Higher-order Fourier analysis studies functions using nonlinear phase patterns and Gowers uniformity norms, extending the linear characters used in ordinary Fourier analysis. Its motivation includes counting long arithmetic progressions: controlling ordinary Fourier coefficients does not control every higher-order additive pattern.^[ref]

For example, let $p$ be an odd prime number and put $f(x)=\exp (2\pi i{x}^2/p)$ on the finite field ${\mathbb{F}}_p$. Its multiplicative difference $f(x+h)\overline{f(x)}=\exp (2\pi i(2hx+h^2)/p)$ has a linear phase; three successive multiplicative differences are identically one. Consequently its $U^3$ norm is one, although each normalized ordinary Fourier coefficient has magnitude $p^{-1/2}$. The quadratic phase is highly structured even when the linear probes are small. Higher-order inverse theorems describe the structured objects detected by large Gowers uniformity norms.

Saltatory conduction is propagation along a myelinated axon in which action potentials are regenerated mainly at successive nodes of Ranvier. Beneath the myelin, local electric current spreads to the next node and brings its excitable cell membrane toward threshold. The signal does not teleport across an electrically inactive gap: passive spread between nodes and active regeneration at nodes work together.^[ref]

The myelin reduces leakage and the effective membrane capacitance that must be charged along an insulated segment. This allows rapid transmission without requiring an enormous axon diameter. Removing the insulation can slow propagation or prevent the next node reaching threshold; axon continuity alone therefore does not guarantee reliable signaling.^[ref]

Praat is software for analyzing, synthesizing, annotating and manipulating speech, created by Paul Boersma and David Weenink. It makes the connection between a recorded acoustic signal and measurable speech properties explicit: a waveform, a spectrogram, a pitch contour and estimated formants describe different aspects of the same recording.^[ref]

For a sustained vowel, one can compare the repeating structure of the waveform with the estimated fundamental frequency, then examine how the frequency spectrum is shaped by the vocal tract. A change in vowel quality need not be a change in pitch. Annotations can align intervals with phonemes or syllables without treating every acoustic boundary as an unambiguous linguistic boundary.

Measurements depend on the analysis settings and the signal. In particular, a displayed pitch contour can contain octave errors, while estimated formants depend on the analysis model. Inspecting the recorded sound and analysis assumptions is therefore part of the experiment, rather than accepting every displayed trajectory as ground truth.

A formant is a resonant spectral feature associated with the filtering of speech by the vocal tract. Its centre frequency and bandwidth describe different aspects of the feature; the first few formants are conventionally labeled $F_1,F_2,\dots$. Praat represents estimated formant structure as time-sampled centre frequencies and bandwidths.^[ref]

A simple physical model treats the vocal tract as a uniform tube, closed at one end and open at the other. Under those ideal boundary conditions its resonance frequencies arewhere $c$ is the speed of sound and $L$ the tube length. For $L=0.17\mathrm{m}$ and $c=340\mathrm{m}{\mathrm{s}}^{-1}$, the first three model resonances are 500, 1500 and 2500 hertz. A real vocal tract has a varying cross-section, losses and moving articulators, so this illustrates the mechanism rather than predicting every vowel.

These resonances are distinct from the fundamental frequency of vocal fold vibration. The source-filter model separates the repeating excitation from the filter's spectral envelope, explaining how a speaker can change vowel quality without making the same change in pitch.