The Representation of Triadic Pitch

Previously we have seen that it is impossible to achieve just M3 and P5 in any diatonic tuning. It was suggested that multiple tunings of the same pitch, i.e. dynamic intonation, be allowed in order to admit this possibility. But we cannot allow this since our theory requires a tuning to be a function, which requires the production of a unique output from any given input. The answer to this dilemma is to switch to a richer description of pitch class. This allows us to discuss dynamic intonation while still using most of the theory we have developed for static intonation (tuning).

Diatonic pitch classes (traditional or FRV) are identified by a distance in P5. We would like to be able to identify triadic pitch classes in terms of a two-dimensional distance in P5 and M3_-1, where M3_-1 is a new kind of M3 not defined in terms of P5. We will first enrich traditional representation with this extra information and then enrich FRV similarly.

Let's look at an example of what this representation will need to do and then present a way to accomplish it. The diatonic pitch class E identifies itself as 4P5 away from C. We need to differentiate this E from the one that is M3_-1 away from C. To accomplish this differentiation, let us call C + 4P5 ``E<sub>0</sub>'' and C + M3<sub>-1</sub> ``E_-1.''

The general rule of this new representation is that the familiar diatonic pitch classes are subscripted by zero, and all the classes that are M3_-1 above them retain the letter name one would expect but are subscripted by -1. For example, the F that is M3_-1 above D₀ is F_-1, and a major triad based on G3 would consist of G3₀, B3_-1, and D4₀. This notation extends to subscripts other than -1. For example the pitch class M3_-1 below C is A₁.

Other theoreticians use a system of subscripting that looks similar to this but means something quite different. This other system has its roots at least as far back has Helmholtz and Ellis [12, 276-8]. New to this work is the idea that a subscripted pitch is a triadic pitch, not a frequency ratio. Conventionally, X_n means $\ensuremath{\ensuremath{\underline{\tau}}_{{}}\!\left(X\right)} + nC_S$ where is the Pythagorean tuning of diatonic pitch X and C_S is the syntonic comma. The notion of triadic pitch is far more theoretically powerful because it does not imply a specific tuning. As we shall see below, the conventional meaning of subscripting implies just triadic tuning. Our use of subscripting is more closely related to Lindley and Turner-Smith's concept of the pitch class relation for M3 (``III'' in their notation) [14, 24].

Traditional notation will be enriched to describe triadic intervals in the following manner. The triadic interval from triadic pitch X_i to Y_j is the diatonic interval from X to Y subscripted by j-i. For example, the interval from D3₀ to F3₁ is m3₁.

We will enrich the FRV representation by adding an initial ``thirds'' component to the vector, forming a third-fifth-register vector (TFRV) that represents the interval

$$x_t\mbox{M3}_{-1} + x\hspace{-1pt}_f\mbox{P5} + x_r\mbox{P8}$$

or the pitch that is that distance away from C0₀. For example, A major triad based on C0₀ would consist of ,, and . The three-dimensional vector representation of TFRV represents a significant and original extension to Regener's work in two-dimensional pitch representations.

A traditionally represented triadic pitch or interval X_i can be converted to a TFRV y in the following manner. Consider X_i as

$$\mbox{X}_i = \mbox{X}_0 + i\mbox{P1}_1.$$

Note how the subscript i functions very similarly to the way sharps do in diatonic notation: it defines a multiple of the smallest positive prime interval. Diatonically, this is Å1, and triadically it is P1₁. By the way, what exactly is this strange interval P1₁? Well, for example, it is the interval from E_-1 to E₀.

To convert $\mbox{X}_0 + i\mbox{P1}_1$ to a TFRV, we need to express X₀ and $i\mbox{P1}_1$ as TFRV. X₀ is easy to express as a TFRV; if X has FRV , it is just . An easy way to calculate P1₁ in TFRV form is to reckon it as the difference between M3₀ (our old, P5-based notion of M3) and M3_-1 (our new notion of M3). By this method,

$$\mbox{P1}_1 = \mbox{M3}_0 - \mbox{M3}_{-1} = \ensuremath{[0\... ...}\ {0}]^\mathrm{T}} = \ensuremath{[-1\ {4}\ {-2}]^\mathrm{T}}.$$

Therefore, the TFRV y for the traditional triadic pitch or interval X_i can be expressed as follows:

$$\mbox{\boldmath$y$} = \ensuremath{[0\ {x\hspace{-1pt}_f}\ {x_r}]^\mathrm{T}} + i\ensuremath{[-1\ {4}\ {-2}]^\mathrm{T}}.$$