The problem of determining the pitch center or the perceived pitch of frequency modulated sounds has been studied over a long period of time by a number of workers. The problem is of interest to psychoacousticians for insight into mechanisms of pitch perception. An understanding is also necessary for the study of intonation choices by string performers since most of their notes are played with vibrato. In fact for a meaningful study of intonation, the following questions must be answered:
What pitch is perceived by experienced musical performers and listeners when a musical sound with vibrato is presented?How do the accuracy and standard deviation of the responses of these experts compare for modulated and unmodulated sounds?
More fundamentally, in order to even pose the above questions or to study any question related to perception of musical sounds, it is essential to know:
How constant a frequency is it possible to produce with a musical instrument?
In this study we will address these questions with an emphasis on the first two. As a partial answer to the last question we report results on one performer.
The results of previous studies of pitch perception of frequency modulated sounds are presented in inverse chronological order in Table 1. The three most recent studies (Iwamiya et al, 1983, Shonle and Horan, 1980, and Sundberg, 1978) in Table 1 appear to agree that the mean pitch is perceived although there is some discussion as to whether it is the geometric or arithmetic mean. All three experiments were conducted using the method of adjustment which has certain problems in its rational underpinings (Hake and Rodwan, 1966).
Little attention has been paid to the use of musically trained subjects known to have ``good ears''. One study uses a group of "novices" which were undergraduate students picked at random and so-called ``experts" who were graduate students in music, physics, mathematics, or engineering; the tacit assumption being that scientific talent translates into expertise in musical perception. Insufficient emphasis has been paid to the inherent accuracy of the experiments. All were conducted using synthesized sounds. Shonle and Horan (1980) include an excellent review of early work in this area.
In some of these studies (entries 4,5,6 in Table 1) sounds were produced with and without vibrato by a performer, and these were considered to be his or her choice of the same pitch. We include in the Experimental Results Section a similar study, which is an extensive analysis of sounds produced by a well-known concert violist.
In more recent studies, Iwamiya et al (1985, 1989) have studied the effect of simultaneous amplitude modulation and frequency modulation of the components of two and three component complexes. They find that there is a pitch shift which depends on the relative phases of the AM and FM components. These results are not included in Table 1 since this is a different experiment from those reviewed.
d'Alessandro and Castellengo (1994) have studied the perception of very short time segments of synthesized vocal vibrato. They conclude that the pitch perceived depends on the final phase of the vibrato, and propose a weighted time average theory where the end of the tone is weighted.
In his 1970 review of musical perception, Ward states ``Studies matching the pitch of frequency-modulated sinusoids (or of actual musical tones) to that of steady tones are badly needed.'' Two and one half decades later there have been no studies using musical sounds produced by real instruments although it is on these sounds that performers have spent vast amounts of time training as they tune their instruments. They are thus the most experienced subjects ever to enter a booth as well as subjects with a known profiency in listening tasks.
It was also felt that a study with a musical emphasis is a more meaningful way to address the question of the JND of natural sounds since it is on these sounds that experts are trained. The current study differs from the previous ones in that it is conducted with actual musical sounds. The subjects all have experience performing with musical instruments where tuning is a necessary part of performance (in contrast to playing keyboard instruments where the performer is not responsible for the intonation). The experimental method is that of two interval forced choice (2IFC) which has a more solid rational basis than the method of adjustment.
In reporting the results the performers are divided into two groups according to their musical experience. The first group are non professional performers; whereas the second group consists of graduate students studying violin at a nearby conservatory.