Calculating the Pythagorean Scale — Teacher’s Guide & Student Handout (Age 13)

Step-by-step teacher guide for calculating the Pythagorean C scale from a monochord (middle C = 261.63 Hz). Includes clear calculations, common mistakes, classroom activities and mapping to ACARA v9 learning goals for 13‑year‑olds.

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Ally McBeal‑style Brief: Calculating the Pythagorean Scale (for a 13‑year‑old)

Issue: How did Pythagoras use the monochord and simple ratios to create a 7‑note scale inside one octave, and how can we calculate each note’s frequency when middle C = 261.63 Hz?

Facts:

  • The monochord is one string. Changing the vibrating length changes pitch.
  • If the string length is halved, the frequency doubles: that interval is an octave (ratio 1:2 in length, 2:1 in frequency).
  • Pythagoras used the ratio 2:3 in length (which gives 3:2 in frequency) to make the interval called the perfect fifth. Repeating fifths and shifting by octaves builds the scale.

Analysis / Calculations (step‑by‑step):

Rule to remember: multiplying a frequency by 3/2 moves up a perfect fifth. If a frequency goes above the octave (above 523.26 Hz for middle C), divide by 2 to bring it down into the same octave. If it falls below the octave, multiply by 2.

  1. Start: C
    C = 261.63 Hz (given). The octave above is 2 × 261.63 = 523.26 Hz.
  2. G (a fifth above C)
    G = C × 3/2 = 261.63 × 1.5 = 392.445 → round to 392.45 Hz.
  3. D (a fifth above G)
    D_raw = G × 3/2 = 392.445 × 1.5 = 588.6675 Hz. This is > 523.26, so bring it down one octave: D = 588.6675 ÷ 2 = 294.33375 → 294.33 Hz.
  4. A (a fifth above D)
    A_raw = D × 3/2 = 294.33375 × 1.5 = 441.500625 → 441.50 Hz.
  5. E (a fifth above A)
    E_raw = A × 3/2 = 441.500625 × 1.5 = 662.2509375 Hz. This is > 523.26, so divide by 2: E = 662.2509375 ÷ 2 = 331.12546875 → 331.13 Hz.
  6. B (a fifth above E)
    B_raw = E × 3/2 = 331.12546875 × 1.5 = 496.688203125 → 496.69 Hz (already inside the octave).
  7. F (the fourth relative to C)
    To get F you can go down a fifth from C: F_raw = C ÷ (3/2) = C × 2/3 = 261.63 × 0.666... = 174.42 Hz (this is below the octave), so multiply by 2 to bring it up: F = 174.42 × 2 = 348.84 Hz. (Equivalent to F = C × 4/3 = 348.84 Hz.)
  8. Octave (top C)
    Top C = 2 × 261.63 = 523.26 Hz.

Final Pythagorean C scale (frequencies rounded):

  • C = 261.63 Hz
  • D = 294.33 Hz
  • E = 331.13 Hz
  • F = 348.84 Hz
  • G = 392.45 Hz
  • A = 441.50 Hz
  • B = 496.69 Hz
  • C (octave) = 523.26 Hz

Common student errors to watch for:

  • Mixing length ratios and frequency ratios. (Shorter length → higher frequency; if length ratio is 2:3 the frequency ratio is 3:2.)
  • Forgetting to divide by 2 when a calculated frequency falls above the octave, or forgetting to multiply by 2 if it falls below the target octave.
  • Rounding too early — keep extra decimal places until final rounding to avoid small errors.

Classroom activities (teacher suggestions):

  • Give students the starting C and let them compute G, D, A, E, B, and F with the rule multiply by 3/2, adjusting octaves as needed. Ask them to show work for each step.
  • Have students compare the Pythagorean frequencies to the equal‑tempered frequencies for the same notes and listen to the differences (if you have a keyboard or tuning app).
  • Ask students to calculate the size of the intervals in cents between Pythagorean and equal‑tempered notes (extension).

Short Conclusion / Ruling: Using repeated 3:2 frequency multiplications (perfect fifths) and bringing results into the single octave by powers of two produces the Pythagorean C scale shown above. The method is consistent and exposes the relationship between ratios, pitch, and musical intervals.

ACARA v9 mapping (plain language):

  • Mathematics: Number and Algebra — understand and use ratios, multiplying and dividing by whole numbers and fractions, and working with decimals. (Students practise calculating with fractions like 3/2 and powers of two.)
  • Mathematics: Measurement and Geometry — use multiplicative reasoning to change scale (doubling/halving frequencies to move octaves).
  • Music / Arts: Explore how pitch is related to vibration frequency and learn historical context for tuning systems (Pythagorean tuning and how it differs from equal temperament).

If you want, I can provide: (a) a printable student worksheet with blank spaces for calculations, (b) a one‑page teacher answer key, or (c) a short listening activity comparing Pythagorean vs equal‑tempered C scale tones.

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