Quick explanation (for a 13-year-old)
Sound frequency is measured in hertz (Hz). An interval ratio compares two frequencies by dividing the lower (C = 261.63 Hz) by the higher note's frequency. That decimal can match a simple fraction (like 2/3 or 3/4) that musicians use to describe musical intervals.
Step-by-step method
- Take the root note frequency (C = 261.63 Hz).
- Divide C by the other note's frequency: C / note.
- Use the rounding rule required (your list gives four options). The clearest results often come from Rule 1 (do not round) because these frequencies were made to match Pythagorean ratios.
- Recognize the decimal as a known fraction (or convert the decimal to a fraction mathematically) and simplify it.
How to convert decimals back to fractions (short)
If the decimal repeats like 0.666..., that equals 2/3. If it equals 0.750..., that equals 3/4. For other repeating decimals (like 0.790123...), the fraction may be a ratio like 64/81. You can also multiply numerator and denominator by powers of ten and solve for the repeating block to make an exact fraction.
Completed table: Pythagorean C Scale Interval Ratios
| Root Note | Compliment (Hz) | Exact Division | Simplified Interval Ratio (C : note) |
|---|---|---|---|
| C (261.63 Hz) | D (294.33 Hz) | 261.63 ÷ 294.33 ≈ 0.888888... | 8/9 |
| C (261.63 Hz) | E (331.13 Hz) | 261.63 ÷ 331.13 ≈ 0.790123... | 64/81 |
| C (261.63 Hz) | F (348.84 Hz) | 261.63 ÷ 348.84 ≈ 0.7500 | 3/4 |
| C (261.63 Hz) | G (392.45 Hz) | 261.63 ÷ 392.45 ≈ 0.666666... | 2/3 |
| C (261.63 Hz) | A (441.51 Hz) | 261.63 ÷ 441.51 ≈ 0.592592... | 16/27 |
| C (261.63 Hz) | B (496.70 Hz) | 261.63 ÷ 496.70 ≈ 0.526748... | 128/243 |
| C (261.63 Hz) | C (523.26 Hz) | 261.63 ÷ 523.26 = 0.5 | 1/2 |
Notes on the table
- These simplified fractions are the standard Pythagorean ratios for those intervals. For example: C to G is a perfect fifth; its ratio (G/C) is 3/2, so C/G = 2/3.
- If you must apply one of the other rounding rules, the decimal will change slightly and your converted fraction will be an approximation. Rule 1 (no rounding) gives the exact Pythagorean fractions above.
- How to check: Multiply the fraction (like 2/3) by the higher note frequency (392.45 × 2/3 ≈ 261.63) to verify it returns the root frequency.
Short worked example (C to G)
- Divide: 261.63 ÷ 392.45 ≈ 0.666666... .
- Recognize repeating decimal 0.666... = 2/3.
- So the simplified interval ratio (C : G) is 2/3.
150-word homeschool teacher feedback (Ally McBeal cadence and prose)
You did a wonderful job exploring the Pythagorean C scale. I liked how your student listened, measured frequencies, and used ratios. A few small fixes would help: label columns, pick a rounding rule and use it consistently, and match decimals to exact fractions (for example, 0.666... = 2/3 and 0.888... = 8/9). Encourage practice converting repeating decimals into fractions and double-checking octave math (C to high C = 1/2). Celebrate their connection of math and music; it is the heart of Pythagorean tuning. Next, ask them to compute ratios both as decimals and fractions, show each step, and compare rounded answers with exact fractions. Keep tasks short, playful, and specific to build confidence. I’m excited to see their next scale exploration. Offer a brief challenge: pick a new root note, calculate intervals, and explain why simple fractions sound stable. Provide gentle praise and clear correction as they practice. Thank you.