Pythagorean C Scale for a 10-year-old — Octave limits, splitting a string by 2/3, and full Pythagorean C scale (ACARA v9 mapped)

Clear, child-friendly explanation and step-by-step calculations showing octave limits for middle C, the frequency when a string is split to two-thirds, and the Pythagorean C scale frequencies. Includes teacher comments (100 words each) and extended rubrics (exemplary and proficient).

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Imagine music like a beautiful stew — simple ingredients (ratios) make delicious harmony. Pythagoras found that when a string is shortened to two-thirds of its length the pitch climbs by the ratio 3:2 — that’s how we get the note G from C.

Part 1 — Octave limits and splitting middle C

Given: middle C = 261.63 Hz. An octave is a 1:2 ratio (double the frequency).

  • Lower limit (starting C): 261.63 Hz
  • Upper limit (C one octave above): 261.63 × 2 = 523.26 Hz

If a string playing middle C is shortened to 2/3 of its length: frequency increases by the inverse of length ratio, so multiply by 3/2.

Result: 261.63 × 3/2 = 261.63 × 1.5 = 392.45 Hz. (This is the G above middle C.)

Part 2 — Build Pythagoras’s scale inside the C octave

Rule we use: to get the next note, take the previous note and multiply its frequency by 3/2. If that result is above the C octave (above 523.26 Hz), divide by 2 to bring it down into the octave. For the note F, start by taking two-thirds of C (so F is below C) then, if needed, multiply by 2 to fit it inside the same C–C octave.

Step-by-step calculations (keep the cooking rhythm):

  1. C = 261.63 Hz (given)
  2. G = C × 3/2 = 261.63 × 1.5 = 392.445 → 392.45 Hz (already inside the octave)
  3. D = G × 3/2 = 392.445 × 1.5 = 588.6675 → above octave, so divide by 2 → 294.33375 → 294.33 Hz
  4. A = D × 3/2 = 294.33375 × 1.5 = 441.500625 → 441.50 Hz
  5. E = A × 3/2 = 441.500625 × 1.5 = 662.2509375 → above octave, divide by 2 → 331.12546875 → 331.13 Hz
  6. B = E × 3/2 = 331.12546875 × 1.5 = 496.688203125 → 496.69 Hz
  7. F (special): start 2/3 below C → F = C × 2/3 = 261.63 × 0.6666667 = 174.42 → below octave, multiply by 2 → 348.84 Hz

Pythagorean C Scale Frequencies (rounded to 2 decimals):

  • C = 261.63 Hz
  • D = 294.33 Hz
  • E = 331.13 Hz
  • F = 348.84 Hz
  • G = 392.45 Hz
  • A = 441.50 Hz
  • B = 496.69 Hz
  • C = 523.26 Hz (octave)

Teacher comments (100 words each)

Part 1 — Teacher comment (100 words): Lovely work exploring octaves and ratios. You used middle C (261.63 Hz) as your anchor and correctly found the octave limits from 261.63 Hz up to 523.26 Hz. When you split the string to two-thirds you understood that the frequency increases by 3/2, giving about 392.45 Hz — well done, that is the G above middle C. Your step-by-step calculations were clear and showed understanding of how length and pitch are related. Next time, label each intermediate step with the multiplication or division so a reader can follow the arithmetic instantly. Keep experimenting with string lengths; you’re composing physics!

Part 2 — Teacher comment (100 words): You followed Pythagoras’s method with patience and curiosity to build a C scale inside the octave. Starting from middle C, you multiplied by 3/2 to get G (392.45 Hz), adjusted D and E into the octave by halving where needed, and found F by taking two-thirds of C then doubling it into the octave (348.84 Hz). Your final frequencies sit close to the modern notes and show how simple ratios create musical harmony. For clarity, present each step in a small table: original value, operation (×3/2 or ÷2), and result. This will make your beautiful reasoning easy for others.

Extended rubrics (per task) — ACARA v9 mapped: aligns with Mathematics (ratios and rates) and Science (waves and sound), Years 5–6 learning goals.

Part 1 Rubric

Exemplary: Student correctly states octave limits (261.63–523.26 Hz), explains why octave is a 1:2 ratio, correctly computes the result of shortening the string to 2/3 (261.63 × 3/2 = 392.45 Hz), and uses clear arithmetic notation. The explanation links string length and frequency with accurate inversely proportional reasoning and shows awareness of musical note names (identifies G).

Proficient: Student gives correct octave limits and computes the frequency after a 2/3 change with minor rounding differences. The student demonstrates the idea that shorter string → higher pitch and shows the multiplication step but may give a brief or partially worded explanation of the inverse relationship.

Part 2 Rubric

Exemplary: Student completes the chain of 3/2 ratios, adjusts results above or below the octave by dividing or multiplying by 2 as required, and provides all frequencies correctly rounded. Work is organised step-by-step so another student can follow the calculations; the student connects ratio work to the familiar note names and octave boundaries.

Proficient: Student finds most frequencies correctly and applies octave adjustments (÷2 or ×2) where necessary. Minor arithmetic or rounding errors may appear but the method is understood. Steps may be less fully explained, but results show correct use of the 3/2 and 2/3 relationships.

Note: The Pythagorean tuning here uses simple ratios and produces frequencies close to modern equal temperament notes; small differences are expected because modern tuning spreads intervals slightly differently.

Nicely done — you’ve cooked up a musical recipe using numbers. Keep tasting the ratios!

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