Quick answers
Lower and upper limits for a C-scale built within one octave: 261.63 Hz (middle C) and 523.26 Hz (the C one octave above).
If a string playing middle C (261.63 Hz) is shortened to 2/3 of its length, the frequency becomes 3/2 times larger: 261.63 × 3/2 = 392.445 Hz (this is the note G).
How we build the Pythagorean scale (step-by-step)
Rule used: shortening the string to 2/3 of its length raises the pitch by a ratio of 3:2 (multiply frequency by 3/2). If a calculated frequency lies above the C-octave (above 523.26 Hz), divide by 2. If it lies below the octave, multiply by 2 to bring it between 261.63 and 523.26 Hz.
- C = 261.63 Hz (given)
- 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: Start from C and find the note 2/3 below (so frequency = C × 2/3), then bring into the octave: C × 2/3 = 261.63 × 0.666... = 174.42 Hz → ×2 to fit the octave → 348.84 Hz
Pythagorean C Scale Frequencies (in order)
- C = 261.63 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) = 523.26 Hz
Notes on rounding and temperament
These values come from pure Pythagorean (3:2 fifth) ratios. They differ slightly from modern equal-temperament frequencies (e.g., A = 440 Hz is a modern standard), but this method shows how Pythagoras built scales from simple number ratios.
Overall comments and evaluation (ACARA v9 — exemplary outcome)
Excellent, disciplined work. You followed the Pythagorean method precisely: you used the 3:2 (string 2/3) relationship, applied octave adjustments correctly, and kept clear arithmetic steps. Your answers show strong procedural fluency (accurate multiplication and sensible rounding) and conceptual understanding (why we divide or multiply by 2 to keep pitches within an octave). For an exemplary outcome under ACARA v9, you demonstrated: (1) clear mathematical reasoning, (2) careful recording of working steps, (3) correct application of musical-ratio concepts, and (4) tidy numerical answers with correct units (Hz). Keep pushing: label each intermediate step in future work (this makes checking easier), and compare your Pythagorean values with equal-tempered ones to deepen musical understanding. Overall: precise, methodical, and confidently handled — very well done.