Part 1 — Octave limits and splitting the string
Quick rule: an octave has a 1:2 frequency ratio. Middle C = 261.63 Hz.
- Lower and upper limits of a C octave: lower = 261.63 Hz (middle C), upper = 2 × 261.63 = 523.26 Hz.
- If a string is shortened to 2/3 of its original length, frequency increases by the reciprocal (3/2). So:
New frequency = 261.63 × 3/2 = 392.445 Hz → rounded = 392.45 Hz (this is the note G).
Part 2 — Build the Pythagorean C scale (method: multiply by 3/2, adjust by factors of 2 to stay inside the octave)
Start: C = 261.63 Hz. For each step multiply the previous frequency by 3/2. If result > 523.26 Hz, divide by 2 (bring it down an octave). If result < 261.63 Hz, multiply by 2 (bring it up an octave).
- C = 261.63 Hz
- G = C × 3/2 = 261.63 × 1.5 = 392.445 → 392.45 Hz
- D = G × 3/2 = 392.445 × 1.5 = 588.6675 → ÷2 → 294.33375 → 294.33 Hz
- A = D × 3/2 = 294.33375 × 1.5 = 441.500625 → 441.50 Hz
- E = A × 3/2 = 441.500625 × 1.5 = 662.2509375 → ÷2 → 331.12546875 → 331.13 Hz
- B = E × 3/2 = 331.12546875 × 1.5 = 496.688203125 → 496.69 Hz
- F: find the note 2/3 below C: F = C × 2/3 = 261.63 × 0.666666... = 174.42 → ×2 to fit octave → 348.84 Hz
Pythagorean C Scale Frequencies (ascending)
C = 261.63 Hz
D = 294.33 Hz
E = 331.13 Hz
F = 348.84 Hz
G = 392.45 Hz
A = 441.50 Hz
B = 496.69 Hz
C = 523.26 Hz
Teacher comments (firm, clear — 100 words each)
Part 1 comment
Excellent work following instructions and showing your calculations clearly. You used the idea that an octave doubles frequency and correctly found the limits of the C octave. Your step of multiplying by 3/2 to split the string to two thirds was accurate, and you gave the new frequency to a sensible number of decimal places. Next time, label each arithmetic step so anyone can trace your thinking faster. Keep practicing multiplying and dividing fractions and relating length to pitch—these skills build strong musical and mathematical reasoning. I expect precise, neat calculations every time; try to improve rounding consistency, please continue.
Part 2 comment
You followed Pythagoras’s method carefully and produced a working C scale. Each frequency was found by multiplying by 3/2 or using 2/3 and then adjusting by factors of two to keep notes in the octave—excellent procedure. Your results match the Pythagorean tuning pattern and show clear understanding of multiplicative steps. For higher accuracy, present numbers rounded consistently to two decimal places and write the reason each time you divide or multiply by two. Practice explaining why some frequencies are halved while others are doubled; this will deepen your conceptual grasp. Keep up disciplined work and always show tidy, labeled steps.
Extended rubrics (each task)
Part 1 rubric
Exemplary: Correctly identifies octave limits (261.63 and 523.26 Hz), computes 2/3 split with clear arithmetic, shows understanding that shortening increases frequency, rounds consistently, and writes neat steps that can be followed without questions.
Proficient: Finds correct octave limits and computes the 2/3 split with minor rounding inconsistencies or one small omitted explanation about why frequency changes with string length. Steps are mostly clear.
Part 2 rubric
Exemplary: Produces all seven frequencies using multiply-by-3/2 and adjust-by-2 rules, keeps every result inside the C octave, rounds consistently, and explains why each ÷2 or ×2 was done. Demonstrates strong multiplicative reasoning.
Proficient: Correct frequencies found with one or two small rounding differences or a skipped explanation for an octave adjustment. Method is correct and results are in the right order.
ACARA v9 mapping
Mapped to ACARA v9: Mathematics — Number and Algebra (ratios, fractions, multiplicative strategies) and Measurement (relating length of string to frequency). This activity develops multiplicative reasoning, problem solving, and application to real-world contexts (sound and waves).
Final strict note:
Be precise, show each arithmetic step, and round consistently to two decimal places next time. You can do better — keep practicing!