13-year-old Pythagorean C Scale Frequencies — octave limits, split-by-2/3 result, and full Pythagorean C scale (ACARA v9, Sailor Moon cadence)

Step-by-step Pythagorean scale calculations for a 13-year-old: octave limits for middle C, the frequency when a string is split to 2/3, and the full Pythagorean C scale frequencies with clear working and ACARA v9-aligned evaluation.

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Octave limits for a C scale (using middle C = 261.63 Hz)
Lower limit: 261.63 Hz (middle C). Upper limit: 2 × 261.63 = 523.26 Hz (the C one octave above).

If the string playing middle C is split to 2/3 of its length:
Frequency increases by the inverse of the length change, so multiply by 3/2.
261.63 × 3/2 = 392.445 Hz (this is G).

Pythagorean method (each new note is a 3:2 frequency ratio above the previous note; if result is above the octave, divide by 2; for F use 2/3 below C then multiply by 2 to fit the octave):

  • Start: C = 261.63 Hz
  • q1. C → G: 261.63 × 3/2 = 392.445 Hz
  • q2. G → D: 392.445 × 3/2 = 588.6675 → ÷2 = 294.33375 Hz
  • q3. D → A: 294.33375 × 3/2 = 441.500625 Hz
  • q4. A → E: 441.500625 × 3/2 = 662.2509375 → ÷2 = 331.12546875 Hz
  • q5. E → B: 331.12546875 × 3/2 = 496.688203125 Hz
  • q6. C → F (special rule): C × 2/3 = 174.42 Hz → ×2 = 348.84 Hz (fits the C octave)

Pythagorean C Scale Frequencies (rounded to 3 decimal places):
C = 261.630 Hz
D = 294.334 Hz
E = 331.125 Hz
F = 348.840 Hz
G = 392.445 Hz
A = 441.501 Hz
B = 496.688 Hz
C = 523.260 Hz

Overall comments and evaluation (ACARA v9, Sailor Moon cadence — exemplary student outcome, 200 words):

Radiant work! You methodically used Pythagoras’s 2:3 string rule to build the C octave and showed excellent numerical reasoning. Your calculations are accurate (C = 261.63 Hz; G = 392.445 Hz; D = 294.334 Hz; A = 441.501 Hz; E = 331.125 Hz; B = 496.688 Hz; F = 348.840 Hz; upper C = 523.260 Hz) and your use of halving or doubling to keep notes inside the C octave shows clear understanding of octave constraints. You demonstrated multiplicative thinking and unit awareness when converting string ratios to frequency changes, matching ACARA v9 outcomes for Years 7–8 in Number and Algebra (ratio and proportion) and v9 Science (waves and their properties). To reach exemplary standard you: showed every intermediate step with labels (good), explained why frequencies are halved/doubled (excellent), and checked your results against known pitch values (strong practice). For extension, graph the frequencies on a number line to visualise spacing, and compare this Pythagorean scale to equal temperament to hear differences. As a next step, describe in one sentence how the 3:2 ratio produces consonance. Overall, this work is precise, well-organised, and demonstrates deep conceptual understanding — shine on, soldier of music! Keep exploring sound and ratios with curiosity.

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