In the name of the Moon — let’s make music with math!
Quick rules (so Sailor Moon and you can stay magical):
- An octave is a 1:2 frequency ratio. If middle C is 261.63 Hz, one octave above is 261.63 × 2 = 523.26 Hz.
- When a string’s length becomes 2/3 of the original, the pitch (frequency) becomes 3/2 times the original (frequency ∝ 1/length).
- Pythagoras builds notes by repeatedly using a 2:3 (or 3:2) ratio. If a frequency goes above the octave, divide by 2; if it falls below the octave, multiply by 2.
Part 1 — Question 2
What are the lower and upper limits, in Hz, to build a C scale?
Using middle C = 261.63 Hz, the C octave runs from:
- Lower limit (middle C): 261.63 Hz
- Upper limit (one octave above): 261.63 × 2 = 523.26 Hz
Calculate the resulting frequency if a string playing middle C (261.63 Hz) is split into 2/3.
Length becomes 2/3 → frequency becomes 3/2 times:
Frequency = 261.63 × (3/2) = 261.63 × 1.5 = 392.445 Hz ≈ 392.45 Hz (this is the note G).
Part 2 — Build the Pythagorean scale (step-by-step)
Start: C = 261.63 Hz. For each step multiply the previous note by 3/2, then if the result is outside the C octave (below 261.63 or above 523.26) shift it by a factor of 2 to bring it inside the octave.
- C → G: G = 261.63 × 3/2 = 392.445 Hz (inside octave) → 392.45 Hz
- G → D: D = 392.445 × 3/2 = 588.6675 Hz (above octave) → divide by 2 → 588.6675 / 2 = 294.33375 Hz → 294.33 Hz
- D → A: A = 294.33375 × 3/2 = 441.500625 Hz (inside octave) → 441.50 Hz
- A → E: E = 441.500625 × 3/2 = 662.2509375 Hz (above octave) → divide by 2 → 331.12546875 Hz → 331.13 Hz
- E → B: B = 331.12546875 × 3/2 = 496.688203125 Hz (inside octave) → 496.69 Hz
- C → F (special rule): F is 2/3 below C, so F = 261.63 × 2/3 = 174.42 Hz (below octave) → multiply by 2 to bring into the C octave → 174.42 × 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 = 523.26 Hz (one octave above start)
Teacher comments (about Part 1) — Sailor Moon cadence (≈100 words)
In the name of the Moon, you discovered that an octave holds a secret: it is a perfect 2:1 family of notes! You correctly used middle C (261.63 Hz) as the lower limit and doubled it to find the upper limit (523.26 Hz). Your reasoning about length and frequency was brave and correct — when a string is shortened to two-thirds, the pitch climbs by three-halves. Keep practicing the idea that frequency is the opposite of length. Try humming C then G to hear the 3:2 jump — your ears will feel the magic!
Teacher comments (about Part 2) — Sailor Moon cadence (≈100 words)
Moon Prism Power helped you follow Pythagoras’ footsteps! You multiplied by 3/2 each time and applied octave shifts when notes left the C octave — that’s exactly how the Pythagorean scale is made. Your numbers are accurate and neatly adjusted into the octave range. Remember: sometimes we move notes up or down by factors of 2 to keep them inside one octave. For extra practice, try comparing these frequencies to piano notes — you’ll hear small differences because Pythagorean tuning sounds different from modern equal temperament. Fantastic logical thinking — keep shining!
Extended rubrics (per task)
Task: Part 1 — Octave limits & 2/3 string split
Exemplary — Correctly identifies limits (261.63 Hz and 523.26 Hz), explains why an octave is a 1:2 ratio, and shows reasoning that frequency ∝ 1/length so a 2/3 length → frequency × 3/2 with clear numerical work (392.45 Hz). Work is neat and could be communicated to classmates.
Proficient — Gives correct limits and result of the split with a brief explanation or correct calculation but with minimal reasoning steps shown or small rounding differences.
Task: Part 2 — Pythagorean scale construction
Exemplary — Accurately applies ×3/2 repeatedly, octave-reduces (÷2) when needed, handles the special C→F (×2/3 then ×2) rule, and lists all eight frequencies correctly rounded. Shows understanding of why octave shifts were applied.
Proficient — Produces the correct sequence with mostly correct numbers and uses octave shifts, but may show minor arithmetic rounding or incomplete explanation of octave adjustments.
ACARA v9 mapping (teacher-friendly)
- Mathematics (Years 5–6): Ratios, proportional reasoning and multiplication strategies — using multiplicative factors (3/2, 2) and understanding how ratios change values.
- Science (Years 5–6): Physical world — properties of waves and sound (pitch relates to frequency), investigating how frequency changes with string length.
- The Arts (Music): Practical music concepts — intervals (octave, perfect fifth) and tuning systems (Pythagorean tuning vs modern equal temperament).
If you want, I can make a printable worksheet with the steps, or create a little listening activity where you hear these Pythagorean notes compared to equal-tempered notes — Sailor Moon style!