Teacher orientation
This week asks: Why can a mathematically measured pitch still require musical judgment? The three meetings move from shared experience to explicit theory and then transfer. Require students to predict before measuring or labeling; use performance and explanation together as evidence.
The tuner determines the only correct pitch; +10 cents means 10 Hz; equal temperament makes every interval acoustically pure.
Three-meeting sequence
Frequency and ratio
Connect frequency ratios to intervals
Measure A3, A4, and A5 and test the octave prediction.
Review logarithmic pitch, octave doubling, and simple-ratio consonances without overstating a single tuning system.
Lab teams predict, measure, graph, and explain octave and fifth relationships.
Why is pitch spacing logarithmic rather than linear?
Solve frequency-ratio problems.
Cents and beating
Measure cents deviation
Sound two nearly matched tones and listen for pulses.
Define 100 cents per equal-tempered semitone; connect cents sign to sharp/flat and beating to interference.
Students use CREPE to approach target notes from both directions and record stable versus unstable readings.
What does –18 cents mean?
Complete a tuning trial table and reflection.
Temperament is compromise
Distinguish tuning from ensemble intonation
Compare a pure fifth and equal-tempered fifth where equipment allows.
Introduce just intonation, equal temperament, chord context, and why ensemble intonation is relational.
Sections tune roots, thirds, and fifths in contrasting chord roles, then discuss detector limits.
Why should a tuner not replace listening?
Write a lab conclusion with limitations.
Student synthesis task
Claim–evidence–application: Students submit one notated or analytical artifact from the week, a 150–250 word explanation using at least four vocabulary terms, and a brief performance, recording, or demonstration. The explanation must identify a decision, cite musical evidence, and connect the idea to unfamiliar material.
Assessment and answer guidance
| Prompt | Value |
|---|---|
| 1. How many cents are in an equal-tempered octave? | 2 points |
| 2. What causes audible beating? | 3 points |
| 3. Why may a major third be adjusted in ensemble playing? | 3 points |
| 4. Apply the week’s central idea to a new four-to-eight-measure example and explain the reasoning. | 7 points |
Teacher answer guidance
1. 1200.
2. Interference between nearby frequencies creates periodic amplitude fluctuation.
3. Players may favor a purer ratio and chord blend rather than equal-tempered placement, depending on context.
4. Award 2 points for accurate identification or construction, 2 for correct notation/terminology, 2 for evidence tied to the example, and 1 for a clear conclusion. Accept alternate solutions when the musical reasoning is defensible.
Access, differentiation, and extension
Support without lowering the target
Keep calculations to octaves and use the visual meter before logarithms. Preteach vocabulary, model one complete think-aloud, reduce the number of simultaneous variables, and allow a second attempt after feedback.
Extension
Calculate equal-tempered frequency with f=440×2^(n/12).
Evidence to retain
Keep the exit tickets, synthesis artifact, and scored application prompt. Together they show factual fluency, transfer, and the student’s ability to explain music—not only select an answer.
Supplementary laboratory chapters
These references provide extended acoustics and intonation work.