Scene: Ally McBeal meets Pythagoras — a little theatrical, but precise. We’ll walk step-by-step so you (age 13) can see the pattern, do the calculations, and fix the fraction confusions.
Lesson 1 — The monochord, halves and octaves
On a monochord, shorter string = higher pitch. If you divide the string in half, the pitch (frequency) doubles. So:
- Middle C (open string) = 261.63 Hz.
- If the string is half the original length, frequency = 261.63 × 2 = 523.26 Hz. That interval is an octave.
- In words: when you halve string length, frequency doubles and pitch goes up one octave.
How Pythagoras builds the scale (the 2/3 trick)
Pythagoras used the 2/3 length on the string. A length of 2/3 gives frequency multiply by 3/2 (because frequency ∝ 1/length). Stacking that 3/2 "fifth" up and bringing results into one octave (divide or multiply by 2 when needed) gives the Pythagorean C scale.
Start: C = 261.63 Hz. Use the known Pythagorean fractions for each note (these are exact fractions from stacking fifths):
- D = C × 9/8 = 261.63 × 1.125 = 294.36 Hz (rounded)
- E = C × 81/64 = 261.63 × 1.265625 = 331.13 Hz
- F = C × 4/3 = 261.63 × 1.333333... = 348.84 Hz
- G = C × 3/2 = 261.63 × 1.5 = 392.45 Hz
- A = C × 27/16 = 261.63 × 1.6875 = 441.51 Hz
- B = C × 243/128 = 261.63 × 1.8984375 = 496.70 Hz
- C (octave) = C × 2 = 523.26 Hz
So the Pythagorean C scale (rounded to two decimals) is:
C 261.63, D 294.36, E 331.13, F 348.84, G 392.45, A 441.51, B 496.70, C 523.26.
Lesson 2 — Interval ratios (C : other)
We want the ratio C : note, so divide 261.63 by each note frequency and simplify to a fraction (the exact Pythagorean fractions are rational because they come from 3/2 powers).
Exact simplified interval ratios (C : note) — these are the reciprocals of the fractions used above:
- C : C = 261.63 : 261.63 = 1 = 1/1
- C : D = 261.63 : 294.36 ≈ 0.888888... = 8/9
- C : E = 261.63 : 331.13 ≈ 0.79012345679 = 64/81
- C : F = 261.63 : 348.84 = 0.75 = 3/4
- C : G = 261.63 : 392.45 ≈ 0.6666667 = 2/3
- C : A = 261.63 : 441.51 ≈ 0.5925926 = 16/27
- C : B = 261.63 : 496.70 ≈ 0.52674897 = 128/243
- C : C (octave) = 261.63 : 523.26 = 0.5 = 1/2
Why those fractions?
Because Pythagorean tuning uses simple fractions made from powers of 3 and 2 (for example, 3/2, 9/8, 81/64, etc.). The note frequency = C × (Pythagorean fraction). The interval C:note is the reciprocal of that Pythagorean fraction.
Rounding rules — what they do to your fraction
Example: C:D = 261.63/294.36 ≈ 0.888888... (this is 8/9.)
- Rounding Rule 1 (do not round): keep 0.888888... → convert 0.888... to fraction → 8/9 (exact).
- Rounding Rule 2 (nearest tenth): 0.888... → 0.9 → fraction 9/10 (approximation).
- Rounding Rule 3 (nearest hundredth): 0.888... → 0.89 → fraction 89/100 (approximation).
- Rounding Rule 4 (truncate at thousandths, treat as repeating): 0.888 → treat as 0.888888... → 8/9 (recovers exact).
Conclusion: if you want the exact Pythagorean interval ratios, use the exact fractions (no rounding) or use rounding rule 4 appropriately; rounding to simple decimal places can produce different approximate fractions.
Corrections to the student answers
- D: correct interval ratio = 8/9 (not 4/5).
- E: correct interval ratio = 64/81 (not 4/5).
- G: correct interval ratio = 2/3 (student left blank; curriculum shows 2/3).
- A: correct interval ratio = 16/27 (not 3/5).
- B: correct interval ratio = 128/243 (not 13/25 or 53/100).
- F and octave were correct (3/4 and 1/2 respectively).
Tip for working: when you see a frequency that came from a known Pythagorean fraction (like 3/2 or 9/8), either keep the fraction form or use high-precision decimals before converting to a fraction. That avoids mismatched simplified fractions.
Quick practice you can try
- Start with C = 261.63. Multiply by 3/2 to get G; then multiply G by 3/2 and divide by 2 to get D; check those numbers against the fractions above.
- Take C and divide by each note frequency to get decimals, then convert the decimal to a fraction — compare with the reciprocal of the Pythagorean fractions.
Teacher / Homeschool feedback (200 words) — Ally McBeal cadence
Dear homeschool parent/teacher — in the delightful cadence of Ally McBeal, I watch Pythagoras twirl the monochord and whisper: well done for guiding your 13‑year‑old through ratios and sound. This task maps neatly to ACARA v9 outcomes: calculating the Pythagorean scale and interval ratios. Strengths: your student used the monochord method and produced sensible frequencies; their scale closely matches Pythagorean tuning. Areas to tighten: converting decimals back into simplified fractions — several answers used approximate fractions (4/5, 3/5) instead of the exact reciprocals of Pythagorean ratios (8/9, 16/27, 128/243). Teaching tips: show the student how each musical frequency comes from multiplying C by a simple Pythagorean fraction (3/2, 9/8, 81/64, etc.) and then take reciprocals to get C:note interval ratios. Use one clear rounding rule for the whole exercise — Rounding Rule 1 (no rounding) preserves exact fractions when possible. Assessment suggestion: a short quiz asking students to produce three frequencies from C using 3/2 steps and to simplify three decimal ratios into fractions will confirm mastery. Encouragement: celebrate the pattern recognition — it’s exactly what mathematicians and musicians love. Keep the learning playful, and let curiosity lead the calculations; report back with questions and triumphs. Bravo, and stay curious.