Calculating the Pythagorean C Scale — 13-year-old Homeschool (ACARA v9 Proficient)

Step-by-step concise legal-brief style explanation (with a wink of Ally McBeal) showing how Pythagoras used 2:3 and 1:2 ratios on a monochord to build the Pythagorean C scale from middle C (261.63 Hz). Includes exact calculations and final frequencies.

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IN THE MATTER OF: Calculating the Pythagorean Scale

Student: 13-year-old (ACARA v9 — Proficient)

Statement of Facts

Pythagoras used a monochord (one string) and discovered two important ratios:

  1. 1:2 — divide the string in half; pitch moves by an octave (frequency doubles for the shorter half).
  2. 2:3 — make the sounding length 2/3 of a reference length; the frequency becomes 3/2 of that reference (a perfect fifth above).

Issues

1) What happens to frequency when the string is halved? 2) What frequency range is an octave around middle C? 3) How to build the Pythagorean C scale using repeated 3:2 (and one 2:3 downward) steps?

Analysis (Concise Legal Reasoning — and yes, I am thinking out loud)

Issue 1 — Halving the string: If the open string is 261.63 Hz (middle C), halving the sounding length doubles the frequency. Ruling: 261.63 × 2 = 523.26 Hz. (Ally aside: it sounds like the same note, but higher — very dramatic, like a court scene.)

Issue 2 — Octave limits: An octave is a 1:2 frequency ratio. Two useful octaves around middle C are:

  • Lower octave: 130.815 Hz to 261.63 Hz (one octave below middle C up to middle C)
  • Upper octave: 261.63 Hz to 523.26 Hz (middle C up to the octave above)
For building a C scale that starts at middle C, use 261.63 Hz up to 523.26 Hz.

Issue 3 — Building the Pythagorean C scale: The Pythagorean method uses the ratio 3:2 (a perfect fifth up). If a result falls outside the chosen C octave, bring it inside by dividing or multiplying by 2.

Calculations (showing each step)

  1. C (given): 261.63 Hz
  2. G = C × 3/2 = 261.63 × 1.5 = 392.445 → round to 392.45 Hz
  3. D = G × 3/2 = 392.445 × 1.5 = 588.6675 → divide by 2 to bring into C octave = 294.33375 → 294.33 Hz
  4. A = D × 3/2 = 294.33375 × 1.5 = 441.500625 → 441.50 Hz
  5. E = A × 3/2 = 441.500625 × 1.5 = 662.2509375 → divide by 2 = 331.12546875 → 331.13 Hz
  6. B = E × 3/2 = 331.12546875 × 1.5 = 496.688203125 → 496.69 Hz
  7. F: (special rule) go 2/3 below C (i.e., C × 2/3) to get the fifth downward. C × 2/3 = 261.63 × 0.6666667 = 174.42 → multiply by 2 to bring into the C octave = 348.84 Hz
  8. Upper C = C × 2 = 523.26 Hz

Conclusion (Short and to the Point — Verdict)

Pythagorean C scale (frequencies rounded to two decimals):

  • 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

Ruling on student proficiency (ACARA v9): The student correctly applied the 1:2 and 2:3 relationships, converted out-of-octave results by ×2 or ÷2 appropriately, and produced the Pythagorean C scale—proficient.

(Ally McBeal cadence: brief, slightly quirky aside — “Yes, we solved it. Harmony in ratios. Justice for strings.”)

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