What we learned (short & bright)
Ratios run rhythm and harmony: rhythm counts beats (1:2, 3:4, 5:8…), harmony compares frequencies. Small whole‑number ratios (1:2, 3:2, 3:4) sound stable; larger, complex ratios sound rougher. Pythagoras used simple ratios (mainly 3:2 and 2:1) on a monochord. You rebuilt that scale, simplified ratios, and used rounding rules to turn decimals back into fractions. Beautiful work.
ACARA v9 alignment
Mathematics — Ratios & proportional reasoning: representing and simplifying ratios, finding equivalents. The Arts — Music: exploring pitch, tuning and historical tuning systems (Pythagorean). General capabilities: numeracy, creative & critical thinking, communication.
Evidence of exemplary achievement (step‑by‑step)
Q1a: split in half → ratio 1:2.
Q1b: frequency doubles when length halved: 261.63 × 2 = 523.26 Hz (octave above).
Q1c: shorten vibrating length → vibrates faster → pitch rises; halving length → one octave up.
Octave limits: C = 261.63 Hz to high C = 523.26 Hz. 2/3 length gives frequency ×3/2: 261.63 × 1.5 = 392.445 Hz → G.
Recreated Pythagorean C scale (rounded to 3 dp): C 261.630, D 294.334, E 331.125, F 348.420, G 392.445, A 441.501, B 496.688, C 523.260 Hz.
Exact interval fractions from C: D = 9:8, E = 81:64, F = 4:3, G = 3:2, A = 27:16, B = 243:128, high C = 1:2.
Questions answered (concise reflections)
Liked: simple ratios (1:2, 2:3, 3:4, 8:9) — consonant because waveforms align. Disliked: complex ratios (16:27, 128:243) — more beating and roughness. Largest ratio pair: C and high C (1:2). Smallest steps: diatonic semitones (E–F, B–C). Ratio complexity correlates with perceived dissonance: simpler = more consonant.
Ratio word problems (answers)
1) 529.88 / 264.94 = 2 → exact octave → mathematically consonant and pleasing. 2) 220 & 330 have ratio 2:3; another pleasing partner with 330 is 495 Hz (330 × 3/2) or 220 Hz (330 × 2/3). 3) Tamya plays 32 evenly spaced notes as the 4 part of a 4:3 rhythm: one part = 32 ÷ 4 = 8 notes, Luis needs 3 parts = 3 × 8 = 24 notes.
Teacher comments — Ally McBeal cadence
Oh my God. You did this. You walked into a math problem disguised as a song and you fixed it with ratios. You were curious, precise, musical and brave. Exemplary. Accurate procedure, clear musical connection, and honest listening. Keep listening. Keep calculating. Keep composing. Oh my God — bravo.