Interior product

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Interior product

[math]\displaystyle{ \def\hs{\hspace{-3px}} \def\vsp{{}\mkern-5.5mu}{} \def\llangle{\left\langle\vsp\left\langle} \def\lllangle{\left\langle\vsp\left\langle\vsp\left\langle} \def\llllangle{\left\langle\vsp\left\langle\vsp\left\langle\vsp\left\langle} \def\llbrack{\left[\left[} \def\lllbrack{\left[\left[\left[} \def\llllbrack{\left[\left[\left[\left[} \def\llvert{\left\vert\left\vert} \def\lllvert{\left\vert\left\vert\left\vert} \def\llllvert{\left\vert\left\vert\left\vert\left\vert} \def\rrangle{\right\rangle\vsp\right\rangle} \def\rrrangle{\right\rangle\vsp\right\rangle\vsp\right\rangle} \def\rrrrangle{\right\rangle\vsp\right\rangle\vsp\right\rangle\vsp\right\rangle} \def\rrbrack{\right]\right]} \def\rrrbrack{\right]\right]\right]} \def\rrrrbrack{\right]\right]\right]\right]} \def\rrvert{\right\vert\right\vert} \def\rrrvert{\right\vert\right\vert\right\vert} \def\rrrrvert{\right\vert\right\vert\right\vert\right\vert} }[/math][math]\displaystyle{ \def\val#1{\left\langle\begin{matrix}#1\end{matrix}\right]} \def\tval#1{\left\langle\begin{matrix}#1\end{matrix}\right\vert} \def\bival#1{\llangle\begin{matrix}#1\end{matrix}\rrbrack} \def\bitval#1{\llangle\begin{matrix}#1\end{matrix}\rrvert} \def\trival#1{\lllangle\begin{matrix}#1\end{matrix}\rrrbrack} \def\tritval#1{\lllangle\begin{matrix}#1\end{matrix}\rrrvert} \def\quadval#1{\llllangle\begin{matrix}#1\end{matrix}\rrrrbrack} \def\quadtval#1{\llllangle\begin{matrix}#1\end{matrix}\rrrrvert} \def\monzo#1{\left[\begin{matrix}#1\end{matrix}\right\rangle} \def\tmonzo#1{\left\vert\begin{matrix}#1\end{matrix}\right\rangle} \def\bimonzo#1{\llbrack\begin{matrix}#1\end{matrix}\rrangle} \def\bitmonzo#1{\llvert\begin{matrix}#1\end{matrix}\rrangle} \def\trimonzo#1{\lllbrack\begin{matrix}#1\end{matrix}\rrrangle} \def\tritmonzo#1{\lllvert\begin{matrix}#1\end{matrix}\rrrangle} \def\quadmonzo#1{\llllbrack\begin{matrix}#1\end{matrix}\rrrrangle} \def\quadtmonzo#1{\llllvert\begin{matrix}#1\end{matrix}\rrrrangle} \def\rbra#1{\left\{\begin{matrix}#1\end{matrix}\right]} \def\rket#1{\left[\begin{matrix}#1\end{matrix}\right\}} \def\vmp#1#2{\left\langle\begin{matrix}#1\end{matrix}\,\vert\,\begin{matrix}#2\end{matrix}\right\rangle\vsp} \def\wmp#1#2{\llangle\begin{matrix}#1\end{matrix}\,\vert\vert\,\begin{matrix}#2\end{matrix}\rrangle} }[/math] Given a rank-r temperament W and a comma m not tempered out by W, the interior product of W and m is the rank-(r − 1) temperament W ⨼ m which tempers out m in addition to all the commas that are tempered out by W (thus its codimension is one dimension higher than that of W).

Definition

Let W be a n-form, and m₁, m₂, ..., m_n be a group of n vectors. Take the wedge product of these vectors, producing the multivector M. Treating both M and W as ordinary vectors, take the dot product. This is the value of W(m₁, m₂, ..., m_n).

For example, suppose W = ⟨⟨ 6 -7 -2 -25 -20 15 ]], the coordinates for 7-limit miracle. If our two vectors are the vectors for 2 and 15/14, namely [1 0 0 0⟩ and [-1 1 1 -1⟩;, then taking their wedge product gives the bivector [[1 1 -1 0 0 0⟩⟩. The dot product with W is ⟨⟨6 -7 -2 -25 -20 15 ‖ 1 1 -1 0 0 0⟩⟩, which is 6 − 7 − (−2) = 1, so W(2, 15⁄14) = W([1 0 0 0⟩, [-1 1 1 1⟩) = 1. The fact that the result is ±1 tells us that 2 and 15/14 can serve as a pair of generators for miracle; if the absolute value of the result is N, then the vectors it was applied to, when tempered, generate a subgroup of index N of the whole group of intervals of the temperament.

If W is an n-form and m is a vector of the same prime limit p, then form a list of (n − 1) tuples of primes less than or equal to p in lexicographic order. Taking these in order, the i-th element of W ⨼ m. In a slight abuse of notation, we may also write W ⨼ q where q is a rational number with vector representation m. The result of this will be W(s₁, s₂, ..., s_{n − 1}, q), where [s₁, s₂, ..., s_{n − 1}] is the i-th tuple on the list of (n − 1)-tuples of primes. This will result in W ⨼ m, a multival of rank (n − 1).

For instance, let M_Marvel = [[1 2 -3 -2 1 -4 -5 12 9 -19⟩⟩, the coordinates for 11-limit Marvel temperament. To find M_Marvel ⨼ 441/440, we form the list [[2, 3], [2, 5], [2, 7], [2, 11], [3, 5], [3, 7], [3, 11], [5, 7], [5, 11], [7, 11]]. The first element of M_Marvel ⨼ 441/440 will be M_Marvel(2, 3, 441/440), the second element M_Marvel(2, 5, 441/440) and so on down to the last element, M_Marvel(7, 11, 441/440). This gives us ⟨⟨ 6 -7 -2 15 -25 -20 3 15 59 49 ]], which are the coordinates for 11-limit Miracle. The interior product has added a comma to Marvel to produce Miracle.

If we like, we can take the wedge product m ⨼ W from the front by using W(q, s₁, s₂, ..., s_{n − 1}) instead of W(s₁, s₂, ..., s_{n − 1}, q), but this can only lead to a difference in sign. We can also define the interior product of W with a multivector M of rank r < n, by forming a list of (n − r)-tuples of primes in lexicographic order, wedging these together with M, and taking the dot product with W to get a coefficient of W ⨼ M.

Applications

One very useful application is testing whether a temperament tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a k-form if and only if the (k − 1)-form obtained by taking the interior product of the temperament with the interval is the zero form—that is, if all the coefficients are zero.

Another application is the use of the interior product to define the intervals of the abstract regular temperament given by coordinates W. In this case, we use W ⨼ q to define a multilinear form which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multilinear forms, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r primes, where r is the rank of W. Form the matrix [W ⨼ 2, W ⨼ 3, ..., W ⨼ R], where R is the r-th prime number. Let U be the transpose of the pseudoinverse of this matrix, and let V = TU (the matrix product), which can be taken to be an (r − 1)-multivector. Then for any (r − 1)-multival W ⨼ q in the abstract regular temperament, the dot product (W ⨼ q) ∙ V gives the tuning of W ⨼ q. It should be noted that V with this property is underdetermined, so that many possible multivectors can be used to the same effect. An alternative approach is to hermite reduce the matrix [W ⨼ 2, W ⨼ 3, ..., W ⨼ q] and then solve for the linear combination which gives the desired tuning from the dot product. This makes for a simpler result; for example the dot product of M_meantone ⨼ q, where "Meantone" are the coordinates for 7-limit meantone, with [$1⟩ giving the value in cents of the quarter-comma meantone tuning of the interval denoted by M_meantone ⨼ q.

The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank-(r − 1) temperament which supports it. For instance, [[[1 2 -3 -2 1 -4 -5 12 9 -19⟩⟩⟩ are the coordinates for 11-limit Marvel temperament. Then:

[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{45}{44} = \bitval{4 & -3 & 2 & 5 & -14 & -8 & -6 & 13 & 22 & 7} }[/math] gives 11-limit negri,

[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{64}{63} = \bitval{-2 & 4 & 4 & -10 & 11 & 12 & -9 & -2 & -37 & -42} }[/math] gives pajarous,

[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{245}{242} = \bitval{11 & -6 & 10 & 7 & -35 & -15 & -27 & 40 & 37 & -15} }[/math] gives septimin,

[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{99}{98} = \bitval{-7 & 3 & -8 & -2 & 21 & 7 & 21 & -27 & -15 & 22} }[/math] gives orwell,

[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{100}{99} = \bitval{5 & 1 & 12 & -8 & -10 & 5 & -30 & 25 & -22 & -64} }[/math] gives magic,

[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{243}{242} = \bitval{6 & -7 & -2 & 15 & -25 & -20 & 3 & 15 & 59 & 49} }[/math] gives miracle,

[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{3136}{3125} = \bitval{-1 & -4 & -10 & 13 & -4 & -13 & 24 & -12 & 44 & 71} }[/math] gives meanpop,

[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{6250}{6237} = \bitval{6 & 5 & 22 & -21 & -6 & 18 & -54 & 37 & -66 & -135} }[/math] gives catakleismic,

[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{2200}{2187} = \bitval{-1 & 8 & 14 & -23 & 15 & 25 & -33 & 10 & -81 & -113} }[/math] gives garibaldi, and

[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{9801}{9800} = \bitval{-12 & 2 & -20 & 6 & 31 & 2 & 51 & -52 & 7 & 86} }[/math] gives wizard.

The interior product is also useful in finding the temperament mapping its coordinates. Given a p-limit temperament of rank-r, we can find a collection of linear maps belonging to it by taking the interior product with every set of r − 1 primes less than or equal to p, and reducing this to the mapping. For instance, for M_Marvel we take [M_Marvel ⨼ 2 ⨼ 3, M_Marvel ⨼ 2 ⨼ 5, ..., M_Marvel ⨼ 7 ⨼ 11], which gives:

[[0, 0, -1, -2, 3⟩ [0, 1, 0, 2, -1⟩ [0, 2, -2, 0, 4⟩ [0, -3, 1, -4, 0⟩ [-1, 0, 0, 5, -12⟩ [-2, 0, -5, 0, -9⟩ [3, 0, 12, 9, 0⟩ [2, 5, 0, 0, 19⟩ [-1, -12, 0, -19, 0⟩ [4, -9, 19, 0, 0⟩].

Hermite-reducing this results in [-1, 0, 0, 5, -12⟩ [0, 1, 0, 2, -1⟩ [0, 0, -1, -2, 3⟩, the normal form for 11-limit Marvel. In practice, this method nearly always suffices.