ACARA v9 Homeschool Report (Age 13) — Calculating the Pythagorean Scale: Legal‑Brief Style Assessment and Ally McBeal‑Cadence Parent‑Teacher Comments

IN THE MATTER OF: Student (Age 13) — Calculation of the Pythagorean Scale

Case Caption

Student v. Task: Determine the effect of string division on pitch and reproduce the Pythagorean C scale from Middle C (261.63 Hz) using 2/3 length splits (perfect fifths) with octave adjustments.

Issues

1. What is the frequency when the string is halved (1:2 length ratio) given Middle C = 261.63 Hz?
2. How do we interpret a 2/3 length ratio on pitch (frequency)?
3. Using repeated 2/3 splits (i.e. 3:2 frequency ratios), what are the correct frequencies for the Pythagorean C scale between C and the octave C?

Findings of Fact (observations from the student work)

• The student correctly identified the length ratio 1:2 and that halving the string doubles frequency. (Correct)
• The student computed the half‑string frequency as 523.26 Hz for Middle C doubled. (Correct: 261.63 × 2 = 523.26 Hz.)
• The student understood that a 2/3 length split produces a higher pitch but showed some algebraic confusion in the written working for that single step. The numeric results for the final scale were largely correct, though several intermediate worksheet entries had arithmetic/rounding errors (these are corrected below).

Law (Principles used)

Frequency is inversely proportional to string length. If new length = p × original length, new frequency = original frequency × (1/p). Thus:

• Halving length (p = 1/2) → frequency × (1/(1/2)) = frequency × 2 (an octave up).
• Splitting to 2/3 length (p = 2/3) → frequency × (1/(2/3)) = frequency × (3/2) (a perfect fifth up).

Analysis (step‑by‑step calculations and corrections)

Start: C = 261.63 Hz.

1) Halving the string (1:2 length ratio):

Frequency = 261.63 × 2 = 523.26 Hz (upper C). Student: 523.26 Hz (correct).

2) Splitting the string to 2/3 length (makes a perfect fifth):

Frequency multiplier = 3/2 = 1.5.

Compute the chain of fifths, bringing any result outside the C→C octave (261.63–523.26 Hz) into the octave by multiplying or dividing by 2 as needed:

1. G = C × 3/2 = 261.63 × 1.5 = 392.445 → round to 392.45 Hz.
2. D = G × 3/2 = 392.445 × 1.5 = 588.6675 → bring down an octave: 588.6675 ÷ 2 = 294.33375 → 294.33 Hz.
3. A = D × 3/2 = 294.33375 × 1.5 = 441.500625 → 441.50 Hz.
4. E = A × 3/2 = 441.500625 × 1.5 = 662.2509375 → ÷ 2 = 331.12546875 → 331.13 Hz.
5. B = E × 3/2 = 331.12546875 × 1.5 = 496.688203125 → 496.69 Hz.
6. Upper C (octave) = 2 × 261.63 = 523.26 Hz (or B × 3/2 gives a tone above the octave and is adjusted back into octave; the octave is 523.26 Hz).
7. F is found by going down a fifth from C (or equivalently multiplying C by 2/3 then bringing into the octave): C × (2/3) = 261.63 × 0.6666667 = 174.42 → ×2 to bring into octave = 348.84 Hz.

Correct Pythagorean C Scale (frequencies rounded to 2 decimal places)

1. C = 261.63 Hz
2. D = 294.33 Hz
3. E = 331.13 Hz
4. F = 348.84 Hz
5. G = 392.45 Hz
6. A = 441.50 Hz
7. B = 496.69 Hz
8. C (octave) = 523.26 Hz

Comparison with the student submission

The student final ordered scale is essentially correct (their final list was C 261.63, D 294.34, E 331.1, F 348.84, G 392.45, A 441.51, B 496.7, C 523.26). Minor differences are rounding differences (e.g. 294.33 vs 294.34). Some intermediate worksheet numbers (e.g. a listed 587.918 for G→D) appear to be arithmetic or copying mistakes; the correct intermediate product before octave reduction is 588.6675, then halved to 294.33375.

Conclusion (Verdict)

Verdict: Exemplary understanding. The student demonstrated correct conceptual understanding (inverse relation of length and frequency; halving doubles frequency; 2/3 length gives a 3/2 frequency multiplier), produced a correct final Pythagorean C scale within acceptable rounding, and only needs minor attention to clear algebraic presentation and careful arithmetic notation in intermediate steps.

Recommendations (Orders)

• Practice writing the algebraic relation explicitly: if new length = p × original, then new frequency = original × (1/p). Showing this eliminates the small algebraic confusion seen earlier.
• Keep at least 4–6 significant figures during chain calculations, then round at the end to avoid cumulative rounding error.
• Extension: compare these Pythagorean frequencies to equal temperament frequencies for Middle C and see where intervals differ.

Parent‑Teacher Comments (Ally McBeal cadence — approx. 300 words)

She counted. He hummed. The string sang. You, dear student, found the secret handshake of sound — the little numerical waltz that makes C become G and G become D and so on, like friends calling to friends across the musical neighbourhood. I read your work and I feel proud. Yes, proud, and a little delighted — because the math is mostly in tune and the thinking is right. You knew: halve the string, double the sound; make the string two‑thirds, the note leans up by a perfect fifth. That is a neat, clear idea. You showed the right answers at the finish line. Bravo. But in the middle — the messy algebra, the scribbly multiplication — that’s where practice will help. Clean steps. Boxed work. A little patience with decimals. The scale you built is a ladder of friends: C to D to E, small steps, big jumps, all kept between two Cs that hug each other an octave apart. Your ears are learning to read the language of ratios. That is rare and beautiful. I see curiosity. I see you reaching for the rule and testing it, catching it, and then playing with it. Next, let us compare. Let us listen to equal temperament and Pythagoras side by side. Hear the small differences. Let us draw the ratios. Let us tidy the work so your math sings as clearly as the note. Until then — well done. Keep asking, keep calculating, keep listening. I’ll bring coffee. You bring the monochord. We’ll argue numbers and hum harmonies. End scene.

Teacher: Evidence of exemplary achievement in proportional reasoning, application of inverse relationships and sequence generation — ACARA v9 aligned.