Trig-Free Recursive Reading of Beat Phenomena

Reading Position

This page sits after the circular phase geometry note and before the careful acoustic beat note. It belongs in the mathematics part of the library because it demonstrates a recurring RSG habit: start with recursive state updates, preserve the state norm, then interpret observable structure as projection, overlap, or closure rather than as a new object introduced by algebraic rewriting.

The PDF is compact, about 1,200 extracted words over 5 pages. Its central point is not that the textbook trigonometric identity is wrong. The identity is correct. The warning is interpretive: the average frequency in the product form is a carrier-like factor in one time-domain factorisation, not automatically a new Fourier component.

Purpose

Textbook beat explanations often begin by rewriting a sum of two cosines as a product of a slow factor and a carrier-like factor. That rewrite is useful, but it can make the average-frequency term look like a newly created physical component.

cos(2*pi*f1*t) + cos(2*pi*f2*t)
= 2*cos(pi*(f1 - f2)*t)*cos(2*pi*((f1 + f2)/2)*t)

The note keeps the original generators at f1 and f2. The beat is then described as the changing overlap of two recursive phase states, not as the creation of an extra oscillator at the average frequency.

Recursive Phase State

A tone is represented by a two-component phase state. Instead of calling sine or cosine as primary functions, the state is advanced by a rational, norm-preserving rotation update.

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]]

The update preserves the phase-state norm, so the state remains on the same circle. The observed scalar sound signal is then a projection of this recursive state.

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

Two Tones

For two tones, the measured pressure is simply the projected sum. The two phase generators remain distinct; the beat comes from how their phase states align and misalign over time.

p_n = A_1 x_1,n + A_2 x_2,n

u_1,n+1 = R(a_1) u_1,n
u_2,n+1 = R(a_2) u_2,n

U_n = A_1 u_1,n + A_2 u_2,n

The squared beat envelope is the squared norm of the combined phase vector. When the phase states are normalised, the only changing term is their dot product.

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

Relative Update

The beat rate is governed by the relative phase update between the two tone generators. Since the updates are norm-preserving rotations of the same type, the relative update also preserves norm.

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

The audible beat is controlled by the closure cycle of Q. When the relative update returns close to the identity, the relative alignment repeats.

Q^N_b ~= I

f_beat = 1 / (N_b * Delta_t)

Factor-Of-Two Trap

The usual product identity introduces a signed modulation factor. If this signed factor is called f_mod, it is half the audible beat rate counted as loudness maxima.

f_avg = (f1 + f2) / 2

f_mod = abs(f1 - f2) / 2

f_beat = abs(f1 - f2)

f_beat = 2*f_mod

In recursive language, the audible beat is the relative closure rate of Q. The half-rate term is a signed-envelope convention that returns to the same signed state only after twice the audible beat interval.

Long Chain Of Tones

The same construction generalises to many tones. Each tone has its own recursive phase state, the observed pressure is the sum of projected components, and the envelope is built from pairwise overlaps.

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)

In the note's compact chain, individual recursive phase states become a projected pressure sum, which becomes pairwise phase overlaps, which become audible beat structure.

Interpretation

The source does not reject the trigonometric product identity. It rejects an over-reading of that identity. The original spectral generators remain the two tones. The average-frequency factor belongs to one algebraic representation of their time-domain sum.

generators: f1, f2
beat mechanism: changing phase-state overlap
not required: new oscillator at (f1 + f2)/2

RSG Fit

This page is useful for RSG because it turns an ordinary wave phenomenon into recursive language without inflating the claim. The object that closes is not a new average-frequency tone; it is the relative update. The observable envelope comes from projection and overlap.

recursive state -> norm-preserving update
projection -> observed scalar signal
relative update -> closure cycle
overlap -> beat envelope

That makes it a good companion to the visualisation language as well: smooth wave appearances can be read as projections of underlying recursive phase states, while beat-like structure is a relation among generators rather than a separate thing added by the drawing.

Terms To Carry Forward

Recursive phase state

A two-component state advanced by a norm-preserving update rather than by primitive sine or cosine calls.

Projection

The scalar observable obtained from the phase state, such as p_i,n = A_i x_i,n.

Overlap

The dot product between phase states; this is the changing term that drives the beat envelope.

Relative update

The update Q = R(a_1)^(-1) R(a_2) comparing the two generators.

Relative closure rate

The audible beat rate read from the step count needed for the relative update to return near identity.

Signed-envelope convention

The half-rate modulation in the textbook product identity; useful algebra, but not the same as counted loudness maxima.

Copyable Core

These compact anchors are written in ASCII for reuse in notes and live theory pages.

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

R(a_i)^T R(a_i) = I

p_n = A_1 x_1,n + A_2 x_2,n

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

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

Q^N_b ~= I
f_beat = 1 / (N_b * Delta_t)

f_mod = f_beat / 2

Claim Discipline

The note is a pedagogical and mathematical reframing. It does not deny Fourier analysis, and it does not say the product identity is false. It says that the average-frequency factor in the product form is a representation inside an algebraic rewrite, not automatically a newly created spectral line.

• Keep f1 and f2 as the original generators.
• Read the beat envelope as overlap, not as an added average-frequency oscillator.
• Distinguish audible beat maxima from signed-envelope recurrence.
• Use relative closure of Q as the recursive statement of beat rate.

How It Fits The Library

This is reading-order item 20. It follows the circular phase geometry note because both pages help the library speak in recursive update language without losing the standard mathematical reference point. It is followed by the more careful acoustic beat note, which separates the algebraic identity, the Fourier spectrum, finite-window displays, and perception before the route moves on to Koide-style algebra.

Recommended Reading Move

Read the two-tone section first, then the relative update, then the factor-of-two trap. Those three sections carry the core lesson: beats are relative closure of recursive phase states, while the textbook average-frequency term is a carrier-like factor in a chosen representation.