Ratios in Music for a 13-year-old: Simplifying Ratios, Finding Equivalents, and Building the Pythagorean 7‑Note Scale (ACARA v9 mapped)

A pinch of math, a dash of music — the recipe

Imagine you are a chef in a warm kitchen. Instead of salt and pepper, you use beats and pitches. Ratios are your measuring spoons. This lesson shows, step‑by‑step, how to simplify and compare those measuring spoons, how rhythm and harmony are built from ratios, and how to recreate the Pythagorean 7‑note scale — with delicious, neat numbers so you can taste the math.

1) What is a ratio? How to simplify it

A ratio compares two quantities. We write it as a:b, a/b, or "a to b." To simplify, divide both numbers by their greatest common divisor (GCD).

1. Example: simplify 48:32. GCD(48,32)=16. Divide both by 16: 48/16 : 32/16 = 3:2. So 48:32 is equivalent to the simple ratio 3:2.
2. Why care? Simpler ratios often describe clearer relationships in music (e.g., 2:1 is an octave, 3:2 is a perfect fifth).

2) Finding equivalent ratios using proportions

Two ratios a:b and c:d are equivalent when a/b = c/d. Use cross multiplication to check or to find a missing number.

• Check: Are 3:2 and 9:6 equivalent? Cross multiply: 3*6 = 18 and 2*9 = 18  → yes, equivalent.
• Find missing: If 3:2 = x:10, then 3*10 = 2*x → 30 = 2x → x = 15. So 15:10 is equivalent to 3:2.

3) Rhythm as ratios — a quick, tasty exercise

Rhythm divides time. If a teacher claps steady beats, students can clap at a different ratio to that beat.

• 4:1 — students clap once for every 4 teacher beats (slow, roomy).
• 1:4 — students clap four times while the teacher claps once (fast subdivision).
• 3:2 — play a polyrhythm where one person plays 3 evenly spaced notes in the same time another plays 2. That gives a slightly ‘spicey’ cross‑rhythm.

Activity: Count out loud (1,2,3,4...) and try the 4:2 case: clap together on counts 1 and 3. That shows how subdivisions sit evenly inside the beat.

4) Pythagoras, the monochord, and building a scale

Pythagoras discovered pleasing pitch relationships by dividing a single string (a monochord). He used the ratio 3:2, the perfect fifth, and stacked fifths to make a seven‑note scale. The method:

1. Start with a root note (call it 1:1). We will use C = 256 Hz because it makes numbers tidy — and its a nice, simple base to multiply by powers of 2.
2. Use the perfect fifth ratio 3:2. To get other notes, multiply or divide by 3/2 repeatedly and then adjust by octaves (divide or multiply by 2) to bring the result into the same octave range (between 1 and 2 times the root).

The seven Pythagorean ratios (relative to C = 1)

• C (1) = 1/1
• D (2) = 9/8
• E (3) = 81/64
• F (4) = 4/3
• G (5) = 3/2
• A (6) = 27/16
• B (7) = 243/128
• High C = 2/1 (octave)

These come from stacking fifths (3/2) and moving results into the base octave by multiplying or dividing by 2 until each ratio lies between 1 and 2. For example, D = (3/2)^2 / 2 = 9/8.

If C = 256 Hz, compute actual frequencies

• C = 256 * 1 = 256 Hz
• D = 256 * 9/8 = 256 * 1.125 = 288 Hz
• E = 256 * 81/64 = 256 * 1.265625 = 324 Hz
• F = 256 * 4/3 = 256 * 1.333... = 341.333... Hz (≈341.33 Hz)
• G = 256 * 3/2 = 256 * 1.5 = 384 Hz
• A = 256 * 27/16 = 256 * 1.6875 = 432 Hz
• B = 256 * 243/128 = 256 * 1.8984375 = 486.5 Hz
• High C = 256 * 2 = 512 Hz

Note how some results are whole numbers (because 256 divides many denominators), which makes it lovely and easy to compute by hand.

5) Calculating the ratio between two pitches (intervals)

To find the interval between two notes, divide the higher frequency by the lower frequency, then simplify the fraction.

1. Example: C (256 Hz) and G (384 Hz). Ratio = 384 / 256 = 3/2  → perfect fifth, a very simple and consonant ratio.
2. Example: C (256) and E (324). Ratio = 324 / 256 = 81/64. That is the Pythagorean major third. Its a little sharper than the "just" major third (which is 5/4 = 1.25), so the Pythagorean third sounds slightly different.

In general, the simpler the numbers in the ratio (like 2:1 or 3:2), the more the interval tends to sound consonant or "pleasant" to most listeners. More complex ratios (large numbers) often sound more dissonant or tense.

6) Classroom activities you can follow (brief)

1. Rhythm clapping: Teacher claps steady pulse. Students practice 4:1, 1:4, 3:2. Discuss which felt "simple" or "busy."
2. Pythagorean handout: Students compute the seven ratios as above, using C = 256 or any root frequency they choose.
3. Interval listening: Using a keyboard or online tool, hold down C and compare each scale note. Students describe the character (bright, warm, tense) and then compute the ratio and simplify it.
4. Reflection vote: Which intervals sounded most pleasant? Compare the vote to the simplicity of the ratios.

7) Quick answers to likely questions

• Why pick 256 Hz for C? Its a pedagogic choice — powers of two keep denominators neat so students can compute by hand. In the real world, concert A is often tuned near 440 Hz and middle C ≈ 261.63 Hz.
• Is Pythagorean tuning the same as modern tuning? No — modern equal temperament spreads small differences across the 12 notes so instruments can play in every key. Pythagorean tuning emphasizes perfect fifths, so some thirds sound different than in equal temperament.

Mapped to ACARA v9 (brief)

This lesson links to Mathematics (Number and Algebra: ratios and rates — simplifying ratios, equivalent ratios, using proportions) and to the Music curriculum (elements of music — rhythm, pitch and tuning). It helps students practise arithmetic with fractions, simplification, and proportional reasoning while exploring musical concepts.

Final taste: Ratios are tiny recipes that shape rhythm and harmony. Simplify them, compare them, stack fifths like layers in a cake, and youll find the music inside the math — and, perhaps, a little more beauty in both.