We present numerical methods for studying the relationship between the shape of the vocal tract and its acoustic output. For a stationary vocal tract, the articulatory-acoustic relationship can be represented as a multidimensional function of a multidimensional argument: y=f(x), where x, y are vectors describing the vocal-tract shape and the resulting acoustic output, respectively. Assuming that y may be computed for any x, we develop a procedure for inverting f(x). Inversion by computer sorting consists of computing y for many values of x and sorting the resulting (y,x) pairs into a convenient order according to y; x for a given y is then obtained by looking up y in the sorted data. Application of this method for determining parameters of an articulatory model corresponding to a given set of formant frequencies is presented. A method is also described for finding articulatory regions (fibers) which map into a single point in the acoustic space. The local nature of f(x) is determined by linearization in a small neighborhood. Larger regions are explored by extending the linear neighborhoods in small steps. This method was applied for the study of compensatory articulation. Sounds produced by various articulations along a fiber were synthesized and were compared by informal listening tests. These tests show that, in many cases of interest, a given sound could be produced by many different vocal-tract shapes.

A musical scale based on the 3:5:7:9 tetrachord is described. It has clearly audible harmonic properties that are derived from the harmonies of the tetrachord. In its equal-tempered version, the nine tones of the scale are selected from a set of 13 equal steps, which have a frequency ratio equal to 31/13. The tenth note of the scale, here called the tritave, has a 1:3 frequency ratio with respect to the first note and has a role similar to that of the traditional octave. Both ‘‘major’’ and ‘‘minor’’ chords can be formed from the notes of the scale. Musicians and untrained listeners rated the consonance of all possible triads formed from a 1-tritave range of the equal-tempered chromatic scale. A wide range of consonance ratings was observed, and chords were judged as most dissonant when they had 1-step intervals. A critical bandwidth dissonance model fits the data well. The same subjects rated the consonance of a harmonized passage played in different tunings. Both groups judged the equal-tempered version as most consonant, but only the musicians were influenced by previous exposure to these sounds. Subjects also judged the similarity of chords and their inversions, both for chords from the new scale and for traditional chords. Musicians were influenced by key relationships, inversions, and chord type in their ratings of the traditional chord set but judged the new chords only by pitch height. Untrained listeners relied on pitch height for both chord sets. This suggests that the ability to abstract more complex information depends on training.

A computer can be programmed to play "instrumental" music, to aid the composer, or to compose unaided. M. V. MathewsWith the aid of suitable output equipment, the numbers which a modern digital computer generates can be directly converted to sound waves. The process is completely general, and any perceivable sound can be so produced. This potentiality of the computer has been of considerable use at the Bell Telephone Laboratories in generating stimuli for experiments in the field of hearing, and for generating speech sounds and connected speech in investigations of the factors which contribute to the intelligibility and naturalness of speech.The quality of sound is of great importance in two fields-that of speech and communication and that of music. Our studies at the Bell Laboratories in the first of these fields have led us, over the past few years, to related studies in the production of musical sounds and their organization into musical compositions. I believe that this by-product of our work on speech and hearing may be of considerable value in the world of music, and that further work in this direction will be of substantial value in furthering our understanding of psychoacoustics.There are no theoretical limitations to the performance of the computer as a source of musical sounds, in contrast to the performance of ordinary instruments. At present, the range of computer music is limited principally by cost and by our knowledge of psychoacoustics. These limits are rapidly receding.In addition to generating sound, the computer can also function as a ma-chine for composing music. It can either compose pieces based entirely on random numbers generated by itself or it can cooperate with a human composer. It can play its own compositions.Here I first describe the process for converting numbers to sounds, then I describe a program for playing music. Next I consider a psychoacoustic problem which is typical of those posed in attempts to make more interesting sounds. Finally, I look to the future, to the time when the computer is itself the composer. Sound from NumbersHow can the numbers with which a computer deals be converted into sounds the ear can hear? The most general conversion is based upon the use of the numbers as samples of the sound pressure wave. A schematic diagram of this process is shown in Fig. 1. Here a sequence of numbers from the computer is put into an analog-to-digital converter, which generates a se-' quence of electric pulses whose amplitudes are proportional to the numbers. These pulses are smoothed with a filter and then converted to a sound wave by means of an ordinary loudspeaker. Intuitively, we feel that if a high enough pulse rate is used and the amplitudes of the pulses are generated with sufficient precision, then any sound wave can be closely approximated by this process. Mathematically, it has been established (1) that this conclusion is correct. A sound wave with frequencies from 0 to B cycles per second can be generated from a sequence of two B pulses per second. Thus, for

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POWERFUL NEW METHODS are now available for the study of tone quality of both existing and new instruments. The new methods involve both new tools (primarily computers) and new applications of experimental psychology (psychophysics and subjective judgments).