Quick idea (for a 10-year-old): Pythagoras used string lengths to make notes. If he made a string 2/3 as long, the sound it made was higher by a ratio of 3:2 (a perfect fifth). To build the Pythagorean scale inside the C-octave we keep every note between middle C (261.63 Hz) and the next C above it (523.26 Hz).
Part A — Octave limits
Middle C = 261.63 Hz.
- Lower limit (one octave below middle C): 261.63 ÷ 2 = 130.815 Hz
- Upper limit (one octave above middle C): 261.63 × 2 = 523.26 Hz
Evaluation: Student answer
Student wrote: "261.63hz ÷2 and 261.63hz x 2" — That is correct as an expression. Teacher note: write the numbers too: lower = 130.815 Hz, upper = 523.26 Hz.
Part B — If a string is split into 2/3
To find the frequency when the string is made 2/3 as long: frequency scales by the inverse, so multiply by 2/3 (if you want a lower note) or multiply by 3/2 (if you shorten the string to 2/3 of its length and want a higher note).
Student wrote: "261.63 ÷ 3 x 2" — that is the same as 261.63 × (2/3). Compute it:
261.63 × 2/3 = 174.42 Hz
Evaluation: Student answer
Operation is correct. Final value = 174.42 Hz. Teacher note: show the final decimal.
Part C — Build the Pythagorean C scale (step-by-step)
Rule used: to go from one note to the next a perfect fifth up = multiply frequency by 3/2. If a result is above the C-octave (above 523.26 Hz), divide by 2 to bring it down one octave. If a result is below the C-octave (below 261.63 Hz), multiply by 2 to bring it up.
- Start: C = 261.63 Hz
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C → G: multiply by 3/2 (perfect fifth up)
261.63 × 3/2 = 392.445 Hz
Teacher comment: student did an expression that was almost right but used 391.945 — correct value is 392.445 Hz. -
G → D: 392.445 × 3/2 = 588.6675 Hz. This is above the C octave, so divide by 2 to bring it down one octave:
588.6675 ÷ 2 = 294.33375 → round to 294.334 Hz (D) - D → A: 294.33375 × 3/2 = 441.500625 → round to 441.501 Hz (A)
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A → E: 441.500625 × 3/2 = 662.2509375 Hz → above the octave, divide by 2:
662.2509375 ÷ 2 = 331.12546875 → round to 331.125 Hz (E) - E → B: 331.12546875 × 3/2 = 496.688203125 → round to 496.688 Hz (B)
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Find F: the worksheet says "start from middle C and go 2/3 below C". That means multiply C by 2/3, then, because that value is below the C-octave, multiply by 2 to bring it back up into the C-octave.
261.63 × 2/3 = 174.42 Hz (below the octave) → × 2 = 348.84 Hz → F = 348.84 Hz
(Note: 348.84 Hz is the same as 261.63 × 4/3.)
Final Pythagorean C scale (frequencies rounded to 3 decimals)
- 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 (octave) = 523.260 Hz
Teacher comments and student evaluation summary
- Octave limits: student knew the correct operations (÷2 and ×2). Award full marks but remind to write numeric answers: 130.815 Hz and 523.26 Hz.
- Splitting into 2/3: student wrote the correct operation (261.63 ÷ 3 × 2). The computed answer should be 174.42 Hz — full credit for method.
- First fifth (C → G): student used (C + C/2) which is an equivalent idea, but their final number was 391.945 Hz — small arithmetic error. Correct value is 392.445 Hz. Encourage careful calculator checking and writing intermediate steps.
- Remaining notes: follow the multiply-by-3/2 rule and move octaves (× or ÷ 2) as needed. Final list above is correct for the Pythagorean tuning inside the C octave.
Suggested ACARA v9 curriculum links (topics to tick off)
- Music: exploring pitch, intervals and scales — understanding how ratios produce musical intervals (perfect fifth, perfect fourth, octave). (Link to the music strand: explore and respond to elements of music such as pitch and intervals.)
- Mathematics: ratios and proportional reasoning — using multiplication and division to work with ratios (3:2 and 2:3), and converting between operations and decimal values.
If you want, I can make a neat worksheet with blank boxes and these answers hidden so the student can fill them in and then check.