Turning Trigonometric Formulas into Sound

― Visualizing and Hearing Math with Python (Google Colab)

Author: Math & Sound Engineering Lab
Date: November 13, 2025

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1. Overview

This project converts trigonometric identities into audible sine-wave tones, allowing students and engineers to “hear” the difference between formulas such as the addition theorem, double-angle, or half-angle formulas.

Using NumPy, matplotlib, and soundfile, the program synthesizes short tones whose waveforms directly reflect the structure of each trigonometric equation.
All code runs natively in Google Colab without any external dependencies.

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2. Concept

Every trigonometric formula represents a relationship between waves.
For instance:

Formula	Mathematical Meaning	Acoustic Analogy
sin(A + B)	Interference between two signals	Beat tone
sin(A − B)	Phase difference	Slight detuning
sin 2θ	Frequency doubling	2× harmonic
sin(θ⁄2)	Frequency halving	One octave below
sin A cos B	Product–sum mixing	AM-type modulation
sin A + sin B	Sum–product blending	Polyphonic beating

By mapping these formulas to time-varying sine waves, mathematics becomes directly perceivable through sound.

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3. Implementation (Python)

# Program Name: trig_allformulas_sineplayer_colab_en_singleton.py
# Overview: Sequential playback of all trigonometric formula tones + one pure sine reference tone.

!pip install numpy scipy soundfile matplotlib --quiet

import numpy as np, soundfile as sf, matplotlib.pyplot as plt, IPython.display as ipd, re

SR, DURATION, AMP, BASE_FREQ = 44100, 2.0, 0.4, 440.0

def safe_filename(name):
    return re.sub(r'[^A-Za-z0-9_\-]', '_', name)

def formula_sine(name, func, freq_factor):
    t = np.linspace(0, DURATION, int(SR*DURATION))
    A = 2*np.pi*BASE_FREQ*freq_factor[0]*t
    B = 2*np.pi*BASE_FREQ*freq_factor[1]*t
    y = AMP*func(A,B,t)/np.max(np.abs(func(A,B,t)))
    fname = safe_filename(name)+".wav"
    sf.write(fname, y, SR)
    print(f"▶ {name}")
    plt.figure(figsize=(8,1.8)); plt.plot(t[:2000], y[:2000]); plt.title(name)
    plt.xlabel("Time [s]"); plt.ylabel("Amplitude"); plt.show()
    ipd.display(ipd.Audio(fname, rate=SR))

formulas = {
    "Addition Formula sin(A+B)": (lambda A,B,t: np.sin(A+B), (1.0,1.5)),
    "Subtraction Formula sin(A-B)": (lambda A,B,t: np.sin(A-B), (1.0,0.75)),
    "Double Angle Formula sin(2θ)": (lambda A,B,t: np.sin(2*A), (0.5,1.0)),
    "Half Angle Formula sin(θ/2)": (lambda A,B,t: np.sin(A/2), (2.0,1.0)),
    "Product-to-Sum Formula": (lambda A,B,t: (np.sin(A+B)+np.sin(A-B))/2, (1.0,1.25)),
    "Sum-to-Product Formula": (lambda A,B,t: 2*np.sin((A+B)/2)*np.cos((A-B)/2), (1.0,1.5))
}

for name,(func,factors) in formulas.items():
    formula_sine(name, func, factors)

print("▶ Pure Sine Reference Tone (A4 = 440 Hz)")
t = np.linspace(0, DURATION, int(SR*DURATION))
y = AMP*np.sin(2*np.pi*BASE_FREQ*t)
sf.write("Pure_Sine_440Hz.wav", y, SR)
plt.figure(figsize=(8,1.8)); plt.plot(t[:2000], y[:2000])
plt.title("Pure Sine Reference (440 Hz)"); plt.xlabel("Time [s]"); plt.ylabel("Amplitude"); plt.show()
ipd.display(ipd.Audio("Pure_Sine_440Hz.wav", rate=SR))

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4. How It Works

1. Wave Generation
   Each formula defines a mathematical transformation of sine waves A and B.
   Different frequency ratios (1.0, 1.5, etc.) simulate harmonic intervals such as octaves or fifths.

2. Normalization & Output
   The synthesized waveform is normalized to ±0.4 amplitude and written as a 44.1 kHz WAV file.

3. Visualization
   A short time window (2 000 samples) is plotted to show the oscillatory pattern of each equation.

4. Playback
   IPython.display.Audio() embeds a real-time player in Colab so users can listen directly.

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5. Results

Running the cell in Colab plays six successive tones:

1. Addition Formula → Complex beat
2. Subtraction Formula → Slightly detuned oscillation
3. Double Angle → Higher frequency (2× pitch)
4. Half Angle → Lower pitch (½ frequency)
5. Product-to-Sum → Mixed modulated tone
6. Sum-to-Product → Rich amplitude-modulated pattern
7. Pure Sine (440 Hz) → Reference tone for comparison

Each sound corresponds to the mathematical structure of the equation—an audible demonstration of trigonometric synthesis.

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6. Educational Significance

• Mathematics + Music Integration: Demonstrates that trigonometric identities underpin musical acoustics and signal theory.
• STEAM Education: Bridges abstract formulas with experiential sound.
• Signal Processing Insight: Visualizes frequency addition, subtraction, and modulation using simple math.

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7. Possible Extensions

1. Spectrogram Visualization
   Add librosa.display.specshow() to visualize frequency content in time.
2. MIDI Export
   Convert sine tones into MIDI notes for use in DAWs.
3. Interactive GUI
   Allow real-time formula switching and pitch selection with ipywidgets.
4. Polyphonic Combination
   Mix multiple formulas simultaneously to explore harmonic interference.

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Conclusion

This experiment shows that trigonometric formulas are not just symbols—they are sound generators.
By turning math into music, learners can intuitively grasp how addition, subtraction, and modulation shape the world of waves, audio, and beyond.