IN THIS PAPER, WE INTRODUCE A SMALL FAMILY OFnovel bottom-up (sensory) models of the Krumhansl and Kessler (1982) probe tone data. The models are based on the spectral pitch class similarities between all twelve pitch classes and the tonic degree and tonic triad. Cross-validation tests of a wide selection of models show ours to have amongst the highest fits to the data. We then extend one of our models to predict the tonics of a variety of different scales such as the harmonic minor, melodic minor, and harmonic major. The model produces sensible predictions for these scales. Furthermore, we also predict the tonics of a small selection of microtonal scales-scales that do not form part of any musical culture. These latter predictions may be tested when suitable empirical data have been collected.

Models of the perceived distance between pairs of pitch collections are a core component of broader models of music cognition. Numerous distance measures have been proposed, including voice-leading [1], psychoacoustic [2][3][4], and pitch and interval class distances [5]; but, so far, there has been no attempt to bind these different measures into a single mathematical or conceptual framework, nor to incorporate the uncertain or probabilistic nature of pitch perception. This paper embeds pitch collections in expectation tensors and shows how metrics between such tensors can model their perceived dissimilarity. Expectation tensors indicate the expected number of tones, ordered pairs of tones, ordered triples of tones, etc., that are heard as having any given pitch, dyad of pitches, triad of pitches, etc.. The pitches can be either absolute or relative (in which case the tensors are invariant with respect to transposition). Examples are given to show how the metrics accord with musical intuition.

Tonal affinity is the perceived goodness of fit of successive tones. It is important because a preference for certain intervals over others would likely influence preferences for, and prevalences of, "higher-order" musical structures such as scales and chord progressions. We hypothesize that two psychoacoustic (spectral) factors-harmonicity and spectral pitch similarity-have an impact on affinity. The harmonicity of a single tone is the extent to which its partials (frequency components) correspond to those of a harmonic complex tone (whose partials are a multiple of a single fundamental frequency). The spectral pitch similarity of two tones is the extent to which they have partials with corresponding, or close, frequencies. To ascertain the unique effect sizes of harmonicity and spectral pitch similarity, we constructed a computational model to numerically quantify them. The model was tested against data obtained from 44 participants who ranked the overall affinity of tones in melodies played in a variety of tunings (some microtonal) with a variety of spectra (some inharmonic). The data indicate the two factors have similar, but independent, effect sizes: in combination, they explain a sizeable portion of the variance in the data (the model-data squared correlation is r 2 = .64). Neither harmonicity nor spectral pitch similarity require prior knowledge of musical structure, so they provide a potentially universal bottom-up explanation for tonal affinity. We show how the model-as optimized to these data-can explain scale structures commonly found in music, both historical and contemporary, and we discuss its implications for experimental microtonal and spectral music.

The transformed form of the Webster equation is investigated. Usually described as analogous to the Schrödinger equation of quantum mechanics, it is noted that the second-order time dependency defines a Klein-Gordon problem. This "acoustical Klein-Gordon equation" is analyzed with particular reference to the acoustical properties of wave-mechanical potential functions, U(x), that give rise to geometry-dependent dispersions at rapid variations in tract cross section. Such dispersions are not elucidated by other one-dimensional--cylindrical or conical--duct models. Since Sturm-Liouville analysis is not appropriate for inhomogeneous boundary conditions, the exact solution of the Klein-Gordon equation is achieved through a Green's-function methodology referring to the transfer matrix of an arbitrary string of square potential functions, including a square barrier equivalent to a radiation impedance. The general conclusion of the paper is that, in the absence of precise knowledge of initial conditions on the area function, any given potential function will map to a multiplicity of area functions of identical relative resonance characteristics. Since the potential function maps uniquely to the acoustical output, it is suggested that the one-dimensional wave physics is both most accurately and most compactly described within the Klein-Gordon framework.

Acoustic pulse reflectometry is a noninvasive technique for measuring the internal dimensions of tubular objects. A sound pulse is produced using a loudspeaker and injected into the object under investigation via a source tube. The resultant object reflections are recorded by a microphone embedded in the wall of the source tube. Analysis of the reflections yields information about the bore profile of the object. At present, there is a restriction on the length of object that can be measured using the standard reflectometer. This restriction arises because after the object reflections pass the microphone they undergo further reflection at the loudspeaker. These source reflections return to the microphone as unwanted signal. Hence, there is only a finite time over which the object reflections can be accurately recorded which, in turn, limits the length of object that can be measured. In this paper, a method for removing the length restriction is described and preliminary results are presented. The method involves the cancellation of the incoming object reflections at the loudspeaker, thus preventing the formation of source reflections. [Work supported by EPSRC.]