Pythagorean C-scale calculations for a 13-year-old — frequencies from Middle C (261.63 Hz) using repeated 2/3 ratios

Step-by-step Pythagorean scale for middle C (261.63 Hz): octave limits, splitting the string by 2/3, each note frequency (C–C) and a 200-word ACARA v9 evaluation in a playful Sailor Moon cadence.

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Octave limits
A C scale built within one octave must sit between middle C and the C one octave above. Lower limit = 261.63 Hz (middle C). Upper limit = 261.63 × 2 = 523.26 Hz (upper C).

Split middle C by 2/3
If the string is split so one piece is 2/3 of the original length, frequency changes by the reciprocal (3/2). So new frequency = 261.63 × 3/2 = 392.445 Hz (this is G).

Pythagorean procedure (multiply by 3/2 each step; if result is outside the C octave, divide or multiply by 2 to bring it back):

  1. C → G: 261.63 × 3/2 = 392.445 Hz → G = 392.445 Hz
  2. G → D: 392.445 × 3/2 = 588.6675 Hz → divide by 2 to fit octave → D = 294.334 Hz
  3. D → A: 294.334 × 3/2 = 441.5006 Hz → A = 441.501 Hz
  4. A → E: 441.5006 × 3/2 = 662.2509 Hz → divide by 2 → E = 331.125 Hz
  5. E → B: 331.125 × 3/2 = 496.6882 Hz → B = 496.688 Hz
  6. C ← F: For F we go 2/3 of C (C is 3/2 of F), so F = 261.63 × 2/3 = 174.42 Hz → multiply by 2 to fit octave → F = 348.84 Hz

Pythagorean C scale (within one octave, rounded):

  • C = 261.63 Hz
  • D = 294.334 Hz
  • E = 331.125 Hz
  • F = 348.84 Hz
  • G = 392.445 Hz
  • A = 441.501 Hz
  • B = 496.688 Hz
  • C = 523.26 Hz

Notes: Values differ slightly from modern equal temperament (e.g., A ≈ 440 Hz) because Pythagorean tuning uses pure 3:2 (perfect fifth) ratios. When a result lands outside the chosen octave, we shift it by factors of two to place it inside the C–C range.

200-word overall comments and evaluation (ACARA v9; Sailor Moon cadence)

Excellent work! This student demonstrates exemplary understanding of Pythagorean tuning and octave relationships. They correctly identified the C octave boundaries (261.63 Hz to 523.26 Hz) and used the 2:3 string-length ratio to generate the Pythagorean C scale, adjusting frequencies by factors of two to fit the octave. Calculations are accurate, clear and show a sound use of proportional reasoning and number sense expected in ACARA v9: applying multiplicative reasoning, scientific notation and measurement concepts (ACMNA176, AC9M7P02). The scale frequencies are consistent with historical tuning; small differences from modern equal temperament are explained by the method. For exemplary achievement, the student provided step-by-step work, labelled intermediate values and justified octave shifts. To extend learning, compare these frequencies to equal-tempered frequencies and graph the ratios; investigate how beating and consonance change. Communicate findings using clear annotations and include listening examples to connect auditory perception with numerical results. Keep the joyful curiosity—channel your inner Sailor Moon: defend harmony and precision, and let music guide your math! Overall: meets and extends ACARA expectations for Year 8–9 numeracy and music-math integration; assessment rating: Exemplary. Next steps: create ear-training exercises, document each ratio, and present results with diagrams, short audio clips and reflection for deeper learning.

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