Diatonic Pitches as Fifth-Register Vectors

In addition to the traditional representation of diatonic pitch, this work will use a mathematical representation called the fifth-register vector (FRV). The fifth-register vector is the pitch $x\hspace{-1pt}_f\mbox{P5} + x_r\mbox{P8}$ away from C0. For instance, G0 is since it is $1\mbox{P5} + 0\mbox{P8}$ away from C0. Table 3.1 shows the location of some traditionally-represented pitches in FRV vector space.

 

 

$$x\hspace{-1pt}_f$$

x_r -4 -3 -2 -1 0 1 2 3 4

4 F3 C4 G4 D5 A5 E6

3 F2 C3 G3 D4 A4 E5

Readers familiar with the notion of the circle of fifths may find it helpful to think about the ``fifths'' axis of FRV as being an unwound circle of fifths. (The notion of the circle of fifths only makes sense in the context of tunings in which all pitches separated by a d2 are tuned to the same frequency. Therefore, the circle of fifths has no place in a tuning-independent representation of pitch, and we must unwind it into a line (axis) of fifths.)

The FRV representation of pitch was discovered independently by the author, who then later found that Regener [20] had discovered it as well.

Like traditional pitch representation, the FRV is a registral representation. The use of ``register'' to describe the second component of an FRV may seem odd, as the range of pitches in any one register is very large (in fact, infinite) and overlaps with other registers. These spread-out registers are not consistent with the traditional, dense sense of the term ``register,'' but they are wholly consistent with the definition of register and register-doubling tunings given previously in this work.

FRV can be interpreted as pitches or intervals. For example, can be interpreted as the pitch P5 away from C0 (G0) but it can also just be interpreted as P5. Pitches and intervals were actually always the same thing, but traditionally they have been represented differently. This difference in representation can avoid confusion; for example, it would sound a bit weird to say that G3 is E0 above E3. But for mathematical purposes, it makes sense to use the same representation for pitches and intervals.

The familiar mathematical operations of vector addition and negation on FRV are equivalent to the corresponding operations on traditionally represented intervals and/or pitches. For example,

$$\mbox{P5} - \mbox{M2} = \ensuremath{\begin{bmatrix} 1 \\ 0 ... ...nsuremath{\begin{bmatrix} -1 \\ 1 \end{bmatrix}} = \mbox{P4}.$$

An interesting fact is that the FRV is not the only vector representation of interval possible. Regener [20, 58-85] explores the idea of alternate interval spaces, although unfortunately within his unwieldy formalism of diatones and quints. Bases can be changed from P5 and P8 to a variety of other intervals via standard matrix transformation. For instance, to find the transformation matrix to switch to a space whose bases are M2 and Å1, we invert the matrix whose columns are M2 and Å1 expressed as FRV.

$$M = \begin{bmatrix} \vert & \vert \\ \mbox{M2} & \mbox{\AA1... ...matrix}^{-1} = \begin{bmatrix} 4 & 7 \\ -1 & -2 \end{bmatrix}$$

Another interpretation of the transformation matrix is a matrix whose columns are P5 and P8 expressed in the new bases of M2 and Å1.

Not just any two intervals can form a basis of an interval space. In particular, two FRV x and y are valid bases if and only if the matrix

$$M = \begin{bmatrix} x\hspace{-1pt}_f& y\hspace{-1pt}_f\\ x_r & y_r \end{bmatrix}^{-1}$$

exists and has only integer components.