Quick overview (for a 13‑year‑old)
Pythagoras used a single string (a monochord) to discover musical ratios. Shortening the string raises the pitch; doubling the frequency makes the pitch an octave higher. We can use ratios to build the Pythagorean 7‑note C scale and then compare each note to the root (C) as interval ratios.
Step‑by‑step math and corrections to the student answers
Question 1 — Halving the string
a) Ratio of one part to total when string is halved: 1:2. (Correct.)
b) Frequency when string is halved: middle C = 261.63 Hz. Halving the string length doubles the frequency, so new frequency = 261.63 × 2 = 523.26 Hz. (Correct.)
c) What happens to pitch when string is halved? The pitch doubles — you hear the octave above. (Correct.)
Question 2 — Octave limits
You build a C scale inside the octave from C to the next C. Using middle C and its octave: lower limit = 261.63 Hz, upper limit = 523.26 Hz. (Student: correct.)
When the string is 2/3 of its original length
Important: frequency is inversely proportional to length. If the new length = (2/3) × original length, then frequency multiplies by 1/(2/3) = 3/2. So:
261.63 × (3/2) = 392.445 Hz → rounded sensibly to 392.45 Hz. (Student wrote 392.45 Hz — correct.)
How to build the Pythagorean C scale (method and corrected table)
Classic Pythagorean tuning uses stacking perfect fifths (ratio 3:2) and bringing results into the octave (divide or multiply by 2 as needed). The well‑known theoretical fractions for notes relative to C are:
- C = 1
- D = 9/8
- E = 81/64
- F = 4/3
- G = 3/2
- A = 27/16
- B = 243/128
- Upper C = 2 (an octave above)
Using middle C = 261.63 Hz, multiply by each fraction to get the frequency. Here are the calculated values (rounded to two decimals):
| Note | Fraction (relative to C) | Frequency (Hz) |
|---|---|---|
| C | 1 | 261.63 |
| D | 9/8 | 261.63 × 1.125 = 294.33 |
| E | 81/64 | 261.63 × 1.265625 = 331.13 |
| F | 4/3 | 261.63 × 1.333333 = 348.84 |
| G | 3/2 | 261.63 × 1.5 = 392.45 |
| A | 27/16 | 261.63 × 1.6875 = 441.50 |
| B | 243/128 | 261.63 × 1.8984375 = 496.70 |
| Upper C | 2 | 523.26 |
Comments on the student's scale: The student's frequency list matches the correct Pythagorean frequencies (small rounding differences only). So the scale table the student wrote is essentially correct.
Interval ratios (root C compared to each other note)
Task: write ratio C : note and simplify. The simplest exact method is to use the Pythagorean fractions above and invert them when needed.
| Root | Compliment | Exact decimal (C ÷ note) | Exact simplified fraction |
|---|---|---|---|
| C | D (294.33 Hz) | 261.63 / 294.33 ≈ 0.8888889 | 8/9 |
| C | E (331.13 Hz) | ≈ 0.768 | 64/81 |
| C | F (348.84 Hz) | ≈ 0.75 | 3/4 |
| C | G (392.45 Hz) | ≈ 0.6666667 | 2/3 |
| C | A (441.50 Hz) | ≈ 0.5925926 | 16/27 |
| C | B (496.70 Hz) | ≈ 0.527 | 128/243 |
| C | Upper C (523.26 Hz) | 0.5 | 1/2 |
Corrections to the student's interval answers:
- C:D student wrote 4/5 — correct answer is 8/9. (4/5 = 0.8, too small.)
- C:E student wrote 4/5 — correct answer is 64/81 ≈ 0.7901 (student's 4/5 = 0.8 is a different approximation; the exact Pythagorean fraction is 64/81 and the student's decimal for E was slightly different.)
- C:F student wrote 3/4 — correct.
- C:G student left blank — correct is 2/3. (Student earlier found G correctly as 392.45 Hz.)
- C:A student wrote 3/5 — correct is 16/27 ≈ 0.5926 (3/5 = 0.6, close but not correct as the simplified fraction is 16/27 and does not reduce to 3/5.)
- C:B student wrote 13/25 (or curriculum sheet showed 53/100) — both are incorrect. Correct is 128/243 ≈ 0.5267. This fraction cannot reduce further.
- C:C student wrote 1/2 — correct.
About the rounding rules (how to go from decimal to fraction)
The activity asked you to divide the two note frequencies, round according to a specific rule, then convert to a fraction and simplify. That is a multi‑step, error‑prone process unless you use the theoretical fractions directly (which are exact). Still, here's how to follow the rounding instructions using one example (C and D):
- Exact decimal: 261.63 / 294.33 ≈ 0.8888888889 (this is repeating 8s because exact theory gives 8/9).
- Rounding Rule 1: Do not round → 0.8888889... Convert repeating decimal 0.888... → 8/9 directly.
- Rounding Rule 2: Round to nearest 10ths → 0.9 → as fraction 9/10 (this loses precision).
- Rounding Rule 3: Round to nearest 100ths → 0.89 → convert to 89/100 and simplify if possible (here it doesn't simplify further than 89/100).
- Rounding Rule 4: Truncate at 1000ths and treat like repeating decimal → e.g. 0.888 truncated to 0.888 → treat as 0.888888... and convert to fraction (this is more advanced algebraically). Better: use the exact theoretical fraction when available.
Summary: using the Pythagorean theoretical fractions (9/8, 81/64, 3/2, etc.) gives exact interval fractions (8/9, 64/81, 2/3, etc.). If the task forces rounding the decimal first, you must be consistent with the specified rounding rule; rounding changes the fraction you will get.
ACARA v9 alignment (high‑level mapping)
Learning area: Mathematics — Number and Algebra & Measurement
- Key ideas: solving problems involving ratios and proportional reasoning; converting between decimals and fractions; applying multiplicative thinking (multiply by 3/2, divide by 2 to stay in the octave).
- Suggested alignment (age 13 / early secondary): aligns to content descriptors about ratios and rates, and using multiplicative reasoning to solve real‑world problems (music and frequency used as context).
- Skills practised: calculating and simplifying ratios, fractional multiplication, applying inverse proportionality, rounding decimals and converting to fractions — all consistent with ACARA emphasis on proportional reasoning and number manipulation at Years 7–8.
Homeschool feedback in Ally McBeal cadence — per item (parent + teacher rubric style)
Question 1a (ratio 1:2)
Ally McBeal parent tone: "Oh! You said one to two — crisp, like a neat note in court. Gorgeous start — spot on, darling."
Rubric/teacher quick note: Correct. Showed understanding of the halved string ratio (meets expectation for ratio recognition).
Question 1b (523.26 Hz)
Ally McBeal parent tone: "You doubled the number and my heart doubled with it — bravo. The octave sang true."
Teacher rubric comment: Correct numerical result and explanation of relationship between length and frequency. Evidence of multiplicative reasoning.
Question 1c (pitch doubles)
Ally McBeal parent tone: "Shorten the string, raise the song — you understood the magic. Charming and precise."
Teacher rubric comment: Clear conceptual explanation — student understands inverse relation between length and frequency.
Scale construction (frequencies)
Ally McBeal parent tone: "Oh, the melody of numbers — you lined them up like a chorus. Very nearly flawless — those tiny decimals, we forgive!"
Teacher rubric comment: Correct application of the Pythagorean method. Frequencies match expected ratios; rounding is acceptable at two decimal places. Meets and in places exceeds expectations for procedural fluency.
Interval ratios (C to each note)
Ally McBeal parent tone: "Now here came the hiccup — you guessed a few faces wrong at the party of fractions. But you knew the guests; we just need names on the invitations."
Teacher rubric comment: Student found the frequencies correctly but made a few errors when converting decimals to simplified fractions (e.g., C:D should be 8/9; C:E 64/81; C:A 16/27; C:B 128/243). This indicates a need for focused practice on converting decimal quotients to exact fractional forms and on using the theoretical fractions directly.
Homeschool teacher overall report (Ally McBeal cadence)
"Darling, what a delightful mathematical aria. You built the C scale like a careful composer — each frequency sounded true and clear, the arithmetic harmonised. A few misprinted notes when you turned decimals into tidy fractions — little slips in the finale, but the performance was musical and brave. Keep leaning on the exact ratios (9/8, 3/2, 27/16 — they are your sheet music), and your conversions will follow like a chorus."
Homeschool parent final comment (Ally McBeal cadence)
"Bravo, my young Pythagoras! You sang the right frequencies, and the errors were mere whispers. A tiny review on fraction conversion and rounding and we’ll be ready for a standing ovation. I’m proud. Encore?"
Suggested next steps (practical, short)
- Practice converting decimal quotients to exact fractions using the known Pythagorean fractions — this avoids rounding mistakes.
- Work one example per rounding rule so the student sees how rounding changes the fraction outcome (pick C:D and try each rule).
- Try a small activity: tune a cheap electronic tone generator to the computed frequencies and listen for how the intervals sound. Hearing helps solidify the fraction idea.
If you want, I can: (a) generate a worksheet that converts the decimals under each rounding rule step‑by‑step, or (b) produce an audio file of the Pythagorean C scale so you can listen and compare. Which would you like next?