Chuang, Bell, Tseng and Baayen (2026): Word-specific tonal realizations in Mandarin

The pitch contour of a Mandarin disyllabic word is usually taken to realize the lexical tones of its syllables under constraints such as intonation, coarticulation and speech rate. Chuang, Bell, Tseng and Baayen add the token's meaning in context as a further determinant. On the 3,778 rise–fall (T2–T4) tokens of 51 word types in Fon's corpus of spontaneous Taiwan Mandarin, a generalized additive model with a by-word smooth beats the segmental controls combined (§2.6). A discriminative lexicon trained on pairs of 50-sample normalized contours (§3.2) and 768-dimensional contextualized embeddings (§3.1) predicts word type from contour and contour from embedding an order of magnitude above chance (§3.3–3.4).

The contour that the linear production map produces from a word's centroid embedding closely resembles the average of that word's training contours (Fig. 18), and the centroid of all tokens produces the unmodulated rise–fall contour (§3.4.2). The paper reads this as isomorphy between the space of contours and the space of embeddings, against the dual articulation it attributes to Martinet (§4). The file accordingly works on continuous f0 vectors rather than on discrete tones.

Main statements

• ChuangEtAl2026.comprehension_comp_production_ne_id: no linear map takes the fifty-dimensional contour space onto the 768-dimensional embedding space, so no comprehension map inverts production.
• ChuangEtAl2026.production_centroid_eq_of_decodable: a least-squares-trained production map sends the centroid of a word type's embeddings to the average of its training contours whenever membership in the word type is a linear functional of the embeddings.
• ChuangEtAl2026.production_centroid_univ_eq: the centroid of all tokens produces the grand-mean training contour.

Implementation notes

• A word type is the set of tokens carrying its corpus label (§1, §2.2). Linear decodability idealizes the distinct word-type regions of Fig. 14.
• The additive models, the nearest-neighbour accuracies and the residual network ResLDL are not represented.

References

• Y.-Y. Chuang, M. J. Bell, Y.-H. Tseng and R. H. Baayen, Word-specific tonal realizations in Mandarin (2026)
• R. H. Baayen, Y.-Y. Chuang, E. Shafaei-Bajestan and J. P. Blevins, The discriminative lexicon (2019)
• M. Heitmeier, Y.-Y. Chuang and R. H. Baayen, The Discriminative Lexicon (2026)
• J. Fon, A preliminary construction of Taiwan Southern Min spontaneous speech corpus (2004)

@[reducible, inline]

abbrev ChuangEtAl2026.TaiwanMandarinPitchSampleCount : ℕ

The fifty f0 values sampled per token at equally spaced points of normalized time (§3.2).

Equations

• ChuangEtAl2026.TaiwanMandarinPitchSampleCount = 50

@[reducible, inline]

abbrev ChuangEtAl2026.CKIPGPT2HiddenDim : ℕ

The dimension of CKIP's GPT-2 character vectors (§3.1).

Equations

• ChuangEtAl2026.CKIPGPT2HiddenDim = 768

@[reducible, inline]

abbrev ChuangEtAl2026.PitchContour : Type

A pitch contour consists of the fifty samples, centered by the token's mean and scaled by its range (§3.2), so it is a shape rather than a pitch or an amplitude.

Equations

• ChuangEtAl2026.PitchContour = DiscriminativeLexicon.FormVec ChuangEtAl2026.TaiwanMandarinPitchSampleCount

@[reducible, inline]

abbrev ChuangEtAl2026.ContextualEmbedding : Type

A contextualized embedding is the mean of the GPT-2 vectors of a token's two characters given the preceding context, the current utterance up to the token and the one before it (§3.1).

Equations

• ChuangEtAl2026.ContextualEmbedding = DiscriminativeLexicon.MeaningVec ChuangEtAl2026.CKIPGPT2HiddenDim

@[reducible, inline]

abbrev ChuangEtAl2026.TaiwanMandarinRFDLM : Type

The paper's DLM consists of linear comprehension and production maps between pitch contours and contextualized embeddings (§3.3–3.4).

Equations

• ChuangEtAl2026.TaiwanMandarinRFDLM = DiscriminativeLexicon.Linear ℝ ChuangEtAl2026.PitchContour ChuangEtAl2026.ContextualEmbedding

Comprehension from fifty dimensions (§3.3)

theorem ChuangEtAl2026.comprehension_not_surjective (D : TaiwanMandarinRFDLM) : ¬Function.Surjective ⇑D.comprehension

No linear map takes the fifty-dimensional contour space onto the 768-dimensional embedding space, the paper's ground for finding ResLDL's comprehension advantage over LDL unsurprising (§3.3.2, fn. 29).

theorem ChuangEtAl2026.comprehension_comp_production_ne_id (D : TaiwanMandarinRFDLM) : D.comprehension ∘ₗ D.production ≠ LinearMap.id

Points of a cube projected onto a plane cannot be recovered from the projection, the picture of fn. 29, so no comprehension network inverts production.

Word-centroid contours (§3.4)

theorem ChuangEtAl2026.production_centroid_eq_of_decodable {D : TaiwanMandarinRFDLM} {m : ℕ} {W : Type u_1} [DecidableEq W] {data : DiscriminativeLexicon.TrainingExperience m TaiwanMandarinPitchSampleCount CKIPGPT2HiddenDim} (hD : DiscriminativeLexicon.Linear.IsELTrainedOn data D) (word : Fin m → W) {w : W} (hw : ∃ (i : Fin m), word i = w) {ℓ : ContextualEmbedding →ₗ[ℝ] ℝ} (hℓ : ∀ (i : Fin m), ℓ (data.S i) = if word i = w then 1 else 0) : D.production (Finset.centroid ℝ {x : Fin m | word x = w} data.S.row) = Finset.centroid ℝ {x : Fin m | word x = w} data.C.row

A word type being the set of tokens carrying its corpus label (§1, §2.2), a least-squares-trained production map sends the centroid of its embeddings to the average of its training contours whenever membership in it is a linear functional of the embeddings. The paper reports the match as remarkable similarity (Fig. 18), Fig. 14's distinct word-type regions being its evidence for the hypothesis.

theorem ChuangEtAl2026.production_centroid_univ_eq {D : TaiwanMandarinRFDLM} {m : ℕ} {data : DiscriminativeLexicon.TrainingExperience m TaiwanMandarinPitchSampleCount CKIPGPT2HiddenDim} [NeZero m] (hD : DiscriminativeLexicon.Linear.IsELTrainedOn data D) {ℓ : ContextualEmbedding →ₗ[ℝ] ℝ} (hℓ : ∀ (i : Fin m), ℓ (data.S i) = 1) : D.production (Finset.centroid ℝ Finset.univ data.S.row) = Finset.centroid ℝ Finset.univ data.C.row

The centroid of all tokens' embeddings, which the paper reads as the meaning of the unmodulated rise–fall contour (§3.4.2, Fig. 1), produces the grand-mean training contour when some linear functional is constant on the training embeddings.