Quick idea: An octave is a 1:2 frequency ratio. If middle C is 261.63 Hz, the octave above is double that (523.26 Hz). Pythagoras built his scale by using 2:3 ratios between neighboring notes, which on frequency means multiplying by 3/2 (if you shorten the string to 2/3 its length, frequency becomes 3/2 times larger).
1) Lower and upper limits of a C octave
- Lower limit (middle C): 261.63 Hz
- Upper limit (one octave above): 261.63 × 2 = 523.26 Hz
2) If the string playing middle C is shortened to 2/3 of its length
Frequency scales inversely with string length, so f_new = f_old × (1 ÷ (2/3)) = f_old × 3/2.
Result: 261.63 × 3/2 = 392.445 Hz (this is G).
3) Pythagorean step-by-step (use ×3/2 each time, then move by octaves by ×/÷ 2 to keep within C to C)
- C = 261.63 Hz
- G = C × 3/2 = 261.63 × 1.5 = 392.445 Hz
- D = G × 3/2 = 392.445 × 1.5 = 588.6675 → divide by 2 to fit octave → 294.33375 ≈ 294.334 Hz
- A = D × 3/2 = 294.33375 × 1.5 = 441.500625 ≈ 441.501 Hz
- E = A × 3/2 = 441.500625 × 1.5 = 662.2509375 → divide by 2 → 331.12546875 ≈ 331.125 Hz
- B = E × 3/2 = 331.12546875 × 1.5 = 496.688203125 ≈ 496.688 Hz
- F: special rule — start from C and find the note 2/3 below, so F = C × 2/3 = 261.63 × 0.666666... = 174.42 Hz → bring into octave by ×2 → 348.84 Hz
- Upper C = 261.63 × 2 = 523.26 Hz
4) Final Pythagorean C Scale (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 = 523.260 Hz
Overall comments and evaluation (exemplary student outcome — ACARA v9 style, Nigella-Lawson cadence)
You have produced a careful, confident piece of work that shows real musical curiosity and mathematical precision. Your understanding that an octave is a 1:2 ratio is rock solid, and you’ve correctly used the inverse relationship between string length and frequency to change lengths into frequency changes — that is the crucial physics idea here. The repeated use of the 2/3 string-length step (which becomes a 3/2 frequency increase) is applied consistently, and you also correctly moved notes up or down by octaves (×2 or ÷2) so every pitch fits neatly between middle C and its octave above. Your final numbers are accurate and rounded sensibly for musical use.
To lift this from excellent to exemplary, you might: label which notes required octave adjustment and explain why (that tightens your reasoning), compare the Pythagorean frequencies to modern equal-tempered pitches to notice small tuning differences, and try producing the notes on an instrument or a tuning app to hear the Pythagorean character. Overall, you’ve demonstrated high-level work: clear procedure, correct calculations, and thoughtful presentation. Well done — keep following that warm curiosity and exactness; your maths and music sense are growing beautifully.