Calculating the Pythagorean Scale — ACARA v9 Homeschool Report for a 13-year-old (Exemplary Outcome)

Clear, step‑by‑step ACARA v9 homeschool report (age 13) explaining how to compute the Pythagorean 7‑note C scale from a monochord. Includes corrected calculations, method (3:2 fifths and 2:3 inverse), and final frequencies with brief legal‑brief style analysis and Ally McBeal cadence.

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IN THE MATTER OF: Calculating the Pythagorean C Scale (Student age 13)

Issue. Did the student correctly apply Pythagoras' monochord ratios to produce the 7‑note Pythagorean C scale?

Facts. Middle C = 261.63 Hz. On a monochord, a string length ratio of 2:3 produces a pitch whose frequency is 3:2 times the original (frequency ∝ 1/length). Notes above the octave are halved (÷2) to fall inside the C octave; notes below are doubled (×2).

Analysis. (Short, precise, with a little spring — Ally McBeal cadence: oh! delightful symmetry.)

  1. Task 1a. Student answer 1:2 for halving the string. Correct. (Half the length = 1:2 of total.)
  2. Task 1b. Middle C 261.63 Hz halved in length → frequency doubles: 261.63 × 2 = 523.26 Hz. Student answer 523.26 Hz is correct.
  3. Task 1c. What happens to pitch when string length is halved? The frequency doubles, so pitch goes up by one octave. Student: "The pitch doubles." Correct (frequency doubles → octave).
  4. Question 2 (octave limits). A C scale sits between C and the next C (1:2 frequency ratio). Using middle C = 261.63 Hz, the lower limit (one octave below) is 261.63 ÷ 2 = 130.815 Hz and the upper limit is 261.63 × 2 = 523.26 Hz. Student gave the correct idea (they listed 130.315 which is a minor digit slip — correct lower limit is 130.815 Hz).)
  5. 2/3 split explanation. When the string is set to length 2/3 of original, the frequency becomes 3/2 times the original frequency (because frequency ∝ 1/length). So splitting middle C's string to 2/3 length yields: 261.63 × 3/2 = 392.445 Hz (G). Student computed 392.45 Hz — good rounding.

Core method for the Pythagorean scale (clear rule):

  • To go up a perfect fifth (the Pythagorean step): multiply the current frequency by 3/2.
  • If that result is above the C octave (>523.26 Hz), divide by 2 to bring it down one octave.
  • To go down a perfect fifth: multiply by 2/3 (or find the note 2/3 of the string length); then if that is below the C octave, multiply by 2.
  • Follow the given worksheet order: starting at C, find G (a fifth above), then D, A, E, B by repeated 3/2 steps; find F by taking 2/3 of C and bringing it up into the octave if needed.

Step‑by‑step calculations (showing arithmetic and rounding to two decimal places):

C     = 261.63 Hz (given)
G     = C × 3/2 = 261.63 × 1.5 = 392.445 → 392.45 Hz
D     = G × 3/2 = 392.445 × 1.5 = 588.6675 → ÷2 = 294.33375 → 294.33 Hz
A     = D × 3/2 = 294.33375 × 1.5 = 441.500625 → 441.50 Hz
E     = A × 3/2 = 441.500625 × 1.5 = 662.2509375 → ÷2 = 331.12546875 → 331.13 Hz
B     = E × 3/2 = 331.12546875 × 1.5 = 496.688203125 → 496.69 Hz
F (special instruction) = C × 2/3 = 261.63 × 0.6666667 = 174.42 → ×2 = 348.84 Hz
Upper C = C × 2 = 523.26 Hz

Final Pythagorean C scale (ascending, rounded to two decimal places):

  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 (upper) — 523.26 Hz

Assessment / Conclusion. The student demonstrated understanding of the core idea: halving length doubles frequency (octave) and using 2/3 (string length) leads to a 3/2 frequency ratio (perfect fifth). The final scale the student produced matches these values closely; minor issues were present in the intermediate worksheet numbers (some arithmetic or place‑value slips like 39.945 which appears to be a miswritten or mis‑placed number). Overall — exemplary work: clear method, mostly accurate results, correct musical reasoning. (Score: exemplary — bravo!)

Filed respectfully in the classroom court of music and math — Ally McBeal would say: "It’s harmonically compelling."

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