Overview & octave limits
We start with middle C = 261.63 Hz. An octave is a 1:2 relationship, so the upper limit is 2 × 261.63 = 523.26 Hz. The C octave runs from 261.63 Hz (lower C) up to 523.26 Hz (upper C).
Split a string by 2/3 (resulting frequency)
When a string length becomes 2/3 of its original length, the frequency increases by the reciprocal factor 3/2. So:
New frequency = 261.63 × (3/2) = 261.63 × 1.5 = 392.445 Hz (round to 392.45 Hz). This is the pitch G.
Pythagorean procedure (multiply by 3/2 each step; if result is outside the C octave, divide or multiply by 2 to bring it between 261.63 and 523.26 Hz)
- Start: C = 261.63 Hz
- q1. C → G: 261.63 × 3/2 = 392.445 → G = 392.45 Hz
- q2. G → D: 392.445 × 3/2 = 588.6675; divide by 2 to fit the octave → D = 588.6675 / 2 = 294.33375 → 294.33 Hz
- q3. D → A: 294.33375 × 3/2 = 441.500625 → A = 441.50 Hz
- q4. A → E: 441.500625 × 3/2 = 662.2509375; divide by 2 → E = 662.2509375 / 2 = 331.12546875 → 331.13 Hz
- q5. E → B: 331.12546875 × 3/2 = 496.688203125 → B = 496.69 Hz
- q6. C → F (note: F is found as 2/3 of C, then moved into the octave): 261.63 × 2/3 = 174.42 Hz; multiply by 2 to raise into the C octave → F = 174.42 × 2 = 348.42 Hz
Pythagorean C Scale Frequencies (rounded to 2 decimal places)
- C = 261.63 Hz
- D = 294.33 Hz
- E = 331.13 Hz
- F = 348.42 Hz
- G = 392.45 Hz
- A = 441.50 Hz
- B = 496.69 Hz
- C = 523.26 Hz
Notes on rounding and tuning
These are Pythagorean frequencies derived from repeated 3:2 ratios. They differ slightly from equal-tempered concert tuning (A4 = 440 Hz) — for example A here is 441.50 Hz. For classroom work, round to two decimals and show the multiplication or halving step each time.
100-word evaluative comment (firm, disciplined cadence)
Excellent work. You followed the Pythagorean method precisely, showing clear calculations and careful octave adjustments. Your G (392.45 Hz) and subsequent notes are correctly derived using the 3:2 ratio, and you correctly halved or doubled frequencies to keep them inside the C–C octave (261.63–523.26 Hz). Your arithmetic is neat, units are consistent, and you noted why F is found by taking two-thirds of C then raising it into the octave. For further refinement, check rounding conventions (report to two decimal places) and label each step with multiplication/division by 3/2 or 2/3. Very good — keep disciplined practice. Stay focused daily. Always.