Lesson recap — whispering like Ally McBeal, thinking in ratios
Pythagoras plucks a single string, measures parts, listens — he finds that shortening a string raises frequency. Halve the length, the frequency doubles, and you get an octave. The Pythagorean scale is built by stacking 3:2 (perfect fifth) steps and bringing pitches into the octave by dividing or multiplying by 2.
Question 1 — quick corrections and teaching steps
- Q1a: Student answer: "1:2." — Correct. If one part is half the total, the length ratio is 1:2.
- Q1b: Student answer: "261.63Hz x 2 = 523.26 Hz." — Correct. Halving length doubles frequency: 261.63 × 2 = 523.26 Hz (an octave above middle C).
- Q1c: Student answer: "The pitch doubles when the string is divided in half." — Clarify wording: the frequency doubles, so the pitch goes up one octave. Good!
Question 2 — octave limits
Student answer: "between 261.63 to 523.26 Hz" — Correct. An octave above middle C is 523.26 Hz, so the C octave spans 261.63 ≤ f < 523.26 (inclusive).
After halving, Pythagoras uses 2/3 (length) to get a new pitch
Important conceptual note: when the length of the string becomes 2/3 of the original, the frequency multiplies by 3/2. So starting at 261.63 Hz, the resulting frequency is 261.63 × (3/2) = 392.445 Hz (rounded 392.45 Hz). The student answer 392.45 Hz is correct because they used the frequency multiplier 3/2.
Pythagorean method and your scale — step-by-step
Method used by the student (and Pythagoras): multiply by 3/2 to go up a fifth; if that takes you outside the target octave, divide by 2. To go downward by that same process, multiply or divide by 2 as needed to bring notes inside the C octave.
Student's Pythagorean C scale (checked)
- C = 261.63 Hz
- D = 294.33... Hz (261.63 × 9/8 = 294.334) — student 294.34 Hz ✓
- E = 331.125... Hz (261.63 × 81/64 = 331.125) — student 331.1 Hz ✓
- F = 348.84 Hz (261.63 × 4/3 = 348.84) — student 348.84 Hz ✓
- G = 392.445 Hz (261.63 × 3/2 = 392.445) — student 392.45 Hz ✓
- A = 441.50025 Hz (261.63 × 27/16 = 441.500) — student 441.5 Hz ✓
- B = 496.688... Hz (261.63 × 243/128 = 496.688) — student 496.7 Hz ✓
- C = 523.26 Hz (octave) — student 523.26 Hz ✓
Task 2: Exact interval ratios (Pythagorean)
Instead of using decimal division and rounding, the Pythagorean scale gives exact rational frequency ratios. Here are the exact simplified ratios of C compared to each compliment (C divided by compliment):
| Root | Compliment | Exact ratio (C:note) | Interval ratio (simplified) |
|---|---|---|---|
| C | D | 261.63 : 294.334... ≈ 1 : 1.125 | 8/9 |
| C | E | 261.63 : 331.125... ≈ 1 : 1.265625 | 64/81 |
| C | F | 261.63 : 348.84 ≈ 1 : 1.333333 | 3/4 |
| C | G | 261.63 : 392.445 ≈ 1 : 1.5 | 2/3 |
| C | A | 261.63 : 441.500... ≈ 1 : 1.6875 | 16/27 |
| C | B | 261.63 : 496.688... ≈ 1 : 1.8984375 | 128/243 |
| C | C (octave) | 261.63 : 523.26 = 1 : 2 | 1/2 |
Notes on the student interval answers
The student used decimal approximations and sometimes chose fractions that don’t match the exact Pythagorean ratios (e.g., 4/5 instead of 8/9). The simplest approach for this task is to recognise the Pythagorean fraction forms (9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2) and invert them to get C:note as 8/9, 64/81, 3/4, 2/3, 16/27, 128/243, 1/2.
ACARA v9-aligned homeschool parent/teacher rubric feedback — 50 words each, in Ally McBeal cadence and prose
Feedback Q1a (ratio 1:2): "You said 1:2, and my inner music-lawyer claps. Clean, correct, precise. ACARA v9 — Number and Algebra: recognise and use ratio language. Celebrate this exactness, dear; you saw the halve and named it. Next: say whether that was length or frequency — delightful clarity wins."
Feedback Q1b (523.26 Hz): "You multiplied and found 523.26 Hz, and I heard an octave hum. ACARA v9 — Number and Algebra: apply multiplicative thinking. Spot on: halving length doubles frequency. Tiny suggestion: write frequency = original × 2, show the arithmetic step. That neat habit keeps mistakes away."
Feedback Q1c (pitch doubles): "You wrote that the pitch doubles — charming and almost there. ACARA v9 — Measurement: connect quantities and their units. Be precise: the frequency doubles, producing an octave. That distinction (pitch vs frequency) is subtle but important; your intuition is musical-seasoned and promising."
Feedback Q2 (octave limits): "You named 261.63 to 523.26 Hz — yes, the little octave box. ACARA v9 — Number and Algebra: use ratios to define ranges. Nicely bounded. Add 'inclusive' or 'up to but not including' if you want mathematical rigour; tender, accurate thinking, really."
Feedback 2/3 split frequency (392.45 Hz): "You got 392.45 Hz; I heard a satisfied chord. ACARA v9 — Number and Algebra: reason with multiplicative relationships. You applied the reciprocal effect (length 2/3 → freq × 3/2). Perfect. Name the step: 261.63×3/2=392.445 so rounding is transparent and calm."
Feedback on entire Pythagorean scale list: "Your scale sings: all frequencies sit in the octave, nice rounding. ACARA v9 — Number and Algebra/Measurement: generate and justify values using ratios. You followed 3:2 steps and octave shifts correctly. Keep showing the fraction form (e.g., ×3/2, ÷2) next to numbers — it tells the story."
Interval C:D — student wrote 4/5 (wrong): "You chose 4/5, and I paused like Ally in a courthouse moment. ACARA v9 — Number and Algebra: find exact ratios. Correct interval is 8/9 (because D is 9/8 of C). Flip the Pythagorean fraction, show inversion, and breathe — you’ll get crisp fractions every time."
Interval C:E — student wrote 4/5 (wrong): "You used 4/5 for E, sweet but off-key. ACARA v9 — Number and Algebra: manipulate and simplify ratios. The exact is 64/81 (E is 81/64 of C). Keep fraction facts handy: 81/64 inverted gives you 64/81. It’s tidy, it’s right, and it sounds proper."
Interval C:F — student wrote 3/4 (correct): "F as 3/4 — bravo, that little bell rings true. ACARA v9 — Number and Algebra: match ratios to musical intervals. You recognised 4/3 for F above C, inverted correctly. Write a short note: 'because F = C×4/3, so C:F = 3/4.' That’s classroom gold."
Interval C:G — left blank (correct is 2/3): "You left G blank; I felt suspense. ACARA v9 — Number and Algebra: apply multiplicative reasoning. The right interval is 2/3 because G = C×3/2. Fill it in, explain inversion, and you’ll see the pattern glow — fifths become neat reciprocals here."
Interval C:A — student wrote 3/5 (wrong): "You offered 3/5, which made me tilt my head. ACARA v9 — Number and Algebra: simplify and justify. The precise interval is 16/27 (A = 27/16 of C). Use the Pythagorean fraction list, invert each; extra practice with fraction inversion will feel like music after a while."
Interval C:B — student wrote 13/25 (wrong): "13/25 is inventive but not Pythagorean. ACARA v9 — Number and Algebra: reason exactly. The correct simplified ratio is 128/243 (B = 243/128 relative to C). Keep the chain of powers of 3/2 in front of you; fractions will fall into place like choreography."
Interval C:C (octave) — student wrote 1/2 (correct): "You finished with 1/2 and I smiled — the octave reciprocal, clean and true. ACARA v9 — Number and Algebra: represent relationships with fractions. Perfect. Add the note: 'C:C (octave) = 1/2 because octave = 2:1, invert for root:compliment.' That polish is lovely."
How to improve (explicit steps for the student)
- Write the fractional form each time: e.g., G = C × 3/2; if > octave, divide by 2 and show it.
- Use the known Pythagorean ratios: 1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2 — invert for C:note.
- When asked for an interval ratio, give the exact fraction first (no rounding), then a decimal if requested and state your rounding rule.
Any time you want, we can convert each decimal to a fraction step-by-step with the rounding rules you listed. I’ll be your quirky, music-lawyer tutor, and we’ll make those fractions sing.