Finnish Vowel Harmony @cite{karlsson-2017}

@cite{goldsmith-1976} @cite{rose-walker-2011}

Finnish has palatal vowel harmony: the [±back] feature of root vowels propagates rightward through all suffixes. This is a textbook case of autosegmental feature spreading.

Vowel classes (Karlsson §2.1)

• Back vowels: a [+back, +low], o [+back, +round], u [+back, +high]
• Front vowels: ä [−back, +low], ö [−back, +round], y [−back, +high]
• Neutral vowels: e [−back, −round, −low], i [−back, +high, −round] — transparent to harmony (do not trigger or block spreading)

Suffix alternation

Most suffixes contain an archiphonemic vowel /A/ that surfaces as [a] after back-vowel stems and [ä] after front-vowel stems:

• Partitive: kirja-a ('book') vs. käsi-ä ('hand')
• Inessive: talo-ssa ('in the house') vs. metsä-ssä ('in the forest')

Formalization (@cite{rose-walker-2011})

Finnish VH is a single HarmonySystem with [back] as the spreading feature. Neutral vowels /e/ and /i/ are transparent: they satisfy isTransparent and are skipped by triggerValue (not triggers) and harmonizeOne (not targets).

def Fragments.Finnish.VowelHarmony.a_vowel : Theories.Phonology.Segment

Finnish back vowel /a/: [+syll, +low, +back, +dorsal, −high, −round].

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def Fragments.Finnish.VowelHarmony.ä_vowel : Theories.Phonology.Segment

Finnish front vowel /ä/: [+syll, +low, −back, +dorsal, −high, −round].

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def Fragments.Finnish.VowelHarmony.o_vowel : Theories.Phonology.Segment

Finnish back vowel /o/: [+syll, +round, +back, +dorsal, −high, −low].

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def Fragments.Finnish.VowelHarmony.ö_vowel : Theories.Phonology.Segment

Finnish front vowel /ö/: [+syll, +round, −back, +dorsal, −high, −low].

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def Fragments.Finnish.VowelHarmony.u_vowel : Theories.Phonology.Segment

Finnish back vowel /u/: [+syll, +high, +back, +round, +dorsal, −low].

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def Fragments.Finnish.VowelHarmony.y_vowel : Theories.Phonology.Segment

Finnish front vowel /y/: [+syll, +high, −back, +round, +dorsal, −low].

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def Fragments.Finnish.VowelHarmony.e_vowel : Theories.Phonology.Segment

Neutral vowel /e/: [+syll, −back, −round, −high, −low, +dorsal].

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def Fragments.Finnish.VowelHarmony.i_vowel : Theories.Phonology.Segment

Neutral vowel /i/: [+syll, +high, −back, −round, +dorsal, −low].

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def Fragments.Finnish.VowelHarmony.isBackVowel (s : Theories.Phonology.Segment) : Bool

Is a segment a back vowel? [+syll, +back].

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• Fragments.Finnish.VowelHarmony.isBackVowel s = (s.hasValue Theories.Phonology.Feature.syllabic true && s.hasValue Theories.Phonology.Feature.back true)

def Fragments.Finnish.VowelHarmony.isFrontVowel (s : Theories.Phonology.Segment) : Bool

Is a segment a front vowel? [+syll, −back].

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• Fragments.Finnish.VowelHarmony.isFrontVowel s = (s.hasValue Theories.Phonology.Feature.syllabic true && s.hasValue Theories.Phonology.Feature.back false)

def Fragments.Finnish.VowelHarmony.isNeutral (s : Theories.Phonology.Segment) : Bool

Is a segment a neutral vowel? Front vowels /e/ and /i/ that are transparent to harmony: [+syll, −back, −round, −low].

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inductive Fragments.Finnish.VowelHarmony.HarmonyClass : Type

The harmony class of a vowel: back, front, or neutral.

• back : HarmonyClass
• front : HarmonyClass
• neutral : HarmonyClass

instance Fragments.Finnish.VowelHarmony.instDecidableEqHarmonyClass : DecidableEq HarmonyClass

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• Fragments.Finnish.VowelHarmony.instDecidableEqHarmonyClass x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Fragments.Finnish.VowelHarmony.instBEqHarmonyClass.beq : HarmonyClass → HarmonyClass → Bool

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• Fragments.Finnish.VowelHarmony.instBEqHarmonyClass.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Fragments.Finnish.VowelHarmony.instBEqHarmonyClass : BEq HarmonyClass

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• Fragments.Finnish.VowelHarmony.instBEqHarmonyClass = { beq := Fragments.Finnish.VowelHarmony.instBEqHarmonyClass.beq }

def Fragments.Finnish.VowelHarmony.instReprHarmonyClass.repr : HarmonyClass → Nat → Std.Format

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instance Fragments.Finnish.VowelHarmony.instReprHarmonyClass : Repr HarmonyClass

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• Fragments.Finnish.VowelHarmony.instReprHarmonyClass = { reprPrec := Fragments.Finnish.VowelHarmony.instReprHarmonyClass.repr }

def Fragments.Finnish.VowelHarmony.classifyVowel (s : Theories.Phonology.Segment) : HarmonyClass

Classify a vowel segment.

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def Fragments.Finnish.VowelHarmony.finnishHarmony : Theories.Phonology.Harmony.HarmonySystem

Finnish palatal harmony: [back] spreads from the last harmonic (non-neutral) stem vowel to non-neutral suffix vowels. Neutral vowels /e/, /i/ are transparent — they neither trigger nor undergo harmony.

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theorem Fragments.Finnish.VowelHarmony.a_is_back : isBackVowel a_vowel = true

/a/ is a back vowel.

theorem Fragments.Finnish.VowelHarmony.ä_is_front : isFrontVowel ä_vowel = true

/ä/ is a front vowel.

theorem Fragments.Finnish.VowelHarmony.e_is_neutral : isNeutral e_vowel = true

/e/ is neutral.

theorem Fragments.Finnish.VowelHarmony.i_is_neutral : isNeutral i_vowel = true

/i/ is neutral.

theorem Fragments.Finnish.VowelHarmony.o_not_neutral : isNeutral o_vowel = false

/o/ is not neutral (it's a harmonic back vowel).

theorem Fragments.Finnish.VowelHarmony.ö_not_neutral : isNeutral ö_vowel = false

/ö/ is not neutral (it's a harmonic front vowel).

theorem Fragments.Finnish.VowelHarmony.back_stem_harmony : Theories.Phonology.Harmony.triggerValue finnishHarmony [a_vowel] = some true

Back-vowel stems yield back harmony.

theorem Fragments.Finnish.VowelHarmony.front_stem_harmony : Theories.Phonology.Harmony.triggerValue finnishHarmony [ä_vowel] = some false

Front-vowel stems yield front harmony.

theorem Fragments.Finnish.VowelHarmony.neutral_only_no_trigger : Theories.Phonology.Harmony.triggerValue finnishHarmony [e_vowel, i_vowel] = none

Neutral-only stems have no trigger (default to front harmony).

theorem Fragments.Finnish.VowelHarmony.back_with_neutral : Theories.Phonology.Harmony.triggerValue finnishHarmony [a_vowel, i_vowel] = some true

A back stem with a neutral vowel still yields back harmony (the neutral vowel is not a trigger, so triggerValue finds /a/).

theorem Fragments.Finnish.VowelHarmony.a_ä_dorsal_disagree : Theories.Phonology.Autosegmental.agreeAt a_vowel ä_vowel Theories.Phonology.FeatureGeometry.GeomNode.dorsal = false

The /a/–/ä/ pair differs only in [back]: dorsal agreement fails between them, confirming they belong to different harmony classes.

theorem Fragments.Finnish.VowelHarmony.a_o_dorsal_agree_on_back : a_vowel.hasValue Theories.Phonology.Feature.back true = true ∧ o_vowel.hasValue Theories.Phonology.Feature.back true = true

Dorsal agreement holds between /a/ and /o/ (both [+back]).