Accent Assignment and Tone Derivation

@cite{hayes-1995} @cite{kawahara-2015}

Language-general accent assignment rules and the accent-to-tone derivation for pitch accent systems.

Accent Assignment Rules

Two rules that derive default accent position from syllable weight:

• Antepenultimate Accent Rule (AAR): accent falls on the syllable containing the antepenultimate (3rd-from-last) mora. This is the traditional characterization of the default accent pattern in Japanese (@cite{mccawley-1968}).

• Latin Stress Rule (LSR): accent falls on the penultimate syllable if heavy, otherwise on the antepenultimate. @cite{kubozono-2008} argues the LSR better characterizes Japanese loanword accentuation, capitalizing on the cross-linguistic parallel with Latin stress (@cite{hayes-1995}).

Accent-to-Tone Derivation

For pitch-accent languages like Japanese, surface tonal patterns are fully determined by accent location. The derivation follows @cite{kawahara-2015} §1.4:

1. Accentual HL: H on the accented mora, L on the following mora.
2. Initial rise: L on the first mora, H on the second (blocked when the first mora is accented).
3. Spreading: unspecified moras copy the rightmost specified tone.

NonFinality

Two NonFinality constraints from @cite{prince-smolensky-1993}:

• NonFinality(σ): penalizes accent on the word-final syllable.
• NonFinality(Ft): penalizes the head foot in word-final position.

def Phonology.Accent.findSyllable (weights : List Syllable.SyllWeight) (targetMora : ℕ) : Option ℕ

Find the 0-indexed syllable containing the k-th mora (0-indexed). Returns none if the target mora exceeds the total mora count.

Equations

• Phonology.Accent.findSyllable weights targetMora = Phonology.Accent.findSyllable.go weights targetMora 0

def Phonology.Accent.findSyllable.go : List Syllable.SyllWeight → ℕ → ℕ → Option ℕ

Equations

• Phonology.Accent.findSyllable.go [] a✝¹ a✝ = none
• Phonology.Accent.findSyllable.go (w :: ws) a✝¹ a✝ = if a✝¹ < w.morae then some a✝ else Phonology.Accent.findSyllable.go ws (a✝¹ - w.morae) (a✝ + 1)

def Phonology.Accent.defaultAccentAAR (weights : List Syllable.SyllWeight) : Option ℕ

Antepenultimate Accent Rule (@cite{mccawley-1968}): accent falls on the syllable containing the antepenultimate (3rd-from-last) mora.

For words with fewer than 3 morae, accent falls on the initial syllable (the only possibility for bimoraic minimal words).

Returns the 0-indexed syllable position.

Equations

• One or more equations did not get rendered due to their size.
• Phonology.Accent.defaultAccentAAR [] = none

@[irreducible]

def Phonology.Accent.latinStressRule : List Syllable.SyllWeight → Option ℕ

Latin Stress Rule (@cite{hayes-1995}): accent the penultimate syllable if it is heavy (≥ 2μ), otherwise accent the antepenultimate.

For monosyllables, accent the only syllable. For disyllables, always accent the penultimate (= initial) syllable.

@cite{kubozono-2008} argues this rule better characterizes Japanese default accentuation than the AAR.

Equations

• Phonology.Accent.latinStressRule [] = none
• Phonology.Accent.latinStressRule [head] = some 0
• Phonology.Accent.latinStressRule [head, head_1] = some 0
• Phonology.Accent.latinStressRule [head, { morae := 0 }, head_1] = some 0
• Phonology.Accent.latinStressRule [head, { morae := 1 }, head_1] = some 0
• Phonology.Accent.latinStressRule [head, head_1, head_2] = some 1
• Phonology.Accent.latinStressRule (head :: head_1 :: head_2 :: tail) = Option.map (fun (x : ℕ) => x + 1) (Phonology.Accent.latinStressRule (head_1 :: head_2 :: tail))

inductive Phonology.Accent.LevelTone : Type

Level tone for pitch accent systems. Japanese uses only H and L at the lexical level (@cite{kawahara-2015} §1.3).

• H : LevelTone
• L : LevelTone

@[implicit_reducible]

instance Phonology.Accent.instDecidableEqLevelTone : DecidableEq LevelTone

Equations

• Phonology.Accent.instDecidableEqLevelTone x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance Phonology.Accent.instReprLevelTone : Repr LevelTone

Equations

• Phonology.Accent.instReprLevelTone = { reprPrec := Phonology.Accent.instReprLevelTone.repr }

def Phonology.Accent.instReprLevelTone.repr : LevelTone → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance Phonology.Accent.instBEqLevelTone : BEq LevelTone

Equations

• Phonology.Accent.instBEqLevelTone = { beq := Phonology.Accent.instBEqLevelTone.beq }

def Phonology.Accent.instBEqLevelTone.beq : LevelTone → LevelTone → Bool

Equations

• Phonology.Accent.instBEqLevelTone.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

def Phonology.Accent.accentToTones (accentMora : Option ℕ) (nMorae : ℕ) : List LevelTone

Derive surface tones from accent position and mora count.

Implements the 4-step derivation of @cite{kawahara-2015} §1.4:

1. Accentual HL assignment (H on accented mora, L on next)
2. Initial rise (L on mora 0, H on mora 1 — blocked by initial accent)
3. Spreading (unspecified moras copy rightmost specified tone)

Returns a list of LevelTone with one entry per mora.

Equations

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

def Phonology.Accent.accentToTones.spread : List (Option LevelTone) → LevelTone → List LevelTone

Equations

• Phonology.Accent.accentToTones.spread [] a✝ = []
• Phonology.Accent.accentToTones.spread (some t :: rest) a✝ = t :: Phonology.Accent.accentToTones.spread rest t
• Phonology.Accent.accentToTones.spread (none :: rest) a✝ = a✝ :: Phonology.Accent.accentToTones.spread rest a✝

def Phonology.Accent.nonFinalitySigma (accentSyll : Option ℕ) (nSyll : ℕ) : ℕ

NonFinality(σ) (@cite{prince-smolensky-1993}): penalizes accent on the word-final syllable. Returns 1 if accent is final, 0 otherwise.

Drives the avoidance of final accent observed in Japanese compound formation and loanword adaptation (@cite{kawahara-2015} §4).

Equations

• Phonology.Accent.nonFinalitySigma (some pos) nSyll = if (pos + 1 == nSyll) = true then 1 else 0
• Phonology.Accent.nonFinalitySigma none nSyll = 0

def Phonology.Accent.nonFinalityFoot (parse : Syllable.MetricalParse) (accentSyll : Option ℕ) : ℕ

NonFinality(Ft) (@cite{prince-smolensky-1993}): penalizes the head foot in word-final position. Returns 1 if the rightmost foot in the parse is the head foot (contains the accent), 0 otherwise.

Distinct from NonFinality(σ): accent on the final syllable may be tolerated if the final foot is not the head foot.

Equations

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

def Phonology.Accent.shortN2CompoundAccent (n1Morae : ℕ) (n2Accent : Option ℕ) (preAccenting : Bool) : Option ℕ

Short N2 compound accent: when the second member of a compound (N2) is short (≤ 2 morae), accent may fall on the last syllable of N1 (pre-accenting pattern) or N2 may retain its own accent.

@cite{kawahara-2015} §4.1: pre-accenting N2s are those that lose their accent to NonFinality(Ft) and receive new accent via a compound accent rule analogous to dominant pre-accenting suffixes.

Equations

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

def Phonology.Accent.longN2CompoundAccent (n1Morae : ℕ) (n2Accent : Option ℕ) (n2Morae : ℕ) : Option ℕ

Long N2 compound accent: when N2 is long (≥ 3 morae), if N2 is unaccented or has final accent, accent falls on N2-initial syllable. Otherwise, N2 accent is retained.

@cite{kawahara-2015} §4.2.

Equations

• Phonology.Accent.longN2CompoundAccent n1Morae none n2Morae = some n1Morae
• Phonology.Accent.longN2CompoundAccent n1Morae (some pos) n2Morae = if (pos + 1 == n2Morae) = true then some n1Morae else some (pos + n1Morae)

theorem Phonology.Accent.aar_bimoraic_hl : defaultAccentAAR [Syllable.SyllWeight.heavy, Syllable.SyllWeight.light] = some 0

theorem Phonology.Accent.aar_bimoraic_ll : defaultAccentAAR [Syllable.SyllWeight.light, Syllable.SyllWeight.light] = some 0

theorem Phonology.Accent.aar_lll : defaultAccentAAR [Syllable.SyllWeight.light, Syllable.SyllWeight.light, Syllable.SyllWeight.light] = some 0

theorem Phonology.Accent.aar_hll : defaultAccentAAR [Syllable.SyllWeight.heavy, Syllable.SyllWeight.light, Syllable.SyllWeight.light] = some 0

theorem Phonology.Accent.lsr_heavy_penult : latinStressRule [Syllable.SyllWeight.light, Syllable.SyllWeight.heavy, Syllable.SyllWeight.light] = some 1

theorem Phonology.Accent.lsr_light_penult : latinStressRule [Syllable.SyllWeight.light, Syllable.SyllWeight.light, Syllable.SyllWeight.light] = some 0

theorem Phonology.Accent.lsr_disyllable : latinStressRule [Syllable.SyllWeight.light, Syllable.SyllWeight.light] = some 0

theorem Phonology.Accent.tones_unaccented_3 : accentToTones none 3 = [LevelTone.L, LevelTone.H, LevelTone.H]

Unaccented trisyllable → LHH (initial rise, H spreads).

theorem Phonology.Accent.tones_initial_3 : accentToTones (some 0) 3 = [LevelTone.H, LevelTone.L, LevelTone.L]

Initial accent trisyllable → HLL (accent HL, L spreads).

theorem Phonology.Accent.tones_medial_3 : accentToTones (some 1) 3 = [LevelTone.L, LevelTone.H, LevelTone.L]

Medial accent trisyllable → LHL (initial rise L, accent H, post-accent L).

theorem Phonology.Accent.tones_unaccented_4 : accentToTones none 4 = [LevelTone.L, LevelTone.H, LevelTone.H, LevelTone.H]

Unaccented 4-mora → LHHH. Kawahara (7).

theorem Phonology.Accent.tones_initial_4 : accentToTones (some 0) 4 = [LevelTone.H, LevelTone.L, LevelTone.L, LevelTone.L]

Initial accent 4-mora → HLLL. Kawahara (8).

theorem Phonology.Accent.tones_antepenult_4 : accentToTones (some 1) 4 = [LevelTone.L, LevelTone.H, LevelTone.L, LevelTone.L]

Antepenultimate accent 4-mora → LHLL.

theorem Phonology.Accent.tones_antepenult_5 : accentToTones (some 2) 5 = [LevelTone.L, LevelTone.H, LevelTone.H, LevelTone.L, LevelTone.L]

Antepenultimate accent 5-mora → LHHLL. Kawahara (9).

theorem Phonology.Accent.nonfin_sigma_final : nonFinalitySigma (some 2) 3 = 1

theorem Phonology.Accent.nonfin_sigma_nonfinal : nonFinalitySigma (some 1) 3 = 0

theorem Phonology.Accent.nonfin_sigma_unaccented : nonFinalitySigma none 3 = 0