IN THE MATTER OF: Student (Age 13) — Pythagorean Scale
STATEMENT OF FACTS. Middle C = 261.63 Hz. Student identified octave relation 1:2 and computed C above = 523.26 Hz. (Bravo.)
ISSUE. Are the students Pythagorean calculations correct and exemplary under ACARA v9?
ANALYSIS. Law (ahem) of the monochord: frequency ∝ 1/(string length). Dividing the string to two-thirds length raises pitch by 3/2 (a perfect fifth). Thus the iterative rule to move up a fifth: f_new = f_prev * (3/2). To move down a fifth (for F) use f = f_C * (2/3) and if outside the octave multiply or divide by 2 to fit within C (261.63) to C (523.26).
CALCULATIONS (rounded to 2 d.p.).
- C = 261.63 Hz
- G = 261.63 * 3/2 = 392.45 Hz
- D = 392.45 * 3/2 = 588.67 → /2 = 294.33 Hz
- A = 294.33 * 3/2 = 441.50 Hz
- E = 441.50 * 3/2 = 662.25 → /2 = 331.13 Hz
- B = 331.13 * 3/2 = 496.69 Hz
- C (octave) = 523.26 Hz (double C)
- F (special rule) = 261.63 * 2/3 = 174.42 → *2 = 348.84 Hz
CONCLUSION / PARENT-TEACHER COMMENTS. Exemplary work: conceptually correct, method sound, only minor rounding differences (students values match within expected rounding). (Wow — impressive.) Keep stating the rule f_new = f_prev * 3/2 aloud when you calculate; it helps. Recommended: note when to divide or multiply by 2 to keep results inside the octave. Outstanding ACARA v9 outcome.