Handout — Calculating the Pythagorean Scale

Read this like a little recipe: warm, precise and rather delicious — the music of numbers.

Background (short)

Pythagoras and his friends used a monochord (one string) to explore how dividing string length changes pitch. The pitch heard depends on the ratio of the divided length to the original. If the string length is halved (1:2), the pitch rises by an octave. If the string length is 2/3 of the original, the pitch rises by a 3:2 frequency ratio — a consonant fifth.

Question 1

1. a. Look at the monochord above. What is the ratio of one of the parts of the string to the total number of parts?

   Answer: If the string is divided in half, the ratio of one part to the whole is 1:2 (or 1/2).

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

   Reason: Frequency is inversely proportional to length. Halving the length doubles the frequency.

   Calculation: 261.63 Hz × 2 = 523.26 Hz.

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

   Answer: The pitch rises by an octave — it sounds like the same note but higher; specifically, the frequency doubles.

Question 2 — Octave limits

Scales are built within an octave, so pitches must fall between a note and its 1:2 partner.

Based on Question 1, the lower and upper limits of a C scale are:

• Lower limit (C): 261.63 Hz
• Upper limit (C an octave above): 523.26 Hz

Next: 2/3 split (creating a perfect fifth)

If the string length becomes 2/3 of the original, the frequency is multiplied by 3/2 (because frequency ∝ 1/length).

Starting from Middle C (261.63 Hz):

Calculation: 261.63 × (3/2) = 261.63 × 1.5 = 392.45 Hz (rounded to two decimal places). This note is G.

Pythagorean Scale Construction — Rules

We will build the 7 notes using the 3:2 frequency multiplier repeatedly. After each multiplication, if the result lies above the C-octave (above 523.26 Hz), divide it by 2 (drop an octave). If it's below the C-octave (below 261.63 Hz), multiply by 2 (raise an octave).

1. C → G: multiply C by 3/2

   261.63 × 1.5 = 392.445 → G = 392.45 Hz (rounded)

2. G → D: multiply G by 1.5

   392.445 × 1.5 = 588.6675 → above octave, so divide by 2 → 294.33375 → D = 294.33 Hz

3. D → A: multiply D by 1.5

   294.33375 × 1.5 = 441.500625 → A = 441.50 Hz

4. A → E: multiply A by 1.5

   441.500625 × 1.5 = 662.2509375 → above octave, divide by 2 → 331.12546875 → E = 331.13 Hz

5. E → B: multiply E by 1.5

   331.12546875 × 1.5 = 496.688203125 → B = 496.69 Hz

6. To find F: start from Middle C and take the frequency that is 2/3 below C (so multiply C by 2/3), then if needed raise an octave

   C × (2/3) = 261.63 × 0.6666667 = 174.42 → below octave, so ×2 to bring into C octave → F = 348.42 Hz

Pythagorean C Scale Frequencies (rounded to two decimals)

• C = 261.63 Hz
• D = 294.33 Hz
• E = 331.13 Hz
• F = 348.42 Hz
• G = 392.45 Hz
• A = 441.50 Hz
• B = 496.69 Hz
• C = 523.26 Hz

Workspace (for your calculations)

(Write your steps here when you do this on paper: show each multiplication by 3/2, and every time you divide or multiply by 2 to place the note in the C octave.)

C = 261.63 Hz
G = C × 3/2 = ______ Hz
D = G × 3/2 (adjust octave) = ______ Hz
A = D × 3/2 = ______ Hz
E = A × 3/2 (adjust octave) = ______ Hz
B = E × 3/2 = ______ Hz
F = C × 2/3 (adjust octave) = ______ Hz

Teacher feedback — Task 1 (100 words, Nigella Lawson cadence)

Dear musician-chef, your calculations for the Pythagorean scale are a delicious blend of curiosity and precision. You neatly found the octave and understood that halving the string doubles the frequency — marvelous. Your work on the 2:3 ratios showed clear procedure and sensible octave shifting; the arithmetic is tidy and musical. Next time, label each step and show the multiplication or division explicitly so a reader can follow your feast of numbers. Try rounding consistently to two decimal places for clarity. With a touch more explanation of why we divide or multiply by two, your understanding will sing. Well done, chef.

Teacher feedback — Task 2 (100 words, Nigella Lawson cadence)

You have crafted a thoughtful rubric and mapped learning outcomes with warmth and clarity. The exemplary descriptors shimmer: precise calculations, clear justification, and musical insight. The proficient standards are generous and achievable: correct methods, sensible rounding, and demonstration of octave shifting. Consider adding a short checklist for students — show method, show arithmetic, adjust octave, and interpret pitch. Anchor each outcome to the relevant curriculum strand so teachers can quickly see alignment. Offer one or two extension tasks for advanced students, such as comparing Pythagorean and equal temperament frequencies. This will make your assessment a beautifully nourishing lesson. Bravo and merci.

Extended Rubric (ACARA v9-aligned guidance)

Mapped learning areas: Mathematics (Number and Algebra: ratios, proportional reasoning and multiplicative thinking) and The Arts — Music (understanding pitch, intervals, and scales). Use this rubric to judge student work on the worksheet.

Criteria

• Accuracy of calculations (multiplying by 3/2, dividing/multiplying by 2 to fit octave)
• Method & reasoning (shows step-by-step arithmetic and explains octave adjustments)
• Interpretation (explains what happens to pitch and identifies interval names, e.g. octave, fifth)
• Presentation (clear labels, consistent rounding, readable workspace)

Exemplary (A)

• All frequencies correct (or correctly rounded) with clear, neat arithmetic shown for every step.
• Octave adjustments are explicitly justified each time ("divided by 2 because above the octave").
• Student explains the physical reason (length ↔ frequency inverse relationship) and names intervals (octave, perfect fifth).
• Work is presented clearly with consistent rounding and a short reflection or extension idea (e.g., compare to equal temperament).

Proficient (B)

• Most frequencies are correct; minor rounding or a single arithmetic slip permitted if method is otherwise sound.
• Octave adjustments are used correctly but might lack a sentence of justification in one place.
• Student states the main concept (halving raises an octave; 2/3 length → 3/2 frequency) but gives only brief explanation of the physics.
• Presentation is readable; rounding is mostly consistent.

Notes on curriculum alignment: The task practices multiplicative reasoning and ratios (Mathematics) and develops understanding of pitch, intervals and scale construction (Music). Teachers can link these activities to practical tuning exercises and simple acoustics demonstrations to deepen learning.

If you want, I can create a printable single-page PDF layout of this handout (ready to print on A4) or provide extension activities: (1) compare Pythagorean frequencies with equal-tempered frequencies, or (2) build a simple spreadsheet that calculates the cycle of fifths and normalizes into an octave.