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Tutoring abilities

How did classic 1980s synthesizers produce razor-sharp electric pianos, punchy basses, and shimmering brass using only pure, dull sine waves?

In 1967, composer John Chowning discovered that speeding up vibrato into the audio spectrum creates entirely new sonic overtones instead of pitch wobbles.

**Frequency Modulation (FM) synthesis** generates complex audio spectra by modulating the frequency of a **carrier** oscillator using another oscillator known as the **modulator**.

When modulation occurs at audio rates ($f_m \ge 20\text{ Hz}$), our ears stop hearing pitch fluctuations and instead perceive a rich change in **timbre** (tone color).

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>FM Synthesis: Carrier & Modulator</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      min-height: 100%;
      user-select: none;
    }

    /* Top Flow Architecture */
    .flow-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 8px;
    }
    .node {
      flex: 1;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 8px 10px;
      text-align: center;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
    }
    .node-mod { border-top: 4px solid #f59e0b; }
    .node-car { border-top: 4px solid #0284c7; }
    .node-title {
      font-size: 13px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }
    .node-mod .node-title { color: #d97706; }
    .node-car .node-title { color: #0284c7; }
    .node-val {
      font-size: 15px;
      font-weight: 700;
      color: #1e2945;
      margin-top: 2px;
    }
    .arrow-sign {
      font-size: 18px;
      font-weight: bold;
      color: #94a3b8;
    }

    /* Screen Display Panel */
    .display-panel {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px;
      display: flex;
      flex-direction: column;
      gap: 8px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
    }
    .canvas-wrap {
      width: 100%;
      height: 120px;
      background: #0f172a;
      border-radius: 6px;
      overflow: hidden;
      position: relative;
    }
    canvas {
      width: 100%;
      height: 100%;
      display: block;
    }
    .spectrum-legend {
      display: flex;
      justify-content: space-between;
      align-items: center;
      font-size: 12px;
      font-weight: 600;
      color: #64748b;
      padding: 0 2px;
    }
    .legend-tag {
      display: inline-flex;
      align-items: center;
      gap: 4px;
    }
    .dot {
      width: 8px;
      height: 8px;
      border-radius: 50%;
    }
    .dot-car { background: #38bdf8; }
    .dot-side { background: #f59e0b; }

    /* Controls Panel */
    .controls {
      display: flex;
      flex-direction: column;
      gap: 8px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px;
    }
    .ctrl-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 8px;
    }
    .ctrl-label {
      font-size: 13px;
      font-weight: 600;
      color: #475569;
      min-width: 90px;
    }
    .btn-group {
      display: flex;
      gap: 6px;
      flex: 1;
    }
    .btn-ratio {
      flex: 1;
      padding: 6px 4px;
      font-size: 12px;
      font-weight: 600;
      background: #f1f5f9;
      color: #334155;
      border: 1px solid #cbd5e1;
      border-radius: 6px;
      cursor: pointer;
    }
    .btn-ratio.active {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
    }
    .slider-wrap {
      flex: 1;
      display: flex;
      align-items: center;
      gap: 8px;
    }
    input[type="range"] {
      flex: 1;
      accent-color: #0284c7;
      cursor: pointer;
      height: 6px;
    }
    .val-tag {
      font-size: 13px;
      font-weight: 700;
      color: #0284c7;
      min-width: 28px;
      text-align: right;
    }
    .play-btn {
      width: 100%;
      padding: 8px;
      font-size: 14px;
      font-weight: 600;
      background: #0284c7;
      color: #ffffff;
      border: none;
      border-radius: 6px;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
    }
    .play-btn:active { background: #0369a1; }
  </style>
</head>
<body>

  
  <div class="flow-row">
    <div class="node node-mod">
      <div class="node-title">Modulator</div>
      <div class="node-val" id="modFreqTxt">220 Hz</div>
    </div>
    <div class="arrow-sign">→</div>
    <div class="node node-car">
      <div class="node-title">Carrier</div>
      <div class="node-val" id="carFreqTxt">220 Hz</div>
    </div>
    <div class="arrow-sign">→</div>
    <div class="node" style="border-top: 4px solid #10b981;">
      <div class="node-title" style="color: #10b981;">Timbre</div>
      <div class="node-val" id="timbreTxt">1 : 1 Harmonic</div>
    </div>
  </div>

  
  <div class="display-panel">
    <div class="canvas-wrap">
      <canvas id="scopeCanvas"></canvas>
    </div>
    <div class="spectrum-legend">
      <span class="legend-tag"><span class="dot dot-car"></span> Carrier ($f_c$)</span>
      <span class="legend-tag"><span class="dot dot-side"></span> Sidebands ($f_c \pm n f_m$)</span>
      <span id="sidebandCount">Sidebands: 2</span>
    </div>
  </div>

  
  <div class="controls">
    <div class="ctrl-row">
      <span class="ctrl-label">C : M Ratio</span>
      <div class="btn-group">
        <button class="btn-ratio active" data-ratio="1" data-type="Harmonic">1 : 1</button>
        <button class="btn-ratio" data-ratio="2" data-type="Bright">1 : 2</button>
        <button class="btn-ratio" data-ratio="3" data-type="Hollow">1 : 3</button>
        <button class="btn-ratio" data-ratio="1.414" data-type="Metallic">1 : 1.41</button>
      </div>
    </div>

    <div class="ctrl-row">
      <span class="ctrl-label">FM Index ($I$)</span>
      <div class="slider-wrap">
        <input type="range" id="indexSlider" min="0" max="6" step="0.1" value="2">
        <span class="val-tag" id="indexVal">2.0</span>
      </div>
    </div>

    <button class="play-btn" id="audioBtn">
      <span>▶</span> Preview FM Tone
    </button>
  </div>

  <script>
    const canvas = document.getElementById('scopeCanvas');
    const ctx = canvas.getContext('2d');
    const slider = document.getElementById('indexSlider');
    const indexVal = document.getElementById('indexVal');
    const modFreqTxt = document.getElementById('modFreqTxt');
    const timbreTxt = document.getElementById('timbreTxt');
    const sidebandCount = document.getElementById('sidebandCount');
    const ratioBtns = document.querySelectorAll('.btn-ratio');
    const audioBtn = document.getElementById('audioBtn');

    const BASE_FC = 220;
    let ratio = 1.0;
    let index = 2.0;
    let timbreName = 'Harmonic';
    let phase = 0;

    function resizeCanvas() {
      const dpr = window.devicePixelRatio || 1;
      const rect = canvas.getBoundingClientRect();
      canvas.width = rect.width * dpr;
      canvas.height = rect.height * dpr;
      ctx.scale(dpr, dpr);
    }
    window.addEventListener('resize', resizeCanvas);
    resizeCanvas();

    function updateState() {
      const fm = BASE_FC * ratio;
      modFreqTxt.textContent = `${Math.round(fm)} Hz`;
      timbreTxt.textContent = `1 : ${ratio === 1.414 ? '1.41' : ratio} ${timbreName}`;
      indexVal.textContent = index.toFixed(1);
      
      const activeSidebands = Math.max(0, Math.round(index * 2));
      sidebandCount.textContent = `Sidebands: ${activeSidebands}`;
    }

    ratioBtns.forEach(btn => {
      btn.addEventListener('click', () => {
        ratioBtns.forEach(b => b.classList.remove('active'));
        btn.classList.add('active');
        ratio = parseFloat(btn.dataset.ratio);
        timbreName = btn.dataset.type;
        updateState();
      });
    });

    slider.addEventListener('input', (e) => {
      index = parseFloat(e.target.value);
      updateState();
    });

    // Drawing Loop: Oscilloscope (left half) + Spectrum Bars (right half)
    function draw() {
      const w = canvas.getBoundingClientRect().width;
      const h = canvas.getBoundingClientRect().height;

      ctx.clearRect(0, 0, w, h);

      // Grid center divider
      ctx.strokeStyle = '#1e293b';
      ctx.lineWidth = 1;
      ctx.beginPath();
      ctx.moveTo(w * 0.52, 10);
      ctx.lineTo(w * 0.52, h - 10);
      ctx.stroke();

      // --- Left: FM Waveform ---
      ctx.save();
      ctx.beginPath();
      ctx.rect(0, 0, w * 0.5, h);
      ctx.clip();

      ctx.strokeStyle = '#38bdf8';
      ctx.lineWidth = 2.5;
      ctx.beginPath();

      const waveW = w * 0.5;
      const midY = h * 0.5;
      const amp = h * 0.35;
      const cycles = 3;

      for (let x = 0; x < waveW; x++) {
        const t = (x / waveW) * cycles * Math.PI * 2;
        const mod = index * Math.sin(t * ratio + phase);
        const y = midY + amp * Math.sin(t + mod);
        if (x === 0) ctx.moveTo(x, y);
        else ctx.lineTo(x, y);
      }
      ctx.stroke();
      ctx.restore();

      // --- Right: Spectrum (Sidebands) ---
      const specX = w * 0.56;
      const specW = w * 0.40;
      const baseLineY = h - 20;

      // Axis
      ctx.strokeStyle = '#334155';
      ctx.lineWidth = 1.5;
      ctx.beginPath();
      ctx.moveTo(specX, baseLineY);
      ctx.lineTo(specX + specW, baseLineY);
      ctx.stroke();

      // Plot Carrier & Sidebands
      const maxOrder = 4;
      const spacing = specW / (maxOrder * 2 + 2);
      const centerX = specX + specW * 0.45;

      for (let n = -maxOrder; n <= maxOrder; n++) {
        const barX = centerX + n * spacing * (ratio > 2 ? 1.2 : 0.9);
        if (barX < specX || barX > specX + specW) continue;

        // Approximate Bessel decay for height
        let barHeight = 0;
        if (n === 0) {
          barHeight = Math.max(10, (h - 40) * (1 / (1 + index * 0.4)));
        } else {
          const dist = Math.abs(n);
          const weight = Math.max(0, (index - dist * 0.5) / 3);
          barHeight = Math.min(h - 45, (h - 45) * weight);
        }

        if (barHeight > 3) {
          ctx.strokeStyle = (n === 0) ? '#38bdf8' : '#f59e0b';
          ctx.fillStyle = (n === 0) ? '#38bdf8' : '#f59e0b';
          ctx.lineWidth = 3.5;
          ctx.beginPath();
          ctx.moveTo(barX, baseLineY);
          ctx.lineTo(barX, baseLineY - barHeight);
          ctx.stroke();

          // Peak dot
          ctx.beginPath();
          ctx.arc(barX, baseLineY - barHeight, 2.5, 0, Math.PI * 2);
          ctx.fill();
        }
      }

      phase += 0.04;
      requestAnimationFrame(draw);
    }

    // Web Audio FM Synthesizer Preview
    let audioCtx = null;
    audioBtn.addEventListener('click', () => {
      if (!audioCtx) {
        audioCtx = new (window.AudioContext || window.webkitAudioContext)();
      }
      if (audioCtx.state === 'suspended') {
        audioCtx.resume();
      }

      const now = audioCtx.currentTime;
      const duration = 1.0;

      // Carrier & Modulator Oscillators
      const carrier = audioCtx.createOscillator();
      const modulator = audioCtx.createOscillator();
      const modGain = audioCtx.createGain();
      const masterGain = audioCtx.createGain();

      const fc = 220;
      const fm = fc * ratio;
      const modDepth = index * fm;

      carrier.frequency.setValueAtTime(fc, now);
      modulator.frequency.setValueAtTime(fm, now);

      // Depth scaling
      modGain.gain.setValueAtTime(modDepth, now);
      modGain.gain.exponentialRampToValueAtTime(Math.max(1, modDepth * 0.2), now + duration);

      // Connections: Modulator -> ModGain -> Carrier.frequency
      modulator.connect(modGain);
      modGain.connect(carrier.frequency);

      // Master Envelope
      carrier.connect(masterGain);
      masterGain.connect(audioCtx.destination);

      masterGain.gain.setValueAtTime(0.2, now);
      masterGain.gain.exponentialRampToValueAtTime(0.001, now + duration);

      modulator.start(now);
      carrier.start(now);
      modulator.stop(now + duration);
      carrier.stop(now + duration);

      audioBtn.style.background = '#0369a1';
      setTimeout(() => { audioBtn.style.background = '#0284c7'; }, duration * 1000);
    });

    updateState();
    draw();
  </script>
</body>
</html>

What determines the exact frequencies of these new overtones? It all comes down to basic addition and subtraction.

Audio-rate modulation generates **sidebands**, which are pairs of frequencies spaced symmetrically above and below the carrier frequency ($f_c$).

Each sideband appears at intervals of the modulator frequency ($f_m$), mathematically governed by $f_n = f_c \pm n \cdot f_m$, where $n$ is an integer ($1, 2, 3, \dots$).

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      user-select: none;
    }

    .toolbar {
      display: flex;
      flex-wrap: wrap;
      align-items: center;
      justify-content: space-between;
      gap: 8px;
      background: #ffffff;
      padding: 8px 12px;
      border-radius: 10px;
      border: 1px solid #e2e8f0;
    }

    .btn-group {
      display: flex;
      gap: 6px;
      flex-wrap: wrap;
    }

    .preset-btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      border-radius: 6px;
      padding: 4px 10px;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      transition: all 0.15s ease;
    }

    .preset-btn.active {
      background: #0284c7;
      border-color: #0284c7;
      color: #ffffff;
    }

    .audio-btn {
      background: #059669;
      border: none;
      border-radius: 6px;
      padding: 5px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #ffffff;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 4px;
    }
    .audio-btn:active { opacity: 0.85; }

    .sliders {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 12px;
      background: #ffffff;
      padding: 8px 12px;
      border-radius: 10px;
      border: 1px solid #e2e8f0;
      font-size: 12px;
      font-weight: 600;
    }

    @media (max-width: 480px) {
      .sliders { grid-template-columns: 1fr; }
    }

    .slider-row {
      display: flex;
      align-items: center;
      gap: 8px;
    }
    .slider-row label {
      min-width: 80px;
      color: #64748b;
    }
    .slider-row input[type=range] {
      flex: 1;
      accent-color: #0284c7;
      cursor: pointer;
    }
    .slider-val {
      min-width: 48px;
      text-align: right;
      font-variant-numeric: tabular-nums;
      color: #0f172a;
    }

    .display-card {
      background: #ffffff;
      border-radius: 10px;
      border: 1px solid #e2e8f0;
      padding: 8px 8px 4px 8px;
      display: flex;
      flex-direction: column;
      position: relative;
    }

    svg {
      width: 100%;
      height: 220px;
      overflow: visible;
    }

    .axis-line { stroke: #cbd5e1; stroke-width: 1.5; }
    .grid-line { stroke: #f1f5f9; stroke-width: 1; }
    .axis-text { font-size: 11px; fill: #94a3b8; text-anchor: middle; font-family: monospace; }
    
    .peak-carrier { stroke: #0284c7; stroke-width: 4; stroke-linecap: round; }
    .peak-carrier-dot { fill: #0284c7; }
    .peak-sideband { stroke: #8b5cf6; stroke-width: 3; stroke-linecap: round; }
    .peak-sideband-dot { fill: #8b5cf6; }

    .legend {
      display: flex;
      justify-content: center;
      gap: 16px;
      padding: 4px;
      font-size: 12px;
      font-weight: 600;
    }
    .legend-item {
      display: flex;
      align-items: center;
      gap: 6px;
    }
    .badge-c { width: 10px; height: 10px; border-radius: 50%; background: #0284c7; }
    .badge-m { width: 10px; height: 10px; border-radius: 50%; background: #8b5cf6; }

    .spacing-arrow { stroke: #8b5cf6; stroke-width: 1.2; stroke-dasharray: 2 2; }
    .spacing-text { font-size: 10px; fill: #8b5cf6; font-weight: 700; text-anchor: middle; }
  </style>
</head>
<body>

  <div class="toolbar">
    <div class="btn-group">
      <button class="preset-btn active" data-fc="200" data-fm="200" data-i="2.0">1 : 1</button>
      <button class="preset-btn" data-fc="200" data-fm="400" data-i="2.0">1 : 2</button>
      <button class="preset-btn" data-fc="200" data-fm="100" data-i="2.5">2 : 1</button>
      <button class="preset-btn" data-fc="200" data-fm="282" data-i="2.2">1 : 1.41 (Bell)</button>
    </div>
    <button class="audio-btn" id="playBtn">🔊 Listen</button>
  </div>

  <div class="sliders">
    <div class="slider-row">
      <label>Ratio (C:M)</label>
      <input type="range" id="modSlider" min="50" max="600" step="10" value="200">
      <span class="slider-val" id="fmVal">200 Hz</span>
    </div>
    <div class="slider-row">
      <label>Index (I)</label>
      <input type="range" id="indexSlider" min="0" max="4.5" step="0.1" value="2.0">
      <span class="slider-val" id="iVal">2.0</span>
    </div>
  </div>

  <div class="display-card">
    <svg id="spectrumSvg" viewBox="0 0 600 200">
      
      <line x1="40" y1="170" x2="570" y2="170" class="axis-line" />
      <g id="axisLabels"></g>

      
      <g id="peaksGroup"></g>
      
      
      <g id="markersGroup"></g>
    </svg>
    <div class="legend">
      <div class="legend-item"><div class="badge-c"></div>Carrier (fc = 200 Hz)</div>
      <div class="legend-item"><div class="badge-m"></div>Sidebands (fc ± n·fm)</div>
    </div>
  </div>

  <script>
    const fc = 200;
    const maxFreq = 1600;
    const svgW = 600, svgH = 200;
    const padL = 50, padR = 30, baseY = 170, maxH = 120;

    const modSlider = document.getElementById('modSlider');
    const indexSlider = document.getElementById('indexSlider');
    const fmVal = document.getElementById('fmVal');
    const iVal = document.getElementById('iVal');
    const peaksGroup = document.getElementById('peaksGroup');
    const markersGroup = document.getElementById('markersGroup');
    const axisLabels = document.getElementById('axisLabels');
    const presetBtns = document.querySelectorAll('.preset-btn');
    const playBtn = document.getElementById('playBtn');

    // Draw static axis markers
    for (let f = 0; f <= maxFreq; f += 200) {
      const x = padL + (f / maxFreq) * (svgW - padL - padR);
      const text = document.createElementNS('http://www.w3.org/2000/svg', 'text');
      text.setAttribute('x', x);
      text.setAttribute('y', baseY + 18);
      text.setAttribute('class', 'axis-text');
      text.textContent = f;
      axisLabels.appendChild(text);

      const tick = document.createElementNS('http://www.w3.org/2000/svg', 'line');
      tick.setAttribute('x1', x);
      tick.setAttribute('y1', baseY);
      tick.setAttribute('x2', x);
      tick.setAttribute('y2', baseY + 4);
      tick.setAttribute('stroke', '#cbd5e1');
      axisLabels.appendChild(tick);
    }

    // Bessel approximation for sideband amplitudes (J_n(beta))
    function besselJ(n, x) {
      n = Math.abs(n);
      let sum = 0;
      for (let k = 0; k < 12; k++) {
        let num = Math.pow(-1, k) * Math.pow(x / 2, 2 * k + n);
        let den = fact(k) * fact(k + n);
        sum += num / den;
      }
      return sum;
    }

    function fact(n) {
      let r = 1;
      for (let i = 2; i <= n; i++) r *= i;
      return r;
    }

    function xFromFreq(f) {
      return padL + (f / maxFreq) * (svgW - padL - padR);
    }

    function render() {
      const fm = parseFloat(modSlider.value);
      const idx = parseFloat(indexSlider.value);

      fmVal.textContent = Math.round(fm) + ' Hz';
      iVal.textContent = idx.toFixed(1);

      peaksGroup.innerHTML = '';
      markersGroup.innerHTML = '';

      // Calculate sideband frequencies and spectral amplitudes
      const spectrum = {};

      const maxOrder = 8;
      for (let n = -maxOrder; n <= maxOrder; n++) {
        let f = fc + n * fm;
        let amp = besselJ(n, idx);

        // Fold negative frequencies (phase inversion)
        if (f < 0) {
          f = -f;
          amp = -amp; // 180° phase inversion
        }

        const mag = Math.abs(amp);
        if (mag > 0.03) {
          const key = Math.round(f * 10) / 10;
          spectrum[key] = (spectrum[key] || 0) + mag;
        }
      }

      // Draw peaks
      Object.keys(spectrum).forEach(freqStr => {
        const freq = parseFloat(freqStr);
        if (freq > maxFreq) return;

        const amp = Math.min(1.0, spectrum[freqStr]);
        const x = xFromFreq(freq);
        const y = baseY - amp * maxH;
        const isCarrier = Math.abs(freq - fc) < 1;

        // Peak line
        const line = document.createElementNS('http://www.w3.org/2000/svg', 'line');
        line.setAttribute('x1', x);
        line.setAttribute('y1', baseY);
        line.setAttribute('x2', x);
        line.setAttribute('y2', y);
        line.setAttribute('class', isCarrier ? 'peak-carrier' : 'peak-sideband');
        peaksGroup.appendChild(line);

        // Peak top dot
        const circle = document.createElementNS('http://www.w3.org/2000/svg', 'circle');
        circle.setAttribute('cx', x);
        circle.setAttribute('cy', y);
        circle.setAttribute('r', isCarrier ? 4.5 : 3.5);
        circle.setAttribute('class', isCarrier ? 'peak-carrier-dot' : 'peak-sideband-dot');
        peaksGroup.appendChild(circle);
      });

      // Interval distance bracket between fc and fc+fm
      if (fc + fm <= maxFreq) {
        const x1 = xFromFreq(fc);
        const x2 = xFromFreq(fc + fm);
        const arrowY = 32;

        const arrowLine = document.createElementNS('http://www.w3.org/2000/svg', 'line');
        arrowLine.setAttribute('x1', x1);
        arrowLine.setAttribute('y1', arrowY);
        arrowLine.setAttribute('x2', x2);
        arrowLine.setAttribute('y2', arrowY);
        arrowLine.setAttribute('class', 'spacing-arrow');
        markersGroup.appendChild(arrowLine);

        const txt = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        txt.setAttribute('x', (x1 + x2) / 2);
        txt.setAttribute('y', arrowY - 6);
        txt.setAttribute('class', 'spacing-text');
        txt.textContent = 'Δ = fm (' + Math.round(fm) + 'Hz)';
        markersGroup.appendChild(txt);
      }
    }

    modSlider.addEventListener('input', () => {
      presetBtns.forEach(b => b.classList.remove('active'));
      render();
    });

    indexSlider.addEventListener('input', () => {
      render();
    });

    presetBtns.forEach(btn => {
      btn.addEventListener('click', () => {
        presetBtns.forEach(b => b.classList.remove('active'));
        btn.classList.add('active');
        modSlider.value = btn.dataset.fm;
        indexSlider.value = btn.dataset.i;
        render();
      });
    });

    // Sound generation
    let audioCtx = null;
    playBtn.addEventListener('click', () => {
      if (!audioCtx) audioCtx = new (window.AudioContext || window.webkitAudioContext)();
      if (audioCtx.state === 'suspended') audioCtx.resume();

      const now = audioCtx.currentTime;
      const duration = 1.0;

      const carrierOsc = audioCtx.createOscillator();
      const modOsc = audioCtx.createOscillator();
      const modGain = audioCtx.createGain();
      const masterGain = audioCtx.createGain();

      const fm = parseFloat(modSlider.value);
      const idx = parseFloat(indexSlider.value);
      const modIndexFreq = idx * fm;

      carrierOsc.frequency.setValueAtTime(fc, now);
      modOsc.frequency.setValueAtTime(fm, now);

      modGain.gain.setValueAtTime(modIndexFreq, now);
      modGain.gain.exponentialRampToValueAtTime(0.01, now + duration);

      modOsc.connect(modGain);
      modGain.connect(carrierOsc.frequency);

      masterGain.gain.setValueAtTime(0.2, now);
      masterGain.gain.exponentialRampToValueAtTime(0.001, now + duration);

      carrierOsc.connect(masterGain);
      masterGain.connect(audioCtx.destination);

      modOsc.start(now);
      carrierOsc.start(now);
      modOsc.stop(now + duration);
      carrierOsc.stop(now + duration);
    });

    render();
  </script>
</body>
</html>

The instantaneous voltage of an FM signal is defined by $y(t) = A \sin(2\pi f_c t + I \sin(2\pi f_m t))$, where $I$ represents modulation depth.

Using trigonometric angle-sum identities, this formula expands into an infinite series of sine and cosine terms at frequencies $f_c + n f_m$ and $f_c - n f_m$ scaled by **Bessel functions**.

How do sound designers ensure these sidebands sound like a musical note rather than unpitched noise?

## Frequency Ratios: Harmonic vs Inharmonic

Instead of setting fixed Hertz values, FM synthesizers link the two oscillators using a **frequency ratio** ($C:M$, or carrier to modulator).

When $C:M$ is a simple whole-number ratio (such as $1:1$, $1:2$, or $1:3$), the sidebands land directly on the **harmonic series**, creating pitched musical sounds.

Non-integer ratios (like $1:1.414$ or $1:3.17$) generate **inharmonic sidebands**, producing metallic tones like bells, gongs, and cymbals.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>FM Synthesis Ratios & Spectrum</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
    }

    /* Preset selector tabs */
    .tabs {
      display: flex;
      gap: 6px;
      flex-wrap: wrap;
    }
    .tab-btn {
      flex: 1;
      min-width: 130px;
      padding: 8px 10px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      text-align: center;
      transition: background 0.2s, color 0.2s, box-shadow 0.2s;
    }
    .tab-btn.active {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
      box-shadow: 0 2px 6px rgba(2, 132, 199, 0.25);
    }

    /* Main visual grid */
    .display-box {
      background: #ffffff;
      border-radius: 12px;
      padding: 12px;
      border: 1px solid #e2e8f0;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    /* Upper schematic: Modulator -> Carrier */
    .signal-flow {
      display: flex;
      align-items: center;
      justify-content: space-around;
      padding: 6px 0;
      background: #f1f5f9;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
    }
    .block {
      padding: 6px 12px;
      border-radius: 6px;
      text-align: center;
    }
    .mod-block {
      background: #e0f2fe;
      color: #0369a1;
      border: 1px solid #bae6fd;
    }
    .arrow {
      color: #64748b;
      font-size: 16px;
    }
    .car-block {
      background: #fef3c7;
      color: #b45309;
      border: 1px solid #fde68a;
    }

    /* SVG Spectrum Area */
    .spectrum-svg {
      width: 100%;
      height: 140px;
      overflow: visible;
    }

    /* Bottom Control Bar */
    .controls {
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 12px;
      flex-wrap: wrap;
    }
    .tag {
      font-size: 13px;
      font-weight: 700;
      padding: 4px 10px;
      border-radius: 6px;
    }
    .tag.harmonic {
      background: #dcfce7;
      color: #15803d;
    }
    .tag.inharmonic {
      background: #fee2e2;
      color: #b91c1c;
    }
    .play-btn {
      display: inline-flex;
      align-items: center;
      gap: 6px;
      padding: 8px 16px;
      background: #0284c7;
      color: #ffffff;
      border: none;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
    }
    .play-btn:active {
      opacity: 0.85;
    }
  </style>
</head>
<body>

  
  <div class="tabs">
    <button class="tab-btn active" id="tab0" onclick="setPreset(0)">1 : 1 (Harmonic)</button>
    <button class="tab-btn" id="tab1" onclick="setPreset(1)">1 : 2 (Harmonic)</button>
    <button class="tab-btn" id="tab2" onclick="setPreset(2)">1 : 1.414 (Bell)</button>
    <button class="tab-btn" id="tab3" onclick="setPreset(3)">1 : 3.17 (Metallic)</button>
  </div>

  
  <div class="display-box">
    <div class="signal-flow">
      <div class="block mod-block" id="modLabel">Modulator: 200 Hz</div>
      <div class="arrow">Modulates Frequency →</div>
      <div class="block car-block" id="carLabel">Carrier: 200 Hz</div>
    </div>

    
    <svg class="spectrum-svg" viewBox="0 0 500 140" id="specSvg">
      
      <line x1="40" y1="110" x2="480" y2="110" stroke="#cbd5e1" stroke-width="1.5" />
      <text x="40" y="128" fill="#64748b" font-size="11" font-weight="600">0 Hz</text>
      <text x="260" y="128" fill="#64748b" font-size="11" font-weight="600">500 Hz</text>
      <text x="470" y="128" fill="#64748b" font-size="11" font-weight="600" text-anchor="end">1000 Hz</text>
      
      
      <g id="spectralLines"></g>
    </svg>
  </div>

  
  <div class="controls">
    <span class="tag harmonic" id="natureTag">Harmonic Series</span>
    <button class="play-btn" onclick="playCurrentTone()">
      <span>Play Sound</span> 🔊
    </button>
  </div>

  <script>
    const presets = [
      {
        name: "1:1 Harmonic",
        ratioText: "1 : 1",
        fc: 200,
        fm: 200,
        type: "Harmonic Series",
        isHarmonic: true,
        lines: [
          { f: 200, h: 80, label: "fc (200)" },
          { f: 400, h: 50, label: "+fm (400)" },
          { f: 600, h: 30, label: "+2fm (600)" },
          { f: 800, h: 18, label: "+3fm (800)" }
        ]
      },
      {
        name: "1:2 Harmonic",
        ratioText: "1 : 2",
        fc: 200,
        fm: 400,
        type: "Harmonic (Odd/Octave)",
        isHarmonic: true,
        lines: [
          { f: 200, h: 80, label: "fc (200)" },
          { f: 600, h: 55, label: "+fm (600)" },
          { f: 1000, h: 25, label: "+2fm (1000)" }
        ]
      },
      {
        name: "1:1.414 Bell",
        ratioText: "1 : 1.414",
        fc: 200,
        fm: 282.8,
        type: "Inharmonic (Metallic / Bell)",
        isHarmonic: false,
        lines: [
          { f: 82.8, h: 45, label: "83 Hz" },
          { f: 200, h: 80, label: "fc (200)" },
          { f: 482.8, h: 60, label: "483 Hz" },
          { f: 765.6, h: 35, label: "766 Hz" }
        ]
      },
      {
        name: "1:3.17 Metallic",
        ratioText: "1 : 3.17",
        fc: 200,
        fm: 634,
        type: "Inharmonic (Gong / Cymbal)",
        isHarmonic: false,
        lines: [
          { f: 200, h: 80, label: "fc (200)" },
          { f: 434, h: 50, label: "434 Hz" },
          { f: 834, h: 55, label: "834 Hz" }
        ]
      }
    ];

    let activeIdx = 0;
    let audioCtx = null;

    function renderSpectrum() {
      const p = presets[activeIdx];
      
      // Update UI Text
      document.getElementById('carLabel').textContent = `Carrier: ${p.fc} Hz`;
      document.getElementById('modLabel').textContent = `Modulator: ${Math.round(p.fm)} Hz (${p.ratioText})`;
      
      const tag = document.getElementById('natureTag');
      tag.textContent = p.type;
      tag.className = p.isHarmonic ? "tag harmonic" : "tag inharmonic";

      // Render Spectrum Bars (0 to 1000 Hz mapped to 40px - 470px)
      const container = document.getElementById('spectralLines');
      const minX = 40, maxX = 470, maxFreq = 1000;
      const baseY = 110;

      let html = '';
      p.lines.forEach((line) => {
        const x = minX + (line.f / maxFreq) * (maxX - minX);
        const yTop = baseY - line.h;
        const color = p.isHarmonic ? "#0284c7" : "#e11d48";

        html += `
          <g>
            <line x1="${x}" y1="${baseY}" x2="${x}" y2="${yTop}" stroke="${color}" stroke-width="4" stroke-linecap="round" />
            <circle cx="${x}" cy="${yTop}" r="4" fill="${color}" />
            <text x="${x}" y="${yTop - 8}" fill="${color}" font-size="11" font-weight="700" text-anchor="middle">${line.label}</text>
          </g>
        `;
      });
      container.innerHTML = html;
    }

    function setPreset(idx) {
      activeIdx = idx;
      for (let i = 0; i < 4; i++) {
        document.getElementById(`tab${i}`).classList.toggle('active', i === idx);
      }
      renderSpectrum();
    }

    function playCurrentTone() {
      if (!audioCtx) {
        audioCtx = new (window.AudioContext || window.webkitAudioContext)();
      }
      if (audioCtx.state === 'suspended') {
        audioCtx.resume();
      }

      const p = presets[activeIdx];
      const now = audioCtx.currentTime;
      const dur = 1.2;

      // FM Synthesis: Carrier + Modulator
      const carrier = audioCtx.createOscillator();
      const modulator = audioCtx.createOscillator();
      const modGain = audioCtx.createGain();
      const masterGain = audioCtx.createGain();

      carrier.frequency.setValueAtTime(p.fc, now);
      modulator.frequency.setValueAtTime(p.fm, now);

      // Modulation Index / Depth
      const modIndex = 250;
      modGain.gain.setValueAtTime(modIndex, now);
      modGain.gain.exponentialRampToValueAtTime(10, now + dur);

      modulator.connect(modGain);
      modGain.connect(carrier.frequency);

      // Master Envelope
      masterGain.gain.setValueAtTime(0.25, now);
      masterGain.gain.exponentialRampToValueAtTime(0.001, now + dur);

      carrier.connect(masterGain);
      masterGain.connect(audioCtx.destination);

      modulator.start(now);
      carrier.start(now);
      modulator.stop(now + dur);
      carrier.stop(now + dur);
    }

    // Initial render
    renderSpectrum();
  </script>
</body>
</html>

Let's trace a concrete example from scratch to see how sidebands populate the spectrum.

## Worked Example: Calculating a 1:2 Ratio

Suppose you play concert A ($f_c = 440\text{ Hz}$) with a ratio of $C:M = 1:2$, meaning the modulator frequency $f_m = 880\text{ Hz}$.

The upper sidebands ($f_c + n f_m$) yield: $440 + 880 = 1320\text{ Hz}$ ($n=1$), $440 + 1760 = 2200\text{ Hz}$ ($n=2$), and $440 + 2640 = 3080\text{ Hz}$ ($n=3$).

The first lower sideband ($n=1$) gives $440 - 880 = -440\text{ Hz}$. In acoustics, a **negative frequency** reflects into the positive spectrum with an inverted 180° phase ($+440\text{ Hz}$).

All resulting peaks ($440, 1320, 2200, 3080\text{ Hz}$) are odd harmonics ($1f, 3f, 5f, 7f$), generating a hollow, clarinet-like square wave tone.

A common mistake is assuming that increasing the modulator frequency makes the sound brighter.

Changing the modulator frequency actually changes the harmonic spacing (timbral quality), while increasing the modulator amplitude (**modulation index**) adds more sidebands to make the tone brighter.

Try setting your synth carrier to $1$ and modulator to $2$ with a quick amplitude envelope decay on the modulator to hear how naturally an FM electric piano snaps to life.

music-technology

Frequency Modulation (FM) Synthesis: Carriers, Modulators, and Ratios. Modulate carrier oscillator frequency with modulator sine waves at audio rates, generating sideband harmonics defined by integer frequency ratios.

Synth Secrets: An Introduction to Frequency Modulation

Ableton Learning Synths: Audio-rate Modulation & FM Basics

The Synthesis of Complex Audio Spectra by Means of Frequency Modulation

Teaching FM synthesis at A2 Level (Year 13) requires connecting foundational physics concepts (wave interference, frequency domain vs. time domain, and sideband mathematics) directly to practical DAW synthesis workflows (MUSTECH-4.3). Introduce the transition from sub-audio LFO vibrato to audio-rate sidebands dynamically using software FM synthesizers (e.g., Logic Retro Synth FM mode, Ableton Operator, or Native Instruments FM8). Reinforce mathematical prediction of sideband frequencies via $f_c \pm n f_m$ before students adjust parameters by ear, ensuring clear cognitive links between integer $C:M$ ratios and harmonic musical tones versus decimal/irrational ratios and metallic/percussive textures.

Frequency Modulation (FM) Synthesis: Carriers, Modulators, and Ratios

Audio-Rate Modulation

Sidebands and the FM Formula

Why Sidebands Appear