Pythagorean scale with middle C (age 10): calculating octave limits, 2/3 string split, and building the Pythagorean C scale (ACARA v9 mapped)

Step-by-step, student-friendly instructions (age 10) to find octave limits around middle C, compute the frequency when a string is reduced to 2/3, and build the Pythagorean C scale by repeatedly using the 3:2 (or 2:3) ratio. Includes teacher comments, ACARA v9 mapping and rubrics.

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Imagine the warm, slow stirring of sound like a pot of caramel: Pythagoras listened and found neat ratios. Let us do the same, gently and precisely.

Part 1 — Question 2 (a): Lower and upper limits for a C scale

In musical terms an octave is a 1:2 frequency ratio. With middle C = 261.63 Hz, the octave above is exactly double: 261.63 × 2 = 523.26 Hz. So, any C scale built within one octave has pitches between the lower limit 261.63 Hz (that C) and the upper limit 523.26 Hz (the C above). All notes used for the C-scale must have frequencies that fall inside that interval. This keeps the whole scale snugly inside one pleasing span of pitch — like a single shelf of delicate spices.

Part 1 — Question 2 (b): If the string is split into 2/3, what is the resulting frequency?

When a string’s length becomes two-thirds of its original length, the pitch rises. Frequency is inversely proportional to length, so f_new = f_original × (1 ÷ (2/3)) = f_original × (3/2). For middle C: 261.63 Hz × 3/2 = 392.445 Hz. That note is the G above middle C (often written as G4) — bright and clear. So, splitting the string to two-thirds produces 392.445 Hz. If the value were outside the octave we would halve or double it to bring it between 261.63 and 523.26 Hz.

Teacher comment — Part 1 (100 words)

Gently praise curiosity: this exercise unites number sense with listening. Encourage students to see ratios as recipes — shortening a string by 2/3 amplifies pitch by 3/2. Ask learners to show steps and explain why we divide or multiply by two when notes leave the octave. Use a tuning app or keyboard so they can hear 261.63 and 392.45 Hz. Challenge fast finishers to predict other octave relationships (for example, what happens at 1/2 or 2x length). Remind them to keep units (Hz) visible in every step. Concrete audio makes the maths sing.

Rubric — Part 1

  • Exemplary: Correctly identifies octave limits (261.63–523.26 Hz), correctly computes 261.63×3/2 = 392.445 Hz, shows reasoning about inverse relation of length and frequency, explains octave-halving/doubling where needed, and links result to musical note (G).
  • Proficient: Gives correct numeric limits and computes the 2/3 split frequency with correct multiplication or reciprocal reasoning; minor rounding differences acceptable; brief explanation of octave adjustment.

Part 2 — Question 1: Build the Pythagorean scale using 3:2 (from 2/3 string length)

We start with middle C = 261.63 Hz. Pythagoras built notes by forming a 2:3 length ratio (which raises frequency by 3:2). That means multiply by 3/2 for each successive fifth, and then move the result into the C octave (261.63–523.26 Hz) by doubling or halving as needed. Calculations (keep significant digits):

  • C = 261.63 Hz
  • G = 261.63 × 3/2 = 392.445 Hz
  • D = 392.445 × 3/2 = 588.6675 → divide by 2 to bring into octave → 294.3338 Hz
  • A = 294.3338 × 3/2 = 441.5007 Hz
  • E = 441.5007 × 3/2 = 662.2511 → ÷2 → 331.1255 Hz
  • B = 331.1255 × 3/2 = 496.6883 Hz
  • F: use the down-fifth from C: C × 2/3 = 174.42 Hz → ×2 to bring into octave → 348.84 Hz
Keep frequencies to three decimal places as needed; each step requires checking whether the result lies between 261.63 and 523.26, and halving or doubling to fit.

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

Now order the notes within the octave from C up to C using our found values. The Pythagorean C scale (frequencies rounded) is:

  • 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 — 523.260 Hz (octave)
These are the Pythagorean pitches within the C octave made by stacking perfect fifths and adjusting by octaves as needed. The results are close to, but not identical with, modern equal temperament frequencies — that difference is a delicious historical detail to explore.

Teacher comment — Part 2 (100 words)

Encourage neat working: remind students to multiply by 3/2 for each successive fifth and to check octave bounds, halving or doubling as required. Use a labelled worksheet: columns for raw product, adjustment (÷2 or ×2), and final in-octave value. Offer a keyboard or tuner so learners can match computed frequencies to sounds — listening reinforces understanding. For those who finish early, ask: why does B (496.69 Hz) differ slightly from the modern B (≈493.88 Hz)? This opens inquiry into tuning systems. Celebrate correct rounding and correct unit labelling (Hz).

Rubric — Part 2

  • Exemplary: Correctly computes every note with clear arithmetic, correctly adjusts octave by halving/doubling, lists final frequencies in order, explains why F was found by descending fifth, and connects results to tuning differences.
  • Proficient: Computes most notes correctly with correct octave adjustments and lists the scale in order; minor rounding variances acceptable; explains one reason for octave adjustment.

ACARA v9 mapping

Mapped to ACARA v9 priorities: Mathematics — Number and Algebra: using multiplicative reasoning, ratio and proportion to scale quantities; The Arts (Music) — understanding pitch, musical elements, and how tuning systems are constructed. Use this activity to meet learning intents for applying ratios and exploring sound in the music curriculum.

There — like a pinch of sea salt to finish — you have numbers that sing.

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