Effect of Performance Tempo on the Level of Bass Notes

Abstract

The effect of a significant (up to 20 dB) decrease in the audibility level of low-frequency piano sounds (accompaniment in the bass register) against a background of high-frequency sounds (melody) when music is played at a fast tempo was discovered and investigated. This indicates that the performance tempo of a piece significantly affects a listener’s perception of the sound sequence of a piece of music. The effect is explained based on an analysis of the physical characteristics of sound formation in a grand piano depending on the performance tempo and perception of sounds by the human ear. The effect is clearly demonstrated by the example of performance of the first movement (allegro di molto e con brio) of Beethoven’s Sonata No. 8 in C minor (Pathetique).

Appendices

APPENDIX 1

LET’S LISTEN

Let’s hear how it sounds. All pianists play the Allegro di molto e con brio of Beethoven’s Sonata Pathetique at the fastest possible tempo, in which Chopin’s fast études are performed, ignoring the composer’s instructions for the Allegro tempo. Study [11] shows this Allegro performed by star pianist Tiffany Poon.

It is interesting to hear how this piece sounds if you play it at half tempo. In [12], an amateur performance of the first movement of Pathetique is demonstrated at half tempo, to which the author of this article was forced to refer, since there is no slow performance of this sonata on the Internet.

Despite obvious imperfections in the recording and reproduction of the performance, due to which, in particular, the frequency range of the instrument’s sounding is narrowed, it is noteworthy that in [11], the accompaniment is all but inaudible, and in [12], it is heard much more strongly. In Chopin’s études, too, exactly the same fast tempo and same time low frequencies are used, which for some reason are clearly audible. Why? Because in Chopin’s études, the low frequencies sound separately, and in Pathetique, the accompaniment accompanies the melody at the same time. Here another feature of human hearing is manifested, namely, its sensitivity is a variable value: it increases with a decrease in the level of audible sound, but only if this weak sound is heard separately. If a weak sound is perceived against another strong sound, then the hearing sensitivity does not increase. Therefore, in [11], the bass accompaniment is barely audible. If only low notes sound, albeit at a small level, then the listener’s hearing increases in sensitivity and amplifies these sounds.

Note that halving the performance tempo did not lead to the fact that the performance [12] began to seem deliberately slowed, since the accompaniment in the left hand began to play a much greater role, and it has twice the tempo, which compensates for the overall loss of tempo. However, fast passages, which simply slip through in [11], acquire expressiveness in [12].

This performance also raises a question that goes beyond the bounds of physics. Why does the pianist himself, pressing all the necessary keys diligently and on time and not hearing almost half the notes, calmly continue to play, pretending that everything is perfectly audible? I only have an answer to this question because I myself play. The pianist also has a certain inner ear: the sensation of sound arises as a result of pressing a key with a finger without exciting sound. For example, I can play my instrument for myself at night when everyone is asleep. I do not plug my instrument in and it does not make any sounds when I play it. But I can clearly hear myself playing. A virtuoso experiences the same then when performing Beethoven’s Sonata 8 at Presto, and for him there is no problem with the accompaniment. Internal hearing does not work according to physics! However, the virtuoso plays not for himself, but for the audience, and they perceive everything differently, as described above in the article.

APPENDIX 2

The editorial board considers it expedient and interesting for readers of Acoustical Physics to quote excerpts from a review of the article, which supplement and develop this topic.

“The article touches on how the duration of sounding of the chorus (string) of the piano (mainly, in the bass register of the piano) influences the time of Fourier analysis at a given resolution of the analyzer. It is concluded that a small number of oscillations in a pulse corresponding to brief pressing of a key (e.g., when playing at a fast tempo) leads not only to the frequency uncertainty of the fundamental sound frequency perceived by the listener, but also to a decrease in the listener’s perceived sound level compared to a longer sound pulse, which leads to a decrease in the perceived level of sounds in the bass register. Hence, it was concluded that many pianists play music at an excessively fast pace (with the example of Beethoven’s Piano Sonata No. 8).

In itself, musical works played by performers at too fast a tempo with poor control over the resulting sound of the instrument seems very relevant to us. For example, the author of this review had to repeatedly note, using the example of an organ performance, that some organists, in trying to demonstrate a virtuoso technique, take bass notes on the organ pedal keyboard so short that the sound of the organ’s bass pipes simply does not have time to form, being cut off at the attack stage; i.e., it is a transient process (the attack time in the sounding of bass organ pipes can reach 1 s or more).

However, for the piano, which the present article discusses, this problem is not as simple as it might seem, due to the psychophysiology of human perception of musical sounds.

First, perception of piano sound, including the sound frequency, is determined not only by the level of the fundamental frequency, but also by overtones (of which there are quite a few for a piano), and in its the bass register, the listener determines the fundamental frequency (more precisely, a subjective characteristic of the fundamental frequency, the pitch) not so much by the sound of the fundamental frequency as by the difference in frequencies of the overtones. The frequency of the fundamental tone can be almost completely absent; nevertheless, the listener identifies it from the difference in frequencies of the overtones. A.S. Galembo in his book “Piano. Sound quality” (Moscow, 1987, p. 21; Russian translation) indicates: “If there is no fundamental tone in a musical sound rich in overtones, its pitch does not change (see: Taylor, 1976); this is due to the nonlinear properties of hearing, which contribute to the sensation of difference tones, in particular, tones with a frequency equal to the difference in frequencies of adjacent overtones; this difference is equal to (or approximately equal to) the frequency of the fundamental tone. Such complex pitch perception characterizes, e.g., a piano’s bass sounds, the spectrum of which does not contain an audible pitch, sometimes in one or two of the lowest overtones.” Strictly speaking, this largely eliminates the problem formulated by the author of the article in relation to the piano.

The following points can be noted in the article, which are beyond the scope of the simplified model used:

—the author claims that human hearing has a resolution of 1/12 of an octave (obviously, focusing on the interval between the steps of the modern even-tempered scale), but in reality, the resolution of human hearing is much higher;

—sound attenuation (damping decrement) is considered in the article to be independent of frequency;

—amplification of sound by the piano deck cannot be independent of frequency, because the deck is a multiresonance system and formants are present in the amplified sound;

—the author refers to the performance tempo of the sonata indicated by Beethoven (on the metronome scale) and rightly notes that in fact this tempo corresponds to Presto, not Allegro, which Beethoven expressly indicated at the beginning of the sonata. However, it is known that in practice, orchestras rarely play Beethoven’s works according to the metronome he indicated, since this leads to an overly fast pace, and in the first movement of the Ninth Symphony, Beethoven indicated two metronomes at once (108 and 120). According to researchers, apparently Beethoven mistakenly indicated the tempo by the number not at the upper, but at the lower limit of the weight of the metronome, which led to overestimation of the tempo, while in the Ninth Symphony, he indicated both numbers at once. This may explain the problem of the fast pace in Beethoven’s works (see A. Martin-Castro, I. Ucar. Conductors’ tempo choices shed light over Beethoven’s metronome//Plos One. 2020. December).

In conclusion, we emphasize that the article describes one of the real factors that can affect the perception of piano sound or that of another instrument with a short attack.”

P.N. Kravchun, Moscow State University, Faculty of Physics