Completed Handout — Calculating the Pythagorean Scale
| Root Note | Compliment (note & frequency) | Exact Ratio (C / note) | Simplified Interval Ratio (C : note) |
|---|---|---|---|
| C — 261.63 Hz | D — 294.33 Hz | 261.63 / 294.33 | 8/9 |
| C — 261.63 Hz | E — 331.13 Hz | 261.63 / 331.13 | 64/81 |
| C — 261.63 Hz | F — 348.84 Hz | 261.63 / 348.84 | 3/4 |
| C — 261.63 Hz | G — 392.45 Hz | 261.63 / 392.45 | 2/3 |
| C — 261.63 Hz | A — 441.51 Hz | 261.63 / 441.51 | 16/27 |
| C — 261.63 Hz | B — 496.70 Hz | 261.63 / 496.70 | 128/243 |
| C — 261.63 Hz | C (octave) — 523.26 Hz | 261.63 / 523.26 | 1/2 |
Notes: The simplified interval ratios above come from Pythagorean tuning (built by stacking perfect fifths). Example: D is 9/8 above C, so C/D = 8/9. E is 81/64 above C, so C/E = 64/81, and so on. Multiplying C (261.63 Hz) by those Pythagorean ratios gives the listed frequencies.
Homeschool Music Lesson — in Ally McBeal Cadence and Prose (for a 13‑year‑old)
Okay — picture this: you, me, a tiny courtroom in your head where notes gossip about ratios. C strolls in at 261.63 Hz, very composed. D enters, slightly sharper, at 294.33 Hz. They dont fight; they compare fractions like polite neighbors. We make the scale by multiplying C by neat Pythagorean fractions: 9/8 for D, 81/64 for E, 4/3 for F, 3/2 for G, 27/16 for A, 243/128 for B, and 2/1 for the high C. Step one: listen and hum C, then D — notice how D sounds a little higher. Step two: write the frequency and multiply by the fraction to get the next note. Step three: flip the fraction (C divided by the note) to see how small numbers describe their relationship (thats how we get 8/9, 64/81, etc.). Practice: sing C, then D, then G; clap while you sing the fraction names. Make it playful — imagine each fraction as a quirky character. Thats how math becomes melody.
150‑Word Parent/Teacher Feedback — Ally McBeal Cadence (150 words)
Dear Parent and Teacher,
I watched our 13-year-old leap into Pythagorean ratios like a curious sparrow dancing on a filing cabinet — ridiculous, delightful, oddly precise. They listened, they hummed D to G, and their face lit when C became a fraction, not just a note. They measured frequencies, checked ratios, and solved simplifications with steady patience. Next steps: practice finding intervals by ear using simple songs (Twinkle, Happy Birthday), clap the rhythm, label ratios aloud. Encourage short daily sessions, celebrate correct interval guesses, and gently correct leaps. Add a quick quiz: identify C to E and C to A intervals. Keep lessons playful — sing the fractions, call them characters (C is cosy, G is grand). Overall: strong curiosity, improving accuracy, and growing confidence. With warmth and a pinch of drama (Ally would approve), I recommend continued exploration of Pythagorean tuning and ear training. Stay curious and sing often. Play happily.