Calculating the Pythagorean Scale — Age 13: ACARA v9 Homeschool Exemplary Outcome (Concise Legal Brief with Ally McBeal Cadence)

Clear, step-by-step explanation and corrected calculations for the Pythagorean C scale using a monochord and 2/3 (3:2) ratios. Perfect for a 13‑year‑old student — shows where answers were correct, where small calculation or ratio mistakes occurred, and presents the final scale frequencies.

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Brief — Issue

Did the student correctly calculate the Pythagorean C scale, given middle C = 261.63 Hz and the Pythagorean method of using the 2/3 string division (which produces a frequency ratio of 3/2)?

Facts (fast, legal-style — and yes, a little Ally McBeal: "Oooh — rhythm!")

  • Monochord: changing string length changes frequency inversely. Halving the length doubles the frequency.
  • Middle C (C) = 261.63 Hz.
  • Pythagorean step: cutting the string so its length is 2/3 of the previous length raises the pitch by a frequency factor of 3/2 (a perfect fifth).
  • Keep every resulting pitch within the C–C octave (261.63 to 523.26 Hz) by multiplying or dividing by 2 as needed.

Law / Rule (short)

If new length = (2/3) × old length, then new frequency = (old frequency) × (1 / (2/3)) = (old frequency) × (3/2) = ×1.5. If a computed frequency lies above the octave, divide by 2; if below, multiply by 2.

Analysis — check the student answers and show correct math

  1. Task 1a — Ratio of one part to total when string is divided in half: student answered 1:2. Correct. (Halving length = length ratio 1:2.)
  2. Task 1b — Frequency when string is halved (middle C → octave): 261.63 Hz × 2 = 523.26 Hz. Student answered 523.26 Hz. Correct.
  3. Task 1c — What happens to pitch when string is halved? Student: "The pitch doubles." Clarification: the frequency doubles, so the pitch goes up one octave. Correct idea and wording for this age.
  4. Question 2 — Limits for a C scale (one octave): the usual C octave runs from 261.63 Hz (middle C) up to 523.26 Hz (the next C). The student also correctly noted the lower octave pair (130.815 to 261.63) — both statements are true; for the C scale we normally use 261.63–523.26 Hz.
  5. Finding a note when the string is 2/3 of the original length — student gave work that arrived at ~392.45 Hz. The clean explanation: if length is 2/3 of original, frequency = original × 3/2.
    So: 261.63 × 3/2 = 261.63 × 1.5 = 392.445 Hz (round ≈ 392.45 Hz). Student’s numerical result is fine; algebra in the student’s work was muddled but final number is essentially correct.

Constructing the Pythagorean C scale — step by step (clean math)

Start at C = 261.63 Hz. Use ×3/2 to go up a fifth; if the result leaves the C–C octave, divide by 2 to bring it back.

C261.63 Hz (given)
G = C × 3/2261.63 × 1.5 = 392.445 → 392.45 Hz
D = G × 3/2392.445 × 1.5 = 588.6675 → ÷2 to fit octave = 294.33375 → 294.33 Hz
A = D × 3/2294.33375 × 1.5 = 441.500625 → 441.50 Hz
E = A × 3/2441.500625 × 1.5 = 662.2509375 → ÷2 = 331.12546875 → 331.13 Hz
B = E × 3/2331.12546875 × 1.5 = 496.688203125 → 496.69 Hz
F (special rule) = C × 2/3 (length) → frequency = C × 3/2? Wait — clearer: to find F the worksheet said: "starting note for F is middle C and is 2/3 below C (not above)." That means the length giving F is 2/3 of C's length below (so frequency = C × 2/3? No — the safe direct calculation is below)Better: use the instruction: F is 2/3 below C in length: frequency = C × (1 / (2/3))? That would make it above — easier to apply the worksheet instruction exactly: the frequency 2/3 below C in length is C × 2/3 = 174.42 Hz, then bring up one octave (×2) to fit inside C octave: 174.42 × 2 = 348.84 Hz → F = 348.84 Hz
Top C261.63 × 2 = 523.26 Hz

Final Pythagorean C scale (rounded to 2 d.p.)

  1. C — 261.63 Hz
  2. D — 294.33 Hz
  3. E — 331.13 Hz
  4. F — 348.84 Hz
  5. G — 392.45 Hz
  6. A — 441.50 Hz
  7. B — 496.69 Hz
  8. C — 523.26 Hz

Conclusion — verdict (brief, stylish)

The student demonstrated correct understanding of the ratio concepts and most final numbers are close. Minor arithmetic rounding and a little symbolic confusion appeared in intermediate work (the 2/3 algebra was written awkwardly), but the final scale the student wrote out essentially matches the correct Pythagorean values above (I corrected tiny rounding differences and explained the F calculation). Exemplary outcome: conceptually sound and practically accurate — well done. (Cue the soft chime — Ally would be proud.)

If you want, I can show the same scale with fractions/ratios instead of decimals, or show how these Pythagorean frequencies compare with equal temperament (modern tuning) — quick, neat, and musical.

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