### Vocal Tract Analog Models

There is one speech-synthesis thread that clearly classifies under
computational physical modeling, and that is the topic of *vocal
tract analog models*. In these models, the vocal tract is regarded as
a *piecewise cylindrical acoustic tube*. The first mechanical
analogue of an acoustic-tube model appears to be a hand-manipulated
leather tube built by Wolfgang von Kempelen in 1791, reproduced with
improvements by Sir Charles Wheatstone [140].
In *electrical* vocal-tract analog models, the piecewise
cylindrical acoustic tube is modeled as a cascade of electrical
*transmission line* segments, with each cylindrical segment being
modeled as a transmission line at some fixed characteristic impedance.
An early model employing four cylindrical sections was developed by
Hugh K. Dunn in the late 1940s [120]. An even earlier model
based on two cylinders joined by a conical section was published by
T. Chiba and M. Kajiyama in 1941 [120]. Cylinder
cross-sectional areas were determined based on X-ray images of
the vocal tract, and the corresponding characteristic impedances were
proportional to . An impedance-based, lumped-parameter
approximation to the transmission-line sections was used in order that
analog LC ladders could be used to implement the model electronically.
By the 1950s, LC vocal-tract analog models included a side-branch for
nasal simulation [131].

The theory of transmission lines is credited to applied mathematician
Oliver Heaviside (1850-1925), who worked out the *telegrapher's
equations* (sometime after 1874) as an application of Maxwell's
equations, which he simplified (sometime after 1880) from the original
20 equations of Maxwell to the modern vector formulation.^{A.12} Additionally,
Heaviside is credited with introducing complex numbers into circuit
analysis, inventing essentially Laplace-transform methods for solving
circuits (sometime between 1880 and 1887), and coining the terms
`impedance' (1886), `admittance' (1887), `electret', `conductance'
(1885), and `permeability' (1885). A little later, Lord Rayleigh
worked out the theory of waveguides (1897), including multiple
propagating modes and the cut-off phenomenon.^{A.13}

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Singing Kelly-Lochbaum Vocal Tract

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Dudley's Vocoder