Short explanation: Pythagoras found musical intervals by changing string length. If a string is shortened to 2/3 of its length, the frequency increases by 3/2 (because frequency ∝ 1/length). We build the scale inside one octave (between C and the next C = 2×C).
Octave limits: lower = 261.63 Hz (middle C), upper = 523.26 Hz (one octave above middle C).
If the string for middle C is split to 2/3 of its length:
frequency = 261.63 × 3/2 = 392.445 → 392.45 Hz (rounded).
Pythagorean chain of fifths (use ×3/2 for an upward fifth; adjust by dividing or multiplying by 2 to fit the C–C octave):
- q1. C → G: G = 261.63 × 3/2 = 392.445 → 392.45 Hz
- q2. G → D: raw D = 392.445 × 3/2 = 588.6675 → /2 to fit octave = 294.33375 → 294.33 Hz
- q3. D → A: A = 294.33375 × 3/2 = 441.500625 → 441.50 Hz
- q4. A → E: raw E = 441.500625 × 3/2 = 662.2509375 → /2 = 331.12546875 → 331.13 Hz
- q5. E → B: B = 331.12546875 × 3/2 = 496.688203125 → 496.69 Hz
- q6. C → F (a fifth below C): F = 261.63 × 2/3 = 174.42 → ×2 to bring into octave = 348.84 Hz (equivalently C × 4/3)
Pythagorean C scale frequencies (within one C–C octave):
- 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
Teacher comment / exemplary evaluation (ACARA v9, for a 13-year-old)
Exemplary work — a delightful, precise exploration of Pythagoras’s tuning that reads like a recipe executed with care. You correctly identified the octave limits (261.63 Hz to 523.26 Hz) and showed understanding that dividing a string to two‑thirds raises the pitch by a factor of 3/2 (middle C → 392.45 Hz). Your step‑by‑step chain of perfect fifths (C→G→D→A→E→B) uses the 3:2 ratio, applying octave adjustments (divide or multiply by 2) so every frequency fits within the C–C octave. Calculations are accurate and neatly rounded, and you correctly found F by taking a fifth below C then adjusting into the octave. This demonstrates strong number sense, proportional reasoning and attention to unit limits — all ACARA v9 achievement indicators for Years 7–8. To refine further: label whether each fifth was 'up' (×3/2) or 'down' (×2/3) and compare your Pythagorean pitch numbers to equal‑tempered ones to notice small tuning differences. For deeper inquiry, try plotting the frequencies or listening to audio examples to hear why Pythagorean tuning’s thirds sound different. Overall, your method, accuracy and clear notation would be an exemplar in a classroom—elegant, confident and thoroughly understandable. Keep documenting your steps; your neat work makes learning musical maths truly delicious indeed.