Goal (plain): Use the given Pythagorean C‑scale frequencies, divide C by each compliment to get a decimal, then convert that decimal back into a simplified fraction (the interval ratio). Because these frequencies come from the Pythagorean tuning system, the interval ratios are exact rational numbers. We will show the decimals and the exact simplified fractions and explain common mistakes.
Step‑by‑step method
- Take the root frequency (C = 261.63 Hz). For each compliment frequency f, compute decimal = 261.63 ÷ f.
- Write the decimal to enough places to see a pattern (many Pythagorean ratios repeat). Example: 0.888888... shows a repeating pattern 8/9.
- Convert the repeating decimal to a simplified fraction. For these frequencies the fractions are standard Pythagorean ratios (see table).
- If you must round, apply the chosen rounding rule consistently. Rounding to different places will give different fractions when you convert back to a fraction.
Completed table (decimals and exact simplified interval ratios)
| Root | Compliment | Freq (Hz) | Decimal (C ÷ note) | Exact simplified interval ratio (fraction) |
|---|---|---|---|---|
| C | D | 294.33 | 0.888888... (≈0.8889) | 8/9 |
| C | E | 331.13 | 0.79012345679... | 64/81 |
| C | F | 348.84 | 0.75 | 3/4 |
| C | G | 392.45 | 0.666666... (≈0.6667) | 2/3 |
| C | A | 441.51 | 0.59259259259... | 16/27 |
| C | B | 496.70 | 0.52679487179... | 128/243 |
| C | C (octave) | 523.26 | 0.5 | 1/2 |
Why these fractions are the ones you should expect
These frequencies were generated from Pythagorean tuning multipliers (D = C×9/8, E = C×81/64, F = C×4/3, etc.). Dividing C by each note produces the reciprocal of those multipliers, which are the fractions shown above. For example D: 261.63 ÷ 294.33 ≈ 0.888888... which is 8/9, because D was 9/8 of C.
Why the student/parent answers were off
- Inconsistent rounding: the student used different rounding places for different notes (tenths for some, hundredths for others). Different rounding places produce different decimal values and therefore different fractions when converted back.
- Converting rounded decimals to fractions incorrectly: e.g. rounding 0.592592... to 0.6 and converting to 3/5 loses the repeating pattern 592 which corresponds to 16/27.
- Some fractions the curriculum printed (like 53/100 for B) are the result of rounding the decimal 0.52679 to the nearest hundredth (0.53 → 53/100). That is not the exact Pythagorean fraction (128/243) but it is a valid rounded approximation if the teacher asked for rounding to hundredths.
How to use the four rounding rules (brief)
- Rule 1 (do not round): if the exact fraction is available (as here), use it (e.g. 64/81).
- Rule 2 (nearest 10ths): 16/27 ≈ 0.5926 → 0.6 → 3/5. Good for quick estimations but loses exact tuning info.
- Rule 3 (nearest 100ths): 128/243 ≈ 0.52679 → 0.53 → 53/100. A better approximation but still not exact.
- Rule 4 (truncate at 1000ths and treat like repeating): because many Pythagorean decimals are repeating blocks (e.g. 0.592592...), truncating to three places and treating that pattern as repeating will often recover the exact fraction (0.592 → 0.592592... → 16/27).
Teaching tip: When the given frequencies come from a known tuning system (Pythagorean here), look for the theoretical rational multipliers (9/8, 81/64, 3/2, etc.). That will give you exact fractions without guesswork. If the task specifically asks you to apply a rounding rule, pick and apply that rule consistently across all notes and then convert to a fraction from the rounded decimal.
150‑word homeschool parent/teacher feedback (Ally McBeal cadence and prose)
Oh, I hear you—the numbers sang a little off‑key at first. You and your student did the right work: divided, looked, chose fractions. The tiny difference was this: the tuning wasn't random; it was Pythagorean, so those decimals hide neat repeating patterns and exact fractions (8/9, 64/81, 3/4...). When rounding wandered from exercise to exercise, so did the answers. That’s an easy fix — choose the rounding rule the worksheet asked for and use it consistently. When I teach this, I show the repeating decimal first, then show the neat fraction behind it. Lovely to see curiosity; even better to see you checking the curriculum notes. Keep encouraging careful rounding and the habit of asking, “Was this frequency made from a rational tuning?” That one question will clear up many future musical mysteries.