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Tutoring abilities

How do mixing engineers remove an ear-piercing ringing frequency from a snare drum without dulling the snap and body of the rest of the hit?

In 1972, audio pioneer George Massenburg solved this by designing the **parametric equalizer**, granting audio engineers independent control over frequency, amplitude, and curve width.

What components make this surgical precision possible?

Every parametric bell filter is defined by three fundamental parameters: **center frequency** ($f_c$), **gain** ($G$ in decibels), and the **Q factor** (quality factor).

<!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: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 12px;
      user-select: none;
    }
    .graph-container {
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e2e8f0;
      box-shadow: 0 2px 6px rgba(0, 0, 0, 0.04);
      position: relative;
      width: 100%;
      height: 250px;
      overflow: hidden;
    }
    svg {
      width: 100%;
      height: 100%;
      display: block;
    }
    .controls {
      display: grid;
      grid-template-columns: repeat(3, 1fr);
      gap: 8px;
    }
    .ctrl-card {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 8px 10px;
      display: flex;
      flex-direction: column;
      gap: 4px;
      box-shadow: 0 1px 3px rgba(0, 0, 0, 0.02);
    }
    .ctrl-head {
      display: flex;
      justify-content: space-between;
      align-items: baseline;
      font-size: 13px;
      font-weight: 600;
    }
    .param-val {
      font-size: 14px;
      font-weight: 700;
      color: #0284c7;
    }
    input[type=range] {
      width: 100%;
      height: 6px;
      border-radius: 3px;
      background: #e2e8f0;
      outline: none;
      -webkit-appearance: none;
      cursor: pointer;
    }
    input[type=range]::-webkit-slider-thumb {
      -webkit-appearance: none;
      width: 18px;
      height: 18px;
      border-radius: 50%;
      background: #0284c7;
      cursor: pointer;
      border: 2px solid #ffffff;
      box-shadow: 0 1px 4px rgba(0, 0, 0, 0.2);
    }
    .presets {
      display: flex;
      gap: 6px;
    }
    .btn-preset {
      flex: 1;
      padding: 6px 4px;
      background: #ffffff;
      border: 1px solid #cbd5e1;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      text-align: center;
      transition: all 0.2s;
    }
    .btn-preset:active, .btn-preset.active {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
    }
  </style>
</head>
<body>

  <div class="graph-container">
    <svg id="eqSvg" viewBox="0 0 600 250" preserveAspectRatio="none">
      <defs>
        <linearGradient id="curveFill" x1="0" y1="0" x2="0" y2="1">
          <stop offset="0%" stop-color="#0284c7" stop-opacity="0.28"/>
          <stop offset="100%" stop-color="#0284c7" stop-opacity="0.02"/>
        </linearGradient>
      </defs>

      
      <g stroke="#f1f5f9" stroke-width="1">
        <line x1="50" y1="35" x2="570" y2="35"/>
        <line x1="50" y1="75" x2="570" y2="75"/>
        <line x1="50" y1="115" x2="570" y2="115"/>
        <line x1="50" y1="155" x2="570" y2="155"/>
        <line x1="50" y1="195" x2="570" y2="195"/>

        <line x1="120" y1="20" x2="120" y2="210"/>
        <line x1="230" y1="20" x2="230" y2="210"/>
        <line x1="340" y1="20" x2="340" y2="210"/>
        <line x1="450" y1="20" x2="450" y2="210"/>
        <line x1="560" y1="20" x2="560" y2="210"/>
      </g>

      
      <line x1="50" y1="115" x2="570" y2="115" stroke="#94a3b8" stroke-width="1.5" stroke-dasharray="4 3"/>

      
      <g fill="#94a3b8" font-size="12" text-anchor="middle">
        <text x="120" y="228">100 Hz</text>
        <text x="340" y="228">1 kHz</text>
        <text x="560" y="228">10 kHz</text>
      </g>

      
      <g fill="#94a3b8" font-size="12" text-anchor="end">
        <text x="44" y="39">+12 dB</text>
        <text x="44" y="79">+6 dB</text>
        <text x="44" y="119">0 dB</text>
        <text x="44" y="159">-6 dB</text>
        <text x="44" y="199">-12 dB</text>
      </g>

      
      <g id="bwGroup">
        <rect id="bwRect" y="20" height="190" fill="#0284c7" opacity="0.07"/>
        <line id="f1Line" stroke="#0284c7" stroke-width="1.5" stroke-dasharray="2 2"/>
        <line id="f2Line" stroke="#0284c7" stroke-width="1.5" stroke-dasharray="2 2"/>
        <line id="bwSpan" stroke="#0284c7" stroke-width="1.5"/>
        <path id="bwArrow1" fill="#0284c7"/>
        <path id="bwArrow2" fill="#0284c7"/>
        <text id="bwLabel" font-size="12" font-weight="700" fill="#0284c7" text-anchor="middle">BW</text>
      </g>

      
      <path id="areaPath" fill="url(#curveFill)"/>
      <path id="curvePath" fill="none" stroke="#0284c7" stroke-width="3" stroke-linecap="round"/>

      
      <g id="fcGroup">
        <line id="fcLine" y1="20" y2="210" stroke="#0369a1" stroke-width="1.5" stroke-dasharray="3 3"/>
        <circle id="peakDot" r="6" fill="#0284c7" stroke="#ffffff" stroke-width="2"/>
        <text id="fcLabel" font-size="13" font-weight="700" fill="#0369a1" text-anchor="middle" y="16">fc</text>
      </g>
    </svg>
  </div>

  <div class="controls">
    <div class="ctrl-card">
      <div class="ctrl-head">
        <span>Center (fc)</span>
        <span class="param-val" id="valFc">1.0 kHz</span>
      </div>
      <input type="range" id="sliderFc" min="100" max="10000" step="10" value="1000">
    </div>

    <div class="ctrl-card">
      <div class="ctrl-head">
        <span>Gain (G)</span>
        <span class="param-val" id="valGain">+9.0 dB</span>
      </div>
      <input type="range" id="sliderGain" min="-12" max="12" step="0.5" value="9">
    </div>

    <div class="ctrl-card">
      <div class="ctrl-head">
        <span>Q Factor</span>
        <span class="param-val" id="valQ">1.4</span>
      </div>
      <input type="range" id="sliderQ" min="0.3" max="8.0" step="0.1" value="1.4">
    </div>
  </div>

  <div class="presets">
    <button class="btn-preset active" data-q="0.5">Wide Band (Low Q = 0.5)</button>
    <button class="btn-preset" data-q="1.41">Standard Bell (Q = 1.4)</button>
    <button class="btn-preset" data-q="5.0">Notch / Narrow (High Q = 5.0)</button>
  </div>

  <script>
    const svgW = 600;
    const minF = 50;
    const maxF = 15000;
    const minX = 50;
    const maxX = 570;
    const baseY = 115; // 0 dB line
    const pxPerDb = 6.66; // 80px for 12dB

    const sliderFc = document.getElementById('sliderFc');
    const sliderGain = document.getElementById('sliderGain');
    const sliderQ = document.getElementById('sliderQ');

    const valFc = document.getElementById('valFc');
    const valGain = document.getElementById('valGain');
    const valQ = document.getElementById('valQ');

    const curvePath = document.getElementById('curvePath');
    const areaPath = document.getElementById('areaPath');
    const peakDot = document.getElementById('peakDot');
    const fcLine = document.getElementById('fcLine');
    const fcLabel = document.getElementById('fcLabel');

    const f1Line = document.getElementById('f1Line');
    const f2Line = document.getElementById('f2Line');
    const bwRect = document.getElementById('bwRect');
    const bwSpan = document.getElementById('bwSpan');
    const bwArrow1 = document.getElementById('bwArrow1');
    const bwArrow2 = document.getElementById('bwArrow2');
    const bwLabel = document.getElementById('bwLabel');

    function fToX(f) {
      const minLog = Math.log10(minF);
      const maxLog = Math.log10(maxF);
      const fLog = Math.log10(f);
      return minX + ((fLog - minLog) / (maxLog - minLog)) * (maxX - minX);
    }

    function xToF(x) {
      const minLog = Math.log10(minF);
      const maxLog = Math.log10(maxF);
      const ratio = (x - minX) / (maxX - minX);
      return Math.pow(10, minLog + ratio * (maxLog - minLog));
    }

    function dbToY(db) {
      return baseY - (db * pxPerDb);
    }

    function calcGainAtF(f, fc, G, Q) {
      if (Math.abs(G) < 0.01) return 0;
      const ratio = (f / fc) - (fc / f);
      return G / (1 + Math.pow(Q * ratio, 2));
    }

    function update() {
      const fc = parseFloat(sliderFc.value);
      const G = parseFloat(sliderGain.value);
      const Q = parseFloat(sliderQ.value);

      // Labels
      valFc.textContent = fc >= 1000 ? (fc / 1000).toFixed(1) + ' kHz' : Math.round(fc) + ' Hz';
      valGain.textContent = (G >= 0 ? '+' : '') + G.toFixed(1) + ' dB';
      valQ.textContent = Q.toFixed(1);

      // Render Curve
      const pts = [];
      const numSteps = 120;
      for (let i = 0; i <= numSteps; i++) {
        const x = minX + (i / numSteps) * (maxX - minX);
        const f = xToF(x);
        const gainDb = calcGainAtF(f, fc, G, Q);
        const y = dbToY(gainDb);
        pts.push({ x, y });
      }

      let d = `M ${pts[0].x} ${pts[0].y}`;
      for (let i = 1; i < pts.length; i++) {
        d += ` L ${pts[i].x} ${pts[i].y}`;
      }
      curvePath.setAttribute('d', d);

      const dArea = `${d} L ${maxX} ${baseY} L ${minX} ${baseY} Z`;
      areaPath.setAttribute('d', dArea);

      // Center Frequency markers
      const fcX = fToX(fc);
      const peakY = dbToY(G);
      peakDot.setAttribute('cx', fcX);
      peakDot.setAttribute('cy', peakY);
      fcLine.setAttribute('x1', fcX);
      fcLine.setAttribute('x2', fcX);
      fcLabel.setAttribute('x', fcX);
      fcLabel.textContent = `fc = ${valFc.textContent}`;

      // Bandwidth (-3dB cutoff points)
      // BW = fc / Q; f1, f2 calculation:
      const f1 = fc * (Math.sqrt(1 + 1 / (4 * Q * Q)) - 1 / (2 * Q));
      const f2 = fc * (Math.sqrt(1 + 1 / (4 * Q * Q)) + 1 / (2 * Q));
      const x1 = Math.max(minX, fToX(f1));
      const x2 = Math.min(maxX, fToX(f2));

      // Display Bandwidth Span
      const halfGainY = dbToY(G * 0.5);
      f1Line.setAttribute('x1', x1);
      f1Line.setAttribute('x2', x1);
      f1Line.setAttribute('y1', baseY);
      f1Line.setAttribute('y2', halfGainY);

      f2Line.setAttribute('x1', x2);
      f2Line.setAttribute('x2', x2);
      f2Line.setAttribute('y1', baseY);
      f2Line.setAttribute('y2', halfGainY);

      bwRect.setAttribute('x', x1);
      bwRect.setAttribute('width', Math.max(0, x2 - x1));

      const spanY = G >= 0 ? Math.min(halfGainY - 14, baseY - 20) : Math.max(halfGainY + 14, baseY + 20);
      bwSpan.setAttribute('x1', x1);
      bwSpan.setAttribute('x2', x2);
      bwSpan.setAttribute('y1', spanY);
      bwSpan.setAttribute('y2', spanY);

      // Arrows
      bwArrow1.setAttribute('d', `M ${x1} ${spanY} L ${x1+4} ${spanY-3} L ${x1+4} ${spanY+3} Z`);
      bwArrow2.setAttribute('d', `M ${x2} ${spanY} L ${x2-4} ${spanY-3} L ${x2-4} ${spanY+3} Z`);

      const bwHz = Math.round(f2 - f1);
      const bwStr = bwHz >= 1000 ? (bwHz / 1000).toFixed(2) + ' kHz' : bwHz + ' Hz';
      bwLabel.setAttribute('x', (x1 + x2) / 2);
      bwLabel.setAttribute('y', spanY - 5);
      bwLabel.textContent = `BW: ${bwStr} (fc / Q)`;
    }

    [sliderFc, sliderGain, sliderQ].forEach(input => {
      input.addEventListener('input', () => {
        document.querySelectorAll('.btn-preset').forEach(b => b.classList.remove('active'));
        update();
      });
    });

    document.querySelectorAll('.btn-preset').forEach(btn => {
      btn.addEventListener('click', (e) => {
        document.querySelectorAll('.btn-preset').forEach(b => b.classList.remove('active'));
        btn.classList.add('active');
        sliderQ.value = btn.dataset.q;
        update();
      });
    });

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

How do we mathematically calculate how narrow or wide that curve actually is in Hertz?

The **bandwidth** ($BW$) of a filter is the span of frequencies between the **-3 dB cutoff points** ($f_1$ and $f_2$) where signal power drops by half.

The **Q factor** represents the filter's selectivity and is defined by the ratio of the center frequency to bandwidth: $$Q = \frac{f_c}{BW_{\text{Hz}}}$$

<!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: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: #ffffff;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      user-select: none;
      -webkit-user-select: none;
    }

    .main-wrap {
      width: 100%;
      max-width: 620px;
      display: flex;
      flex-direction: column;
      gap: 10px;
    }

    /* Controls Bar */
    .controls-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 8px;
    }

    .control-card {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 8px 12px;
      display: flex;
      flex-direction: column;
      gap: 4px;
    }

    .control-header {
      display: flex;
      justify-content: space-between;
      align-items: center;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
    }

    .val-badge {
      font-family: ui-monospace, SFMono-Regular, Menlo, monospace;
      font-size: 13px;
      font-weight: 700;
      color: #0284c7;
    }

    input[type="range"] {
      -webkit-appearance: none;
      appearance: none;
      width: 100%;
      height: 6px;
      background: #e2e8f0;
      border-radius: 3px;
      outline: none;
      cursor: pointer;
    }

    input[type="range"]::-webkit-slider-thumb {
      -webkit-appearance: none;
      appearance: none;
      width: 18px;
      height: 18px;
      border-radius: 50%;
      background: #0284c7;
      border: 2px solid #ffffff;
      box-shadow: 0 1px 4px rgba(0,0,0,0.25);
      cursor: pointer;
    }

    input[type="range"]::-moz-range-thumb {
      width: 18px;
      height: 18px;
      border-radius: 50%;
      background: #0284c7;
      border: 2px solid #ffffff;
      box-shadow: 0 1px 4px rgba(0,0,0,0.25);
      cursor: pointer;
    }

    /* Preset Buttons */
    .presets-row {
      display: flex;
      gap: 6px;
      justify-content: center;
    }

    .btn-preset {
      flex: 1;
      padding: 6px 4px;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      border-radius: 8px;
      font-size: 12px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      text-align: center;
    }

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

    /* Visualizer SVG */
    .graph-container {
      width: 100%;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      position: relative;
      overflow: hidden;
    }

    svg {
      display: block;
      width: 100%;
      height: auto;
    }

    /* Math Summary Bar */
    .formula-bar {
      display: flex;
      justify-content: space-around;
      align-items: center;
      background: #f1f5f9;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 8px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #334155;
    }

    .formula-highlight {
      color: #0284c7;
      font-family: ui-monospace, monospace;
      font-weight: 700;
    }

    .audio-btn {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      background: #ffffff;
      border: 1px solid #0284c7;
      color: #0284c7;
      font-weight: 600;
      font-size: 12px;
      padding: 5px 12px;
      border-radius: 20px;
      cursor: pointer;
      transition: background 0.2s;
    }

    .audio-btn:active {
      background: #e0f2fe;
    }
  </style>
</head>
<body>
  <div class="main-wrap">
    
    
    <div class="controls-grid">
      <div class="control-card">
        <div class="control-header">
          <span>Q Factor</span>
          <span class="val-badge" id="qVal">1.41</span>
        </div>
        <input type="range" id="qSlider" min="0.3" max="10" step="0.05" value="1.41">
      </div>

      <div class="control-card">
        <div class="control-header">
          <span>Center Freq (fc)</span>
          <span class="val-badge" id="fcVal">1000 Hz</span>
        </div>
        <input type="range" id="fcSlider" min="100" max="8000" step="50" value="1000">
      </div>
    </div>

    
    <div class="presets-row">
      <button class="btn-preset" data-q="0.5" data-desc="Broad Tone Shaping">Broad (Q=0.5)</button>
      <button class="btn-preset active" data-q="1.41" data-desc="Musical 1 Octave">Musical (Q=1.4)</button>
      <button class="btn-preset" data-q="5.0" data-desc="Narrow Notch/Problem Fix">Surgical (Q=5.0)</button>
      <button class="audio-btn" id="listenBtn">Preview Tone 🔊</button>
    </div>

    
    <div class="graph-container">
      <svg id="eqSvg" viewBox="0 0 540 240">
        <defs>
          <linearGradient id="curveGrad" x1="0" y1="0" x2="0" y2="1">
            <stop offset="0%" stop-color="#0284c7" stop-opacity="0.35"/>
            <stop offset="100%" stop-color="#0284c7" stop-opacity="0.0"/>
          </linearGradient>
        </defs>

        
        <g stroke="#e2e8f0" stroke-width="1" stroke-dasharray="2,2">
          
          <line x1="75" y1="20" x2="75" y2="200" />
          
          <line x1="270" y1="20" x2="270" y2="200" />
          
          <line x1="465" y1="20" x2="465" y2="200" />
          
          <line x1="30" y1="190" x2="510" y2="190" stroke="#94a3b8" stroke-dasharray="none" />
        </g>

        
        <g fill="#94a3b8" font-size="11" font-weight="600" text-anchor="middle">
          <text x="75" y="216">100 Hz</text>
          <text x="270" y="216">1 kHz</text>
          <text x="465" y="216">10 kHz</text>
          <text x="500" y="186" text-anchor="end">0 dB</text>
        </g>

        
        <line id="minus3Line" x1="30" y1="110" x2="510" y2="110" stroke="#cbd5e1" stroke-width="1.5" stroke-dasharray="4,4" />
        <text id="minus3Label" x="35" y="105" fill="#64748b" font-size="11" font-weight="700">-3 dB (Half Power)</text>

        
        <rect id="bwRect" x="200" y="30" width="140" height="160" fill="#0284c7" fill-opacity="0.08" rx="4" />

        
        <path id="curveArea" fill="url(#curveGrad)" />
        <path id="curveLine" fill="none" stroke="#0284c7" stroke-width="3" stroke-linecap="round" />

        
        <line id="f1Line" x1="200" y1="110" x2="200" y2="190" stroke="#f59e0b" stroke-width="1.5" stroke-dasharray="3,3" />
        <line id="f2Line" x1="340" y1="110" x2="340" y2="190" stroke="#f59e0b" stroke-width="1.5" stroke-dasharray="3,3" />
        <circle id="f1Dot" cx="200" cy="110" r="4.5" fill="#f59e0b" />
        <circle id="f2Dot" cx="340" cy="110" r="4.5" fill="#f59e0b" />
        <text id="f1Text" x="200" y="182" fill="#d97706" font-size="11" font-weight="700" text-anchor="end">f₁</text>
        <text id="f2Text" x="340" y="182" fill="#d97706" font-size="11" font-weight="700" text-anchor="start">f₂</text>

        
        <line id="fcLine" x1="270" y1="30" x2="270" y2="190" stroke="#0284c7" stroke-width="1.5" stroke-dasharray="3,3" />
        <circle id="fcDot" cx="270" cy="30" r="5" fill="#0284c7" />
        <text id="fcPeakText" x="270" y="24" fill="#0284c7" font-size="11" font-weight="700" text-anchor="middle">fc (Peak)</text>

        
        <g id="bwArrowGroup">
          <line id="bwArrowLine" x1="200" y1="65" x2="340" y2="65" stroke="#0f172a" stroke-width="1.5" />
          <polygon id="bwArrL" points="204,61 204,69 198,65" fill="#0f172a" />
          <polygon id="bwArrR" points="336,61 336,69 342,65" fill="#0f172a" />
          <rect id="bwBadgeBg" x="240" y="53" width="60" height="22" fill="#ffffff" stroke="#cbd5e1" rx="4" />
          <text id="bwBadgeText" x="270" y="68" fill="#0f172a" font-size="11" font-weight="700" text-anchor="middle">BW</text>
        </g>
      </svg>
    </div>

    
    <div class="formula-bar">
      <div>BW = <span class="formula-highlight" id="bwValHz">709 Hz</span></div>
      <div>Q = <span class="formula-highlight">fc / BW</span></div>
      <div>BW Octaves ≈ <span class="formula-highlight" id="octVal">1.00 oct</span></div>
    </div>

  </div>

  <script>
    const minFreq = 40;
    const maxFreq = 16000;
    const leftX = 30;
    const rightX = 510;
    const baseY = 190;
    const peakGainY = 30;
    const minus3Y = 110; // -3dB line visually at midpoint of boost

    const qSlider = document.getElementById('qSlider');
    const fcSlider = document.getElementById('fcSlider');
    const qVal = document.getElementById('qVal');
    const fcVal = document.getElementById('fcVal');
    const bwValHz = document.getElementById('bwValHz');
    const octVal = document.getElementById('octVal');

    const curveArea = document.getElementById('curveArea');
    const curveLine = document.getElementById('curveLine');
    const bwRect = document.getElementById('bwRect');
    const f1Line = document.getElementById('f1Line');
    const f2Line = document.getElementById('f2Line');
    const f1Dot = document.getElementById('f1Dot');
    const f2Dot = document.getElementById('f2Dot');
    const f1Text = document.getElementById('f1Text');
    const f2Text = document.getElementById('f2Text');
    const fcLine = document.getElementById('fcLine');
    const fcDot = document.getElementById('fcDot');
    const fcPeakText = document.getElementById('fcPeakText');
    const bwArrowLine = document.getElementById('bwArrowLine');
    const bwArrL = document.getElementById('bwArrL');
    const bwArrR = document.getElementById('bwArrR');
    const bwBadgeBg = document.getElementById('bwBadgeBg');
    const bwBadgeText = document.getElementById('bwBadgeText');

    function freqToX(f) {
      const logMin = Math.log10(minFreq);
      const logMax = Math.log10(maxFreq);
      const logF = Math.log10(Math.max(minFreq, Math.min(maxFreq, f)));
      return leftX + (logF - logMin) / (logMax - logMin) * (rightX - leftX);
    }

    function update() {
      const q = parseFloat(qSlider.value);
      const fc = parseFloat(fcSlider.value);

      // Theoretical Cutoff Frequencies
      // f1, f2 = fc * [ sqrt(1 + 1/(4Q^2)) -+ 1/(2Q) ]
      const term1 = Math.sqrt(1 + 1 / (4 * q * q));
      const term2 = 1 / (2 * q);
      const f1 = fc * (term1 - term2);
      const f2 = fc * (term1 + term2);
      const bw = f2 - f1;
      const octaves = Math.log2(f2 / f1);

      // Display Values
      qVal.textContent = q < 1 ? q.toFixed(2) : q.toFixed(1);
      fcVal.textContent = fc >= 1000 ? (fc / 1000).toFixed(1) + ' kHz' : Math.round(fc) + ' Hz';
      bwValHz.textContent = bw >= 1000 ? (bw / 1000).toFixed(2) + ' kHz' : Math.round(bw) + ' Hz';
      octVal.textContent = octaves.toFixed(2) + ' oct';

      // SVG Curve Path Calculation (Resonant bell response)
      const numPoints = 80;
      let pathD = '';
      for (let i = 0; i <= numPoints; i++) {
        const t = i / numPoints;
        const logMin = Math.log10(minFreq);
        const logMax = Math.log10(maxFreq);
        const f = Math.pow(10, logMin + t * (logMax - logMin));
        
        // Bell shape transfer function magnitude
        const ratio = (f / fc) - (fc / f);
        const response = 1 / (1 + q * q * ratio * ratio); // 1 at peak, 0.5 at f1/f2
        
        const px = leftX + t * (rightX - leftX);
        const py = baseY - response * (baseY - peakGainY);

        if (i === 0) pathD += `M ${px.toFixed(1)} ${py.toFixed(1)}`;
        else pathD += ` L ${px.toFixed(1)} ${py.toFixed(1)}`;
      }

      curveLine.setAttribute('d', pathD);
      curveArea.setAttribute('d', `${pathD} L ${rightX} ${baseY} L ${leftX} ${baseY} Z`);

      // Cutoff X Positions
      const x1 = Math.max(leftX, Math.min(rightX, freqToX(f1)));
      const x2 = Math.max(leftX, Math.min(rightX, freqToX(f2)));
      const xc = Math.max(leftX, Math.min(rightX, freqToX(fc)));

      // Update Center Line & Dot
      fcLine.setAttribute('x1', xc);
      fcLine.setAttribute('x2', xc);
      fcDot.setAttribute('cx', xc);
      fcPeakText.setAttribute('x', xc);

      // Update Cutoff Markers
      f1Line.setAttribute('x1', x1);
      f1Line.setAttribute('x2', x1);
      f1Dot.setAttribute('cx', x1);
      f1Text.setAttribute('x', x1 - 5);

      f2Line.setAttribute('x1', x2);
      f2Line.setAttribute('x2', x2);
      f2Dot.setAttribute('cx', x2);
      f2Text.setAttribute('x', x2 + 5);

      // Update Bandwidth Box & Arrows
      const bwWidth = Math.max(2, x2 - x1);
      bwRect.setAttribute('x', x1);
      bwRect.setAttribute('width', bwWidth);

      const arrowY = 70;
      bwArrowLine.setAttribute('x1', x1);
      bwArrowLine.setAttribute('x2', x2);
      bwArrowLine.setAttribute('y1', arrowY);
      bwArrowLine.setAttribute('y2', arrowY);

      bwArrL.setAttribute('points', `${x1 + 6},${arrowY - 4} ${x1 + 6},${arrowY + 4} ${x1},${arrowY}`);
      bwArrR.setAttribute('points', `${x2 - 6},${arrowY - 4} ${x2 - 6},${arrowY + 4} ${x2},${arrowY}`);

      const badgeMidX = (x1 + x2) / 2;
      bwBadgeBg.setAttribute('x', badgeMidX - 35);
      bwBadgeBg.setAttribute('y', arrowY - 10);
      bwBadgeText.setAttribute('x', badgeMidX);
      bwBadgeText.setAttribute('y', arrowY + 4);
      bwBadgeText.textContent = `BW: ${Math.round(bw)}Hz`;
    }

    qSlider.addEventListener('input', () => {
      clearActivePreset();
      update();
    });

    fcSlider.addEventListener('input', update);

    // Preset buttons
    const presets = document.querySelectorAll('.btn-preset');
    function clearActivePreset() {
      presets.forEach(b => b.classList.remove('active'));
    }

    presets.forEach(btn => {
      btn.addEventListener('click', () => {
        clearActivePreset();
        btn.classList.add('active');
        qSlider.value = btn.getAttribute('data-q');
        update();
      });
    });

    // Audio preview
    let audioCtx = null;
    document.getElementById('listenBtn').addEventListener('click', () => {
      if (!audioCtx) audioCtx = new (window.AudioContext || window.webkitAudioContext)();
      if (audioCtx.state === 'suspended') audioCtx.resume();

      const fc = parseFloat(fcSlider.value);
      const q = parseFloat(qSlider.value);

      // Filtered noise pulse demonstrating Q sharpness
      const dur = 0.5;
      const bufferSize = audioCtx.sampleRate * dur;
      const buffer = audioCtx.createBuffer(1, bufferSize, audioCtx.sampleRate);
      const data = buffer.getChannelData(0);
      for (let i = 0; i < bufferSize; i++) {
        data[i] = (Math.random() * 2 - 1) * 0.4;
      }

      const noise = audioCtx.createBufferSource();
      noise.buffer = buffer;

      const filter = audioCtx.createBiquadFilter();
      filter.type = 'bandpass';
      filter.frequency.value = fc;
      filter.Q.value = q;

      const gain = audioCtx.createGain();
      gain.gain.setValueAtTime(0.8, audioCtx.currentTime);
      gain.gain.exponentialRampToValueAtTime(0.001, audioCtx.currentTime + dur);

      noise.connect(filter);
      filter.connect(gain);
      gain.connect(audioCtx.destination);

      noise.start();
    });

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

Since human hearing perceives pitch logarithmically, how does bandwidth translate into musical octaves?

An **octave** is a doubling of frequency; the bandwidth in octaves ($N$) is defined by the logarithmic ratio of cutoffs: $N = \log_2\left(\frac{f_2}{f_1}\right)$.

Because the center frequency is the geometric mean ($f_c = \sqrt{f_1 \cdot f_2}$), you can convert between octaves ($N$) and Q using the exact formula: $$Q = \frac{\sqrt{2^N}}{2^N - 1}$$

Let's put this into practice on a real studio problem.

## Worked Example: Surgical Resonance Cut

Suppose a harsh whistle resonates at $f_c = 3200\text{ Hz}$ in a vocal track, and you dial in a surgical cut with a Q factor of $Q = 8.0$.

Rearranging $Q = \frac{f_c}{BW}$ gives: $$BW_{\text{Hz}} = \frac{f_c}{Q} = \frac{3200\text{ Hz}}{8.0} = 400\text{ Hz}$$

<!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 {
      width: 100%;
      min-height: 100%;
      background: #ffffff;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      padding: 12px;
    }
    .controls {
      display: flex;
      flex-wrap: wrap;
      gap: 8px;
      justify-content: center;
      align-items: center;
      margin-bottom: 10px;
      width: 100%;
      max-width: 560px;
    }
    .btn-group {
      display: flex;
      background: #f1f5f9;
      padding: 3px;
      border-radius: 8px;
      gap: 4px;
    }
    .btn {
      border: none;
      background: transparent;
      padding: 6px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      border-radius: 6px;
      cursor: pointer;
      transition: all 0.2s ease;
    }
    .btn.active {
      background: #ffffff;
      color: #0284c7;
      box-shadow: 0 1px 3px rgba(0,0,0,0.1);
    }
    .slider-wrap {
      display: flex;
      align-items: center;
      gap: 8px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      padding: 4px 12px;
      border-radius: 8px;
    }
    .slider-wrap label {
      font-size: 13px;
      font-weight: 700;
      color: #475569;
    }
    input[type="range"] {
      accent-color: #0284c7;
      cursor: pointer;
      width: 110px;
    }
    .stat-row {
      display: grid;
      grid-template-columns: repeat(3, 1fr);
      gap: 8px;
      width: 100%;
      max-width: 560px;
      margin-bottom: 8px;
    }
    .stat-box {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 6px 10px;
      text-align: center;
    }
    .stat-lbl {
      font-size: 11px;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      color: #64748b;
      font-weight: 600;
    }
    .stat-val {
      font-size: 15px;
      font-weight: 700;
      color: #0f172a;
      font-feature-settings: "tnum";
    }
    .stat-val.highlight {
      color: #0284c7;
    }
    .svg-container {
      width: 100%;
      max-width: 560px;
      position: relative;
    }
    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }
    .grid-line {
      stroke: #e2e8f0;
      stroke-width: 1;
      stroke-dasharray: 3 3;
    }
    .axis-line {
      stroke: #cbd5e1;
      stroke-width: 1.5;
    }
    .axis-text {
      fill: #94a3b8;
      font-size: 11px;
      font-weight: 500;
    }
    .eq-curve {
      fill: rgba(2, 132, 199, 0.12);
      stroke: #0284c7;
      stroke-width: 3;
      stroke-linejoin: round;
      stroke-linecap: round;
    }
    .bw-span {
      stroke: #0284c7;
      stroke-width: 2;
    }
    .bw-cap {
      stroke: #0284c7;
      stroke-width: 2;
    }
    .bw-fill {
      fill: rgba(2, 132, 199, 0.22);
    }
    .guide-line {
      stroke: #0284c7;
      stroke-width: 1.5;
      stroke-dasharray: 4 4;
      opacity: 0.7;
    }
    .center-line {
      stroke: #f59e0b;
      stroke-width: 1.5;
      stroke-dasharray: 4 4;
    }
  </style>
</head>
<body>

  <div class="controls">
    <div class="btn-group">
      <button class="btn" onclick="setQ(0.5, this)">Q = 0.5</button>
      <button class="btn" onclick="setQ(2.0, this)">Q = 2.0</button>
      <button class="btn active" onclick="setQ(8.0, this)">Q = 8.0</button>
    </div>
    <div class="slider-wrap">
      <label for="qSlider">Q</label>
      <input type="range" id="qSlider" min="0.3" max="12" step="0.1" value="8.0" oninput="onSlider(this.value)">
    </div>
  </div>

  <div class="stat-row">
    <div class="stat-box">
      <div class="stat-lbl">Center (f<sub>c</sub>)</div>
      <div class="stat-val">3200 Hz</div>
    </div>
    <div class="stat-box">
      <div class="stat-lbl">Q Factor</div>
      <div class="stat-val highlight" id="qVal">8.0</div>
    </div>
    <div class="stat-box">
      <div class="stat-lbl">Bandwidth (BW)</div>
      <div class="stat-val highlight" id="bwVal">400 Hz</div>
    </div>
  </div>

  <div class="svg-container">
    <svg viewBox="0 0 560 260">
      
      <line x1="50" y1="30" x2="530" y2="30" class="grid-line" />
      <line x1="50" y1="80" x2="530" y2="80" class="grid-line" />
      <line x1="50" y1="130" x2="530" y2="130" class="grid-line" />
      <line x1="50" y1="180" x2="530" y2="180" class="grid-line" />
      <line x1="50" y1="210" x2="530" y2="210" class="axis-line" />

      
      <text x="42" y="34" class="axis-text" text-anchor="end">+12 dB</text>
      <text x="42" y="84" class="axis-text" text-anchor="end">+9 dB</text>
      <text x="42" y="134" class="axis-text" text-anchor="end">+6 dB</text>
      <text x="42" y="214" class="axis-text" text-anchor="end">0 dB</text>

      
      <line x1="50" y1="80" x2="530" y2="80" stroke="#94a3b8" stroke-width="1" stroke-dasharray="2 4" />
      <text x="535" y="84" class="axis-text" fill="#64748b" text-anchor="start">-3dB</text>

      
      <polygon id="bwBand" class="bw-fill" />
      <line id="f1Line" class="guide-line" />
      <line id="f2Line" class="guide-line" />
      <line id="fcLine" x1="0" y1="30" x2="0" y2="210" class="center-line" />

      
      <path id="eqCurve" class="eq-curve" />

      
      <line id="bwArrow" class="bw-span" />
      <line id="bwCapL" class="bw-cap" />
      <line id="bwCapR" class="bw-cap" />
      <text id="bwLabel" x="0" y="72" font-size="12" font-weight="700" fill="#0284c7" text-anchor="middle"></text>

      
      <circle id="fcDot" r="4" fill="#f59e0b" />

      
      <g id="freqLabels"></g>
    </svg>
  </div>

  <script>
    const FC = 3200;
    const GAIN_DB = 12;
    const F_MIN = 400;
    const F_MAX = 16000;
    const X_MIN = 50;
    const X_MAX = 530;
    const Y_0DB = 210;
    const Y_PEAK = 30;

    function fToX(f) {
      return X_MIN + (X_MAX - X_MIN) * (Math.log10(f) - Math.log10(F_MIN)) / (Math.log10(F_MAX) - Math.log10(F_MIN));
    }

    function calcGainDb(f, Q) {
      const ratio = f / FC - FC / f;
      return GAIN_DB / (1 + (Q * ratio) ** 2);
    }

    function dbToY(db) {
      return Y_0DB - (db / GAIN_DB) * (Y_0DB - Y_PEAK);
    }

    function initAxis() {
      const g = document.getElementById('freqLabels');
      const freqs = [500, 1000, 2000, 3200, 8000, 16000];
      freqs.forEach(f => {
        const x = fToX(f);
        const text = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        text.setAttribute('x', x);
        text.setAttribute('y', 228);
        text.setAttribute('class', 'axis-text');
        text.setAttribute('text-anchor', 'middle');
        text.textContent = f >= 1000 ? (f / 1000 === 3.2 ? '3.2k' : (f / 1000) + 'k') : f;
        if (f === 3200) {
          text.setAttribute('fill', '#f59e0b');
          text.setAttribute('font-weight', '700');
        }
        g.appendChild(text);

        const tick = document.createElementNS('http://www.w3.org/2000/svg', 'line');
        tick.setAttribute('x1', x);
        tick.setAttribute('x2', x);
        tick.setAttribute('y1', 210);
        tick.setAttribute('y2', 215);
        tick.setAttribute('class', 'axis-line');
        g.appendChild(tick);
      });
    }

    function update(Q) {
      document.getElementById('qVal').textContent = Number(Q).toFixed(1);
      const bw = FC / Q;
      document.getElementById('bwVal').textContent = bw >= 1000 ? (bw / 1000).toFixed(2) + ' kHz' : Math.round(bw) + ' Hz';

      const f1 = FC * (Math.sqrt(1 + 1 / (4 * Q * Q)) - 1 / (2 * Q));
      const f2 = FC * (Math.sqrt(1 + 1 / (4 * Q * Q)) + 1 / (2 * Q));

      const x1 = fToX(Math.max(F_MIN, f1));
      const x2 = fToX(Math.min(F_MAX, f2));
      const xC = fToX(FC);
      const y3dB = dbToY(GAIN_DB - 3);

      // Curve
      const steps = 120;
      let d = '';
      const polyPts = [];
      for (let i = 0; i <= steps; i++) {
        const logF = Math.log10(F_MIN) + (i / steps) * (Math.log10(F_MAX) - Math.log10(F_MIN));
        const f = 10 ** logF;
        const x = fToX(f);
        const y = dbToY(calcGainDb(f, Q));
        d += (i === 0 ? `M ${x.toFixed(1)} ${y.toFixed(1)}` : ` L ${x.toFixed(1)} ${y.toFixed(1)}`);
      }
      document.getElementById('eqCurve').setAttribute('d', d);

      // Bandwidth Polygon Fill
      const pts = [];
      pts.push(`${x1.toFixed(1)},${Y_0DB}`);
      for (let i = 0; i <= 40; i++) {
        const f = f1 + (i / 40) * (f2 - f1);
        if (f >= F_MIN && f <= F_MAX) {
          pts.push(`${fToX(f).toFixed(1)},${dbToY(calcGainDb(f, Q)).toFixed(1)}`);
        }
      }
      pts.push(`${x2.toFixed(1)},${Y_0DB}`);
      document.getElementById('bwBand').setAttribute('points', pts.join(' '));

      // Guide lines
      const f1L = document.getElementById('f1Line');
      f1L.setAttribute('x1', x1);
      f1L.setAttribute('y1', y3dB);
      f1L.setAttribute('x2', x1);
      f1L.setAttribute('y2', Y_0DB);

      const f2L = document.getElementById('f2Line');
      f2L.setAttribute('x1', x2);
      f2L.setAttribute('y1', y3dB);
      f2L.setAttribute('x2', x2);
      f2L.setAttribute('y2', Y_0DB);

      const fcL = document.getElementById('fcLine');
      fcL.setAttribute('x1', xC);
      fcL.setAttribute('x2', xC);

      const fcDot = document.getElementById('fcDot');
      fcDot.setAttribute('cx', xC);
      fcDot.setAttribute('cy', Y_PEAK);

      // Bandwidth arrow & label
      const bwArr = document.getElementById('bwArrow');
      bwArr.setAttribute('x1', x1);
      bwArr.setAttribute('y1', y3dB);
      bwArr.setAttribute('x2', x2);
      bwArr.setAttribute('y2', y3dB);

      const capL = document.getElementById('bwCapL');
      capL.setAttribute('x1', x1);
      capL.setAttribute('y1', y3dB - 6);
      capL.setAttribute('x2', x1);
      capL.setAttribute('y2', y3dB + 6);

      const capR = document.getElementById('bwCapR');
      capR.setAttribute('x1', x2);
      capR.setAttribute('y1', y3dB - 6);
      capR.setAttribute('x2', x2);
      capR.setAttribute('y2', y3dB + 6);

      const lbl = document.getElementById('bwLabel');
      lbl.setAttribute('x', (x1 + x2) / 2);
      lbl.textContent = `BW = ${bw >= 1000 ? (bw / 1000).toFixed(1) + 'k' : Math.round(bw)} Hz`;
    }

    function setQ(val, btn) {
      document.getElementById('qSlider').value = val;
      document.querySelectorAll('.btn-group .btn').forEach(b => b.classList.remove('active'));
      if (btn) btn.classList.add('active');
      update(val);
    }

    function onSlider(val) {
      document.querySelectorAll('.btn-group .btn').forEach(b => b.classList.remove('active'));
      update(parseFloat(val));
    }

    initAxis();
    update(8.0);
  </script>
</body>
</html>

A common mistake is assuming that a larger Q value affects a wider range of frequencies. Remember that Q is **inversely proportional** to bandwidth: higher Q means a narrower, more surgical slice.

What other filter shapes do parametric equalizers use to shape sound?

## Filter Topologies: Bell, Notch, and Shelves

Parametric equalizers combine three core topologies: **bell filters** for selective tone shaping, **notch filters** with extreme Q ($Q \ge 10$) for eliminating noise, and **shelving filters** for broad low- or high-end adjustments.

<!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 {
      width: 100%;
      height: 100%;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      user-select: none;
      padding: 8px;
    }

    .toolbar {
      display: flex;
      gap: 6px;
      margin-bottom: 8px;
    }

    .tab-btn {
      flex: 1;
      padding: 6px 4px;
      font-size: 13px;
      font-weight: 600;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      color: #64748b;
      cursor: pointer;
      text-align: center;
      transition: all 0.2s ease;
    }

    .tab-btn.active {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
      box-shadow: 0 2px 6px rgba(2, 132, 199, 0.25);
    }

    .graph-container {
      position: relative;
      flex: 1;
      min-height: 180px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      overflow: hidden;
    }

    svg {
      width: 100%;
      height: 100%;
      display: block;
    }

    .controls-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 8px;
      margin-top: 8px;
    }

    .control-card {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 6px 10px;
      display: flex;
      flex-direction: column;
      gap: 4px;
    }

    .control-header {
      display: flex;
      justify-content: space-between;
      align-items: center;
      font-size: 12px;
      font-weight: 600;
      color: #475569;
    }

    .val-badge {
      font-variant-numeric: tabular-nums;
      color: #0284c7;
      font-weight: 700;
    }

    input[type="range"] {
      -webkit-appearance: none;
      appearance: none;
      width: 100%;
      height: 5px;
      border-radius: 3px;
      background: #e2e8f0;
      outline: none;
      cursor: pointer;
    }

    input[type="range"]::-webkit-slider-thumb {
      -webkit-appearance: none;
      appearance: none;
      width: 16px;
      height: 16px;
      border-radius: 50%;
      background: #0284c7;
      border: 2px solid #ffffff;
      box-shadow: 0 1px 3px rgba(0,0,0,0.25);
    }

    .formula-bar {
      margin-top: 6px;
      background: #f1f5f9;
      border-radius: 6px;
      padding: 4px 8px;
      display: flex;
      justify-content: space-around;
      align-items: center;
      font-size: 12px;
      font-weight: 600;
      color: #334155;
    }

    .formula-item span {
      color: #0284c7;
    }
  </style>
</head>
<body>

  <div class="toolbar">
    <button class="tab-btn active" id="btnBell" onclick="setType('bell')">Bell</button>
    <button class="tab-btn" id="btnNotch" onclick="setType('notch')">Notch (Cut)</button>
    <button class="tab-btn" id="btnShelf" onclick="setType('shelf')">High Shelf</button>
  </div>

  <div class="graph-container">
    <svg id="eqSvg" viewBox="0 0 600 240" preserveAspectRatio="none">
      <defs>
        <linearGradient id="curveGrad" x1="0" y1="0" x2="0" y2="1">
          <stop offset="0%" stop-color="#0284c7" stop-opacity="0.25" />
          <stop offset="100%" stop-color="#0284c7" stop-opacity="0.0" />
        </linearGradient>
      </defs>

      
      <line x1="40" y1="30" x2="570" y2="30" stroke="#f1f5f9" stroke-width="1" />
      <line x1="40" y1="75" x2="570" y2="75" stroke="#f1f5f9" stroke-width="1" />
      <line x1="40" y1="120" x2="570" y2="120" stroke="#cbd5e1" stroke-width="1" stroke-dasharray="3 3" />
      <line x1="40" y1="165" x2="570" y2="165" stroke="#f1f5f9" stroke-width="1" />
      <line x1="40" y1="210" x2="570" y2="210" stroke="#f1f5f9" stroke-width="1" />

      
      <line id="f100" y1="15" y2="215" stroke="#f1f5f9" stroke-width="1" />
      <line id="f1k" y1="15" y2="215" stroke="#f1f5f9" stroke-width="1" />
      <line id="f10k" y1="15" y2="215" stroke="#f1f5f9" stroke-width="1" />

      
      <text x="35" y="34" font-size="10" fill="#94a3b8" text-anchor="end">+12dB</text>
      <text x="35" y="124" font-size="10" fill="#64748b" font-weight="bold" text-anchor="end">0dB</text>
      <text x="35" y="214" font-size="10" fill="#94a3b8" text-anchor="end">-12dB</text>

      <text id="lbl100" y="232" font-size="10" fill="#94a3b8" text-anchor="middle">100Hz</text>
      <text id="lbl1k" y="232" font-size="10" fill="#94a3b8" text-anchor="middle">1kHz</text>
      <text id="lbl10k" y="232" font-size="10" fill="#94a3b8" text-anchor="middle">10kHz</text>

      
      <path id="curveArea" fill="url(#curveGrad)" d="" />
      <path id="curveLine" fill="none" stroke="#0284c7" stroke-width="2.5" d="" />

      
      <g id="bwGroup">
        <line id="bwLine" stroke="#0ea5e9" stroke-width="1.5" stroke-dasharray="2 2" />
        <circle id="p1Dot" r="3.5" fill="#0284c7" />
        <circle id="p2Dot" r="3.5" fill="#0284c7" />
        <text id="bwLabel" font-size="10" font-weight="600" fill="#0284c7" text-anchor="middle" y="105">BW (-3dB)</text>
      </g>

      
      <line id="fcLine" stroke="#64748b" stroke-width="1" stroke-dasharray="2 2" />
      <circle id="centerHandle" r="5" fill="#0284c7" stroke="#ffffff" stroke-width="2" />
      <text id="fcLabel" font-size="10" font-weight="bold" fill="#1e2945" text-anchor="middle">fc</text>
    </svg>
  </div>

  <div class="controls-grid">
    <div class="control-card">
      <div class="control-header">
        <span>Q Factor</span>
        <span class="val-badge" id="qVal">1.0</span>
      </div>
      <input type="range" id="qSlider" min="0.4" max="10" step="0.1" value="1.0" oninput="updatePlot()">
    </div>

    <div class="control-card">
      <div class="control-header">
        <span>Gain</span>
        <span class="val-badge" id="gainVal">+6.0 dB</span>
      </div>
      <input type="range" id="gainSlider" min="-12" max="12" step="0.5" value="6.0" oninput="updatePlot()">
    </div>
  </div>

  <div class="formula-bar">
    <div class="formula-item">Formula: <span>Q = f<sub>c</sub> / BW</span></div>
    <div class="formula-item">BW: <span id="bwVal">1000 Hz</span></div>
  </div>

  <script>
    let filterType = 'bell';
    const fc = 1000;
    const minF = 20, maxF = 20000;
    const minX = 45, maxX = 570;
    const midY = 120, scaleY = 7.5; // 120px = 0dB, ±12dB = ±90px

    function fToX(f) {
      return minX + (Math.log10(f / minF) / Math.log10(maxF / minF)) * (maxX - minX);
    }

    // Static Grid lines alignment
    document.getElementById('f100').setAttribute('x1', fToX(100));
    document.getElementById('f100').setAttribute('x2', fToX(100));
    document.getElementById('lbl100').setAttribute('x', fToX(100));

    document.getElementById('f1k').setAttribute('x1', fToX(1000));
    document.getElementById('f1k').setAttribute('x2', fToX(1000));
    document.getElementById('lbl1k').setAttribute('x', fToX(1000));

    document.getElementById('f10k').setAttribute('x1', fToX(10000));
    document.getElementById('f10k').setAttribute('x2', fToX(10000));
    document.getElementById('lbl10k').setAttribute('x', fToX(10000));

    function setType(type) {
      filterType = type;
      document.querySelectorAll('.tab-btn').forEach(b => b.classList.remove('active'));
      
      const qInput = document.getElementById('qSlider');
      const gainInput = document.getElementById('gainSlider');

      if (type === 'bell') {
        document.getElementById('btnBell').classList.add('active');
        qInput.disabled = false;
        gainInput.disabled = false;
        qInput.value = 1.0;
        gainInput.value = 6.0;
      } else if (type === 'notch') {
        document.getElementById('btnNotch').classList.add('active');
        qInput.disabled = false;
        gainInput.disabled = true;
        qInput.value = 8.0;
        gainInput.value = -12.0;
      } else if (type === 'shelf') {
        document.getElementById('btnShelf').classList.add('active');
        qInput.disabled = true;
        gainInput.disabled = false;
        gainInput.value = 6.0;
      }
      updatePlot();
    }

    function calcGainAt(f, gain, Q) {
      if (filterType === 'shelf') {
        // High shelf approximation
        const norm = Math.atan((f - fc) / (fc * 0.7)) / (Math.PI / 2);
        return gain * (norm + 1) / 2;
      }

      // Bell / Peak / Notch
      const octDiff = Math.log2(f / fc);
      const exponent = -Math.pow(octDiff * Q * 1.2, 2);
      return gain * Math.exp(exponent);
    }

    function updatePlot() {
      const Q = parseFloat(document.getElementById('qSlider').value);
      const gain = parseFloat(document.getElementById('gainSlider').value);

      document.getElementById('qVal').textContent = filterType === 'shelf' ? 'Fixed' : Q.toFixed(1);
      document.getElementById('gainVal').textContent = (gain >= 0 ? '+' : '') + gain.toFixed(1) + ' dB';

      const bw = fc / Q;
      document.getElementById('bwVal').textContent = filterType === 'shelf' ? '—' : Math.round(bw) + ' Hz';

      const pts = [];
      const numSteps = 120;
      for (let i = 0; i <= numSteps; i++) {
        const t = i / numSteps;
        const f = minF * Math.pow(maxF / minF, t);
        const g = calcGainAt(f, gain, Q);
        const x = fToX(f);
        const y = midY - (g * scaleY);
        pts.push({x, y});
      }

      let dLine = `M ${pts[0].x} ${pts[0].y}`;
      for (let i = 1; i < pts.length; i++) {
        dLine += ` L ${pts[i].x.toFixed(1)} ${pts[i].y.toFixed(1)}`;
      }
      document.getElementById('curveLine').setAttribute('d', dLine);

      const dArea = dLine + ` L ${pts[pts.length-1].x} ${midY} L ${pts[0].x} ${midY} Z`;
      document.getElementById('curveArea').setAttribute('d', dArea);

      // Handle center point
      const centerX = fToX(fc);
      const centerY = midY - (calcGainAt(fc, gain, Q) * scaleY);
      
      const handle = document.getElementById('centerHandle');
      handle.setAttribute('cx', centerX);
      handle.setAttribute('cy', centerY);

      const fcLine = document.getElementById('fcLine');
      fcLine.setAttribute('x1', centerX);
      fcLine.setAttribute('y1', centerY);
      fcLine.setAttribute('x2', centerX);
      fcLine.setAttribute('y2', midY);

      const fcLabel = document.getElementById('fcLabel');
      fcLabel.setAttribute('x', centerX);
      fcLabel.setAttribute('y', centerY + (gain >= 0 ? -10 : 18));
      fcLabel.textContent = `${fc}Hz`;

      // Bandwidth visual indicators
      const bwGroup = document.getElementById('bwGroup');
      if (filterType === 'shelf' || Math.abs(gain) < 3.5) {
        bwGroup.style.display = 'none';
      } else {
        bwGroup.style.display = 'block';
        // -3dB relative to peak
        const targetG = gain >= 0 ? (gain - 3) : (gain + 3);
        const f1 = fc * Math.pow(2, -1 / (2 * Q));
        const f2 = fc * Math.pow(2, 1 / (2 * Q));
        
        const x1 = fToX(f1);
        const x2 = fToX(f2);
        const yBand = midY - (targetG * scaleY);

        const bwLine = document.getElementById('bwLine');
        bwLine.setAttribute('x1', x1);
        bwLine.setAttribute('x2', x2);
        bwLine.setAttribute('y1', yBand);
        bwLine.setAttribute('y2', yBand);

        document.getElementById('p1Dot').setAttribute('cx', x1);
        document.getElementById('p1Dot').setAttribute('cy', yBand);
        document.getElementById('p2Dot').setAttribute('cx', x2);
        document.getElementById('p2Dot').setAttribute('cy', yBand);

        const bwLabel = document.getElementById('bwLabel');
        bwLabel.setAttribute('x', (x1 + x2) / 2);
        bwLabel.setAttribute('y', yBand + (gain >= 0 ? 14 : -8));
      }
    }

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

In mixing, use broad bell curves ($Q \approx 0.5$ to $1.4$) for smooth, natural boosts, and reserve high-Q notches ($Q > 5$) strictly for cutting resonant problems.

music-technology

Parametric EQ Architecture: Q Factor, Gain, and Bandwidth Maths. Calculate octave bandwidth from Q factor (Q = fc / BW) and apply surgical bell, notch, and shelving filters.

EQ: What Do All Those Knobs Do? Understanding Frequency, Gain, and Q

Section 7.1.5: Parametric EQ Architecture, Bandwidth, and Filter Topologies

Cookbook Formulae for Audio EQ Biquad Filter Coefficients (Bristow-Johnson)

Pedagogical strategy: Begin by connecting physical acoustics (Hz) to musical perception (octaves and semitones) to solidify why $f_c = \sqrt{f_1 f_2}$ rather than a linear midpoint. Emphasize hands-on DAW experimentation with parametric EQs (e.g., Logic Pro Channel EQ, FabFilter Pro-Q, or Ableton EQ Eight), having students listen to the audible artifacts of high-Q boosting (phase smear and resonant ringing) versus low-Q broadband shaping, followed by rigorous mathematical calculation of $Q$, $BW$, and octave relationships for A-Level exam preparation.

Parametric EQ Architecture: Q Factor, Gain, and Bandwidth Maths

Anatomy of a Parametric Band

The Q Factor and Bandwidth Formula

Octave Bandwidth Maths