Handout — Calculating the Pythagorean Scale
Pythagoras and his followers used a monochord (one string) and simple ratios to discover musical pitches. When you change the length of the vibrating string, the pitch changes. Use middle C = 261.63 Hz as the starting point.
Question 1 — Halving the string
- a. Ratio when the string is divided in half
Answer: 1:2 (one part is one half of the whole). In fraction form: 1/2.
- b. Frequency when the string is divided in half (middle C = 261.63 Hz)
Physics reminder: frequency is inversely proportional to string length. If length is halved, frequency doubles. So:
261.63 Hz × 2 = 523.26 Hz.
Answer: 523.26 Hz (this is the C one octave above middle C).
- c. In your own words — what happens to pitch when the string is divided in half?
Answer (student version): When the string is divided in half, the vibrating portion is shorter so it vibrates faster. The pitch goes higher — specifically it rises by one octave (the same note name, C, but higher).
Question 2 — Limits of a C scale (within one octave)
Scales live inside an octave. From your answer above, the lower and upper limits for building a C scale (one octave) are:
- Lower limit: middle C = 261.63 Hz
- Upper limit: one octave above C = 523.26 Hz
Question 3 — Splitting into 2/3 (finding a perfect fifth)
If the vibrating length is 2/3 of the original, frequency multiplies by 3/2 (because frequency ∝ 1/length). So:
261.63 Hz × (3/2) = 261.63 × 1.5 = 392.445 Hz
Answer: 392.445 Hz (this pitch is the note G above middle C).
Pythagorean Scale Construction — Step by step
Rule used: each new note is found by using the 2:3 length ratio (frequency × 3/2). After you calculate a frequency, if it lies outside the C octave (below 261.63 or above 523.26) bring it into the octave by multiplying or dividing by 2 as needed.
Start: C = 261.63 Hz
- Find G: 261.63 × (3/2) = 392.445 Hz (fits in octave)
G = 392.445 Hz
- Find D: 392.445 × (3/2) = 588.6675 Hz (above octave → divide by 2)
588.6675 ÷ 2 = 294.33375 Hz
D = 294.334 Hz (rounded to 3 decimals)
- Find A: 294.33375 × (3/2) = 441.500625 Hz (fits in octave)
A = 441.501 Hz
- Find E: 441.500625 × (3/2) = 662.2509375 Hz (above octave → ÷ 2)
662.2509375 ÷ 2 = 331.12546875 Hz
E = 331.125 Hz
- Find B: 331.12546875 × (3/2) = 496.688203125 Hz (fits in octave)
B = 496.688 Hz
- Find F: Instruction note — F is found by going down a fifth from C (that is, use 2/3 on C), not by continuing up from B. Compute from C:
C × (2/3) = 261.63 × 0.6666667 = 174.42 Hz (this is below our octave → multiply by 2)
174.42 × 2 = 348.84 Hz
F = 348.84 Hz
Final Pythagorean C scale (frequencies rounded to 3 decimals)
- C = 261.630 Hz
- D = 294.334 Hz
- E = 331.125 Hz
- F = 348.840 Hz
- G = 392.445 Hz
- A = 441.501 Hz
- B = 496.688 Hz
- C (octave) = 523.260 Hz
Student workspace (show your working on the printed sheet)
Write each multiplication and any ÷2 or ×2 octave adjustment beside the note box.
Teacher comments — firm, clear guidance ("Tiger Mother" cadence)
Task 1 (Q1 a–c): Calculating half-string octave
(Teacher comment — ~100 words)
If you got the ratio and frequency correct, fine. You must remember this: halving length doubles frequency. That rule is your anchor — memorize it. If your answer was 523.26 Hz, excellent: you understand octave relationships. If you wrote anything else, redo the calculation now and write it three times until you can do it without thinking. Explain in one clear sentence why the pitch rises when the vibrating length decreases. Use the terms "frequency" and "octave." Precision matters. Neat, labeled work earns full credit. Sloppy work gets redone.
Task 2 (C octave limits): Identifying the octave bounds
(Teacher comment — ~100 words)
You must always state the octave limits when building a scale. The lower bound is middle C (261.63 Hz), the upper bound is 523.26 Hz — memorize these reference points. If you were unsure, practice converting length ratios into frequency multipliers until it is automatic. Write both numbers, label them "lower" and "upper", and explain why notes getting too high or too low are shifted by factors of two. If you can justify the doubling or halving step in one crisp sentence, you are showing real understanding. Keep your calculations tidy and annotated.
Task 3 (2/3 split — perfect fifth)
(Teacher comment — ~100 words)
Good work if you computed 392.445 Hz and recognized this as a perfect fifth (G). This is the central operation behind the Pythagorean method: length 2/3 → frequency ×3/2. If your number is off, check whether you multiplied or divided by 3/2 incorrectly. Explain, in one short sentence, why 2/3 length becomes 3/2 frequency. Use the inverse relationship between length and frequency in your phrasing. Accurate arithmetic and correct rounding to three decimals is expected. If you made a rounding error, fix it and note where you rounded.
Task 4 (Full scale construction and octave adjustments)
(Teacher comment — ~100 words)
The full scale shows you can apply the 3/2 jump repeatedly and bring notes into the octave by ×2 or ÷2. If your final list matches the values above (within rounding), congratulations — you can follow the Pythagorean circle of fifths. If not, check each step: multiply by 3/2, then decide if the result sits inside 261.63–523.26. If it is above, divide by 2; if below, multiply by 2. Show each adjustment. Your final presentation should be ordered C→D→E→F→G→A→B→C and neatly labelled with units (Hz). Redo until flawless.
Extended Rubric (Exemplary and Proficient outcomes)
Criteria
- Accuracy of calculations (multiplications, octave adjustments)
- Understanding of the inverse length-frequency relationship
- Correct ordering and octave placement of notes
- Clarity and neatness of working steps
Exemplary
Student produces all frequencies correctly to at least three decimal places, shows clear stepwise work for each note (including every ×3/2 and any ×2 or ÷2 octave shifts), and writes a concise explanation using terms "frequency," "length," and "octave." The final scale is ordered correctly and annotated. The student can explain why the 2/3 length leads to a 3/2 frequency change and can generalize for other ratios. Presentation is neat, labeled, and ready to be displayed.
Proficient
Student calculates most frequencies correctly, makes one small arithmetic or rounding error but shows correct method. Work shows the use of ×3/2 and octave adjustments, but one or two steps may be lightly annotated. The student demonstrates understanding of inverse relationship between length and frequency and orders the scale correctly. With a brief correction of arithmetic or rounding, work would reach exemplary level.
ACARA v9 curriculum connections (high-level mapping)
Use these learning connections when aligning this lesson to Australian Curriculum v9 outcomes. (Teachers: confirm exact code numbers on the ACARA site for your year level.)
- Mathematics — Number and Algebra: Reasoning with ratios and proportional relationships; multiplicative thinking (use of ×3/2, ÷2 operations and converting ratios to multipliers).
- Science — Physical World / Waves: Understanding sound as vibration, frequency and pitch; how changes in physical parameters (string length) affect frequency.
- Technologies / The Arts (Music): Historical context of tuning systems; applying mathematics to musical tuning and acoustics.
Suggested teaching focus: explicitly link ratio arithmetic to real-world sound examples (play middle C and its Pythagorean G) and have students verify by ear where possible.
Note: All numerical values shown rounded to three decimal places where appropriate. Encourage students to keep at least four decimals in intermediate steps to minimize rounding drift, then round final answers for presentation.