Hausa tone

Hausa has three surface tones: high, unmarked in the orthography (shā 'drink'), low, marked by a grave accent (dà 'with, and'), and falling, marked by a circumflex (tî 'tea'). A fall is a high and a low on one syllable, and occurs only on a heavy syllable. There is no rising tone: a low-high sequence that arises on one syllable, as when a vowel is lost, simplifies to low after a high in the same word and to high elsewhere, so gawàyī 'charcoal' shortens to gawài and tàusàyī 'pity' to tàusái ([New00]).

The definite article is -n, or -r̃ after a feminine singular noun in -a, preceded by a floating low tone. The low turns a final high into a fall, bàkā 'bow' and bàkân 'the bow', and attaches vacuously after a low or a fall, watà 'month' and watàn 'the month'. A few morphemes have polar tone, the opposite of an adjacent tone: the stabilizer nē / cē takes the opposite of the preceding syllable, the possessive markers nā / tā the opposite of a following pronoun, and the weak subject pronouns of the light paradigm the opposite of the adjacent TAM marker. Tone-integrating suffixes give the whole word their melody, as the class 1 plural -ōCī, with a copy of the base-final consonant, makes every tone high: gyàlè 'shawl', gyalōlī 'shawls'.

Main definitions

• Hausa.syllableTones — the three surface tones as melodies on one syllable
• Hausa.simplifyRising, Hausa.simplifyRisingWord — the two rules removing a rise
• Hausa.dockLow — the floating low of the definite article on a word's final syllable
• Hausa.polarOf, Hausa.polarAfter — polar tone, and the polar tone after a syllable
• Hausa.pluralOCi — the class 1 plural as a replacive-dominant grammatical tone ([Rol18])

Main results

• Hausa.simplifyRisingWord_mem — the rules leave every syllable with a surface tone
• Hausa.dockLow_mem, Hausa.getLast?_dockLow — the definite article creates no new tone and leaves the word ending low
• Hausa.polarAfter_fall — a polar tone after a fall is high, as after a low, which follows from analysing the fall as a high and a low

Implementation notes

A syllable's tones are its melody, a list of tonal root nodes, so that the fall is [H, L] and a word is a list of syllable melodies. The polar morphemes and the article are lexical items of other files; their tone is computed here from their host.

References

• [New00]
• [Rol18]

Surface tones and the absence of a rise

def Hausa.syllableTones : Finset (List Tone.TRN)

The surface tones of a syllable: high, low, and the fall.

Equations

• Hausa.syllableTones = {[Tone.TRN.H], [Tone.TRN.L], [Tone.TRN.H, Tone.TRN.L]}

def Hausa.simplifyRising (prev : Option Tone.TRN) (σ : List Tone.TRN) : List Tone.TRN

A low-high sequence on one syllable, given the tone before it in the word, simplifies to low after a high and to high elsewhere; any other melody is unchanged.

Equations

• Hausa.simplifyRising prev σ = if σ = [Tone.TRN.L, Tone.TRN.H] then if prev = some Tone.TRN.H then [Tone.TRN.L] else [Tone.TRN.H] else σ

def Hausa.simplifyRisingWord : Option Tone.TRN → List (List Tone.TRN) → List (List Tone.TRN)

simplifyRising across a word, each syllable seeing the last tone of the one before.

Equations

• Hausa.simplifyRisingWord x✝ [] = []
• Hausa.simplifyRisingWord x✝ (σ :: w) = Hausa.simplifyRising x✝ σ :: Hausa.simplifyRisingWord (Hausa.simplifyRising x✝ σ).getLast? w

theorem Hausa.simplifyRising_mem {prev : Option Tone.TRN} {σ : List Tone.TRN} (h : σ ∈ syllableTones ∨ σ = [Tone.TRN.L, Tone.TRN.H]) : simplifyRising prev σ ∈ syllableTones

theorem Hausa.simplifyRisingWord_mem (prev : Option Tone.TRN) (w : List (List Tone.TRN)) (h : ∀ σ ∈ w, σ ∈ syllableTones ∨ σ = [Tone.TRN.L, Tone.TRN.H]) (σ : List Tone.TRN) : σ ∈ simplifyRisingWord prev w → σ ∈ syllableTones

A word whose syllables carry surface tones or a rise surfaces with surface tones only.

The floating low of the definite article

def Hausa.dockLow (σ : List Tone.TRN) : List Tone.TRN

A floating low docked on a syllable: after a final high it makes a fall; after a low it attaches vacuously.

Equations

• Hausa.dockLow σ = if σ.getLast? = some Tone.TRN.H then σ ++ [Tone.TRN.L] else σ

theorem Hausa.dockLow_mem (σ : List Tone.TRN) : σ ∈ syllableTones → dockLow σ ∈ syllableTones

Docking the low on a surface tone gives a surface tone.

theorem Hausa.getLast?_dockLow (σ : List Tone.TRN) : σ ∈ syllableTones → (dockLow σ).getLast? = some Tone.TRN.L

A syllable with a surface tone ends low once the low has docked.

Polar tone

def Hausa.polarOf : Tone.TRN → Tone.TRN

The tone opposite to a tone: low opposite a high, high opposite anything else.

Equations

• Hausa.polarOf { upper := some true, raised := some false } = Tone.TRN.L
• Hausa.polarOf x✝ = Tone.TRN.H

theorem Hausa.polarOf_polarOf (t : Tone.TRN) : t ∈ [Tone.TRN.H, Tone.TRN.L] → polarOf (polarOf t) = t

Polarity is an involution on high and low.

def Hausa.polarAfter (σ : List Tone.TRN) : Option Tone.TRN

The polar tone after a syllable, opposite to its last tone.

Equations

• Hausa.polarAfter σ = Option.map Hausa.polarOf σ.getLast?

theorem Hausa.polarAfter_fall : polarAfter [Tone.TRN.H, Tone.TRN.L] = polarAfter [Tone.TRN.L]

After a fall a polar tone is high, as after a low: nân nē 'it's here' with zōbè nē 'it's a ring', against nan nè 'it's there'.

Tone-integrating plurals

def Hausa.pluralOCi : Tone.Tune

The class 1 plural -ōCī, whose all-high tune replaces the tones of the whole word: a replacive-dominant grammatical tone accompanying a segmental suffix, [Rol18]'s auxiliary prosodic exponence.

Equations

• Hausa.pluralOCi = { melody := [Tone.TRN.H], anchor := Edge.left }