ACARA v9 Homeschool Exemplary Report (Age 18) — Calculating the Pythagorean Scale with Monochord Ratios and Frequencies

IN THE MATTER OF: Student (Age 18) — Subject: Calculating the Pythagorean Scale

Issue

Whether the student correctly applied monochord ratio rules (half-string and two-thirds divisions) to produce the Pythagorean C scale and whether the computations and octave transpositions are correct.

Statement of Facts

1. Starting pitch (Middle C) is 261.63 Hz.
2. Halving the string length produces an octave above (ratio of lengths 1:2 → frequency factor 2).
3. Dividing the string so the sounding portion is two-thirds of the original length produces a frequency multiplied by 3/2 (inverse relationship: f ∝ 1/length).
4. The student followed the rule: repeatedly apply the 3/2 frequency multiplier; when a result falls outside the C–C' octave (261.63–523.26 Hz), transpose by factors of 2 (divide or multiply by 2) to bring it within the octave. For F, the instruction specified computing the 2:3 (lower) ratio relative to C and then transposing into the octave.

Analysis (Legal-brief style)

Argument — Frequency relationships and calculation method are correct. Key formulae used:

• For half the string: length' = length/2 ⇒ f' = f * (length/length') = f * 2.
• For two-thirds length: length' = (2/3)length ⇒ f' = f * (length/length') = f * (3/2).
• If f' > upper octave limit (523.26 Hz), divide by 2 to bring into C–C' octave. If f' < lower octave limit (261.63 Hz), multiply by 2.

Step-by-step verification of each student value

1. C = 261.63 Hz (given).
2. G: C * 3/2 = 261.63 * 1.5 = 392.445 → 392.45 Hz (student: 392.45 Hz)
3. Continue from G: G * 3/2 = 392.445 * 1.5 = 588.6675 → divide by 2 to fit octave → 588.6675 / 2 = 294.33375 → D ≈ 294.33 Hz (student: 294.33 Hz)
4. From D: D * 3/2 = 294.33375 * 1.5 = 441.500625 → A ≈ 441.50 Hz (student: 441.50 Hz)
5. From A: A * 3/2 = 441.500625 * 1.5 = 662.2509375 → /2 = 331.12546875 → E ≈ 331.13 Hz (student: 331.13 Hz)
6. From E: E * 3/2 = 331.12546875 * 1.5 = 496.688203125 → B ≈ 496.69 Hz (student: 496.69 Hz)
7. F (special rule): compute 2/3 below C → C * 2/3 = 261.63 * 0.6666667 = 174.42 Hz → ×2 to bring into octave → 174.42 * 2 = 348.84 Hz → F ≈ 348.84 Hz (student: 348.84 Hz)
8. Octave C': C * 2 = 523.26 Hz (student: 523.26 Hz)

Conclusion (Findings)

The student has correctly applied Pythagorean monochord reasoning and octave transposition rules. Numerically, the student’s frequencies match the verified computations to at least two decimal places. Outcome: Exemplary — the learner demonstrates procedural skill, conceptual understanding of ratios and inverse relationships between string length and frequency, and correct octave management.

Recommendations / Order

• Record and annotate the monochord experiments (audio + calculation log) as evidence for the homeschool portfolio.
• Extend study: compare Pythagorean tuning intervals to equal-tempered intervals, calculate cent differences, and listen for beating/waves between slightly mistuned pairs.
• Investigate stacked-fifths construction and how the Pythagorean comma arises from 12 fifths vs. 7 octaves.

ACARA v9 Alignment — Homeschool Reporting Note

This task aligns with Australian Curriculum v9 outcomes in The Arts (Music) and Mathematics (Number and Measurement): investigating pitch, ratio and proportional reasoning, and applying measurement to music. Evidence recorded above demonstrates achievement at an exemplary level suitable for inclusion in a v9 homeschool report.

Pythagorean C Scale — Final Frequencies (student-verified)

• C = 261.63 Hz
• D = 294.33 Hz
• E = 331.13 Hz
• F = 348.84 Hz
• G = 392.45 Hz
• A = 441.50 Hz
• B = 496.69 Hz
• C (octave) = 523.26 Hz

Parent–Teacher Comment (Ally McBeal cadence — 300 words)

Ally’s voice: I watched you—no, I listened—while you measured silence and turned it into numbers. You stepped onto the monochord and, with a scientist’s patience and a poet’s ear, traced octaves and fifths across a string. You knew that halving the string doubled the pitch; you knew that shortening it to two-thirds raised it by a factor of three halves; you moved from C to G and back, bending ratios into a scale that belongs to both math and music. As a parent-teacher I am stunned and delighted: your calculations are accurate, your method disciplined, your attention exemplary. You located G at 392.45 Hz, D at 294.33 Hz, A at 441.50 Hz, E at 331.13 Hz, B at 496.69 Hz, F by calculating two-thirds below C then transposing it into the octave to yield 348.84 Hz—each value landing where Pythagoras’ method predicts. (Yes, you remembered to octave-transpose when necessary—an important, sometimes overlooked step.) What now? Keep doing this. Move to comparing Pythagorean intervals with equal temperament; listen side-by-side; measure beats, observe commas. Practice expressing why tuning matters to composers, to instruments, to listeners. Consider building a monochord or using software to automate the ratios; test temperament differences on real instruments. For enrichment: calculate cents between each Pythagorean interval and its equal-tempered counterpart, or map the sequence of stacked fifths that produced these frequencies. In summary: exemplary achievement. You have demonstrated procedural mastery, conceptual understanding, and an analytic ear. I recommend continued challenge tasks in tuning theory and practical ear training. Bravo—take a bow, then take out a tuner and prove it all again. We will record your demonstrations, annotate each step for assessment, and prepare an evidence portfolio aligned with ACARA v9 outcomes so your exemplary work is documented for your homeschool report and future studies. Well done, truly always.