Octave limits
Middle C = 261.63 Hz. An octave is a 1:2 ratio, so the C octave runs from 261.63 Hz up to 261.63 × 2 = 523.26 Hz. Lower limit = 261.63 Hz, upper limit = 523.26 Hz.
Split the string to 2/3
Frequency is inversely proportional to string length. If length becomes 2/3 of the original, frequency multiplies by 3/2.
261.63 × (3/2) = 392.445 Hz (this is G in the C octave).
Pythagorean chain of fifths (step-by-step)
Rule used: new frequency = previous frequency × 3/2. If result is above the C octave, divide by 2. If below the C octave, multiply by 2.
- C = 261.63 Hz (given)
- G = C × 3/2 = 261.63 × 1.5 = 392.445 Hz (within octave)
- D = G × 3/2 = 392.445 × 1.5 = 588.6675 → divide by 2 → 294.33375 Hz
- A = D × 3/2 = 294.33375 × 1.5 = 441.500625 Hz
- E = A × 3/2 = 441.500625 × 1.5 = 662.2509375 → divide by 2 → 331.12546875 Hz
- B = E × 3/2 = 331.12546875 × 1.5 = 496.688203125 Hz
- F: start from C and go down a fifth (multiply by 2/3): F = C × 2/3 = 261.63 × 0.6666667 = 174.42 → multiply by 2 to fit octave → 348.84 Hz
- Top C = 261.63 × 2 = 523.26 Hz
Pythagorean C scale (frequencies rounded)
- 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 = 523.260 Hz
300-word overall comments and evaluation (exemplary student outcome)
You have completed the Pythagorean scale task with excellent technique, clear calculations and careful octave checking. You correctly identified the C octave limits (261.63 Hz to 523.26 Hz) and applied the 3:2 frequency multiplier for each successive fifth. Your G result (392.445 Hz) shows you understand the relationship between string length and frequency. When notes rose above the octave, you divided by two; when they fell below, you doubled them — that shows strong attention to the octave window rule. Your arithmetic was precise and your rounding was sensible for classroom work.
To reach exemplary mastery, continue to: (1) label each intermediate step clearly (write the raw product before dividing or multiplying by 2), (2) compare Pythagorean frequencies to equal-tempered values to understand musical differences, and (3) check units and significant figures each time. You demonstrated persistence and accuracy — exactly what the curriculum expects. For extension work, try generating the full circle of fifths and locating the wolf interval that appears when you cycle twelve 3:2 steps, or plot these frequencies on a logarithmic scale to see equal spacing for equal-temperament versus Pythagorean spacing.
Overall: exemplary. You followed instructions, used correct ratios, adjusted for octave placement, and produced a correct Pythagorean C scale. Keep up this disciplined work — aim for neat annotations and a short reflection showing why some Pythagorean notes differ slightly from modern tuning.