Overview — served warm and simply
Imagine music as a kitchen where notes are ingredients and ratios are the recipes. In this single lesson you will taste how simple fractions and proportions make rhythms breathe and pitches sing. We will clap rhythms, build the Pythagorean seven‑note scale for C, and calculate the ratios that make intervals feel ‘pleasant’ or ‘spicy.’
Essential question
What role do ratios play in the Western musical concepts of rhythm and harmony?
Objectives
- Know what a ratio is and how to write and simplify one.
- Find equivalent ratios using proportions.
- Define rhythm, interval and harmony and explain how ratios relate to each.
- Recreate the Pythagorean seven‑note scale (in C) and calculate interval ratios.
Lesson plan step by step
1. Warm up: watch and taste the idea (5 minutes)
Play the short clip Musical Ratios. Ask: why do different objects make different sounds? Introduce the idea that ratios describe relationships — in music that often means the relationship between two frequencies or two rhythms.
2. Rhythm tasting: clap and count (10 minutes)
- Show the rhythm image that lists ratios like 4:1, 1:4, 4:2 and so on.
- The teacher keeps the steady beat at a slow pulse. Students clap the red part: if the ratio is 4:1 students clap once while the teacher claps four times; if 1:4 students clap four times while the teacher claps once.
- Try polyrhythms: e.g. one group claps 3 per beat while another claps 2 per beat. Notice where they align. This is a 3:2 polyrhythm.
Key math idea for rhythm: ratios tell us how many events happen in the same time span. 4:1 means 4 events for every 1 event of the other part.
3. Polyrhythm listening and reflection (5 minutes)
Play Polyrhythms clip. Have students write which performance sounded more complex. Ask: does a larger or less 'simple' ratio feel more complex?
4. Pitch and the monochord story (5 minutes)
Tell the story: Pythagoras plucked strings on a monochord and found simple string‑length ratios produced pleasing sounds. Introduce the octave as 2:1, the perfect fifth as 3:2, and the perfect fourth as 4:3.
5. Guided math activity: build the Pythagorean scale in C (20 minutes)
Hand out the Calculating the Pythagorean Scale worksheet. Work through this example with the class and then let students finish in pairs.
How the scale is made (a gentle demonstration)
Start with C as 1/1. Pythagoreans stack perfect fifths (ratio 3:2) and move notes into the same octave by multiplying or dividing by 2 as needed.
Stacking fifths gives these raw values:
- C = 1/1
- G = 3/2
- D = (3/2)*(3/2) = 9/4 which reduced by an octave (divide by 2) is 9/8
- A = (3/2)^3 = 27/8 reduced by two octaves becomes 27/16
- E = (3/2)^4 = 81/16 reduced by three octaves becomes 81/64
- B = (3/2)^5 = 243/32 reduced by four octaves becomes 243/128
- F is the fifth below C: (3/2)^{-1} = 2/3, raise an octave to 4/3
- Upper C = 2/1
So the Pythagorean C major scale ratios (relative to C) are:
- C 1/1
- D 9/8
- E 81/64
- F 4/3
- G 3/2
- A 27/16
- B 243/128
- C 2/1
Why these look odd but feel right
Pythagorean tuning privileges perfect fifths (3:2) and constructs other notes by stacking fifths and shifting octaves. The resulting major third 81/64 is slightly sharper than the modern equal tempered third, but it comes from clean 3s and 2s — like honeyed simplicity.
6. Listening lab with the tech tool (10 minutes)
Pairs use the TeachRock tool that plays the notes. One student holds C (1) while the other plays 2, 3, 4 etc. Students describe how each interval sounds and then fill in the Harmony and Interval Chart.
7. Calculate interval ratios (10 minutes)
Using the worksheet, students compute pairwise ratios between C and each other note and simplify them. Example:
Interval C to G: C = 1/1, G = 3/2. The ratio is 3/2 which simplified is 3:2. Interval C to E: 81/64. Ratio 81:64 cannot be simplified further (both divisible by 1 only) so it stays 81:64.
8. Discussion and vote (5 minutes)
For each interval (the seven rows), have a show of hands on which sounded most pleasant. Ask: is there a connection between how simple a ratio is and how pleasant it sounds? Invite students to explain.
9. Exit task and consolidation
Students complete the Ratio Word Problems handout to practise simplifying ratios and finding equivalent ratios by proportional reasoning. As an extension, interested students can look at sound wave graphs and the Golden Ratio exercise.
Worked examples (short and clear)
Simplifying ratios
- Example 1: 4:2. Greatest common divisor is 2, so 4:2 = 2:1.
- Example 2: Are 3:2 and 9:6 equivalent? Multiply 3:2 by 3 to get 9:6, so yes — they are equivalent ratios.
Pythagorean derivation example
Find D from C using fifths:
G = 3/2. D = G * 3/2 = 9/4. Move D down an octave: 9/4 divided by 2 = 9/8. So D = 9/8.
ACARA v9 mapping (clear and useful)
This lesson maps to ACARA v9 learning areas as follows:
- Mathematics: Number and Algebra — Ratios and proportional reasoning; use of equivalent ratios and solving proportional problems appropriate to Year 7 (age 13).
- The Arts: Music — Explore elements of music including rhythm, pitch and harmony; experiment with listening and making sounds and describing musical intervals.
- Cross‑curriculum priorities: Literacy — listening and describing; Using digital tools for listening and experimentation.
Teacher comments in a Nigella Lawson cadence
Exemplary feedback (for a student who has excelled):
My dear, your work shone like a perfectly caramelised onion — sweet, precise and utterly satisfying. You simplified ratios with finesse, explained why 3:2 gives a robust fifth, and your Pythagorean calculations for the C scale were faultless. Your listening notes show a sensitive ear: you named textures and linked them to ratio simplicity. For your next delightful challenge, try comparing how the same intervals feel in equal temperament and Pythagorean tuning.
Proficient feedback (for a student meeting expectations):
Nicely done. Your clapping stayed steady, and your ratios were mostly correct. You identified the octave and fifth and made a good attempt at building the Pythagorean scale. To lift this from tasty to exquisite, double check simplifications (look for common factors) and write one sentence linking why a 2:1 interval sounds similar in pitch but an octave apart.
Assessment rubrics — two levels presented like a recipe
Criteria 1: Understanding and using ratios
Exemplary
- Correctly simplifies and finds equivalent ratios in all examples. Uses proportions fluently to justify answers.
Proficient
- Correctly simplifies most ratios and finds equivalent ratios in straightforward cases. Uses proportions with support.
Criteria 2: Rhythm application and polyrhythms
Exemplary
- Accurately performs and explains multiple rhythms, including a clear description of 3:2 polyrhythms and alignment points.
Proficient
- Performs rhythms correctly and describes simple relationships between them; may need help with identifying alignment points in polyrhythms.
Criteria 3: Construction of the Pythagorean scale and interval calculations
Exemplary
- Constructs the full scale correctly (C, D, E, F, G, A, B, C) with correct ratios (1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1), shows steps of stacking 3/2 and octave adjustment, and simplifies interval ratios correctly.
Proficient
- Constructs most scale notes correctly and gives the main interval ratios (2:1, 3:2, 4:3). May need support showing each arithmetic step when stacking fifths and moving octaves.
Criteria 4: Communication and reflection
Exemplary
- Produces clear written and verbal descriptions of how ratio simplicity relates to perceived pleasantness; makes thoughtful connections to historical Pythagorean ideas.
Proficient
- Explains ideas clearly in most parts and links ratios to musical quality, though some reasoning may be brief or need expansion.
Short checklist for teachers
- Have students write each ratio in three forms: a:b, a/b and words (for example 3:2, 3/2, 'three to two').
- Confirm each student can simplify at least five ratios correctly from the Ratio Word Problems handout.
- Observe each pair in the listening lab and ensure they can identify and label the intervals they hear against the chart.
Final note — a pleasant serving idea
End with Two Melodies. Ask students: using the ratios they calculated today, why might the two melodies feel different? Encourage them to answer with both numbers and sensations: sometimes the maths tastes bright and sparkly, sometimes warm and full-bodied. Both are music.