In a Sailor Moon cadence: "In the name of music and math, I will simplify ratios and reveal the secrets of sound!" — okay, now let's learn how ratios connect rhythm and harmony, step by step.
1. What is a ratio?
A ratio compares two numbers. In music it often compares counts of beats (rhythm) or frequencies/pitches (harmony). Ratios can be written as "a:b", "a to b", or as a fraction a/b.
2. How to simplify a ratio (step-by-step)
- Write the ratio as two whole numbers, e.g. 18:12.
- Find the greatest common divisor (GCD) — the largest number that divides both. For 18 and 12 the GCD is 6.
- Divide both numbers by the GCD: 18 ÷ 6 = 3, 12 ÷ 6 = 2. So 18:12 simplifies to 3:2.
Tip: If you don’t know the GCD, try dividing by small primes (2, 3, 5, etc.) repeatedly until you can’t divide both numbers anymore.
3. Equivalent ratios using proportions
Two ratios a:b and c:d are equivalent if a/b = c/d. You can test this by cross-multiplying: a×d = b×c.
Example: Are 2:3 and 8:12 equivalent? Cross-multiply: 2×12 = 24 and 3×8 = 24. Yes — they are equivalent. To create an equivalent ratio, multiply both parts by the same number: 2:3 → (2×4):(3×4) = 8:12.
4. Ratios in rhythm (easy, clap-ready)
Rhythms are subdivisions of a beat. If the teacher keeps a steady beat and we compare student claps to the teacher’s beat, we use ratios:
- 4:1 — Teacher claps 4 times, student claps once (student notes are slower).
- 1:4 — Teacher claps once, student claps 4 times inside that beat (student subdivides the beat into 4 equal parts).
- 4:2 — Teacher 4, student 2 (student claps on beats 1 and 3 of the teacher’s 4-beat cycle).
Try it: count 1–2–3–4 repeatedly. Clap once on every count (teacher). Now try clapping 4 times in the space of one count (1:4). Notice how the rhythm feels busier — that is the effect of a different ratio.
5. How ratios create harmony: the Pythagorean idea
Pythagoras found that simple fractions of a vibrating string gave pleasing sounds. The simplest ratio is 2:1, the octave (same note, higher). The perfect fifth is 3:2 — another very pleasant interval. By stacking perfect fifths and moving them into one octave, you can build a 7-note scale (the Pythagorean scale).
6. Build the Pythagorean 7-note scale (C major) — step-by-step
Start with C as the root: C = 1/1.
- Make a perfect fifth above C: multiply by 3/2 → G = 3/2.
- Stack another fifth above G: (3/2)×(3/2) = 9/4. To keep notes in the same octave, divide by 2 (if the result is >2). 9/4 ÷ 2 = 9/8 → D.
- Keep stacking fifths and bringing them into the octave (divide by 2, 4, 8… as needed). Doing this gives the following notes (all relative to C = 1):
Pythagorean C scale (ratios vs. C):
- C = 1/1 (ratio 1:1)
- D = 9/8 (ratio 9:8 ≈ 1.125)
- E = 81/64 (ratio 81:64 ≈ 1.2656)
- F = 4/3 (ratio 4:3 ≈ 1.3333)
- G = 3/2 (ratio 3:2 = 1.5)
- A = 27/16 (ratio 27:16 ≈ 1.6875)
- B = 243/128 (ratio 243:128 ≈ 1.8984)
- C (octave) = 2/1
How these came about: every note (except F) above was formed by multiplying by 3/2 repeatedly and reducing by powers of two until the number lies between 1 and 2 (inside one octave).
7. Calculating the ratio between two notes (intervals)
To find the ratio between two notes, divide the higher pitch’s ratio by the lower pitch’s ratio. That gives the interval as a fraction.
Example 1 — G to C: G/C = (3/2) / (1/1) = 3/2. So the interval is 3:2 (a perfect fifth).
Example 2 — G to E: choose the higher over the lower. G (3/2) ÷ E (81/64) = (3/2) × (64/81) = 192/162 = 32/27 after simplifying. So G:E = 32:27 (a Pythagorean minor third in size).
How to simplify the result: find the GCD of numerator and denominator and divide both by it (just like earlier).
8. Quick practice (try these)
- Simplify 45:30. (Answer: 3:2)
- Find equivalent ratio to 2:5 where the first number is 14. (Multiply both parts by 7 → 14:35.)
- Using the Pythagorean list, compute the interval ratio A to C (A = 27/16). So A/C = (27/16)/(1) = 27/16. Simplify? It’s already simplest; as a mixed fraction it’s 1 11/16, or decimal ≈1.6875.
9. Classroom activity ideas (simple)
- Rhythm clapping: Teacher keeps steady 4-beat pulse. Students try clapping 1:4, 2:4, 3:4, 4:4. Notice how the feel changes when rhythms are simultaneous (polyrhythm).
- Scale calculator: Use the ratios above to program or use a tech tool that sets pitch frequencies proportional to those ratios (if C = 264 Hz, multiply by each ratio to hear the Pythagorean scale).
- Vote on pleasant intervals: play pairs (C with each other note) and vote which sounds most pleasant. Discuss whether simpler ratios (like 3:2 or 4:3) tend to sound nicer.
10. Summary (magical finale)
Ratios are the secret travel maps between beats and pitches. For rhythms they tell you how often something happens compared to the pulse. For harmony they tell you how pitches relate by frequency. Simplify ratios by dividing by the GCD, make equivalent ratios by multiplying both parts, and build the Pythagorean scale by stacking 3:2 fifths and bringing notes into a single octave. Now go forth, young musical guardian — simplify with clarity and listen for ratios in every song!