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Chapter 3 The Dynamics of Pitch Contours: Declination, Final Falls and Uptalk
The shape of a sentence’s pitch
When we say a declarative sentence, pitch is not a level line. It is usually higher at the start, rises and falls locally on stressed words, drifts downward overall, and at the end falls further to the low part of the speaker’s range. The overall downward drift is called declination, the extra fall at the end final lowering.
Evidence
Where declination comes from. Declination was long explained as a by-product of breathing: as a breath group goes on, the air pressure in the lungs and below the glottis falls, and the vocal folds vibrate more slowly. But studies found that speakers adjust starting pitch and slope in advance to the length of the sentence, with gentler slopes for longer sentences, showing that declination is at least partly planned and not merely a physiological “running out of air” (Cohen et al. 1982; Ladd 2008).
So the common claim that the natural fall at the end of a sentence “reflects the steady release of subglottal pressure” is only partly right: falling subglottal pressure is one cause of declination, but speakers also actively control it.
The Fujisaki model
Fujisaki and Hirose proposed a widely used model of pitch contours: log fundamental frequency equals a baseline plus smoothed responses to “phrase commands” (impulses) and to “accent commands” (steps) (Fujisaki and Hirose 1984). Phrase commands produce the slow rise-and-fall skeleton of the whole sentence; accent commands add local movements on stressed words.
Calculation
The Fujisaki model generates a 2.4-second declarative sentence (male voice, baseline 85 Hz). The two versions are identical except that the second adds a final accent command, a rise at the end.
# The Fujisaki model of an f0 contour (Fujisaki & Hirose 1984): log f0 is a baseline plus the
# smoothed responses to phrase commands (impulses) and accent commands (steps):
# ln f0(t) = ln Fb + sum Ap * Gp(t - T0) + sum Aa * [Ga(t - T1) - Ga(t - T2)]
# Gp(t) = alpha^2 t exp(-alpha t) (t >= 0, else 0)
# Ga(t) = min(1 - (1 + beta t) exp(-beta t), gamma) (t >= 0, else 0)
# One 2.4-second declarative sentence, a male voice with Fb = 85 Hz. The two versions differ only in
# the last accent: a falling ending versus a final rise (uptalk). Command values are illustrative.
import math
ALPHA, BETA, GAMMA, FB = 3.0, 20.0, 0.9, 85.0
def Gp(t): return ALPHA * ALPHA * t * math.exp(-ALPHA * t) if t >= 0 else 0.0
def Ga(t): return min(1 - (1 + BETA * t) * math.exp(-BETA * t), GAMMA) if t >= 0 else 0.0
def contour(final_rise, T=2.4, dt=0.01):
phrase = [(0.0, 0.30)]
accents = [(0.15, 0.55, 0.22), (0.95, 1.35, 0.18), (1.70, 2.00, 0.12)]
if final_rise:
accents.append((2.05, 2.40, 0.30)) # boundary rise at the end of the sentence
out = []
for i in range(int(T / dt) + 1):
t = i * dt
lnf = math.log(FB) + sum(a * Gp(t - t0) for t0, a in phrase)
lnf += sum(a * (Ga(t - t1) - Ga(t - t2)) for t1, t2, a in accents)
out.append((t, math.exp(lnf)))
return out
if __name__ == "__main__":
for name, rise in (("falling ending", False), ("final rise (uptalk)", True)):
c = contour(rise)
pts = [f"{f:5.0f}" for t, f in c if abs(t * 10 - round(t * 10)) < 1e-6 and round(t * 10) % 4 == 0]
print(f"{name:20s} f0 (Hz) every 0.4 s: " + " ".join(pts))
# falling ending f0 (Hz) every 0.4 s: 85 143 103 109 88 96 85
# final rise (uptalk) f0 (Hz) every 0.4 s: 85 143 103 109 88 96 112
Measuring the final movement
Calculation
The two contours measured as the Chapter 8 checklist does: the declination slope of the whole sentence (in semitones re 100 Hz), the final rise in the last 400 milliseconds, and the fall from the last accent peak to the end.
# Measuring the contours of the previous program the way the Chapter 8 checklist does:
# declination = slope of a straight line fitted to f0 (in semitones re 100 Hz) over the sentence
# final excursion = f0 at the end minus the minimum f0 in the last 400 ms, in semitones
# final fall = f0 at the last accent peak minus f0 at the end, in semitones
# The Fujisaki contours of the previous program, repeated so this one runs on its own.
import math
ALPHA, BETA, GAMMA, FB = 3.0, 20.0, 0.9, 85.0
def Gp(t): return ALPHA * ALPHA * t * math.exp(-ALPHA * t) if t >= 0 else 0.0
def Ga(t): return min(1 - (1 + BETA * t) * math.exp(-BETA * t), GAMMA) if t >= 0 else 0.0
def contour(final_rise, T=2.4, dt=0.01):
phrase = [(0.0, 0.30)]
accents = [(0.15, 0.55, 0.22), (0.95, 1.35, 0.18), (1.70, 2.00, 0.12)]
if final_rise:
accents.append((2.05, 2.40, 0.30)) # boundary rise at the end of the sentence
out = []
for i in range(int(T / dt) + 1):
t = i * dt
lnf = math.log(FB) + sum(a * Gp(t - t0) for t0, a in phrase)
lnf += sum(a * (Ga(t - t1) - Ga(t - t2)) for t1, t2, a in accents)
out.append((t, math.exp(lnf)))
return out
def st(f): return 12 * math.log2(f / 100.0)
def features(c):
ts = [t for t, _ in c]; ys = [st(f) for _, f in c]
n = len(ts); mt = sum(ts) / n; my = sum(ys) / n
slope = sum((t - mt) * (y - my) for t, y in zip(ts, ys)) / sum((t - mt) ** 2 for t in ts)
last = [y for t, y in zip(ts, ys) if t >= ts[-1] - 0.4]
peak = max(y for t, y in zip(ts, ys) if 1.6 <= t <= 2.05)
return slope, ys[-1] - min(last), peak - ys[-1]
for name, rise in (("falling ending", False), ("final rise (uptalk)", True)):
s, exc, fall = features(contour(rise))
kind = "rise" if exc > 1.0 else "fall"
print(f"{name:20s}: declination {s:+.2f} st/s, final excursion {exc:+.2f} st, "
f"fall from last peak {fall:+.2f} st -> classified as final {kind}")
# falling ending : declination -2.93 st/s, final excursion +0.00 st, fall from last peak +2.08 st -> classified as final fall
# final rise (uptalk) : declination -1.77 st/s, final excursion +3.30 st, fall from last peak -2.59 st -> classified as final riseThe falling version drops about 2.1 semitones from the last accent peak; the rising version climbs about 3.3 semitones at the end, finishing about 2.6 semitones above the last accent peak. The declination slopes of the two contours also differ because their endings differ. Measurements like these can be made on one’s own recordings with free software such as Praat.
Does a rise mean uncertainty?
It is often said that a sentence-final rise “reflects tension in the cricothyroid muscle and an uncertain search for approval”. That sentence mixes two things.
First, physiologically, the main way to raise pitch is indeed contraction of the cricothyroid muscle, which lengthens and tenses the vocal folds. But this applies to any rise in pitch, including questions, emphasis and excitement; it does not specifically mean “tension” or “seeking approval”.
Second, socially, the functions of a final rise on a statement (called uptalk in research on English) are much more varied than “uncertainty”.
Evidence
The many functions of uptalk. Warren’s systematic review of rising intonation on English statements shows that it is common in some groups in New Zealand, Australia and North America, and that its functions include checking that the listener is following, signalling that the speaker has not finished, inviting a response and marking new information, not only expressing uncertainty (Warren 2016). Tomlinson and Fox Tree found that rising pitch by itself did not make listeners judge speakers less certain, whereas prolongation did, and that prolonged uptalk sped listeners’ detection of words in the following utterance; context shapes what a rise means (Tomlinson and Fox Tree 2011). A related finding on single words. As Chapter 1 noted, in one reverse-correlation study of the French word bonjour, the prototype judged trustworthy rose at the end of the word while the “dominant” prototype fell (Ponsot et al. 2018); whether this carries over to rises at the end of statements is untested.
A more accurate statement is therefore this: where decisiveness and dominance are called for (announcing a decision, stating a commitment), a final fall may suit better; where warmth and inviting participation are called for, a moderate rise is no weakness. What may really undermine persuasion is perhaps not the rise itself but a mismatch between rise and content: announcing a figure that should be certain in a rising tone may make listeners wonder whether the speaker believes it.
Note
The special case of Chinese speakers. Mandarin is a tone language: every syllable has its own lexical tone, and sentence intonation is laid over the tones; sentence-final particles (ma, ba, ne) also carry part of the work of questioning or suggesting. So the studies above, based on English and French, cannot be transferred directly to Mandarin or Cantonese. Chinese speakers giving talks in English sometimes carry the tonal habits of their first language into English sentence endings, a case that deserves study in its own right.
Hypothesis · not established
The content–intonation consistency hypothesis (not yet established). The effect of a final rise on credibility depends on content: on statements of firm facts or commitments, a rise lowers listeners’ ratings of credibility and of the speaker’s confidence; on invitations or requests for views, a rise does not lower credibility and may raise it. A testable implication: in the same recording, changing only the final movement of “firm statement” sentences and of “invitation” sentences should produce an interaction in credibility ratings. If a rise has the same effect in both kinds of sentence, the hypothesis is weakened.
Summary
The pitch of a declarative sentence usually drifts down and falls at the end; declination comes partly from falling subglottal pressure and partly from the speaker’s planning. The Fujisaki model can generate and decompose such contours, and tools such as Praat can measure the final movement in one’s own recordings. A final rise is produced by the cricothyroid muscle, but its social meaning is not only uncertainty: it can signal that the speaker has not finished, mark new information or invite a response, and in one study of a single word, a final rise was judged more trustworthy. The next chapter turns to another link between speaker and listeners: physiological and neural synchrony.
References
Fan, Y. (2026). Acoustic Persuasion, Prosodic Information Theory and Interpersonal Physiological Synchrony: Pitch Contours, Pause Entropy and Listener Autonomic Coupling in Public Speaking and Leadership [声音说服力、韵律信息论与人际生理同步] (Version 1.0.0). 5D Library. https://doi.org/10.5281/zenodo.23753267 · Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0).