Hey Moon Prism Power — let’s make math sing! Pythagoras worked with string-length ratios (2:3). Because frequency is inversely proportional to string length, a length ratio of 2:3 gives a frequency ratio of 3:2 (a perfect fifth). We use that 3/2 multiplier repeatedly, then shift notes into the C octave by dividing or multiplying by 2 when needed.
Octave limits (Question 1):
Lower limit = middle C = 261.63 Hz. Upper limit = 2 × 261.63 = 523.26 Hz.
Split string into 2/3 (Question):
If a string is shortened to 2/3 of its length, frequency increases by 3/2. So middle C (261.63 Hz) → G = 261.63 × 3/2 = 392.445 Hz.
How we build the Pythagorean notes (rules used):
Multiply the previous note by 3/2. If result is above 523.26 Hz, divide by 2 to bring it into the C octave. If result is below 261.63 Hz, multiply by 2.
Worksheet answers (calculations shown):
- q1. C → G: 261.63 × 3/2 = 392.445 Hz
- q2. G → D: 392.445 × 3/2 = 588.6675 → ÷2 = 294.33375 Hz
- q3. D → A: 294.33375 × 3/2 = 441.500625 Hz
- q4. A → E: 441.500625 × 3/2 = 662.2509375 → ÷2 = 331.12546875 Hz
- q5. E → B: 331.12546875 × 3/2 = 496.688203125 Hz
- q6. C → F: start from C and use 2/3 string-length (frequency × 4/3): 261.63 × 4/3 = 348.84 Hz
Pythagorean C Scale Frequencies (ascending):
C = 261.63 Hz
D = 294.33 Hz (9/8 × C)
E = 331.13 Hz (81/64 × C)
F = 348.84 Hz (4/3 × C)
G = 392.45 Hz (3/2 × C)
A = 441.50 Hz (27/16 × C)
B = 496.69 Hz (243/128 × C)
C = 523.26 Hz (2 × C)
300-word evaluation and overall comments (exemplary outcome)
Excellent work! You followed Pythagoras’ method with care, showing both the musical idea and the arithmetic clearly. You understood that the 2:3 string-length relationship becomes a 3:2 frequency ratio and used it correctly to generate perfect fifths. Your octave limits are correct: the C octave runs from 261.63 Hz up to 523.26 Hz. You demonstrated consistent use of multiplying by 3/2 and then adjusting by factors of 2 to keep notes inside the octave — that shows strong conceptual understanding of both ratios and inverse relationships.
Strengths: accurate arithmetic with fractions and decimals, neat octave-adjustments (divide/multiply by 2), and correct labeling of each note. You used exact fractional ratios implicitly (for example 9/8, 27/16, 81/64, 243/128, 4/3) which connects ancient tuning ideas to modern numbers. Your step-by-step work would be easy for a peer to follow and reproduce.
Next steps and stretch: try comparing these Pythagorean frequencies with equal temperament (A = 440 Hz) to hear the slight pitch differences. Plotting the frequencies on a number line or making a simple audio synthesis will deepen your intuition. Also try building the same scale from a different tonic (like G) to see the same ratios reappear.
ACARA v9 skills demonstrated: ratio and proportion, fraction & decimal computation, inverse relationships and mathematical reasoning. This is an exemplary outcome — accurate, reflective, and ready for creative extension. Keep shining, sailor of sound and numbers!