Interior product

[math] \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] \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

The interior product is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or n-map, a multival of rank n induces on a list of n monzos.

Let W be a multival of rank n, and m₁, m₂, ..., m_n be a group of n monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo 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 [math]W = \bitval{6 & -7 & -2 & -25 & -20 & 15}[/math], the wedgie for 7-limit miracle. If our two monzos are the monzos for 2 and 15/14, namely [math]\tmonzo{1 & 0 & 0 & 0}[/math] and [math]\tmonzo{-1 & 1 & 1 & -1}[/math];, then wedging them together gives the bimonzo [math]\bitmonzo{1 & 1 & -1 & 0 & 0 & 0}[/math]. The dot product with W is [math]\wmp{6 & -7 & -2 & -25 & -20 & 15}{1 & 1 & -1 & 0 & 0 & 0}[/math], which is 6 − 7 − (−2) = 1, so [math]W\left(2, \frac{15}{14}\right) = W\left(\tmonzo{1 & 0 & 0 & 0}, \tmonzo{-1 & 1 & 1 & 1}\right) = 1[/math]. 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 n-map is N, then the monzos it was applied to, when tempered, generate a subgroup of index N of the whole group of intervals of the temperament.

If W is a multival of rank n and m is a monzo of the same prime limit p, then form a list of (n − 1) tuples of primes less than or equal to p in alphabetical order. Taking these in order, the i-th element of W ∨ m, which we may also write W ∨ q where q is the rational number with monzo m, 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 [math]M_\text{Marvel} = \tritmonzo{1 & 2 & -3 & -2 & 1 & -4 & -5 & 12 & 9 & -19}[/math], the wedgie 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 [math]\bitval{6 & -7 & -2 & 15 & -25 & -20 & 3 & 15 & 59 & 49}[/math], which is the wedgie 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 multimonzo M of rank r < n, by forming a list of (n − r)-tuples of primes in alphabetical 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 wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank-r wedgie if and only if the rank r − 1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival—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 a wedgie W. In this case, we use W ∨ q to define a multival 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 multivals, 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 TU (the matrix product), which can be taken to be an (r − 1)-multimonzo. 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 multimonzo vectors 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" is the 7-limit wedgie, with [math]\tmonzo{1200 + 300 \log_{2}(5) & -1200 & 0 & 0}[/math] 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, [math]\tritmonzo{1 & 2 & -3 & -2 & 1 & -4 & -5 & 12 & 9 & -19}[/math] is the wedgie for 11-limit Marvel temperament. Then:

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

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

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

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

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

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

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

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

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

[math]M_\text{Marvel} ∨ \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 given the wedgie. Given a rank-r p-limit wedgie, we can find a collection of vals 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:

[math]\left[\tmonzo{0 & 0 & -1 & -2 & 3},\right.[/math] [math]\tmonzo{0 & 1 & 0 & 2 & -1},[/math] [math]\tmonzo{0 & 2 & -2 & 0 & 4},[/math] [math]\tmonzo{0 & -3 & 1 & -4 & 0},[/math] [math]\tmonzo{-1 & 0 & 0 & 5 & -12},[/math] [math]\tmonzo{-2 & 0 & -5 & 0 & -9},[/math] [math]\tmonzo{3 & 0 & 12 & 9 & 0},[/math] [math]\tmonzo{2 & 5 & 0 & 0 & 19},[/math] [math]\tmonzo{-1 & -12 & 0 & -19 & 0},[/math] [math]\left.\tmonzo{4 & -9 & 19 & 0 & 0}\right].[/math]

Hermite-reducing this to a normal val list results in [math]\left[\tmonzo{-1 & 0 & 0 & 5 & -12},\right.[/math] [math]\tmonzo{0 & 1 & 0 & 2 & -1},[/math] [math]\left.\tmonzo{0 & 0 & -1 & -2 & 3}\right][/math], the normal val list for 11-limit Marvel. In practice, this method nearly always suffices.