On falseness and paradigms for the nature of piano tuning

Brian Capleton PhD

© 2005, Brian Capleton

It is said that (aural) piano tuners tune by “listening to beats”. It is also said that equal temperament tuning (the modern tuning system for the intervals of the musical scale) is achieved in aural piano tuning by “counting beats” or “estimating beat rates”. Both descriptions arise from the idea of “instructions” that are provided for piano tuners to use as a “strategy” for tuning the instrument.

It is indeed necessary for the student of piano tuning to engage in “estimating beat rates” by ear, and with no previous experience of this skill, it may be necessary for the student to literally “count beats” against estimated seconds of time, or against the second hand of a clock. This, of course, reinforces the notion that piano tuning is all about listening to beats, and counting beat rates.

This “strategy” for tuning clearly requires “instructions” that are specific, and that answer the inevitable question “What beat rates should be applied to which intervals?” An answer is conveniently provided by tables of beat rates – tables that purport to show precisely what beat rate should appear in each interval.

In understanding piano tuning in this way, we have reached what some philosophers might call a paradigm:

A theory produces a table of beat rates, that if applied to the intervals, will produce something called equal temperament tuning.

In other words, applying the paradigm, the tuner’s task is presumably to estimate these beat rates by ear, in order to produce an “aural estimation” of the “theoretically correct” tuning.

Later, we might find that the theory used to calculate these beat rates is incomplete (like most scientific theories). We might encounter, in particular, the concept of inharmonicity, which then has to be “added in” to the picture, in order to adjust the beat rates. Then we have a new paradigm:

A theory that takes into account inharmonicity, produces a knowledge of beat rates, that if applied to the intervals, will produce something called equal temperament tuning.

Or, perhaps:

A theory produces a table of beat rates for equal temperament, but we may have to apply somewhat different beat rates, because of inharmonicity.

As a paradigm, this is a “mental picture”, a notion for attempting to understand and explain (a) what is required for piano tuning, and (b) what the master piano tuner does.

The paradigm contains the subtle, or perhaps not so subtle implication, that the beats, especially in the context of beats being counted, are a “tool”, or “indicator” being used by the tuner to “estimate” a “theoretically correct” tuning. To use an analogy, this is rather like “painting by numbers”, except that we may have to vary each number slightly because of inharmonicity.

This view of piano tuning is astonishingly common, but sadly, it entirely misses the point and nature of the art. In the art itself, “beats” are an important phenomenon in their own right that are “adjusted” accordingly, adhering to certain principles such as “progression” of beat rates. In fact, “progression” in aural tuning, is much more important in its own right, than any other notion or definition of “equal temperament”. Progression does not generally come from the application of pre-defined rates or numbers, but has to be achieved empirically, the pre-defined rates being only a starting guide.

The reason for this, or the divide between theoretical “definitions” and the actuality of tuning as an art, is usually presumed to derive from inharmonicity. In fact, even taking inharmonicity into account, the very first premiss of the “tuning by beats” theory, as it stands, is flawed. That premiss is that tuning can be defined in term of beat rates, because beats are by definition adjustable, i.e. they are sensitive to changes in string tension.

The reality is that piano tone contains two types of beating, one whose beat rate is adjustable with string tension, and another whose beat rate is very insensitive to changes of string tension within the normal tuning range. The latter are known to aural tuners as “false” beats, and they are often wrongly imagined to be the exception - a kind of “fault” that occasionally occurs.

“False” beating is not an anomalous “add on” to an otherwise complete picture of piano tone and tuning. Rather, it is an integral and inherent part of natural piano tone behaviour, noticed in the scientific investigation of piano tone at least as far back as 1935 (Wolf and Sette, JASA, 6, 160-168), and mentioned as early as 1907 by the piano tuner Cree Fischer (Piano tuning – a simple and accurate method…). It is at least in part due to the natural behaviour of piano string motion itself, which is much more complicated than as supposed by the standard theory for beats. Piano strings do not simply vibrate polarized in one plane, as the standard theory of beats requires. Nor can the transverse motions of piano strings in situ be adequately described by more generalized elliptical polarization. Rather, in general, the description of transverse piano string motion must allow for parametric loci. As early as 1943 Schuck and Young (JASA, 15, 1, 1-11) suggested that what we would in the context of piano tuning call false beat phenomena, could be explained by the rotation of the plane of vibration of the piano string, and possibly by the transfer of energy from one mode to another.

Today, digital analysis readily shows false beat phenomena to be a ubiquitous feature of piano tone. (and indeed the tone generated from many other musical and quasi-musical sources). When a false beat rate is sufficiently slow that the beat decays at around the same rate as the partial in which it appears, it may simply not be noticed. However, this does not mean it has no influence. False beats are constantly present in piano tuning, and like adjustable beats, constantly affect the tone quality of intervals, notes, and octaves.

The interaction between false and adjustable beats is completely outside the standard theory of beats. The fact that false beat rates are not readily adjustable through changes of string tension, does not mean false beat phenomena and its effects are non-adjustable in the tuning process. In fact, the idea that beat rates are the only adjustable feature of beat phenomena in fine tuning, is an assertion of certain theorists, or theories, rather than an established assertion of master tuners. This is not only relevant in the tuning of the tempered intervals, but is also (perhaps especially) pertinent in the tuning of unisons and octaves.

In addition to the existence of two distinct types of “beating” in piano tuning, is the fact that bridge coupling between strings allows for the adjustment of partial decay rates and amplitudes, at least in unisons, and possibly in other intervals (Weinreich, G, “Coupled piano strings” JASA, 62, 1474-1484, 1977). The existence of false beats in the context of bridge coupling, presents an acoustical system in the actual piano tuning situation, which the standard theory of beats is simply not able to describe. The description provided by the latter, even taking into account inharmonicity, is a crude simplification and approximation for the actual situation. Nonetheless, it has penetrated the psyche of a good many theorists, would-be theorists, and practitioners who claim to be good tuners. It is in the nature of the human mind to attempt to fit the interpretation of experience to presumed knowledge already established in the mind, and in the case of the popular paradigm for tuning, learning from the experience of tuning – even correctly perceiving what is actually there - can be impeded by holding inadequate mental/theoretical models for the phenomena being encountered. The better theorists generally know from experience that a natural phenomenon is always likely to be more complicated than previously supposed, but unfortunately there are many who never question the popular paradigms.        

Provided it is understood that “beating” and “beats” are somewhat generic terms, it is reasonable to say that (aural) piano tuners tune, in part, by “listening to beats”. However, rather than using the paradigm of tuners applying beat rates to “aurally estimate” some theoretically “worked out” tuning, perhaps a more accurate paradigm would be:

The finest piano tuning is an art, carried out by tuners drawing on a deep empirical knowledge of piano tone and tuning behaviour. All current theoretical models for piano tuning are just models. Even if we improved the model to account for the data we now have for piano tone behaviour, in the words of Samuel Karlin, The purpose of models is not to fit the data, but to sharpen the questions.

(Karlin, S, 11^th RA Fischer Memorial Lecture, Royal Society, 20/4/1983, cited in Buchanan, Mark, Ubiquity, Phoenix, 2000)