ACARA v9 Homeschool — Age 13: Calculating the Pythagorean C Scale (monochord, ratios, 3:2 fifths)

Concise step-by-step explanation for a 13-year-old of how Pythagoras used a monochord and 2/3 (3/2) ratios to build the Pythagorean C scale. Includes clear calculations, octave adjustments and final frequencies (Hz).

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Brief (with a little Ally McBeal flourish)

Issue: How did Pythagoras create the 7-note C scale from a monochord, and what are the frequencies (Hz) for each note if middle C = 261.63 Hz?

Short answer (conclusion)

Using the Pythagorean method (stacking perfect fifths with ratio 3:2, and using 2:3 when dividing string length), the C-scale frequencies are:

  • 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

Facts & Rules (Pythagorean method)

  1. The pitch (frequency) of a string is inversely proportional to its length. If the string length becomes 2/3 of original, the frequency becomes 3/2 of original.
  2. A 1:2 length ratio gives an octave (frequency doubles). Dividing the string in half raises pitch one octave.
  3. Pythagoras built the scale by stacking perfect fifths (ratio 3:2). If a result falls outside the target octave, we shift it by factors of 2 (divide or multiply by 2) to bring it inside the octave that starts at middle C (261.63 Hz) and ends at 523.26 Hz.

Step-by-step calculations (work shown)

Start: C = 261.63 Hz

Rule used: To go up a fifth: multiply frequency by 3/2. To go down a fifth (shorten string to 2/3 length): multiply by 2/3. After each multiply, if result lies outside the C-octave (261.63 – 523.26 Hz) shift by factors of 2 into the octave.

  1. G (a fifth above C): G = C × 3/2 = 261.63 × 1.5 = 392.445 → 392.45 Hz
  2. D (a fifth above G): D = G × 3/2 = 392.445 × 1.5 = 588.6675 → divide by 2 to bring into octave = 294.33375 → 294.33 Hz
  3. A (a fifth above D): A = D × 3/2 = 294.33375 × 1.5 = 441.500625 → 441.50 Hz
  4. E (a fifth above A): E = A × 3/2 = 441.500625 × 1.5 = 662.2509375 → divide by 2 = 331.12546875 → 331.13 Hz
  5. B (a fifth above E): B = E × 3/2 = 331.12546875 × 1.5 = 496.688203125 → 496.69 Hz
  6. F (a fifth below C, or equivalently a fourth above C): F = C × 2/3 = 261.63 × 0.6666667 = 174.42 → multiply by 2 to bring into the octave = 348.84 Hz
  7. Upper C (octave): C' = 2 × 261.63 = 523.26 Hz

Explanation in plain words

When you cut the string length in half, the pitch goes up one octave because the frequency doubles. When you shorten the string to two-thirds its length, the frequency becomes 3/2 times larger (so the note is a perfect fifth above). Pythagoras built the scale by repeatedly using these fifths and then shifting by octaves so all notes fit between C and the next C.

Note on rounding

Values above are rounded to two decimal places for clarity; more decimals are available if needed.

Conclusion: The Pythagorean C scale comes from stacking 3:2 fifths (or using 2:3 length cuts) and normalizing by octaves. Music, math, and a monochord walk into a bar — and yes, it sounds like Ally McBeal humming a perfect fifth.

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