Average-Frequency Term in Acoustic Beat Phenomena

Reading Position

This page is the careful acoustic companion to the trig-free recursive beat note. Read it immediately after that page. The trig-free note shows the recursive phase-state mechanism in compact form; this document adds the teaching caution around Fourier spectra, finite-window analyzers, and psychoacoustic reports.

The central correction is modest and strong: the product-form identity is valid, but ideal linear superposition of two pure tones at f1 and f2 does not by itself generate a new independent Fourier spectral line at f_avg = (f1 + f2)/2.

Five-Way Separation

The introduction says the classroom explanation often compresses too many things into one sentence. The note separates them so each can be claimed at the right level.

Equal-Amplitude Identity

For two equal-amplitude pure tones, the standard sum-to-product identity rewrites the time-domain sum as a product. This is correct algebra and still worth teaching.

y(t) = A*cos(2*pi*f1*t) + A*cos(2*pi*f2*t)

f_avg = (f1 + f2) / 2
f_mod = abs(f1 - f2) / 2

y(t) = 2*A*cos(2*pi*f_mod*t)*cos(2*pi*f_avg*t)

The caution is interpretive. This product form reorganizes the same signal. It is not a new physical mechanism and does not prove that a separate generated tone exists at f_avg.

Beat Rate Versus Signed Modulation

The signed modulation term has frequency abs(f1 - f2)/2, but loudness maxima occur twice per signed cycle because both positive and negative peaks have large magnitude. That is why the usual audible beat rate is the full difference frequency.

f_mod = abs(f1 - f2) / 2

f_b = abs(f1 - f2)

f_b = 2*f_mod

This is the factor-of-two trap. A careful teaching version says the signed envelope oscillates at half the difference frequency, while the counted loudness beat rate is normally the full difference frequency.

Recursive Phase-State Reading

The document repeats the recursive formulation from the trig-free note. A tone is represented as a two-component phase state advanced by a rational, norm-preserving update. The observed pressure is a projection.

u_i,n = (x_i,n, y_i,n)^T

u_i,n+1 = R(a_i) u_i,n

R(a_i) = (1 / (1 + a_i^2)) *
         [[1 - a_i^2, -2*a_i],
          [2*a_i,      1 - a_i^2]]

R(a_i)^T R(a_i) = I
||u_i,n+1||^2 = ||u_i,n||^2

p_i,n = A_i x_i,n

Overlap As Beat Mechanism

For two tones, the pressure is the projected sum of two distinct generators. The envelope is the norm of their combined phase vector, and the changing term is the dot product between phase states.

p_n = A_1*x_1,n + A_2*x_2,n

U_n = A_1*u_1,n + A_2*u_2,n

B_n^2 = ||A_1*u_1,n + A_2*u_2,n||^2

B_n^2 = A_1^2 + A_2^2
        + 2*A_1*A_2*(u_1,n dot u_2,n)

u_1,n dot u_2,n = x_1,n*x_2,n + y_1,n*y_2,n

No average-frequency oscillator is introduced. The beat appears because the relative alignment of the two original recursive generators changes slowly.

Relative Closure

The recursive reading also explains the beat-rate distinction. The relative update compares the two phase generators. The audible beat is governed by the closure cycle of this relative update.

Q = R(a_1)^(-1) R(a_2)

m_n = u_1,n dot u_2,n

m_n = u_1,0 dot Q^n u_2,0

Q^N_b ~= I

f_b = 1 / (N_b * Delta_t)

N_signed = 2*N_b
f_mod = f_b / 2

Long Chains Of Tones

The note extends the same logic to many tones. A longer sound should not be explained by inventing new average-frequency oscillators between every pair. Its beat structure is built from pairwise relative phase overlaps among the original recursive generators.

u_i,n+1 = R(a_i) u_i,n

p_n = sum_i A_i*x_i,n
U_n = sum_i A_i*u_i,n

B_n^2 = ||sum_i A_i*u_i,n||^2

B_n^2 = sum_i A_i^2
        + 2*sum_{i<j} A_i*A_j*(u_i,n dot u_j,n)

Fourier Spectrum

For the ideal two-tone signal, the Fourier spectrum contains the source frequencies f1 and f2. In ideal linear superposition, there is no additional independent line at f_avg.

This is not the same as saying the average frequency is meaningless. It exists as a useful algebraic quantity in one representation. What does not follow is a separate spectral line or separately generated audible tone.

Claim Status

The PDF includes a claim-status table. That table is one of the most useful pieces of the document because it prevents the analysis from sliding between algebra, spectrum, perception, and pedagogy.

Finite-Window Analysis

A spectrum analyzer or short-time Fourier transform does not observe an infinite ideal signal. It reads a windowed portion of the signal. The display depends on sampling rate, window function, window length, bin spacing, phase, leakage, and averaging mode.

displayed spectrum
= signal content filtered through window length, leakage, phase, and resolution

A spectrogram can show ridges that blur, split, or vary in apparent intensity. That can be useful, but it should not be treated as direct proof that source amplitudes are physically changing.

Auditory Perception

The absence of a Fourier line at f_avg does not settle what a listener hears. The auditory system is frequency-selective and nonlinear in important ways. Depending on spacing, level, timbre, phase locking, masking, roughness, and context, a listener may report pulsing, roughness, one unresolved pitch, two resolved tones, or a mixed perception.

algebraic identity != Fourier line
Fourier line != analyzer display
analyzer display != listener report

Perception must be tested as perception. The product-form equation alone should not be used as proof that the ear receives or hears a newly generated average-frequency tone.

Numerical Demonstration

The source proposes a simple numerical demonstration using two nearby tones. It is designed to show the time-domain waveform, the ideal long-window spectrum, the recursive phase overlap, and the difference between short and long STFT windows.

fs = 48000 Hz
f1 = 440 Hz
f2 = 444 Hz

f_avg = 442 Hz
f_mod = 2 Hz
f_b = 4 Hz

choose a_i = tan(pi*f_i/fs)
or treat a_i as generator parameter
and calibrate by observed closure cycles

The proposed failure condition is also sensible: if a controlled, low-distortion, linear setup produced a stable spectral line at f_avg above leakage and noise, that would be evidence for a real nonlinear process or measurement artifact to investigate, not a result that follows from the identity.

Teaching Recommendation

The note recommends replacing a common classroom shortcut with a safer sentence. Rather than saying two nearby tones produce a tone at the average frequency that varies in loudness, say that the sum can be rewritten as a carrier-like term at the average frequency multiplied by a signed envelope term, while the ideal spectrum still contains the original two tones.

unsafe shortcut:
two nearby tones produce a tone at the average frequency

safer version:
the average-frequency term is a carrier-like factor
inside one representation of the same waveform

recursive version:
each tone remains its own norm-preserving phase update
the beat is changing overlap
the audible beat rate is relative closure rate

RSG Fit

This page is valuable for RSG because it is a clean example of representation discipline. A visual or algebraic rewrite can be useful without creating a new physical generator. In RSG language, the represented scalar may change because of projection and overlap while the underlying generators remain distinct.

generator layer: f1, f2 or R(a1), R(a2)
representation layer: product form or projected pressure
analysis layer: windowed FFT/STFT
perception layer: listener report

Terms To Carry Forward

Average-frequency term

The arithmetic midpoint factor in the product representation, not automatically a new spectral line.

Signed modulation

The signed envelope factor at half the difference frequency.

Audible beat rate

The usual counted loudness-maxima rate, normally abs(f1 - f2).

Relative closure

The recursive reading of beat rate through the closure cycle of the relative update Q.

Finite-window display

A spectrum display shaped by windowing, leakage, phase, and resolution.

Psychoacoustic report

What the listener hears; a separate empirical question from algebra or ideal spectrum.

Copyable Core

These anchors are written in ASCII so they can be copied into notes or live-theory pages.

y(t) = A*cos(2*pi*f1*t) + A*cos(2*pi*f2*t)

y(t) = 2*A*cos(2*pi*f_mod*t)*cos(2*pi*f_avg*t)

f_avg = (f1 + f2) / 2
f_mod = abs(f1 - f2) / 2
f_b = abs(f1 - f2)

u_i,n+1 = R(a_i) u_i,n
p_i,n = A_i*x_i,n

Q = R(a_1)^(-1) R(a_2)
Q^N_b ~= I
f_b = 1 / (N_b * Delta_t)

ideal linear spectrum:
lines at f1 and f2
no independent line required at f_avg

Claim Discipline

The safest one-sentence version is: the algebraic carrier exists in the representation; the recursive generators remain distinct; the ideal Fourier components remain at the original frequencies; and the perceptual report must be studied separately.

• Do not deny the product identity.
• Do not turn the product identity into proof of a new midpoint spectral line.
• Do not read finite-window display blur as automatic source-amplitude change.
• Do not use algebra alone to decide what the listener hears.

How It Fits The Library

This is reading-order item 21. It follows the trig-free recursive beat note because it makes the same point with broader acoustic caution. It precedes the Koide note because both later pages are about avoiding overinterpretation: a formula can be true and useful without proving the mechanism a reader may be tempted to infer from it.

Recommended Reading Move

Read the identity section, then the beat-rate section, then the Fourier and finite-window sections. That order keeps the source's main correction visible: the identity reorganizes a signal, while spectrum, display, and perception each require their own claim status.