedding of monzos into a real normed vector space automatically induces a dual embedding of vals into a corresponding normed vector space, tuning space, in which vals are lattice points. 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 \lVert V \rVert = \sqrt{\left(\frac{v_1}{\log_2(2)}\right)^2 + \left(\frac{v_2}{\log_2(3)}\right)^2 + \left(\frac{v_3}{\log_2(5)}\right)^2 + \ldots + \left(\frac{v_n}{\log_2(p)}\right)^2} [/math]
It useful to renormalize to the RMS (root mean square) instead, which requires dividing the above by sqrt (n), where n = π (p) is the number of primes up to p. This is the Tenney-Euclidean norm, or TE 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 J = ⟨1 1 1 … 1]. It has the property that if M is a monzo in weighted coordinates, then ⟨J|M⟩ or J (M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, J = ⟨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 patent 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\lt 31 \; \frac{49}{\log_2(3)} \; \frac{72}{\log_2(5)} \; \frac{87}{\log_2(7)}\right| %original was \lt 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; any val closely approximating JI is expected to have the TE norm close to its division of the octave.