Lionnet (2022): The Features and Geometry of Tone in Laal #
This file formalizes the subtonal analysis of the three-height tone system of Laal in
[Lio22]. The heights are bundles of two register features on a tonal root node, in
the two-feature model of [Yip80] and [Pul86] and the geometry of [Sni20]:
H is [+upper, −raised], M is [−upper, +raised], L is [−upper, −raised], and
[+upper, +raised] is the system's gap (featural_analysis, superHigh_is_the_gap). The
analysis unifies the behaviour of the mid tone. It never shares a stem with another height
(M_exclusive); it lowers to L under a [−raised] neighbour, the only [+raised] tone being
the only target (mLowering_from_L, H_stable); and the ventive suffix, a floating
[−raised] taking [upper] from the root, surfaces as H after H and as L after M and L
(ventive_after_M). The tonal root node lets a feature act on its own tier apart from the
whole tone: on the multi-tier autosegmental foundation, delinking the [raised] feature
leaves the association of the tone to its mora untouched (partial_indep_of_full).
Implementation notes #
The substrate TRN bundle carries the two register features, and TRN.assimilate and
TRN.dock are the [−raised] spreading and docking of the paper's derivations. The fusion
of adjacent identical [−raised] autosegments, which the paper mentions as an option it does
not adopt, is the OCP.collapse face of the tier-relative OCP. The full-tone spreading and
deletion processes of the paper's §6, its vowel harmony, and the alternative analyses of its
§7 are not represented.
References #
The subtonal featural analysis (§5.1) #
The featural analysis (ex. 51) as a map into the register-tier TRN encoding.
Equations
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The substrate's TRN.H, TRN.M, TRN.L are the feature matrix of (51).
Paradigmatic pitch (§5.1): [upper] counts two and [raised] one, independently per
node, with no register state.
Equations
- Lionnet2022.absolutePitch t = (if t.upper = some true then 2 else 0) + if t.raised = some true then 1 else 0
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L < M < H, with the gap above H.
M-exclusivity: *MX/XM (§3, §5.4) #
M-exclusivity (*MX/XM): in every attested stem melody, if M occurs then
the melody is all M — M never co-occurs with a different tone at stem level.
M-lowering as [−raised] assimilation (§5.2) #
M-lowering is [−raised] assimilation: an L trigger ([−raised]) spreads its
[raised] value onto M (the only [+raised] tone), turning it into L.
M-lowering from a [−raised] H trigger likewise turns M into L.
Only M is targeted: H is inert under [−raised] assimilation (already
[−raised]), explaining why H- and L-toned roots never lower.
L is likewise inert under [−raised] assimilation.
The ventive suffix (§5.5) #
The ventive suffix (ex. 60) is a floating [−raised] feature with [upper]
inherited from the root: TRN.dock .raised false. It surfaces as H after a
[+upper] (H) root and as L after [−upper] (M or L) roots — the M-lowering
realisation kárá/dàgà/jàrà.
The [+upper, +raised] gap (§5.6) #
The Laal tone inventory, as TRNs.
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The fourth feature combination [+upper, +raised] (TRN.superHigh) is absent:
it is the systematic gap that makes Laal a Table-4 type-A system (§5.6).
The optional OCP-[raised] merger (§5.2, ex. 53–55, 58) #
The paper (exx. 54–55, 58) mentions, without adopting, an optional OCP-[raised]
economy under which two adjacent identical [−raised] autosegments fuse into one
multiply-linked autosegment; for fully identical adjacent tones that fusion is
OCP.collapse.
OCP-[raised] is tier-relative ([CJ19]): it constrains the
[raised]-projected tier (IsCleanOn reading .raised), not whole TRNs. H and L
are distinct tones but both [−raised], so adjacent they violate OCP-[raised]
even though [TRN.H, TRN.L] is clean as a whole-TRN tier.
Under the optional economy, fusing adjacent identical [raised] autosegments leaves
the [raised]-projected tier OCP-clean.
The register-tier geometry on the multi-tier substrate (§5–§6) #
The geometry of (52) is a hub and spokes around the tonal root node: the [±upper] tier,
the [±raised] tier, and the mora tier each associate to it. On the graph foundation this
is a four-tier graph, on which a subtonal feature is a tier of its own and so can be
manipulated apart from the whole node, the paper's partial activity, which the bundled
TRN record cannot express; whole-node operations act on the node-to-mora layer.
The four Laal tone tiers are the [±upper] register, [±raised], the TRN, and the
mora.
Equations
- Lionnet2022.laalTier = ![Option Bool, Option Bool, Unit, Unit]
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The tier words of a one-node M-toned form.
Equations
- Lionnet2022.laalWords = Fin.cons [some false] (Fin.cons [some true] (Fin.cons [()] (Fin.cons [()] finZeroElim)))
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The hub-and-spoke links (ex. 52): register, [raised], and mora each
associate to the TRN.
Equations
- Lionnet2022.laalSpokes i j p q = (((i, j) = (0, 2) ∨ (i, j) = (1, 2) ∨ (i, j) = (2, 3)) ∧ p = 0 ∧ q = 0)
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Equations
- Lionnet2022.instDecidableLaalSpokes i j p q = { decide := decide ((i, j) = (0, 2) ∨ (i, j) = (1, 2) ∨ (i, j) = (2, 3)) && decide (p = 0 ∧ q = 0), reflects_decide := ⋯ }
The M-toned form on the graph foundation.
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Equations
- One or more equations did not get rendered due to their size.
The form obeys the NCC, each spoke being a single association.
Partial activity (§5): delinking a feature acts on one tier-pair layer
alone — here [raised]↔TRN, the pair (1, 2).
Equations
- Lionnet2022.delink L i₀ j₀ i j p q = (¬(i = i₀ ∧ j = j₀) ∧ L i j p q)
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The M form with the subtonal [−raised] feature delinked.
Equations
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Delinking the subtonal [−raised] feature leaves the node-to-mora layer, the tier pair
(2, 3), untouched: partial activity is independent of full activity.
And it does remove the [raised]↔TRN association.