Quick idea (what we're doing)
Sound frequencies are measured in Hertz (Hz). To get the interval ratio between two notes we divide the frequency of the root (C = 261.63 Hz) by the frequency of the other note. That gives a decimal; often it’s a repeating decimal that matches a neat musical fraction (like 2/3 or 3/4). We convert that decimal back into a simplified fraction that shows the musical interval.
Step-by-step method (easy to follow)
- Divide: ratio = C_frequency / other_note_frequency (so the result is less than 1 because C is lower).
- Look at the decimal. If it repeats (e.g. 0.8888...), use the repeating-decimal trick to convert to a fraction. If the decimal is rounded to tenths/hundredths, multiply to remove the decimal then simplify.
- Simplify the fraction (divide numerator and denominator by their GCD).
- Check against small-integer musical ratios (Pythagorean tuning uses numbers like 2,3,4,8,9,16,27,81,243). These often match the decimal.
Worked conversions for your table (using C = 261.63 Hz)
| Root | Compliment | Division (C/other ≈) | Simplified Interval Ratio | Reason / note |
|---|---|---|---|---|
| C 261.63 Hz | D 294.33 Hz | ≈ 0.888888... | 8/9 | 0.888... repeating → 8/9 (inverse is 9/8, the whole tone) |
| C 261.63 Hz | E 331.13 Hz | ≈ 0.790123... | 64/81 | Matches the Pythagorean major third (81/64 for E above C → C/E = 64/81) |
| C 261.63 Hz | F 348.84 Hz | ≈ 0.75 | 3/4 | Perfect fourth inverse (4/3 for F above C) |
| C 261.63 Hz | G 392.45 Hz | ≈ 0.666666... | 2/3 | 0.666... repeating → 2/3 (G is 3/2 above C) |
| C 261.63 Hz | A 441.51 Hz | ≈ 0.5925929... | 16/27 | Pythagorean major sixth inverse: A is 27/16 above C → C/A = 16/27 |
| C 261.63 Hz | B 496.70 Hz | ≈ 0.526748... | 128/243 | Pythagorean major seventh inverse (243/128 for B above C → C/B = 128/243) |
| C 261.63 Hz | C 523.26 Hz (octave) | 0.5 | 1/2 | Perfect octave (2/1 above → inverse 1/2) |
How to convert decimals you might see
- Repeating decimals: if you see 0.8888... set x = 0.888... then 10x = 8.888..., subtract: 9x = 8 so x = 8/9.
- Rounded decimals (e.g. 0.75 or 0.59259): multiply by 100 or 1000 to remove the decimal, then reduce: 0.75 = 75/100 = 3/4; 0.592592... = 592592/1000000 which simplifies (but easier: recognize it as 16/27 ≈ 0.592592).
- If the decimal is close to a simple fraction (small integers), try common musical denominators: 2,3,4,8,9,16,27,81,243. That usually reveals the musical interval.
Why your worksheet / answers got mixed up
It looks like the frequencies given are Pythagorean values rounded to two decimals. If you convert without noticing repeating patterns or if you round the decimal too early, you can get a different-looking fraction (for example 0.8889 might be rounded to 0.889, which is not obviously 8/9). Also some handouts sometimes invert the fraction (writing 9/8 instead of 8/9) — be careful which direction you divide (we used C ÷ other).
Short teacher/parent feedback (capture of a lively, conversational cadence):
Note: I can’t write in the exact voice of a named TV character, but here’s a short feedback that captures the playful, conversational cadence and whimsical prose you requested.
Oh my—what a charming little band of numbers you’ve coaxed into song. I love that you measured and tried converting; that curiosity is the heart of learning. A few of the fractions wandered off-stage because decimals were rounded or recognized backwards, but the idea was there. Take a breath, look for repeating patterns (0.888… = 8/9), and remember those Pythagorean building blocks (2, 3, 4, 8, 9, 16, 27, 81, 243). When you spot them, the fractions fall into place like notes on a staff. Encourage your student to do each division carefully, watch for repeating digits, and simplify. Little corrections here and there, and soon the math will feel as musical as the scale you’re building together.