Afkir & Zellou (2025): The Phonetics of Tarifit

@cite{afkir-zellou-2025}

The Phonetics of Tarifit: Variation and Change in a Moroccan Amazigh Language. Cambridge Elements in Phonetics.

Key empirical findings

1. Two independent schwa processes: prosodic template schwa (C2əC3, morphological) vs intrusive schwa (C1ǎC2, articulatory). They differ in phonetic quality, duration, and conditioning environment.

2. Sonority-conditioned variation: the C1–C2 sonority profile determines whether intrusive schwa is likely:

   • Rising sonority (C1 < C2): intrusive schwa is almost exclusively present (>90% of productions for words with /r/ as C2)
   • Falling/plateauing (C1 ≥ C2): intrusive schwa is never or rarely present

3. Gradient vowelless production: ~5% of tokens surface without any schwa (CCC). This is more accessible for words with low C2/C3 sonority (voiceless obstruents).

4. Sonority granularity matters: the 8-level Parker scale — splitting obstruents by voicing — correctly predicts the gradient pattern, while the coarser 6-level Clements scale collapses critical contrasts.

5. Perception: intrusive schwa boosts auditory discrimination of minimal pairs only for falling-sonority clusters (§5.3.1, Figure 28), consistent with *SONO-PEAK: intrusive schwa in falling onsets creates a new sonority peak that alters perceived syllable structure.

Formal reconstruction

Note: the paper uses mixed effects logistic regression for its statistical analyses, not OT or Harmonic Grammar. The MaxEnt model below is our own formal reconstruction of the paper's empirical generalizations, designed to make the sonority-conditioned predictions verifiable via native_decide. The five constraints and their weights are chosen to capture the paper's main findings about rising/falling/ plateauing onset clusters and gradient vowelless accessibility.

Limitation: the model correctly predicts whether faithful or intrusive wins for each sonority profile, and correctly ranks words by relative vowelless accessibility. However, it overpenalizes vowelless for words with high-sonority C3 (e.g. /ħkəm/, /sχəf/), predicting intrusive > vowelless when empirically vowelless > intrusive for those items. The paper notes these as partly idiosyncratic (Table 7).

inductive AfkirZellou2025.SurfaceForm : Type

The three possible surface realizations of a CCəC word.

• faithful : SurfaceForm

  CCəC: template schwa preserved, no intrusive schwa.

• intrusive : SurfaceForm

  CǎCəC: intrusive schwa inserted between C1 and C2.

• vowelless : SurfaceForm

  CCC: vowelless — no schwa at all.

@[implicit_reducible]

instance AfkirZellou2025.instDecidableEqSurfaceForm : DecidableEq SurfaceForm

Equations

• AfkirZellou2025.instDecidableEqSurfaceForm x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def AfkirZellou2025.instReprSurfaceForm.repr : SurfaceForm → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance AfkirZellou2025.instReprSurfaceForm : Repr SurfaceForm

Equations

• AfkirZellou2025.instReprSurfaceForm = { reprPrec := AfkirZellou2025.instReprSurfaceForm.repr }

structure AfkirZellou2025.TarifitCandidate : Type

A candidate pairs a word's consonant profile with a surface form.

• c1 : Phonology.Syllable.NatClass
• c2 : Phonology.Syllable.NatClass
• c3 : Phonology.Syllable.NatClass
• surface : SurfaceForm

def AfkirZellou2025.instDecidableEqTarifitCandidate.decEq (x✝ x✝¹ : TarifitCandidate) : Decidable (x✝ = x✝¹)

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance AfkirZellou2025.instDecidableEqTarifitCandidate : DecidableEq TarifitCandidate

Equations

• AfkirZellou2025.instDecidableEqTarifitCandidate = AfkirZellou2025.instDecidableEqTarifitCandidate.decEq

@[implicit_reducible]

instance AfkirZellou2025.instReprTarifitCandidate : Repr TarifitCandidate

Equations

• AfkirZellou2025.instReprTarifitCandidate = { reprPrec := AfkirZellou2025.instReprTarifitCandidate.repr }

def AfkirZellou2025.instReprTarifitCandidate.repr : TarifitCandidate → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

def AfkirZellou2025.mkCandidate (w : Fragments.Tarifit.Inventory.TriconWord) (sf : SurfaceForm) : TarifitCandidate

Build a candidate from a word and surface form.

Equations

• AfkirZellou2025.mkCandidate w sf = { c1 := w.c1, c2 := w.c2, c3 := w.c3, surface := sf }

def AfkirZellou2025.maxV : Core.Constraint.WeightedConstraint TarifitCandidate

MAX-V (faithfulness): penalizes deletion of template schwa. Violated once by vowelless production.

Equations

• AfkirZellou2025.maxV = Phonology.Constraints.mkMaxW "MAX-V" (fun (c : AfkirZellou2025.TarifitCandidate) => (c.surface == AfkirZellou2025.SurfaceForm.vowelless) = true) 3

def AfkirZellou2025.sonoCC : Core.Constraint.WeightedConstraint TarifitCandidate

*SONO-CC (markedness): penalizes vowelless clusters proportional to consonant sonority. Higher-sonority consonants in a bare CC cluster are more marked because they expect a vocalic nucleus. Violation count = c2.parkerSonority + c3.parkerSonority. Models the regression finding that lower C2/C3 sonority predicts more vowelless production (@cite{afkir-zellou-2025} Figure 19).

Equations

• One or more equations did not get rendered due to their size.

def AfkirZellou2025.depV : Core.Constraint.WeightedConstraint TarifitCandidate

DEP-V (faithfulness): penalizes insertion of intrusive schwa. Violated once by intrusive production.

Equations

• AfkirZellou2025.depV = Phonology.Constraints.mkDepW "DEP-V" (fun (c : AfkirZellou2025.TarifitCandidate) => (c.surface == AfkirZellou2025.SurfaceForm.intrusive) = true) 2

def AfkirZellou2025.sonoPeak : Core.Constraint.WeightedConstraint TarifitCandidate

*SONO-PEAK (markedness): penalizes intrusive schwa in falling-sonority environments. The penalty is proportional to the sonority drop (c1.son - c2.son), modeling the articulatory implausibility of a vocalic gesture between a more-sonorous C1 and less-sonorous C2. Zero violations for rising or plateauing onsets (Nat subtraction). Captures the regression finding (est. = 0.8, p < 0.001) that rising sonority predicts C1ǎC2 presence (@cite{afkir-zellou-2025} Figure 21).

Equations

• One or more equations did not get rendered due to their size.

def AfkirZellou2025.complexOnset : Core.Constraint.WeightedConstraint TarifitCandidate

*COMPLEX-ONSET (markedness): penalizes complex onsets with rising sonority in the faithful parse. The penalty is proportional to the sonority rise (c2.son - c1.son), modeling the pressure to break up clusters where C2 is much more sonorous than C1. Zero violations for falling or plateauing onsets (Nat subtraction).

Equations

• One or more equations did not get rendered due to their size.

def AfkirZellou2025.tarifitConstraints : List (Core.Constraint.WeightedConstraint TarifitCandidate)

The MaxEnt constraint set for Tarifit schwa variation.

Equations

• AfkirZellou2025.tarifitConstraints = [AfkirZellou2025.maxV, AfkirZellou2025.sonoCC, AfkirZellou2025.depV, AfkirZellou2025.sonoPeak, AfkirZellou2025.complexOnset]

theorem AfkirZellou2025.qreb_intrusive_gt_faithful : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_qreb SurfaceForm.intrusive) (mkCandidate Fragments.Tarifit.Inventory.w_qreb SurfaceForm.faithful)

/qrəβ/ (VLS–liquid, rise=5): intrusive > faithful > vowelless. Table 9 "almost exclusively" C1ǎC2 (@cite{afkir-zellou-2025}).

theorem AfkirZellou2025.qreb_faithful_gt_vowelless : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_qreb SurfaceForm.faithful) (mkCandidate Fragments.Tarifit.Inventory.w_qreb SurfaceForm.vowelless)

theorem AfkirZellou2025.qmes_intrusive_gt_faithful : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_qmes SurfaceForm.intrusive) (mkCandidate Fragments.Tarifit.Inventory.w_qmes SurfaceForm.faithful)

/qməʕ/ (VLS–nasal, rise=4): intrusive > faithful. Table 9 "variably" C1ǎC2 (@cite{afkir-zellou-2025}).

theorem AfkirZellou2025.srem_intrusive_gt_faithful : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_srem SurfaceForm.intrusive) (mkCandidate Fragments.Tarifit.Inventory.w_srem SurfaceForm.faithful)

/srəm/ (VLF–liquid, rise=3): intrusive > faithful. Table 9 "almost exclusively" C1ǎC2 (@cite{afkir-zellou-2025}).

theorem AfkirZellou2025.ntef_faithful_gt_vowelless : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_ntef SurfaceForm.faithful) (mkCandidate Fragments.Tarifit.Inventory.w_ntef SurfaceForm.vowelless)

/ntəf/ (nasal–VLS, fall=4): faithful > vowelless > intrusive. Table 9 "never" C1ǎC2, Table 7 "often vowelless" (@cite{afkir-zellou-2025}).

theorem AfkirZellou2025.ntef_vowelless_gt_intrusive : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_ntef SurfaceForm.vowelless) (mkCandidate Fragments.Tarifit.Inventory.w_ntef SurfaceForm.intrusive)

theorem AfkirZellou2025.nqeb_faithful_gt_vowelless : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_nqeb SurfaceForm.faithful) (mkCandidate Fragments.Tarifit.Inventory.w_nqeb SurfaceForm.vowelless)

/nqəβ/ (nasal–VLS, fall=4): faithful > vowelless. Table 9 "variably" C1ǎC2 — one of the few exceptions to the falling=never pattern (@cite{afkir-zellou-2025} Table 9 note).

theorem AfkirZellou2025.hkem_faithful_gt_intrusive : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_hkem SurfaceForm.faithful) (mkCandidate Fragments.Tarifit.Inventory.w_hkem SurfaceForm.intrusive)

/ħkəm/ (VLF–VLS, fall=2): faithful > intrusive > vowelless (model). Table 9 "never" C1ǎC2; Table 7 "often vowelless" (13–20%) (@cite{afkir-zellou-2025}). The model correctly predicts faithful as winner but overpenalizes vowelless via *SONO-CC (C3=nasal, son=5); empirically vowelless > intrusive, noted as idiosyncratic in Table 7.

theorem AfkirZellou2025.hkem_faithful_gt_vowelless : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_hkem SurfaceForm.faithful) (mkCandidate Fragments.Tarifit.Inventory.w_hkem SurfaceForm.vowelless)

theorem AfkirZellou2025.skhef_faithful_gt_intrusive : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_skhef SurfaceForm.faithful) (mkCandidate Fragments.Tarifit.Inventory.w_skhef SurfaceForm.intrusive)

/sχəf/ (VLF–VLF, plateau): faithful > intrusive > vowelless (model). Table 9 "never" C1ǎC2; Table 7 "often vowelless" (13–20%) (@cite{afkir-zellou-2025}). Like /ħkəm/, the model correctly blocks intrusive but overpenalizes vowelless; empirically this is one of the most frequently vowelless words.

theorem AfkirZellou2025.skhef_faithful_gt_vowelless : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_skhef SurfaceForm.faithful) (mkCandidate Fragments.Tarifit.Inventory.w_skhef SurfaceForm.vowelless)

theorem AfkirZellou2025.sfen_faithful_gt_intrusive : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_sfen SurfaceForm.faithful) (mkCandidate Fragments.Tarifit.Inventory.w_sfen SurfaceForm.intrusive)

/sfən/ (VLF–VLF, plateau): faithful > intrusive > vowelless. Table 9 "never" C1ǎC2 (@cite{afkir-zellou-2025}). Unlike /sχəf/, /sfən/ is not listed as frequently vowelless.

theorem AfkirZellou2025.sfen_faithful_gt_vowelless : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_sfen SurfaceForm.faithful) (mkCandidate Fragments.Tarifit.Inventory.w_sfen SurfaceForm.vowelless)

theorem AfkirZellou2025.vowelless_more_accessible_low_sonority : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_ntef SurfaceForm.vowelless) (mkCandidate Fragments.Tarifit.Inventory.w_qreb SurfaceForm.vowelless)

Low-sonority clusters make vowelless production more accessible: /ntəf/ (C2=VLS, C3=VLF) has higher vowelless harmony than /qrəβ/ (C2=liquid, C3=VDF), because *SONO-CC penalizes high-sonority clusters more heavily. Captures the regression result (C2 est. = -0.7, C3 est. = -1.4) that lower C2/C3 sonority predicts more vowelless production (@cite{afkir-zellou-2025} Figure 19).

theorem AfkirZellou2025.vowelless_obstruent_gt_sonorant : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_skhef SurfaceForm.vowelless) (mkCandidate Fragments.Tarifit.Inventory.w_srem SurfaceForm.vowelless)

Among vowelless candidates, all-obstruent clusters have the highest harmony (least penalized). Consistent with Table 7: /sχəf/ and /skəf/ are the most frequently vowelless words (@cite{afkir-zellou-2025}).

theorem AfkirZellou2025.vowelless_all_obstruent_gt_mixed : Core.Constraint.harmonyDominates tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_skhef SurfaceForm.vowelless) (mkCandidate Fragments.Tarifit.Inventory.w_sfen SurfaceForm.vowelless)

/sχəf/ (all VLF) has higher vowelless harmony than /sfən/ (VLF–VLF–N), because C3=nasal(5) is more sonorous than C3=VLF(3).

theorem AfkirZellou2025.falling_faithful_zero : Core.Constraint.harmonyScore tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_ntef SurfaceForm.faithful) = 0

Falling/plateauing onset: faithful incurs zero constraint violations. *COMPLEX-ONSET only penalizes rising sonority (c2.son - c1.son = 0 for falling/plateauing), and faithful violates no faithfulness constraint.

theorem AfkirZellou2025.plateauing_faithful_zero : Core.Constraint.harmonyScore tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_skhef SurfaceForm.faithful) = 0

theorem AfkirZellou2025.rising_intrusive_base_cost : Core.Constraint.harmonyScore tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_qreb SurfaceForm.intrusive) = -2

Rising onset intrusive: harmony = -2 (only DEP-V violation). *SONO-PEAK contributes 0 for rising onsets (Nat subtraction).

theorem AfkirZellou2025.falling_intrusive_penalized : Core.Constraint.harmonyScore tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_ntef SurfaceForm.intrusive) = -10

Falling onset intrusive: additional *SONO-PEAK penalty. /ntəf/ intrusive: DEP-V (-2) + *SONO-PEAK 4 × (-2) = -10.

theorem AfkirZellou2025.rising_faithful_penalized : Core.Constraint.harmonyScore tarifitConstraints (mkCandidate Fragments.Tarifit.Inventory.w_qreb SurfaceForm.faithful) = -5

Rising onset faithful: *COMPLEX-ONSET penalty proportional to rise. /qrəβ/ faithful: *COMPLEX-ONSET (6-1=5) × (-1) = -5.

theorem AfkirZellou2025.intrusive_disjoint (c : TarifitCandidate) (h : c.surface = SurfaceForm.intrusive) : maxV.eval c = 0 ∧ sonoCC.eval c = 0 ∧ complexOnset.eval c = 0

Intrusive candidates never violate MAX-V, *SONO-CC, or *COMPLEX-ONSET. Their harmony depends only on DEP-V and *SONO-PEAK.

theorem AfkirZellou2025.vowelless_disjoint (c : TarifitCandidate) (h : c.surface = SurfaceForm.vowelless) : depV.eval c = 0 ∧ sonoPeak.eval c = 0 ∧ complexOnset.eval c = 0

Vowelless candidates never violate DEP-V, *SONO-PEAK, or *COMPLEX-ONSET. Their harmony depends only on MAX-V and *SONO-CC.

theorem AfkirZellou2025.faithful_disjoint (c : TarifitCandidate) (h : c.surface = SurfaceForm.faithful) : maxV.eval c = 0 ∧ depV.eval c = 0 ∧ sonoCC.eval c = 0 ∧ sonoPeak.eval c = 0

Faithful candidates never violate MAX-V, DEP-V, *SONO-CC, or *SONO-PEAK. Their harmony depends only on *COMPLEX-ONSET.

@cite{afkir-zellou-2025}'s MaxEnt grammar realised through the generic ConstraintSystem API in Core.Constraint.System. The same softmax decoder used for English onset phonotactics (HayesWilson2008.onsetSystem), French hiatus resolution (Storme2026.stormeSystem), and AAVE t/d-deletion (CoetzeePater2011.aaveSystem) scores Tarifit surface forms here.

@[implicit_reducible]

instance AfkirZellou2025.instFintypeSurfaceForm : Fintype SurfaceForm

Equations

• One or more equations did not get rendered due to their size.

noncomputable def AfkirZellou2025.tarifitSystem (w : Fragments.Tarifit.Inventory.TriconWord) : Core.Constraint.ConstraintSystem SurfaceForm ℝ

The Tarifit MaxEnt grammar as a per-word ConstraintSystem over the three surface candidates (faithful, intrusive, vowelless), decoded by softmax at temperature 1.

Equations

• One or more equations did not get rendered due to their size.

theorem AfkirZellou2025.tarifitSystem_qreb_intrusive_gt_faithful : (tarifitSystem Fragments.Tarifit.Inventory.w_qreb).predict SurfaceForm.faithful < (tarifitSystem Fragments.Tarifit.Inventory.w_qreb).predict SurfaceForm.intrusive

Rising onset /qrəβ/: intrusive schwa (harmony −2) is softmax-preferred over faithful (harmony −5). The system predicts the empirical "almost exclusively C1ǎC2" pattern (Table 9).

theorem AfkirZellou2025.tarifitSystem_ntef_faithful_gt_intrusive : (tarifitSystem Fragments.Tarifit.Inventory.w_ntef).predict SurfaceForm.intrusive < (tarifitSystem Fragments.Tarifit.Inventory.w_ntef).predict SurfaceForm.faithful

Falling onset /ntəf/: faithful is softmax-preferred over intrusive, because *SONO-PEAK heavily penalises a schwa between a more-sonorous C1 (nasal) and a less-sonorous C2 (VLS).

theorem AfkirZellou2025.tarifitSystem_isProb (w : Fragments.Tarifit.Inventory.TriconWord) : ∑ sf : SurfaceForm, (tarifitSystem w).predict sf = 1

The softmax decoder is a probability decoder, so the per-word system's predictions sum to 1 across the three surface forms.