Converting Pitches to FRV

This section will demonstrate that any pitch can be represented as an FRV by giving an explicit algorithm for the conversion from traditional notation to FRV. This conversion algorithm is original to this thesis, i.e. not present in Regener's work.

For our present purposes, it will be useful to consider traditional pitch notation as having three components: letter, accidental section, and register. It is possible to represent these components as a triple of integers $(x_\ell,x_s,x_r)$ where $x_\ell$ is in $[0\ldots6]$ and indexes into the letter array [C,D,E,F,G,A,B], x_s is the number of sharps in the accidental section (where flats count as negative sharps), and x_r is the register. For example, D4 is (1,1,4), G6 is (4,0,6), and E5 is (2,-1,5). Using this triple notation, we can express the interval from C0 to any note $x=(x_\ell,x_s,x_r)$ as

$$\beta(x) = \gamma(x_\ell) + x_s\mbox{\AA1} + x_r\mbox{P8}$$

where $\gamma(x_\ell)$ is as follows:

$$x_\ell$$ 0 1 2 3 4 5 6

$$\gamma(x_\ell)$$ P1 M2 M3 P4 P5 M6 M7

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Note that $\gamma(x_\ell)$ yields the intervals of an ascending major scale.

To convert pitches to FRV, we need to rewrite $\beta(x)$ with all intervals expressed in FRV rather than traditional form. This gives us

$$\mbox{\boldmath$\beta$}(x) = \mbox{\boldmath$\gamma$}(x_\ell... ...atrix}} + x_r\ensuremath{\begin{bmatrix} 0 \\ 1 \end{bmatrix}}$$

where $\gamma(x_\ell)$ is as follows:

$$x_\ell$$ 0 1 2 3 4 5 6

$$\mbox{\boldmath$\gamma$}(x_\ell)$$

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Reviewing the concepts above, Table 3.2 lists some pitches in traditional and FRV form.

 

 

C0 D1 E-1 F0 G-1 A0 B0