John R. Pierce

Current directions in computer music research

An expensive laboratory luxury just 25 years ago, computer music is now entering the public domain. Inexpensive digital keyboards and related hardware are more widely available than the piano. These 21 original contributions by composers, behavioral scientists, engineers, and other specialists from many of the international centers of research in computer music provide an inside report on the most sophisticated aspects of digital synthesis, control and understanding of musical sound, and related work on perception. The use of the computer to analyze and synthesize complex sounds has resulted in faster and more accurate experimental study of this key aspect of human perception. Many of the chapters in this book provide new insights into sounds and hearing. These include the investigation of loudness, masking, and binaural phenomena, and also an extensive study of the perceived pitch of sounds, many of which are remote from sounds ever heard in nature. Among the specific topics covered are speech songs, synthesis of the singing voice, spatial reverberation, Fourier-transform-based timbral manipulations, the simulation of bowed instruments, automatic counterpoint, and a conductor program utilizing a mechanical baton. A chapter authored jointly by Mathews and Pierce describes a new musical scale they have been working on over the past few years. Max V. Mathews and John R. Pierce are members of the music faculty at Stanford University. Both spent the major part of their professional lives at Bell Telephone Laboratories, where Mathews directed the Acoustical & Behavioral Research Center, and Pierce directed the Communication Sciences Division.Current Directions in Computer Music Research isincluded in the System Development Foundation Benchmark series. The System Development Foundation was one of the first major supporters of computer music research.

Harmony and nonharmonic partials

We have explored some musical effects of tones with nonharmonic partials. The spacing of partials can be stretched so that each partial frequency fij present in tones sounded singly or together is given by fij=Ai/12+log2j. Here i is the scale step in semitones, j is the partial number, and A is the frequency ratio of a pseudo‐octave (A=2 for a true octave). We find that subjects can match the keys of stretched (A=2.4) as well as unstretched passages. Stretched cadences (A=2.4) do not seem final. But, a stretched ’’cadence’’ with equally spaced partials that goes from closely spaced (tonally dissonant) to widely spaced (tonally consonant) partials does seem final. Our experiments do not decide finally among three views of harmony: that harmony depends on a fundamental bass or periodicity pitch (Rameau), that harmony depends on the spacing of partials (Helmholtz and Plomp) or that harmony is a matter of brainwashing.

A passive nonlinear digital filter design which facilitates physics-based sound synthesis of highly nonlinear musical instruments

Recent work has led to highly efficient physics-based computational models of wave propagation in strings, acoustic tubes, membranes, plates, and rooms using the digital waveguide filter, the 2-D digital waveguide mesh, and the 3-D tetrahedral digital waveguide mesh, all of which are suitable for real-time musical synthesis applications. A simple first-order nonlinear filter structure derived from a passive nonlinear impedance circuit is described which extends the usefulness of these models, and which avoids the difficulties of energy conservation when memoryless nonlinearities are inserted in resonant feedback systems.

Theoretical and experimental explorations of the Bohlen–Pierce scale

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 ...

Periodicity and pitch perception

There has been experimental evidence pointing to at least two pitch mechanisms, the first involving low-order harmonics that are resolved along the basilar membrane, and the second a periodicity mechanism that depends only on the repetition rate of the time waveform on the basilar membrane. If this time waveform is derived from repeated bursts of sinusoidal tone, the second mechanism might be the sole pitch mechanism. It is found that this can be so up to rates as high as 250 bursts of 4978-Hz tone per second. The stimuli used are periodic patterns of equally spaced tone bursts, with either successive tone bursts in the same phase, or every fourth tone burst 180 degrees out of phase with respect to the rest. Up to a critical transitional rate of tone bursts a second, the two sequences sound exactly the same, despite their different fundamental frequencies and frequency separation of harmonics. Critical rate data are given for sinusoidal bursts of seven different frequencies. Critical rates appear to be closely related to the critical bandwidth. Pitch matching appears to be consistent with these observations; it is on rate below the critical rate and can be on fundamental frequency above the critical rate.

Computer Study of Violin Tones

Recordings were made of tones played by a professional violinist in an anechoic chamber. The sounds were digitized at 10 000 samples/sec and a frequency analysis made by computer. The amplitude of each harmonic was plotted at each pitch period. Two features of the spectrum were noted. The spectral envelope is essentially constant throughout the tone. The spectrum possesses zeros at regular spacing—typically every third harmonic. To explain the zeros, a theory was developed that involves the nature of the excitation by the bow. A computer program was written to synthesize tones with zeros. Tones with zeros characteristic of the violin were judged to have a very stringlike sound. It is hypothesized that these zeros, produced by the bow excitation, may be as important in determining string timbre as resonances introduced by the violin body.

Four new scales based on nonsuccessive‐integer‐ratio chords

Our previous work demonstrated that chords with frequency ratios 3:5:7: and 5:7:9: have many perceptual harmonic properties of major triads (4:5:6 ratios). The traditional diatonic scale can be constructed from three major chords, the tonic, the dominant, and the subdominant chords. Using analogous techniques with the new chords, we have constructed four new scales which we call the M579, M357a, M357b, and the P3579 scales. Pitches of the notes of these scales sound very different from the diatonic scale. All scales have clearly perceptable harmonic structure. All notes can be harmonized with consonant “major” chords. There are strong perceptual differences between consonant and dissonant chords. The M scales are based on an octave with a 2:1 frequency ratio. The P3579 scale has a 3:1 octave and was designed for timbres having only odd harmonics. In addition, P3579 is an equal tempered scale using a subset of 13 equal frequency divisions of the 3:1 octave. We believe these scales will be useful for nontra...

Computer Music, Coming and Going

Computer music is growing so large and various that I am losing touch with it. At this time, it seems worthwhile to review how it started and how it got where it is. Sound examples to the end of the 1960s are available through two records (now out of print), Music from Mathematics (1962, Decca DL9103) and The Voice of the Computer (1970, Decca DL710180). In writing this article, I have used these records to sharpen my recollection. If you have or can get copies of these records you can immerse yourself in computer music from its inception to 1962, and then from 1962 through 1970.

Study of Articulator Dynamics

A program for pitch‐synchronous formant tracking is described which operates semiautomatically on voiced portions of speech. Given the approximate pitch of a section of speech, the program locates the pitch periods, makes a spectral analysis of successive periods, locates 4 formants in each period by spectral fitting, and plots the formant trajectories and pitch for the sequence of periods. The program can also inverse filter to obtain the glottal waveform. The program is applied to study the dynamics of formant transitions as a function of speaking rate. The observed transitions are compared with the transitions which would be obtained from a simple mechanical‐articulator model, in which the vocal‐tract walls are represented by masses with damping driven by a suddenly applied muscular force.

Fletcher and pitch

Preparing remarks on Fletcher’s work on pitch changed drastically. My view of the development of our present understanding. In 1940 Jan Schouten showed that canceling out of the fundamental of a pitched tone did not change pitch. Schouten called the surviving pitch residue pitch. This inspired a spate of papers by people ignorant of earlier work. I had known Fletcher [Phys. Rev. 23(3), 427–437 (1924)] had shown that filtering out the fundamental and lower harmonics of a variety of musical sounds did not change the pitch, and, through synthetic sounds, that mere equal frequency spacing of tones did not give a pitch equal to the spacing. Thus, the pitch heard was evoked by harmonic partials. Unhappily, Fletcher proposed an unsound explanation in terms of production of the fundamental in the ear by nonlinearities. In a latter report, Fletcher [ 311–343 (1930)] gave an incomplete analysis of the functioning of the cochlea. Among his valid conclusions he said ‘‘the pitch of a tone is determined both by the pos...