Acoustic Pedagogy — User Guide
Fifteen interactive demonstrations of tuning, psychoacoustics, synthesis, modulation, and interference. Each run generates the relevant sound examples and can pair them with a measured, process-oriented visualization.
Overview
Acoustic Pedagogy is a set of fifteen self-contained listening demonstrations generated directly in Praat. The demonstrations make specific acoustic relationships audible and, when visualization is enabled, place the generated waveforms beside measured spectra, a compact statement of the acoustic mechanism, and a phenomenon-specific proof diagram.
Demonstration Categories
| Category | Phenomena | Focus |
|---|---|---|
| Tuning and intervals | 1–4 | 3:2 fifth, Pythagorean comma, syntonic comma, Pythagorean wolf fifth |
| Psychoacoustics | 5–8 | Critical-band roughness, difference-tone perception, missing fundamental, binaural beats |
| Synthesis and timbre | 9–13 | Fourier approximation, Shepard-Risset cycle, harmonic series, quadratic combination tones, formant-envelope synthesis |
| Modulation and interference | 14–15 | AM/FM comparison and phase cancellation |
The script synthesizes at 44.1 kHz. Demonstrations are mono unless stereo routing is part of the phenomenon; Demo 8 is the stereo exception, with one tone in each ear.
Quick start
- Open Praat. No pre-existing Sound object is required.
- Run
Acoustic Pedagogy.praat. - Select one of the fifteen phenomena.
- Set Duration, Base frequency, and Amplitude. Not every demonstration uses the base-frequency value; Demos 6, 7, and 12 use fixed frequencies.
- Choose whether to show the Info window, retain generated sounds in the Objects list, show the visualization, or display the built-in help.
- Click OK.
- Demo 8: use headphones. The left and right channels contain different frequencies.
- Demos 1–5: the relevant tones are mixed into the same acoustic channel so interval discrepancy or roughness can be heard directly.
- Demo 10: the script generates one Shepard-Risset ascent cycle. Repetition or crossfading of cycles is needed to extend the cyclic-ascent illusion.
Phenomena Guide
1. Just Intonation (Perfect Fifth)
3:2 frequency ratio
Generated sound: two sine tones at f and 1.5f, mixed into the same mono channel.
Acoustic law: f2/f1 = 3/2, corresponding to approximately 701.96 cents.
Visualization proof: a ParamCurve/Lissajous plot of Tone A against Tone B.
2. Pythagorean Comma
Closure error after twelve fifths
Generated sound: the base frequency and a second tone scaled by (3/2)^12 / 2^7, mixed into the same mono channel.
Acoustic law: the ratio is the Pythagorean comma, about 23.46 cents. The small frequency difference produces a slow acoustic beat.
Visualization proof: a ParamCurve/Lissajous comparison of the two close frequencies.
3. Syntonic Comma
5:4 versus 81:64 major third
Generated sound: a just major third at 5/4 × base and a Pythagorean major third at 81/64 × base, mixed into the same mono channel.
Acoustic law: their ratio is 81/80, about 21.51 cents.
Visualization proof: a ParamCurve/Lissajous comparison of the two tunings.
4. Pythagorean Wolf Fifth
Pure fifth versus comma-narrowed fifth
Generated sound: two harmonic-rich dyads are played in sequence: a pure 3/2 fifth and a Pythagorean wolf defined as (3/2) / Pythagorean comma.
Implementation: each dyad uses up to eight 1/n-weighted harmonics, omitting partials that would exceed the Nyquist limit.
Visualization proof: the pure and wolf interval sizes are shown directly in cents.
5. Critical-Band Roughness
Two tones placed within an ERB-scale auditory band
Generated sound: two same-ear sine tones separated by 0.25 × ERB at the selected base frequency.
ERB estimate: 24.7 × (4.37f/1000 + 1). The spacing therefore changes with centre frequency; it is not a fixed 25-Hz interval.
Visualization proof: both tones are drawn inside a one-ERB frequency region.
6. Tartini Difference Tone
Perceived difference tone with no synthesized difference component
Generated sound: a same-ear mixture of 440 Hz and 660 Hz.
Key distinction: the script synthesizes only the two primaries. A 220-Hz component is absent from the physical source; a perceived difference tone may arise from auditory nonlinearity.
Visualization proof: the 440- and 660-Hz primaries are marked as physical components and 220 Hz is marked as perceived/absent.
7. Missing Fundamental (Phantom)
Periodicity without the fundamental component
Generated sound: 660, 880, and 1100 Hz, corresponding to harmonics 3, 4, and 5 of a 220-Hz fundamental.
Reference: after the complex, the script also plays a separate 220-Hz sine reference.
Key distinction: 220 Hz is absent from the complex spectrum, although pitch can follow the 220-Hz periodicity implied by the present harmonics.
8. Binaural Beats (Headphones)
Separate-frequency stimulation across ears
Generated sound: the left channel receives the selected base frequency and the right channel receives base + 4 Hz.
Key distinction: neither channel contains a 4-Hz acoustic amplitude beat. The 4-Hz difference is produced by comparing the two ear-specific frequencies.
Output: stereo; headphones are required for the intended routing.
9. Fourier Square Wave
Truncated odd-harmonic Fourier approximation
Generated sound: an odd-harmonic sum with coefficients proportional to 1/n.
Implementation: the candidate set is harmonics 1, 3, 5, 7, 9, 11, and 13; only terms below 0.98 × Nyquist are included. At least the fundamental and third harmonic must be available.
Reference: a sine wave at the fundamental is retained for visualization comparison.
10. Shepard-Risset Cyclic Ascent
One octave of cyclic spectral ascent
Generated sound: octave-spaced components rise exponentially by one octave over the selected duration while moving through a fixed log-frequency Gaussian envelope.
Implementation: candidate components span octave offsets −4 to +4 around the base and are retained only when their start and end frequencies remain in the usable range. At least two components are required.
Scope: one ascent cycle is generated. Repeating or crossfading cycles extends the cyclic-ascent illusion.
11. Harmonic Series
Integer harmonics with 1/n amplitudes
Generated sound: harmonics at n × f0 with amplitudes proportional to 1/n.
Implementation: up to 16 harmonics are synthesized, limited by 0.98 × Nyquist. At least two harmonics must remain available.
Reference: a fundamental-only sine is used for comparison in the visualization.
12. Nonlinear Combination Tones
Explicit quadratic nonlinearity
Linear source: 400 Hz + 600 Hz.
Nonlinear model: y = a·x + q·a·(x² − 1) with q = 0.60.
Generated components: the quadratic term creates physical energy at the difference frequency 200 Hz, second harmonics 800 and 1200 Hz, and sum frequency 1000 Hz, in addition to the two primaries.
Key distinction: unlike Demo 6, these additional components are explicitly present in the synthesized signal.
13. Formant Synthesis (Vowel /a/)
Harmonic-source spectral-envelope model
Source: a harmonic series at integer multiples of the selected base frequency, with source amplitudes proportional to 1/n.
Shaping: the amplitudes of those existing harmonics are reweighted by a spectral envelope with peaks near 700, 1220, and 2600 Hz. Harmonic frequencies themselves do not move.
Implementation: up to 40 source harmonics are used, subject to Nyquist; at least three are required.
14. AM vs FM (Tremolo / Vibrato)
Level modulation versus small-deviation frequency modulation
Carrier: normally 2 × base; if that approaches Nyquist, the base frequency itself is used.
AM: 5-Hz modulation with depth 0.8.
FM: 5-Hz modulator with phase-modulation index β = 1, corresponding to a ±5-Hz peak frequency deviation.
Playback: the AM example is played first, followed by the FM example.
15. Phase Cancellation
Equal-amplitude 180° inversion
Generated sounds: a sine at the selected base frequency and an equal-amplitude inverted copy.
Sum: the script explicitly creates x(t) + [−x(t)], whose theoretical result is zero when amplitudes and phase are exact.
Playback: normal signal, inverted signal, then their sum.
Parameters
Playback Controls
| Parameter | Default | Code-enforced behavior |
|---|---|---|
| Duration (s) | 2.0 | Must be positive and at least 0.05 s. There is no hard maximum in the script. |
| Base frequency (Hz) | 220 | Must be positive and below 90% of Nyquist globally. Individual demonstrations impose additional limits when upper tones or required harmonics would exceed the usable frequency range. |
| Amplitude | 0.5 | Must be positive. Values above 0.95 are internally clamped to 0.95. |
Options
| Parameter | Default | Behavior |
|---|---|---|
| Show info window | Enabled | Clears the Info window at the start and writes the selected phenomenon, run parameters, and phenomenon-specific numerical details. |
| Save sounds to list | Disabled | When disabled, generated demonstration and reference objects are removed after playback. When enabled, the generated objects remain in the Objects list. |
| Show visualization | Enabled | Draws the four-panel visualization and summary bar. |
| Show help | Disabled | Displays the built-in v0.5.1 phenomenon summary and exits without running a demonstration. |
Base-frequency dependence
Demos 6, 7, and 12 use fixed demonstration frequencies and therefore do not derive their primaries from the Base frequency field. The other demonstrations use the base frequency directly or as part of their construction.
Visualization
The visualization is organized as a process display rather than a text explanation. It combines generated waveforms and measured spectra with the governing acoustic relation and a compact phenomenon-specific diagram.
| Panel | Content |
|---|---|
| A — TIME DOMAIN | Short waveform view of the compared or combined sounds. Labels identify the sound roles used in the current demonstration. |
| B — MEASURED SPECTRUM | Spectra calculated from the generated Sound objects. The displayed frequency range is selected for the current phenomenon rather than fixed at 5 kHz. |
| C — MECHANISM | Three short statements giving the operative acoustic relation, the key measured or constructed quantity, and the relevant routing or interpretation. |
| D — PROOF | A phenomenon-specific visual construction: ParamCurve, interval markers, ERB region, component map, harmonic stems, octave trajectories, formant envelope, modulation curves, or the cancellation sum. |
| Summary bar | Compact numerical summary for the current run. |
The spectrum panel is based on Praat Spectrum objects created from the generated sounds. For plots that include a multichannel object, a temporary mono copy is used only for visualization.
Teaching Applications
Classroom groupings
Tuning and interval structure: Demos 1–4 compare simple ratios and tuning-system discrepancies directly in sound and cents.
Psychoacoustics: Demos 5–8 separate four different mechanisms: same-ear roughness, perceived difference tones, missing-fundamental periodicity, and binaural comparison across ears.
Synthesis and spectrum: Demos 9–13 connect harmonic construction, cyclic spectral motion, nonlinear component generation, and spectral-envelope shaping to measured spectra.
Modulation and interference: Demos 14–15 contrast amplitude/frequency modulation and demonstrate exact destructive interference.
Useful comparisons
- Demo 6 vs Demo 12: compare a perceived difference tone that is absent from the synthesized source with combination components deliberately created by a quadratic nonlinearity.
- Demo 7 vs Demo 11: compare an incomplete harmonic set whose fundamental is absent with an explicitly synthesized harmonic series whose fundamental is present.
- Demo 8 vs same-ear beating: use headphones to distinguish binaural frequency difference from an acoustic amplitude beat in a mono mixture.
- Demo 9 vs Demo 11: compare an odd-only
1/nseries with a full integer harmonic series. - Demo 14: compare level fluctuation in AM with instantaneous-frequency fluctuation in small-deviation FM while keeping the modulator at 5 Hz.