Pitch Space

Let \(X\) be the continuous space of pitchclasses given by logarithmic (base 2) frequencies mod 1. Then \(X\) is a continuous space representing picthclasses under octave equivalency. Furthermore, \(X\) is homotopy equivalent to the circle \(S^1\). Take \(C \cong \mathbb{R}\) regarded as the logarithmic space of pitches (frequencies). Then \(f : C \rightarrow X\) given by \(f : r \mapsto r \text{ mod }1\) is a covering map. In particular \((C,f)\) gives a universal cover, in the sense that any other covering factors through \(f\).

The universal lifting property for coverings states that, given a covering, every path in the base space lifts to a unique path in the covering space.

Octave Equivalent Mappings

We call a mapping out of the universal covering space of \(X\) into some arbitrary space, \(m : C \to B\), octave equivalent if there exists a map \(e : X \to B\) s.t. \(m = e \circ f\).

Note that not all mappings are octave equivalent. The purpose of this writing is to suggest that there are musically relevant mappings which fail to respect octave equivalency.

Intermodulation Distortion

Let \(A(t) = \sin (f_\beta (t))\) be an audio signal, where \(f_\beta : t \mapsto 2\pi\beta t\). Real world systems are always to some degree non-linear. Suppose we can model this non-linearity as a linear and a quadratic term: \(P(A(t)) = \alpha_0 A(t) + \alpha_1 A(t)^2\). We can then use the trigonometric identity \(\sin (\theta) = \frac{1-\cos(2\theta)}{2}\) to rewrite the system as \(P(A(t)) = \alpha_0 \sin(f_\beta) + \frac{1}{2}\alpha_1 (1-\cos (f_{2\beta}))\). If we neglect the phase shift introduced by the second term containing a cosine instead of a sine, and the DC offset introduced in that term, we can see that \(P\) contains two components: the original signal and a signal at twice the frequency of the original. This is harmonic distortion. Nonlinear systems applied to single pure tones produce (ignoring phase shifts and DC offsets) a linear combination of multiples of the original pure tone.

But what about signals more complicated then a single pure tone? Let \(B(t) = \sin (f_\beta) + \sin (f_\gamma)\). Then

So we can see that pushing two pure tones through this nonlinear system produces: the original tones, tones at double the original frequencies, and tones at the sum and difference of the two input frequencies. In general, upto phase differences, the result of running a collection of tones through a nonlinear system is a collection of harmonics (integer multiples of the input frequencies), and sums and differences of the input tones and their harmonics. This is called intermodulation distortion.

For a component \(\sin(f_{m\beta} \pm f_{n\gamma})\), where \(\beta, \gamma\) are input frequencies, we call \(m+n\) the order of that component. The nonlinearity of most real world systems is mostly low-order. So for a first analysis, we’ll restrict our attention to second-order effects; namely sums and differences of input frequencies, and doublings of input frequencies.

Chords

Given an \(n\) note chord, we have \(n^2\) second order intermodulation products. Of these, \(n\) are simple doublings of the input frequencies. If we assume that octave equivalency has some musical validity, then we can set aside the simple doublings and focus instead on the \(n^2-n = 2\binom{n}{2}\) (musically) nontrivial intermodulation products.

Second order intermodulation products can be computed pairwise. Higher order components can be computed by recursing on the union of the lower order components.

I don’t have a good unifying scheme for analyzing these products from a musical perspective, so for the moment I present here a table of illustrative computations.

Each entry in the table lists the second order intermodulation products which result from combining the row note with the next note up of the column pitchclass. These distortion products are listed as a pitchclass name along with octave number, and a signed error measured in halfsteps; e.g. A4 intermodulated with A5 (the next A ‘up’) results in a somewhat sharp E6 (sharp by ~20 cents), and an exact A4.

Chord Voicings

Consider a CMaj triad in close voicing, \(\{\text{C5},\text{E5},\text{G5}\}\). The second order intermodulation products arrising from this triad are \(\{(\text{D6,1.153e-1}),(\text{E6,-1.486e-1}),(\text{Gb6,-4.351e-1}),(\text{Db3,-3.263e-1}),(\text{C3,-5.872e-2}),(\text{B2,1.765e-1})\}\). By contrast the second order intermodulation products arrising from a wide voiced CMaj triad \(\{\text{C5},\text{E6},\text{G5}\}\) are \(\{(\text{Bb6,-2.139e-1}),(\text{C7,7.837e-2}),(\text{E6,-1.486e-1}),(\text{G5,2.471e-1}),(\text{C5,3.689e-1}),(\text{C3,-5.872e-2})\}\).

If we neglect tuning/temperment issues, we can see that the close voiced major triad produces intermodulation products corresponding to the ninth, the third an octave up, the flat five (flatter than normal) an octave up, the minor second, the root two octaves down, and the major seventh three octaves down.

The wide voiced major traid produces intermodulation products corresponding to the minor seventh, the root two octaves up, the major third one octave up, the fifth, the root, and the root two octaves down.

On this account then, it is no surprise that wide voiced triads are less dissonant than their close voiced cousins. Wide voiced major triads hint at the presence of the major seventh, but are otherwise consonant with themselves. Close voiced major triads on the other hand produce a minor second, flat five, and a major ninth!

Consider now Cmin triads voiced similarly.

A close voiced Cmin triad produces the following intermodulation products \(\{(\text{D6,-4.351e-1}),(\text{E6,-1.486e-1}),(\text{F6,1.153e-1}),(\text{G2,1.765e-1}),(\text{C3,-5.872e-2}),(\text{E3,-3.263e-1})\}\). These correspond to a (particularly flat) major ninth, the major third, the perfect fourth, the fifth three octaves down, the root two octaves down, and the third three octaves down.

A close voiced Cmin triad produces the following intermodulation products \(\{(\text{A6,7.615e-2}),(\text{B6,4.580e-1}),(\text{E6,-1.486e-1}),(\text{Gb5,-4.439e-1}),(\text{Bb4,-2.110e-1}),(\text{C3,5.872e-2})\}\) These correspond to the major thirteenth, major seventh an octave up, major third an octave up, flat five, minor seventh an octave down, and the root two octaves down.