In the name of the Moon — Calculating the Pythagorean Scale

(Student-facing printable — age 14. Read like a calm Sailor Moon chant: bright, rhythmic, and clear.)

Moon power, math power! We use a monochord (one string on wood) and ratios to make the Pythagorean 7-note scale. Follow the steps below and show your work.

Quick facts to remember

• Frequency is inversely proportional to string length. If string length is halved, frequency doubles.
• An octave is a 1:2 frequency ratio (same note name, higher pitch).
• Pythagoras used a 2:3 ratio (lengths 2:3) to move from one note to the next (this produces musical fifths).

Question 1 (Monochord halved)

1. a) What is the ratio of one of the parts of the string to the total number of parts when the string is divided in half?

   Answer: 1:2 (or 1/2). One part is half the whole string.

2. b) If the open string frequency is middle C = 261.63 Hz, what is the frequency when the string is divided in half (1:2)?

   Work: Halving the string doubles the frequency, so new frequency = 2 × 261.63 = 523.26 Hz.

   Answer: 523.26 Hz (the octave above middle C).

3. c) In your own words, what happens to the pitch when the string is divided in half?

   Answer (student style): The pitch goes higher; specifically, when the string length is halved the frequency doubles and the sound moves up one octave.

Question 2 (Limits of a C scale)

Scales live inside an octave. Based on Question 1, if middle C is 261.63 Hz then the upper limit (one octave above) is 523.26 Hz. So a C scale built around middle C has lower limit 261.63 Hz and upper limit 523.26 Hz.

Question 3 (Split into 2/3)

If the string length becomes 2/3 of the original, frequency multiplies by the reciprocal (3/2). So for middle C (261.63 Hz):

Frequency = 261.63 × (3/2) = 261.63 × 1.5 = 392.445 Hz. Rounded: 392.45 Hz.

Answer: 392.45 Hz.

Pythagorean Scale — method and workspace

Rules (follow these exactly):

1. Begin with middle C = 261.63 Hz.
2. To find the next note (G), find the value that makes a 2:3 ratio with C (C/G = 2/3), so G = (3/2) × C.
3. For each new note, make the previous note be 2/3 of the next note (multiply previous by 3/2). If the result is above the octave (above 523.26 Hz), divide by 2. If it's below the octave (below 261.63 Hz), multiply by 2.
4. For F specifically: F is 2/3 below C, so F = (2/3) × C; if that is below the octave multiply by 2 to bring it into the C octave.

Step-by-step calculations (workspace)

• C = 261.63 Hz
• G = (3/2) × C = 1.5 × 261.63 = 392.445 → 392.45 Hz
• D = (3/2) × G = 1.5 × 392.445 = 588.6675 → above octave → divide by 2 → 294.33375 → 294.33 Hz
• A = (3/2) × D = 1.5 × 294.33375 = 441.500625 → 441.50 Hz
• E = (3/2) × A = 1.5 × 441.500625 = 662.2509375 → above octave → divide by 2 → 331.12546875 → 331.13 Hz
• B = (3/2) × E = 1.5 × 331.12546875 = 496.688203125 → 496.69 Hz
• F: compute from C as 2/3 below C → F = (2/3) × C = 0.6666667 × 261.63 = 174.42 Hz → below octave → multiply by 2 → 348.84 Hz

Pythagorean C Scale Frequencies (ordered)

• 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

Note: These are the Pythagorean tuning values produced by repeated 3:2 ratios adjusted into the C octave by multiplying or dividing by 2 as needed. You can listen to a tuner or app to hear how these compare to equal temperament.

Teacher support: ACARA v9 mapping and assessment

Mapped learning connections (ACARA v9 style):

• Mathematics — Number & Algebra: Use ratio, proportion and multiplicative reasoning to solve problems (Year 8–9 level connections).
• Mathematics — Measurement and Geometry: Understand relationships between physical length and frequency in wave phenomena (cross-curricular link to science).
• The Arts (Music): Understand pitch relationships, intervals and scales; historical tuning systems (Pythagorean tuning) — Years 7–9.

Use these links to justify the task: it practices multiplicative reasoning, unit handling (Hz), and connects maths to music and history.

Extended rubric (quick guide)

Criteria: Calculations and accuracy, Octave adjustments, Notation & ordering, Reasoning & explanation, Communication.

Exemplary

• All frequencies calculated correctly with clear step-by-step working and correct use of 3:2 and 1:2 ratios.
• Octave adjustments (×2 or ÷2) applied consistently and correctly so every note fits inside the C octave.
• Notes are correctly ordered and labelled; answers rounded appropriately and units (Hz) shown.
• Student explains why frequency changes (inverse relation to length) and compares Pythagorean tuning to equal temperament.
• Work is neatly presented and could be followed by another student.

Proficient

• Most frequencies are correct; minor arithmetic or rounding errors do not change octave placements.
• Octave adjustments are usually correct, with only occasional oversight.
• Notes are in correct order; units used but some steps may be omitted or brief.
• Student shows understanding of the length-frequency relationship; explanation may be short or missing a connection to tuning systems.
• Presentation is readable and method generally clear.

Teacher comments (100 words each)

Comment for the student task (100 words)

Your work on the Pythagorean scale shows a strong grasp of ratio and how string length affects pitch. You correctly found that halving the string raises frequency to 523.26 Hz and that using a 2:3 ratio yields G at 392.45 Hz. Your step-by-step calculations are clear and your octave adjustments (dividing or multiplying by two) were applied accurately. Next steps: label the notes on a staff or use a tuner app to hear the differences, and compare Pythagorean intervals to equal temperament to deepen musical understanding. Keep showing your reasoning and checking units; excellent curiosity and care! And keep exploring.

Comment for the rubric/task support (100 words)

This rubric and ACARA v9 mapping clearly link mathematics and music learning outcomes. The exemplary descriptor shows mastery: precise calculations, accurate octave adjustments, clear notation, and thoughtful reflection on tuning differences. The proficient descriptor shows solid understanding with minor arithmetic or rounding errors and clear methods. Suggestions for extension: ask students to use a monochord simulator or a keyboard tuner to verify frequencies, graph frequency vs. step number, and research historical context for Pythagorean tuning. Provide feedback that praises correct reasoning, corrects procedural slips, and prompts comparison to equal temperament. Well-structured and useful for assessing conceptual and procedural skills. Explicitly.

Short activity idea: Ask students to sing or play the computed frequencies (or use a virtual keyboard) and note which intervals sound most 'pure' versus slightly different from modern tuned instruments. Sailor Moon cadence: "Together we calculate, together we hear — math and music, loud and clear!"