Investigating Musical Ratios and Birdsong — Age 13

This combined Mathematics, Music and Science learning sequence helps a 13-year-old explore proportional reasoning in music (ACMNA166) and sound analysis in ecology using Raven Lite (ACSIS098). It connects the TeachRock musical ratios activities with real-world recordings of birdsong to deepen understanding of rhythm, pitch, species identification and data collection.

Learning objectives

• Use ratios and proportional reasoning to explain musical intervals (octave, fifth, third) and rhythmic relationships.
• Record, visualise and annotate sounds using Raven Lite; interpret spectrogram features for bird and musical sounds.
• Collect reliable field data and reflect on citizen science principles and ethics.
• Produce a short report that links the mathematics of sound to biological examples (birdsong).

Curriculum alignment

• Mathematics: ACMNA166 — Apply ratio and proportional reasoning to musical intervals and rhythms.
• Science: ACSIS098 — Use sound analysis tools, collect and interpret biological data, and consider data reliability and ethics.

Part A — Musical ratios (TeachRock) with worked examples

Key musical ratios to explore:

• Octave: 2 : 1. If A4 is 440 Hz, the octave above (A5) is 440 × 2 = 880 Hz.
• Pythagorean fifth: 3 : 2. If a note is 300 Hz, the fifth above is 300 × 3/2 = 450 Hz.
• Just major third: 5 : 4. If a note is 400 Hz, the major third above is 400 × 5/4 = 500 Hz.

Worked proportional reasoning question:

1. Start with C4 = 261.63 Hz. What is the frequency of C5 (one octave above)? Calculate: 261.63 × 2 = 523.26 Hz.
2. If E4 is 329.63 Hz, what is the frequency of the just major third above (ratio 5:4)? 329.63 × 5/4 = 412.0375 Hz (approx 412.04 Hz).
3. Rhythm example: If a metronome pulse is 60 beats per minute, a dotted quarter note lasting 1.5 beats equals 1.5 seconds. Using ratios, find time values for other note groupings.

Classroom activity — Applying ratios

1. Use TeachRock materials to listen to intervals, then measure frequencies (or use a tuner) to confirm ratio relationships.
2. Have students pick a starting frequency and calculate frequencies for several intervals using fractional multiplication (example worksheet supplied below).
3. Compare theoretical ratios with actual measured frequencies from recorded instruments or synthesized tones; discuss temperaments vs just intonation.

Part B — Raven Lite guide (step-by-step)

Quick start for Raven Lite (Cornell Lab):

1. Install Raven Lite from the Cornell Lab website and open the program.
2. File - Open to load a WAV or MP3 recording, or record directly if your device supports it.
3. Play the file; Raven Lite shows waveform and spectrogram. Zoom in using the time and frequency sliders to focus on sound of interest.
4. To annotate: Select the selection tool, drag over a sound, then press the annotation button and add a label (species name, note, time span, quality of signal).
5. Measure frequency: Use the spectrogram cursor to read frequency peaks (Hz) and identify fundamentals and harmonics.
6. Save selections as separate files if needed: Selection - Save Selection As. Export annotated images via File - Export Image.
7. Record metadata: For each file/note select, record date, time, location, weather, equipment and any confidence in identification.

Safety & ethics notes: Ask permission to record on private property; avoid disturbing wildlife; follow local wildlife recording guidelines.

Field data-sheet template (printable)

Music ratios worksheet (printable)

Rubric — Descriptive and evaluative (Nigella Lawson cadence)

Style note: each descriptor is written with warmth, clarity and a pinch of sensory pleasure — the Nigella Lawson cadence — to give feedback that is graciously specific and encouraging.

Sample marking grid (rubric to numeric scores)

Each criterion scored 1-4. Maximum score = 20.

Formative feedback phrases — Nigella Lawson cadence

Use these short praises or suggestions to annotate student work. They are written with warmth and clear guidance.

Music / Maths (10 examples)

• "Lovely precision — your ratio work is clear and satisfying, like a perfectly caramelised edge."
• "You found the octave with confidence; your steps were neat and comforting to read."
• "A delightful explanation of the fifth — try showing one more intermediate step so others can savour your thinking."
• "Your calculations sing; a tiny slip in rounding only. Do check your final digit."
• "You made a brave attempt at tempered tuning — well done; label the measured frequencies next time for fuller flavour."
• "Beautiful link between rhythm and beat fractions. Try one more example at a different tempo."
• "Your use of proportional reasoning is confident; show the multiplication fraction explicitly to help your reader."
• "A measured, thoughtful approach; tidy up your working so the logic is as smooth as honey."
• "You’re on the right track — the ratio is correct but the explanation needs a touch more detail."
• "Wonderful curiosity shown; next time, compare calculated and measured frequencies to taste the difference."

Science / Raven Lite (10 examples)

• "What an observant ear — your annotation captures the bird’s phrase so well."
• "The spectrogram notes are deliciously clear; add a frequency reading to complete the picture."
• "Like a good recipe, you recorded useful metadata — now add a note about background noise levels."
• "A careful identification; explain in one sentence why you chose this species to strengthen your case."
• "Brilliant use of Raven Lite tools — your selection boxes are neat and persuasive."
• "Your ecological notes are thoughtful; link one trait of the song to likely habitat for extra credit."
• "You considered ethics — beautifully conscientious. Add any permits or permissions to the data sheet."
• "That spectrogram reading was bold and accurate; include harmonic spacing to show the acoustic structure."
• "Your field notes are evocative; next time include a weather check for data reliability."
• "Excellent care with recordings; a brief reflection on uncertainties would make this exquisite."

Sample student report (filled-in example)

Student: Maya Lopez   Age: 13   Date: 2025-10-01

Project title

How musical ratios explain both instrument pitch and birdsong structure — an analysis using TeachRock and Raven Lite.

Aim

To use proportional reasoning to calculate musical intervals and to use Raven Lite to measure frequency components of a local bird call, comparing mathematical expectations with observed spectra.

Methods

• Used TeachRock samples to listen to an A4 tone (440 Hz) and measured with a tuner to confirm.
• Calculated octave and fifth frequencies from A4 and E4 using ratios (see calculations below).
• Recorded a 30-second field clip of a local robin-like bird using a smartphone, then opened the WAV file in Raven Lite.
• Selected three vocal phrases and annotated them with times and frequency peaks.

Calculations

• Octave above A4: 440 × 2 = 880 Hz.
• Perfect fifth above A4 (3:2): 440 × 3/2 = 660 Hz.
• If E4 = 329.63 Hz, major third above (5:4) = 329.63 × 5/4 = 412.04 Hz.

Raven Lite observations

Selection 1: time 2.1 — 2.8 s; fundamental peak at 280 Hz, harmonics at 560 Hz and 840 Hz. Annotation: "Fundamental 280 Hz, clear harmonic series: 1st harmonic 560 Hz 2nd harmonic 840 Hz"

Selection 2: time 5.5 — 6.0 s; frequency-modulated phrase from 300 Hz down to 250 Hz, strong energy between 250–320 Hz.

Selection 3: time 12.0 — 12.6 s; short staccato notes with fundamentals near 420 Hz, likely a different call type.

Spectrogram annotations (textual)

• Annotation A (Selection 1): Visible vertical bands at 280, 560, 840 Hz indicate a harmonic series consistent with a pitched vocalization. Label: "Bird call — fundamental 280 Hz".
• Annotation B (Selection 2): A descending sweep from 300 to 250 Hz lasting 0.5 s. Label: "Frequency modulation — potential territorial call".
• Annotation C (Selection 3): Rapid, repeated notes around 420 Hz; label as "alarm/flight-call candidate" and mark with high confidence = medium.

Interpretation

The instrument calculations matched expectations: octave and fifth were correctly calculated. The bird recordings show fundamentals and harmonics consistent with voiced sounds (not pure whistles). The harmonic spacing (multiples of the fundamental) suggests the bird's vocal tract filters produce resonant frequencies, similar in concept to instrument overtones.

Evaluation and ethics

Data recorded in a public park with minimal disturbance. Confidence levels: Selection 1 = high, Selection 2 = medium, Selection 3 = medium. I reflected on background noise influencing measurements and noted time of day and weather on the field sheet.

Teacher marking (sample)

Teacher comments (Nigella cadence)

"Maya, this is such a pleasing dish of inquiry — the arithmetic is tidy, your spectrogram notes are appetisingly clear, and your reflections on reliability show care. A touch more frequency labelling on Selection 2 would make this perfect. Keep listening closely — your ear is becoming deliciously discerning."

How to adapt and extend

• Extend to group projects where each student analyses a different species or musical interval and compares harmonic structure.
• Compare just intonation (ratios) with equal temperament by generating tones and measuring frequency differences.
• Introduce simple Fourier ideas: explain how complex tones are sums of sinusoids (harmonics shown in spectrograms).

Resources

• TeachRock musical ratios worksheets and audio samples.
• Raven Lite, Cornell Lab of Ornithology: download and user guide.
• Basic tuner or frequency-measuring app for classroom verification.

If you would like, I can: provide a printable PDF of the data sheet and worksheet, create a classroom slide deck, or generate a set of assessment exemplars with annotated spectrogram images for student use.