Calculating Interval Ratios — Pythagorean C Scale (Age 13) — ACARA v9-aligned homeschool legal brief

A clear, step-by-step ACARA v9-aligned homeschool lesson (age 13) presented in an Ally McBeal legal-brief cadence. Learn how to calculate interval ratios between C (261.63 Hz) and its Pythagorean compliments using specific rounding rules, convert decimals to fractions (including repeating decimals), and simplify each interval ratio.

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IN THE COURT OF MUSICAL INTERVALS — A HOMESCHOOL BRIEF (Ally McBeal cadence)

Case: Calculating interval ratios between Root Note C (261.63 Hz) and its Pythagorean compliments.

Student: Age 13 — ACARA v9-aligned (Number & Algebra: fractions, decimals, ratio & rounding; Reasoning & Problem Solving).

Summary of the task (short, rhythmic)

Present the frequency for each note, write the ratio C:compliment as a decimal, apply the rounding rule provided, convert the rounded decimal back into a fraction using the methods below, then simplify to the final interval ratio.

Pythagorean tuning facts (used in calculations)

  • C (root) = 261.63 Hz
  • D = (9/8)·C = 294.33 Hz
  • E = (81/64)·C = 261.63·(81/64)
  • F = (4/3)·C
  • G = (3/2)·C
  • A = (27/16)·C
  • B = (243/128)·C
  • High C (octave) = 2·C

How we convert decimals back into fractions (short guide)

  1. Terminating decimal (e.g. 0.25 or 0.75): say it aloud, write as fraction of powers of 10, then reduce. Example: 0.75 = "seventy-five hundredths" = 75/100 = 3/4.
  2. Repeating decimal (e.g. 0.888...): use algebra.
    Example for x = 0.888...: 10x = 8.888..., subtract x: 9x = 8, so x = 8/9.
  3. Truncated-to-1000ths then treated as repeating: if truncation gives 0.888, treat as 0.888... (repeating) and use the algebra method to get the fraction.

Workings — each interval with the required rounding rule and final simplified interval ratio

C : D (Root C to D)

Frequencies: C = 261.63 Hz, D = 294.33 Hz.

  1. Exact fraction (conceptual from Pythagorean tuning): C/D = 1 / (9/8) = 8/9.
  2. Decimal form: 8/9 = 0.888888... (repeating).
  3. Rounding Rule: Truncate at 1000ths and treat like repeating. Truncate 0.888888... to 0.888, then treat as repeating 0.888... .
  4. Convert repeating decimal back to fraction (algebra): let x = 0.888...; 10x = 8.888..., subtract x: 9x = 8 → x = 8/9.
  5. Simplified interval ratio: 8/9.

C : E

Conceptual exact fraction: E = (81/64)·C so C/E = 64/81 ≈ 0.79012345679.

  1. Decimal form: ≈ 0.790123...
  2. Rounding Rule: Round to nearest 10ths → 0.8.
  3. Convert 0.8 to fraction: 0.8 = 8/10 = 4/5 (reduce by dividing numerator and denominator by 2).
  4. Simplified interval ratio (after rounding): 4/5. (Note: exact Pythagorean ratio is 64/81, but the rounding rule produces 4/5.)

C : F

Exact: F = (4/3)·C so C/F = 3/4 = 0.75.

  1. Decimal form: 0.75.
  2. Rounding Rule: Do not round (use exact decimal).
  3. Convert 0.75 to fraction: 75/100 = 3/4 after simplifying (divide by 25).
  4. Simplified interval ratio: 3/4.

C : G

Exact: G = (3/2)·C so C/G = 2/3 = 0.666666....

  1. Decimal form: 0.666666... (repeating).
  2. Rounding Rule: Truncate at 1000ths and treat like repeating → 0.666 → treat as 0.666....
  3. Convert repeating decimal back to fraction: x = 0.666...; 10x = 6.666..., subtract: 9x = 6 → x = 6/9 = 2/3 after simplifying.
  4. Simplified interval ratio: 2/3.

C : A

Exact: A = (27/16)·C so C/A = 16/27 ≈ 0.59259259.

  1. Decimal form: ≈ 0.592592...
  2. Rounding Rule: Round to nearest 10ths → 0.6.
  3. Convert 0.6 to fraction: 0.6 = 6/10 = 3/5.
  4. Simplified interval ratio (after rounding): 3/5. (Exact Pythagorean ratio is 16/27, but rounding gives 3/5.)

C : B

Exact: B = (243/128)·C so C/B = 128/243 ≈ 0.52674897.

  1. Decimal form: ≈ 0.526749 (rounded here for demonstration).
  2. Rounding Rule: Round to nearest 100ths → 0.53.
  3. Convert 0.53 to fraction: 53/100 (this is terminating; 53 and 100 share no common factors except 1).
  4. Simplified interval ratio (after rounding): 53/100.

C : C (octave)

Exact: High C = 2·C so C/high-C = 1/2 = 0.5.

  1. Decimal form: 0.5.
  2. Rounding Rule: Do not round.
  3. Convert 0.5 to fraction: 5/10 = 1/2 after simplifying.
  4. Simplified interval ratio: 1/2.

Final table (compact)

  • C:D — decimal treated → 0.888... → fraction 8/9 (truncated 1000ths as repeating) — interval ratio 8:9 (or 8/9).
  • C:E — rounded to 10ths → 0.8 → 4/5.
  • C:F — exact → 0.75 → 3/4.
  • C:G — truncated to 1000ths as repeating → 0.666... → 2/3.
  • C:A — rounded to 10ths → 0.6 → 3/5.
  • C:B — rounded to 100ths → 0.53 → 53/100.
  • C:C (octave) — exact → 0.5 → 1/2.

Teaching notes & ACARA v9 alignment

This lesson practices:

  • Working with fractions and decimals, including converting repeating decimals to fractions (Number—fractions, decimals and rounding).
  • Using ratio and multiplicative reasoning to recognise the relationship between musical frequencies and simple fractions (Ratios and rates).
  • Applying rounding rules and reflecting on how rounding changes exact answers (Problem solving & reasoning in measurement contexts).

ACARA v9 alignment: This activity aligns to number and algebra focus areas where students use equivalence between fractions and decimals, apply rounding conventions, and reason with ratios and proportional relationships (suitable for a Year 8–9 student, age 13).

Closing — the verdict (short)

We followed the rules. We converted decimals, applied rounding exactly as instructed, converted back into fractions (using the algebra method for repeating decimals), and simplified. The musical law (Pythagoras) stands: the exact fractions are the neat rational ratios (8/9, 64/81, 3/4, 2/3, 16/27, 128/243, 1/2). After the rounding rules are applied, you obtain the simplified interval ratios shown above (some exact, some approximated by rounding).

If you want, I can render this as a printable two-page handout with the workspace steps filled in, or produce step-by-step solutions showing every multiplication and decimal calculation for each frequency. Which would you like next?

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