Listen Up — You Will Learn This, and You Will Do It Well.
We are going to learn what ratios are, how to simplify them, how to find equivalent ratios using proportions, and how those ratios make music — in rhythm and in pitch. You will recreate the Pythagorean 7‑note scale and calculate the ratios between notes. No shortcuts. Pay attention, follow the steps, and show your reasoning.
1) What is a ratio? (Simple, clear, and useful)
- A ratio compares two quantities: a:b (read "a to b"), or a/b as a fraction.
- Examples you will use in class: 4:1 (one sound every four beats), 1:4 (four sounds every one beat), 3:2 (three against two, a common polyrhythm).
- To simplify a ratio, divide both parts by their greatest common factor (GCF). Example: 6:8 → divide by 2 → 3:4.
- Equivalent ratios: multiply or divide both terms by the same number. Example: 1:2 = 2:4 = 3:6.
2) Use proportions to find equivalent ratios (Step‑by‑step)
Proportion: a/b = c/d. If you know three pieces, you can find the fourth.
- Example problem: If 2 notes fit into 3 beats, how many notes fit into 6 beats? Set proportion 2/3 = x/6. Cross‑multiply: 2*6 = 3*x → 12 = 3x → x = 4. So 4 notes in 6 beats.
- Practice converting ratios and fractions, simplification, and solving x in proportions. Show your cross‑multiplication each time.
3) Rhythm: how ratios make music move
Rhythm divides time. The beat is the steady pulse. Subdividing the beat creates these ratios:
- 4:1 — one sound every 4 beats (slow relative to the beat)
- 1:4 — four sounds inside one beat (fast subdivision)
- 3:2 — three sounds against two (triplet feel against duple)
When two rhythms happen at once (polyrhythm), the ratio between their subdivisions determines how complex the pattern sounds. Simpler ratios (1:1, 2:1, 3:2) often sound more stable; larger, non‑simple ratios (5:4, 7:4) sound busier.
4) Pitch and harmony: the Pythagorean idea (practical math you must do)
Pythagoras discovered that pleasing pitch relationships are simple ratios of string length or frequency. In Western music the octave = 2:1. A perfect fifth = 3:2. By stacking perfect fifths and adjusting by octaves (factors of 2), Pythagoras and later theorists produced a 7‑note scale.
Build the Pythagorean diatonic scale (step‑by‑step)
Choose a root note and call it 1 (this is a ratio, not a frequency unit). We will produce the C major diatonic set using Pythagorean stacking of fifths:
- C (tonic) = 1/1
- Go up a perfect fifth: multiply by 3/2 → G = 3/2. If result is >2 (bigger than one octave), divide by 2 until it lies between 1 and 2.
- Keep stacking fifths and reduce by octaves (divide by 2 as needed). Then re‑order values so all notes lie between 1 and 2 (the same octave).
Standard Pythagorean ratios for the C scale (relative to C = 1:1)
- C = 1/1
- D = 9/8 (this comes from two stacked fifths: (3/2)*(3/2) = 9/4 → divide by 2 → 9/8)
- E = 81/64 (three stacked fifths reduced by octaves)
- F = 4/3
- G = 3/2
- A = 27/16
- B = 243/128
- Octave C = 2/1
These are exact ratios. Compare them: the interval from C to G is 3:2; from C to E is 81:64; from C to F is 4:3. Simplify and compare fractions when needed.
5) How to calculate an interval ratio between two notes
Take the note ratios (as fractions) and form a ratio: if note1 = a and note2 = b, then interval = b:a (or b/a). Simplify the fraction if possible.
Example: Interval C (1) to E (81/64) → ratio = (81/64) : 1 = 81:64. Simplify? 81 and 64 have no common factors (81 = 3^4, 64 = 2^6) so 81:64 is simplest. You can convert to decimal if helpful: 81/64 = 1.265625.
6) Classroom activities (do these, and document your math)
- Rhythm clapping: teacher keeps steady beat. Students clap subdivisions for 4:1, 1:4, 3:2, 4:3. Write the ratio, simplify, and explain where your claps fall in the beat.
- Pythagorean handout: follow the calculations, show each multiplication by 3/2 and each division by 2. Write final ratios in simplest form.
- Interval listening: hold C (1) and press each other note. Describe sound (consonant, bright, tense). Record the ratio you hear and calculate b/a for each interval.
- Word problems: solve ratio problems by simplifying and using proportions. Show all steps.
7) Quick answers to the motivating questions
- Different objects produce different sounds because they vibrate at different frequencies. Ratios compare those frequencies.
- Ratios describe the relationships between two frequencies (or durations in rhythm).
- 2:1 is an octave.
- Pythagoras used a monochord — a single string with moveable bridge — to measure ratio lengths that produced pleasant intervals.
8) ACARA v9 mapping (clear and classroom‑ready)
Mapped learning areas and outcomes (v9 style summary):
- Mathematics — Number and Algebra: Understand ratio concepts, use ratio notation, simplify ratios, find equivalent ratios and solve problems using proportions.
- The Arts (Music): Explore rhythm, meter, pitch relationships, intervals and tuning systems; develop listening skills to describe intervals and harmonic quality.
- Interdisciplinary: Apply mathematics to real‑world context (music), interpret and represent relationships using numbers and fractions.
9) Teacher comments — strict, high expectations (Tiger‑Mother cadence, but kind)
Excellent work is not negotiable. You will show every step, or you will redo the task until it is flawless. If a student arrives with errors, we correct the reasoning immediately: where did the simplification fail? Where was the proportion set up incorrectly? Never accept an answer without the method.
When students listen and describe intervals, push beyond "nice" or "weird." Ask for comparisons: "How does 3:2 (perfect fifth) feel different than 81:64 (major third in Pythagorean tuning)?" Ask them to justify their descriptions using the ratio simplicity and decimal closeness to small whole number ratios.
10) Extended rubrics — what I expect. Two levels: Exemplary and Proficient
Criteria 1: Mathematical knowledge & accuracy
Exemplary: All ratios are simplified correctly. Equivalent ratios and proportions are solved with correct cross‑multiplication, showing every algebraic step. Pythagorean scale calculations (stacking fifths and octave reductions) are accurate and documented. Work is neat and no arithmetic errors.
Proficient: Most ratios simplified correctly (≤1 minor arithmetic slip). Proportions are set up correctly and solved with correct method but may contain one small calculation error. Pythagorean ratios correct with one or two minor presentation issues.
Criteria 2: Application to music (rhythm & harmony)
Exemplary: Student accurately maps rhythm ratios to clapping examples and explains why polyrhythms sound more/less complex. Student re‑creates the 7‑note Pythagorean scale (ratios for all seven notes) and correctly computes interval ratios from the tonic, explaining sonic qualities in terms of ratio simplicity.
Proficient: Student performs rhythm tasks and can state the ratio relationships. Student lists most Pythagorean ratios correctly (may miss one) and gives reasonable descriptions of interval qualities without detailed ratio‑based justification.
Criteria 3: Reasoning and communication
Exemplary: Student explains every step clearly, justifies choices (why divide by 2 to fit an octave), and connects ratio simplicity to perceived consonance. Responses show confident use of terms: ratio, proportion, interval, octave, perfect fifth.
Proficient: Student gives clear answers but explanations may lack depth. Uses correct terminology but may not fully connect numeric ratio complexity to perceptual effects.
Criteria 4: Evidence of practice and reflection
Exemplary: Student completes extension tasks (e.g. waveform graphs or phi‑point activity), reflects on the golden ratio observation for a chosen song, and can discuss limitations of Pythagorean tuning.
Proficient: Student completes main tasks, attempts at least one extension, and provides a basic reflection; shows awareness that different tuning systems exist.
11) Short sample teacher comments for report or feedback (use as templates)
- Exemplary comment (firm praise): "Outstanding work. You reliably simplify ratios, set up and solve proportions, and your Pythagorean calculations are precise. Your listening descriptions tied clearly to the numeric ratios. Keep this standard — the math and the music meet because you cared about both."
- Proficient comment (encouraging, directive): "Good understanding of ratios and proportions. Your Pythagorean scale is mostly correct; check the arithmetic on the third note. To reach exemplary, show each step when adjusting octaves and explain how ratio simplicity affects consonance in more detail."
12) Quick assessment checklist for the teacher
- Student can simplify ratios and show GCF used.
- Student can solve proportions using cross‑multiplication and show work.
- Student correctly lists the seven Pythagorean ratios for a chosen tonic.
- Student correctly computes interval ratios between tonic and other notes.
- Student gives reasoned descriptions of how interval ratio simplicity relates to consonance/dissonance.
Final note — you must show work
Do not hand me answers without the math. If you make a mistake, trace it. If you are unsure about why 3/2 is important, test it on the monochord or a keyboard and measure. Practice makes exactness. I expect clarity and correct reasoning; your grade follows your precision.