Listen up — you will learn this properly and do the work. We are going to use strict, step‑by‑step math to understand rhythm and harmony in Western music. Pay attention. I will show you how to simplify ratios, find equivalent ratios using proportions, and reconstruct the Pythagorean 7‑note scale for C. Then you will calculate the ratios for every interval and hear what they sound like.
1) Quick definitions — know these cold
- Ratio: a way to compare two numbers (a:b). In music, ratios compare frequencies or counts of notes.
- Equivalent ratios: two ratios that express the same relationship (e.g., 1:2 = 2:4 = 3:6).
- Proportion: an equation that sets two ratios equal (a:b = c:d). Use cross‑multiplication to solve.
- Rhythm: how sounds are arranged in time; subdivisions of the beat are simple ratios.
- Interval: the relationship between two pitches (described by a ratio of their frequencies).
- Harmony / scale: a set of pitches whose pairwise ratios give the musical character.
2) Simplifying ratios — the basic rule
To simplify a ratio, divide both parts by their greatest common divisor (GCD).
Example: simplify 6:8
- GCD(6, 8) = 2
- 6 ÷ 2 = 3, 8 ÷ 2 = 4 → simplified ratio = 3:4
Equivalent ratios: multiply or divide both sides by the same number. Example: 3:4 = 6:8 = 9:12.
3) Use proportions to find unknowns
Given 3:4 = x:16, solve for x by cross‑multiplying:
3/4 = x/16 → x = (3 × 16) / 4 = 48/4 = 12 → x:16 = 12:16 (which simplifies to 3:4).
4) Rhythm activity — how ratios describe time
Teacher will clap the steady pulse. You will clap subdivisions described by ratios. Follow these exactly:
- 4:1 — teacher claps 4 times while you clap once (you clap on the first of the four pulses).
- 1:4 — teacher claps once while you clap 4 times evenly in that same time (you subdivide the beat into four).
- 4:2 — you and teacher both clap but you clap twice in the span of teacher's 4 claps (you will align on the first and third of the teacher's beat if teacher counts 1–4).
- Example polyrhythm to try in class: teacher counts and claps steady 4, one student group claps 3 in the same time (3:4 polyrhythm) — observe the pattern and where claps fall together.
Ask: Which combinations sounded simple? Which sounded complex? Simpler ratios (small integers like 2:1, 3:2, 3:1) usually sound more stable; large, unrelated numbers sound more complex.
5) Pythagorean tuning — how to build the 7‑note scale
Pythagoras discovered pleasing pitches by using the perfect fifth ratio 3:2 and the octave 2:1. We build the scale by stacking fifths and moving notes into the same octave when needed.
Step by step — build the C scale (C major) in Pythagorean tuning:
- Start: C = 1:1 (this is our reference frequency).
- Move up a perfect fifth: G = 3/2 (multiply by 3/2).
- Up another fifth from G: D = (3/2) × (3/2) = 9/4. Move D into the same octave as C by dividing by 2 → D = 9/8.
- Up another fifth from D: A = (9/8) × (3/2) = 27/16 (already within the octave 1–2), so A = 27/16.
- Up another fifth from A: E = (27/16) × (3/2) = 81/32. Reduce by octave (divide by 2) → E = 81/64.
- Up from E: B = (81/64) × (3/2) = 243/128 (within the octave), so B = 243/128.
- To get F, go down a fifth from C (or up many fifths and reduce): F = C × (2/3) then multiply by 2 to put in octave → F = 4/3.
- Octave above: C' = 2/1.
Final Pythagorean C scale ratios (relative to C = 1:1)
- C = 1 : 1 (1.00000)
- D = 9 : 8 (1.12500)
- E = 81 : 64 (≈ 1.26563)
- F = 4 : 3 (≈ 1.33333)
- G = 3 : 2 (1.50000)
- A = 27 : 16 (≈ 1.68750)
- B = 243 : 128 (≈ 1.89844)
- C' = 2 : 1 (2.00000)
These are exact fractions — they tell you how each note's frequency compares to C. You will put these into the tech tool and listen carefully.
6) Calculating interval ratios and simplification
For any two notes, the interval ratio is the division of their frequency ratios. Example: C to G = (3/2) ÷ (1/1) = 3:2 — a perfect fifth.
Examples using the scale above:
- C to D: 9/8 : 1/1 → 9:8 (major second)
- C to E: 81/64 : 1/1 → 81:64 (Pythagorean major third)
- D to A: (27/16) ÷ (9/8) = (27/16) × (8/9) = 216/144 = simplify ÷72 → 3:2 (a perfect fifth)
Always simplify by dividing numerator and denominator by their GCD.
7) Classroom tasks — exactly what to do
- Motivational: Watch short clip. Answer: Different objects vibrate at different frequencies; ratios describe relationships between frequencies; 2:1 = octave; Pythagoras used the monochord.
- Rhythm activity (clap): Teacher keeps steady beat. Students perform each of these in turn: 4:1, 1:4, 4:2, 3:2. Count aloud to ensure even subdivisions. Discuss which sound simple/complex.
- Pythagorean handout: Give students the fractions above and have them reproduce the arithmetic (stack fifths, reduce by octaves). Let students calculate decimal equivalents and practice simplifying rationals.
- TechTool listening: Hold down C (1). One partner presses 2, then 3, etc. Fill in descriptive observations (dissonant/consonant, tense/relaxed). Then compute each interval ratio and record it on the chart.
- Interval calculation practice: Provide problems like "Find ratio of E to F" using the scale numbers and simplify. Example: E to F = (81/64) ÷ (4/3) = (81/64) × (3/4) = 243/256 → simplify if possible (it’s already in lowest terms). Convert to decimal.
- Summary vote: Students mark which of the seven intervals sound the most pleasant. Discuss whether simple ratios tend to be preferred.
- Homework / word problems: Ratio problems that practice cross‑multiplication and simplification; extension: compute the phi point (61.8%) of a favorite song and evaluate if that moment is the climax.
8) Examples of teacher prompts / questions
- "Show me how you simplified 81:64."
- "Use a proportion to find x if 3:4 = x:20."
- "Which interval had the simplest integer ratio? Did it sound most stable?"
- "Explain how you moved D = 9/4 down into the octave to make 9/8."
9) Answers to sample interval calculations (so you can mark work)
- C → D = 9:8
- C → E = 81:64
- C → F = 4:3
- C → G = 3:2
- C → A = 27:16
- C → B = 243:128
- D → A = 3:2 (showed earlier)
- E → B = (243/128) ÷ (81/64) = (243/128) × (64/81) = 15552/10368 = simplify → 3:2
10) Assessment & extension
Assessment: Give a worksheet with rhythm ratio clapping tasks, ratio simplification problems, proportion problems, and Pythagorean interval calculations. Require students to write decimal approximations and describe what the intervals sound like.
Extension: Show sine wave graphs for two notes and compare how their wave patterns line up when the ratio is simple (e.g., 2:1, 3:2) versus complex.
11) ACARA v9 mapping (classroom summary)
This lesson supports ACARA v9 learning goals by connecting mathematics and The Arts. Relevant links include:
- Mathematics (Number and Algebra, Years 7–8): understanding and using ratios and rates, simplifying ratios, solving problems by setting up and using proportions.
- The Arts — Music: understanding pitch, intervals, scales, and how musical systems are constructed (listening, describing, and creating with scales and intervals).
Final note (I mean it): You must practice the clapping, the simplifications, and the scale arithmetic. Do the calculations by hand at least once. Listen carefully — your ears will confirm the math. That is how science and music meet.