Pythagorean Scale Calculations for a 13-year-old: Step-by-step Monochord Workings and C‑major Frequencies

Simple, step-by-step teacher's guide showing how Pythagoras used a monochord and 3:2 and 2:3 ratios to build the Pythagorean C scale. Includes each calculation, octave adjustments and final frequencies (age 13).

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Introduction

Pythagoras used a simple instrument called a monochord (one string) to discover how string length and pitch relate. For a 13-year-old: shorter string = higher pitch. The important math rule is that frequency is inversely proportional to string length. So if the string length is multiplied by a factor, the frequency is multiplied by the inverse factor.

Key facts (easy rules)

  • Halving the string length doubles the frequency (gives an octave up).
  • Shortening the string to 2/3 of its length raises the pitch by a ratio of 3/2 in frequency (this is the perfect fifth).
  • If a calculated frequency falls outside the octave you want (C to next C), divide or multiply by 2 to bring it into the correct octave:
    • If frequency > upper C, divide by 2 until it fits.
    • If frequency < lower C, multiply by 2 until it fits.

Question 1: Half the string (octave)

  1. a) Ratio of part to whole when the string is divided in half: 1 : 2.
  2. b) Middle C frequency given: 261.63 Hz. If you halve the string, frequency doubles: 261.63 × 2 = 523.26 Hz (the C one octave above).
  3. c) In plain words: when the string is divided in half, the pitch goes up by one octave (the frequency doubles).

How to build the Pythagorean C scale (step-by-step)

Pythagoras built notes by using the 3:2 frequency ratio (which comes from shortening length to 2/3). To go up a perfect fifth in frequency: multiply by 3/2. To go down a perfect fifth: multiply by 2/3. After each multiplication, adjust by dividing or multiplying by 2 as needed to keep the result inside the octave from C (261.63 Hz) to the next C (523.26 Hz).

Start: C = 261.63 Hz

  1. G (up a fifth from C): 261.63 × 3/2 = 261.63 × 1.5 = 392.445 → 392.45 Hz.
  2. D (up a fifth from G): 392.445 × 3/2 = 588.6675 → larger than 523.26, so divide by 2 to bring into the C octave: 588.6675 / 2 = 294.33375 → 294.33 Hz.
  3. A (up a fifth from D): 294.33375 × 3/2 = 441.500625 → this is inside the octave: 441.50 Hz.
  4. E (up a fifth from A): 441.500625 × 3/2 = 662.2509375 → divide by 2 to fit: 662.2509375 / 2 = 331.12546875 → 331.13 Hz.
  5. B (up a fifth from E): 331.12546875 × 3/2 = 496.688203125 → inside the octave: 496.69 Hz.
  6. F (this is found by going down a fifth from C, i.e., multiply by 2/3): 261.63 × 2/3 = 174.42 → below the C octave, so multiply by 2 to bring it up one octave: 174.42 × 2 = 348.84 Hz.
  7. Finally the next C (one octave above start): as in Question 1, 523.26 Hz.

Final Pythagorean C scale (rounded to two decimal places)

  • 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 = 523.26 Hz

Short note about tuning

These frequencies are from Pythagorean tuning, which uses pure 3:2 fifths. Modern equal temperament (the tuning used on most pianos today) adjusts those frequencies a little so that every half step is equal. That means some Pythagorean notes (especially E and B) will sound slightly different compared with equal temperament.

If you want, I can show a side-by-side comparison with equal-tempered frequencies or draw a simple monochord diagram to show where the 2/3 or 1/2 divisions are placed on the string.

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