Pythagorean C Scale Calculations for a 10-year-old — Octave Limits, 2/3 String Split, and Full Pythagorean C Scale (ACARA v9 mapped)

Part 1 — Question 1: Octave limits for a C scale

You start with middle C = 261.63 Hz. An octave is a 1:2 ratio, so the higher C is double the frequency and the lower C (the same pitch class an octave lower) is half. If we are building the C scale within the octave that starts at middle C, the lower limit is 261.63 Hz and the upper limit is 261.63 × 2 = 523.26 Hz. So every note in the C scale must lie between 261.63 Hz and 523.26 Hz. This keeps all notes inside one clean octave for making the scale.

Part 1 — Question 2: Split the string to 2/3 (middle C → new frequency)

When a string is shortened to 2/3 of its original length, the frequency rises. Frequency is inversely proportional to length, so shortening to 2/3 multiplies the frequency by 3/2 (because 1 ÷ (2/3) = 3/2). Starting from middle C = 261.63 Hz: new frequency = 261.63 × 3/2 = 261.63 × 1.5 = 392.445 Hz. Rounded sensibly for music, that is 392.45 Hz. That note is the G above middle C in our Pythagorean chain. Remember: shorter string → higher pitch; factor is 3/2 when length becomes 2/3.

Part 2 — Question 1: Build the Pythagorean chain (use 3/2 repeatedly; keep notes in the C octave)

Rules: multiply by 3/2 to move to the next note; if result > 523.26 Hz, divide by 2 to bring it down one octave; for F, use 2/3 below C and then multiply by 2 to bring into the octave. Calculations (rounded):

• C = 261.63 Hz
• G = C × 3/2 = 261.63 × 1.5 = 392.445 → 392.445 Hz
• D = G × 3/2 = 392.445 × 1.5 = 588.6675 → divide by 2 = 294.334 Hz
• A = D × 3/2 = 294.334 × 1.5 = 441.501 Hz
• E = A × 3/2 = 441.501 × 1.5 = 662.252 → divide by 2 = 331.126 Hz
• B = E × 3/2 = 331.126 × 1.5 = 496.689 Hz
• F: start 2/3 below C = 261.63 × 2/3 = 174.42 → multiply by 2 to fit the octave = 348.84 Hz

Notes kept inside 261.63–523.26 Hz as required.

Part 2 — Question 2: Write the Pythagorean C scale in order

Pythagorean C scale frequencies (rounded to three decimals where appropriate):

• C = 261.63 Hz
• D = 294.334 Hz
• E = 331.126 Hz
• F = 348.840 Hz
• G = 392.445 Hz
• A = 441.501 Hz
• B = 496.689 Hz
• C (upper) = 523.260 Hz

These are the scale frequencies created by repeatedly using the 3:2 (or 2:3) ratio and moving notes by octaves when needed to keep every pitch between middle C and the next C.

ACARA v9 mapping (music understanding)

This activity maps to Australian Curriculum (v9) music outcomes: developing understanding of pitch, intervals, scales and the physics of sound (suitable for upper primary: Years 5–6). Students practise measuring pitch, using ratios (mathematics integration), and creating scales — all aligned to cross-curricular numeracy and music learning goals.

Teacher comments (Task 1 — Part 1: 100-word comment)

Good. You must be precise. The octave rule is simple: double for the high C and half for the low C — you used 261.63 Hz for middle C and correctly found 523.26 Hz as the upper limit. That boundary keeps all scale notes in the same octave and makes tuning work. When splitting strings, remember frequency changes inversely with length — shortening to two-thirds multiplies frequency by three-halves. Write down each multiplication and check if the result falls inside the octave; if it’s higher, divide by two. Follow the steps strictly; accuracy in each arithmetic step matters for musical tuning.

Teacher comments (Task 2 — Part 2: 100-word comment)

You followed Pythagoras’ method correctly by repeatedly using the 3:2 factor and shifting by octaves when necessary. The chain C → G → D → A → E → B and the special rule for F (2/3 below C then up an octave) were applied correctly. Your rounding is sensible; keep three decimals for clarity. Emphasise showing each multiply and every octave adjustment; that makes your reasoning traceable and error-proof. Next step: compare these Pythagorean frequencies to equal-tempered notes and listen for the small differences — that’s how you learn why tuning systems sound different.

Rubrics (extended) — Task 1 (Octave limits & 2/3 split)

• Exemplary: Correctly identifies lower and upper octave limits (261.63 Hz and 523.26 Hz). Correctly applies inverse-length rule to compute frequency when length is 2/3, shows multiplication by 3/2 with precise arithmetic (392.445 Hz), explains reasoning clearly, and ties result to musical note name (G). Uses clear steps and units throughout.
• Proficient: Identifies octave limits and computes the 2/3 split result with minor rounding differences, shows method (frequency × 3/2) and gives correct musical interpretation. Explanation is accurate but may be briefer or omit one small arithmetic detail.

Rubrics (extended) — Task 2 (Build and list Pythagorean C scale)

• Exemplary: All six chain calculations are shown, octave adjustments are applied correctly where needed, F is calculated from 2/3 below C then adjusted, final ordered scale lists all frequencies within the octave and matches the computed values (rounded consistently). Shows evidence of checking work and understanding of ratios.
• Proficient: Most chain calculations correct, octave shifts performed when needed, F calculated correctly, final scale mostly correct with small rounding differences. Explanation is complete but may lack one intermediate check.

Strict instruction: show every multiplication and every octave adjustment on your worksheet. If you make a small arithmetic error, fix it immediately — precision matters in tuning.