ACARA v9 mapping (years ~5–6)
Mathematics connections: understanding ratios, fractions and multiplication (apply 2/3 and 3/2). The Arts (Music): explore pitch and scales, how frequency relates to note names. This activity links Maths and Music for upper primary learners.
Part 1 — Question 2a: Octave limits for a C scale
Sailor Moon cadence: "In the name of the octave, let us shine!" A scale built inside an octave must stay between two Cs where the higher C is twice the frequency of the lower C. Starting from middle C at 261.63 Hz, the upper limit is one octave above: 2 × 261.63 = 523.26 Hz. So the usable frequencies for a C scale sit between 261.63 Hz (low C) and 523.26 Hz (high C). Keep notes between these bounds so the full scale is inside a single octave.
Calculation (simple)
Lower limit: 261.63 Hz. Upper limit: 261.63 × 2 = 523.26 Hz.
Teacher comment (Part 1 — 100 words)
What a bright beginning! Emphasise that an octave is a 1:2 frequency ratio — double the frequency is the same note name an octave higher. Ask the student to imagine the low C as a calm sea and the high C as the same sea on a higher wave: same identity, higher pitch. Check they can multiply by 2 and explain why we don't step outside those frequencies if we say "within one octave." Encourage them to hum middle C then imagine it doubled — that helps them hear the mathematical idea. Praise accuracy and curiosity.
Part 1 — Question 2b: If the string for middle C is shortened to 2/3 of its length, what is the frequency?
Sailor Moon cadence: "With the power of ratios, we transform string lengths!" Frequency is inversely proportional to string length. If the string length becomes 2/3 of the original, frequency becomes 1 ÷ (2/3) = 3/2 times the original frequency. So multiply 261.63 Hz by 3/2 (1.5): 261.63 × 1.5 = 392.445 Hz. Rounded sensibly to two decimals: 392.45 Hz. This matches the note G (close to 392 Hz). So shortening to 2/3 raises pitch by a perfect fifth.
Calculation
261.63 × (3/2) = 392.445 ≈ 392.45 Hz (G).
Part 2 — Question 1: Build the Pythagorean scale by repeatedly using the 2/3 string rule
Sailor Moon cadence: "In the name of musical harmony, we find each note!" Start with middle C = 261.63 Hz. To get the next note up (a perfect fifth), shorten the string to 2/3 of its length, which raises frequency by 3/2. For each step (C → G → D → A → E → B), multiply the previous frequency by 3/2. If the result is above the octave ( >523.26 Hz), divide by 2 to bring it down one octave. For F you use C and go 2/3 below (i.e., multiply C by 2/3, then double if needed) to place F inside the same C-octave.
Step-by-step calculations
- C = 261.63 Hz
- G = C × 3/2 = 261.63 × 1.5 = 392.445 → 392.45 Hz
- D = G × 3/2 = 392.445 × 1.5 = 588.6675 → 588.67 / 2 = 294.33375 → 294.33 Hz
- A = D × 3/2 = 294.33375 × 1.5 = 441.500625 → 441.50 Hz
- E = A × 3/2 = 441.500625 × 1.5 = 662.2509375 → 662.25 / 2 = 331.12546875 → 331.13 Hz
- B = E × 3/2 = 331.12546875 × 1.5 = 496.688203125 → 496.69 Hz
- F (special instruction): start from C and take 2/3 below C → 261.63 × 2/3 = 174.42 Hz, then multiply by 2 to bring into the octave → 174.42 × 2 = 348.84 Hz
Part 2 — Question 2: Pythagorean C Scale Frequencies (in-order)
Write the scale from low C up to the next C (rounded to two decimals):
- 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 — 523.26 Hz (octave)
Teacher comment (Part 2 — 100 words)
Wonderful! You used multiplication by 3/2 and octave shifts (×1/2 or ×2) to keep everything inside the single C-octave — exactly what Pythagoras did using string ratios. Ask the student to compare each Pythagorean frequency to modern equal-tempered frequencies (they will be close but not exact). Encourage them to listen to a simple keyboard or online tone generator to hear differences. Praise careful rounding and correct octave moves. Prompt extension: try starting from another note (like G) and build the same chain of fifths to see how the pattern repeats.
Extended rubric (exemplary and proficient outcomes)
Exemplary
- Correctly computes each frequency with clear multiplication/division steps and correct octave adjustments. - Shows understanding of frequency ⇄ string length inverse relationship. - Explains why we use ×3/2 for a shortened string and when to divide or multiply by 2 to fit the octave. - Demonstrates curiosity: compares Pythagorean to equal temperament, listens to tones and describes differences.
Proficient
- Computes most frequencies correctly with reasonable rounding. - Uses ×3/2 for successive notes and applies octave shifts to keep values between 261.63 and 523.26 Hz. - Gives a short explanation of why shortening a string raises pitch and why octave = double frequency. - Might need minor help with the special case for F or a couple of rounding details.
If you want, I can make a printable worksheet with the blank chart and these computed answers hidden so a student can practice filling them in! Shall we transform this into a shiny activity, Sailor Scout?