IN THE COURT OF HARMONY: A Pythagorean BRIEF (Ally McBeal cadence)
Case: The People v. Monochord — Can we build the C (middle C) Pythagorean scale using 2:3 length ratios (which give 3:2 frequency ratios)?
Facts
Pythagoras used a monochord (one string) and discovered that certain simple ratios of string length produce pleasant-sounding notes. Because frequency is inversely proportional to string length, a length ratio of 2:3 gives a frequency ratio of 3:2 (a perfect fifth). Middle C (the starting note) is 261.63 Hz.
Issue
How do we compute each note of the Pythagorean C scale (C–D–E–F–G–A–B–C) by repeatedly using the 3:2 frequency step (and moving notes by octaves as needed to keep all notes inside the C–C octave)?
Law & Key Concept (short and sweet)
- When you halve the string length, frequency doubles (1:2 length → 2:1 frequency). Example: C (261.63 Hz) → upper C (523.26 Hz).
- Pythagoras stacked perfect fifths (frequency × 3/2) to generate the scale. If the product goes above the C octave, divide by 2 (lower by one octave). If it falls below the C octave, multiply by 2 (raise by one octave).
Calculations — step by step (showing formula, raw result, and octave adjustment)
- C (given): 261.63 Hz (middle C). Upper C (octave above) = 261.63 × 2 = 523.26 Hz.
-
G (a perfect fifth above C):
Formula: G = C × 3/2
Calculation: 261.63 × 3/2 = 392.445 → round to 392.45 Hz -
D:
Formula: D = G × 3/2
Calculation: 392.445 × 3/2 = 588.6675. This is above the C–C octave, so divide by 2: 588.6675 ÷ 2 = 294.33375 → round to 294.33 Hz -
A:
Formula: A = D × 3/2
Calculation: 294.33375 × 3/2 = 441.500625 → round to 441.50 Hz -
E:
Formula: E = A × 3/2
Calculation: 441.500625 × 3/2 = 662.2509375. Above the octave → divide by 2: 662.2509375 ÷ 2 = 331.12546875 → round to 331.13 Hz -
B:
Formula: B = E × 3/2
Calculation: 331.12546875 × 3/2 = 496.688203125 → round to 496.69 Hz -
F (found by going a fifth below C or, equivalently, by taking the frequency 2/3 of C and bringing it into the C octave):
Option used: F = C × 2/3 (that gives the fifth below C), then raise by one octave (×2) because 2/3 × C is below the C–C octave.
Calculation: C × 2/3 = 261.63 × 2/3 = 174.42 Hz (below the octave) → raise by one octave: 174.42 × 2 = 348.84 Hz
Result — Pythagorean C scale (frequencies rounded to 2 decimal places)
- 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 (octave) = 523.26 Hz
Findings (what this teaches your students)
- Simple ratios produce musical intervals: 1:2 → octave, 2:3 (length) → 3:2 (frequency) → perfect fifth.
- To keep notes inside one octave, multiply or divide by 2 (moving up or down octaves as needed).
- This method builds the Pythagorean tuning system used in early Western music; modern equal temperament adjusts these slightly to spread tuning across 12 semitones.
Teacher notes & classroom hints
- Ask students to show each multiplication step and to explain why we divide or multiply by 2 when a frequency lies outside the C–C octave.
- Have students compare these Pythagorean frequencies with equal-tempered frequencies for the same notes and listen for differences (student activity: use a keyboard app that shows Hz).
- Use the monochord or a simple string demo to show how shortening the string raises pitch and halving length doubles frequency.
ACARA v9 alignment (summary for teachers)
- Number and Algebra: proportional reasoning and working with ratios and fractions — using real-world frequency and ratio calculations.
- Measurement: using units (Hz), converting across octaves (×2, ÷2) and applying scale factors.
- The Arts/Music connection: understanding tuning systems and how math describes sound.
Order (Conclusion)
By applying the 3:2 frequency step and moving by octaves where required, the Pythagorean C scale frequencies above are correct and demonstrate the mathematical-musical link Pythagoras discovered. Court adjourned (cue quirky inner monologue).