Historical Context

In the penultimate chapter of Helmholtz's On the Sensations of Tone we read: ‘What the singer finds a difficulty in hitting, must naturally also appear an unusual and forced skip to the hearer’.Footnote ¹⁶ What Helmholtz is referring to here is a phenomenon known in his time as ‘unharmonischer Querstand’ or ‘non-harmonic cross position’. The term refers to certain sequences of notes that were forbidden according to the rules of musical composition, namely those in which ‘two tones in consecutive chords, which belong to different voices, form false Octaves or false Fifths’.Footnote ¹⁷ This definition (and especially the designation of the intervals as ‘false’) has its roots in medieval music theory. It stems from the time up to the Renaissance when chromatic steps could occur, but not be notated, at least not without implying additional rules. The rules that clarified how to read the written notes with chromatic steps were known as musica ficta or musica falsa (fictitious or false music).Footnote ¹⁸

‘False relations’ – the correct English term used in the translation of Helmholtz's book – resulted from a breach of one of those rules: mi contra fa est diabolus in musica. That is to say, music that contains two consecutive notes that read as the fa and mi from two different scales in the Guidonian hexachord system constitute an unruly jump between hexachords, resulting in such awkward intervals as the altered octave or the tritone. Neither of these intervals was supposed to occur between two different voices and especially the outer two voices in a polyphonic setting.

The designation of these intervals as ‘false’ had come down to Helmholtz's own time, as can be seen from the continuation of his text:

Starting in the eighteenth century, it struck music theorists as inadequate to call such voice relations ‘false’ or unharmonious. In his Musikalisches Lexikon of 1802, Heinrich Christoph Koch stated under the heading ‘Unharmonischer Querstand’ that ‘no definite rules could be given as to which of the cases of note progression would be objectionable and which admissible’.Footnote ²⁰ If one strictly applied the old rules, then a simple progression of thirds had to be considered a mistake. But as this case showed, many instances of false relations could not be avoided in a harmonic setting, nor did they strike anyone as false or ugly. Some instances, Koch conceded, were indeed unpleasant, but ‘good composers’ (gute Tonsetzer) avoided them.Footnote ²¹ He spent most of the entry, though, on specific cases in which the rule was to be abolished.

Koch's article closely resembles the entry in Johann Georg Sulzer's Allgemeine Theorie der Schönen Künste (1771–74; General Theory of Fine Arts), authored by Johann Philipp Kirnberger and his pupil Johann Abraham Peter Schulz.Footnote ²² Comparing the entries, one notices how graphic markers of performance, such as the treble clef and different-facing note stems, begin to vanish from the music examples, which adumbrates the allegedly neutral piano-inspired setting that later became the standard for music examples in music theory. This indicates a change in the performance medium that was already well under way in the eighteenth century.

More importantly, for the purposes of this discussion, however, we find that listening had gained a new-found status. Both entries prominently emphasize the effect contrapuntal cross relations have on the listener, yet they do so from two different perspectives. The heading for Sulzer's entry presented the qualifier ‘Unharmonisch’, which means both non-harmonic and unharmonious, while the term ‘Querstand’ (cross relation, false relation) remains unexplained in the body of the text. In Sulzer's General Theory of Fine Arts, the clarification of such terminology was not the pressing issue. What was more important was the explanation of what counted as unruly within the confines of good harmony. Readers were instructed that music had its own system of rules, which sometimes clashed with the common sense or the ‘feel’ of listeners. Remarkably, for Kirnberger and Schulz cross relations were the only unharmonious phenomena in music worthy of an entry. The fact that he did not find this unharmonious phenomenon unpleasant makes the entry even more remarkable.

Whereas Sulzer addressed an erudite public without musical expertise, Koch in his lexicon addressed the musical expertise of his readers. His lemma ‘Unharmonischer Querstand’ refers to terminology in the first instance. In addition, Koch discussed a subset of cross relations that he assumed to be known to the reader as non-harmonic from the outset. The fact that Koch, like Sulzer, spent most of his entry on ‘unharmonious cross relations’ by listing examples that did not merit disqualification became more of a reason to change the rules rather than to keep them. Koch's book was thus meant to empower his readers to use correct terminology, and, at the same time, he invited them to redefine the rules that governed composition.

If these eighteenth-century music theorists insisted that most of the cross relations called ‘false’ were in fact tolerable, a reference work from the first half of the nineteenth century does the exact opposite. Ignaz Jeitteles's Aesthetisches Lexikon (1837) has an entry ‘Unharmonischer Querstand’ as does Koch, but he called the phenomenon straightforwardly ‘incorrect’. Notwithstanding its elegant brevity, the explanation he provided rather obscured the matter, for it demonstrated the opacity of musical correctness to the erudite layman.Footnote ²³ The amateur listener, now the one being addressed, should just trust that the rules of composition would be properly executed in the hands and ears of the experts.

As we come to the time of Helmholtz, we see that his explanation of false relations introduced a new perspective that differs from others of the past. Believing that the chromatic step that implied two different tonal contexts was difficult to sing, Helmholtz concluded:

The earlier writers on false relations highlighted the unruliness in strict counterpoint from the medieval writers of theory without any explanation. Their mentioning of any effects on the listener mainly served the discussion of whether the strict rules of counterpoint needed to be updated. Helmholtz, in contrast, did not take the validity of such rules for granted. He looked for an explanation by examining the listening effect that a rule produced during a performance. He assumed that the rule originally had responded to the needs of singers and that, as a consequence of the difficulties in singing cross relations, listeners also felt awkward as they sensed the difficulties of the performers, whereby he insinuated that an empathetic bond now developed between listener and singer.

The speculative mode in which he argued, on the one hand, suggests that Helmholtz did not feel entitled to comment on the rules of counterpoint. The question of what was correct or incorrect, on the other hand, did not matter much for his argument. He observed the situation of performance from an outsider's perspective. The authority for both speculation and observation was no longer granted by the system of rules itself. On the contrary, Helmholtz offered up for discussion the entire system of rules and established a new way of examining them. As a result, speculation of a rule system governed by the effects of a performance on the listener appealed to a thought experiment rather than to an actual performance. Singing became part of an experimental setting: it turned into an activity that had to comply with rules different than those of harmony, namely those of scientific observation in the service of controlled experimentation.

Helmholtz's thought experiment implied a number of new aspects that had been absent in the earlier discussions of false relations. Most strikingly, the earlier authors did not invoke sound in their discussion of the rules that governed false relations. They discussed whether certain instances were objectionable or acceptable, and even when they directly addressed the effects of them, they did so only in a double negation. Thus, Sulzer's entry, for instance, reports that a harmonic progression of major thirds was ‘rarely considered obnoxious’.Footnote ²⁵

Helmholtz, in contrast, considered the acoustic effects of the note. The fact that the outer voice parts were considered more audible was not mentioned by the earlier authors. The rule seemed to be valid for them without such additional explanation to its sonic effect. From his outsider position as a non-music theorist, however, Helmholtz made this a prominent aspect of his explanation of compositional rules. What is more, the depiction of a hypothetical performance highlighted the immediate effects of sound on performers and listeners. If it was difficult for the imagined performer to hit the right note, then this was owing to the fact that another disturbingly resembling note was present in the chord that the singer heard while heading towards the new note. For the composer and the music theorist this may have seemed self-evident, for the observing experimenter, however, this was crucial. Such a reaction would have to occur every time that this conflict between the notes heard and the notes to be sung occurred to make the situation fit into the rules of his thought experiment. In that case the problem caused by false relations could be said to be a repeatable and independent experience.

On the other hand, the discussion of cross relations placed an emphasis on sequences of notes. It is indeed part of the chapter on voice leading in his book, where it figures as the concluding topic. The entire chapter is characterized by a discussion of harmony that reaches far beyond an empirical investigation of the physical activity of hearing. Nevertheless, this discussion is made to fit with the mainstream physics of Helmholtz's time. He subjected musical harmony to a guiding metaphor that he took from fluid mechanics. Thus, music appeared in his text as ‘sound mass’ (Klangmasse), where a stream of liquid matter was to be kept in equilibrium. Harmony was what could, in the end, grant the equilibrium within the liquid, but it could also be used to express ‘stress’ or ‘pressure’ (Kraftanstrengung).Footnote ²⁶

Such terms found in the text betray the scientific disciplines that would come to define Helmholtz as the leading expert of the mid-nineteenth century, namely hydrodynamics and thermodynamics. Helmholtz's contribution to these areas began with his proposal for a law of conservation of energy in 1847, which would eventually lead to his appointment as chair of physics at the University of Berlin in 1871. One notes in his descriptions of music this expertise lurking in the background, especially when he describes, for instance, voice-leading relations to the tonic at the beginning of that same chapter:

The industrial overtones that are presented in this description in the original German are slightly mitigated in the English text, where the translation of ‘a mass of tone’ shifts Helmholtz's neologism ‘Klangmasse’ back into the realm of music and the first-person plural integrates the reader into the organization of sound, which the German reader was confronted with in the passive voice. The grandeur and potential violence, elicited by the motion of sound in music's ‘connected vessels’, comes across despite such mitigation in the translation. In the nineteenth century, new energy supplies increasingly dominated the industrialized world. In order to conduct and exploit the flows of energy, it was essential to remain in control of steam engines, gas systems, and the growing scale of production they enabled. Loss and danger through leakage or turbulence, friction, and uncontrolled change in temperature had to be prevented, and this required the smooth equilibrium that is ascribed here to music.Footnote ²⁸

A related feature in the chapter on voice leading is the striving for the efficiency of expression. Together with notions of effort and equilibrium, Helmholtz was able to cover a wide range of phenomena under this notion of efficiency. The focus was on the effect that they made on a listener without specific training in music theory. He alluded, for instance, to the imitation of speaking, howling or crying. At other times, he commented on the contrapuntal rules for resolving dissonance as a process to overcome resistance, or on the effort voices make for reaching high notes, or even on suspensions pushing each other forward and the best ways of performing such nuances of expression:

In all these statements, Helmholtz's fluid mechanics of sonic expression never embraced music's system of rules. The forces of equilibrium in a moving sound mass were presented as essentially self-sustaining. The sound mass flows where it finds a convincing continuation of its original path; the composer only controls the flow for the sake of expression. Helmholtz thus avoided judging chord progressions as correct or incorrect. To this point, the English translation of Helmholtz is often misleading as it presumes that he is judging the correctness of a chord when he states ‘they appear to be simply mistakes’, when in fact it should be translated as ‘they are simply sounding badly’. The translator thus often inserted cognition, where Helmholtz always spoke of the physicality of the tone.

Deviation

‘False relations’ epitomizes Helmholtz's foray into the realm of sound succession as well as musical aesthetics. As we have seen, for Helmholtz, the thought experiment of false relations consisted of the fact that one singer had to produce a note that did not fit with what he heard in the context of other singers. The auditory sensation felt by the singer hindered his ability to produce the note. The distracting note trapped the singer's mind. The role of sensing the present harmony in a cross relation as described by Helmholtz thus receded to a single moment of exposure where the flow was disrupted. It was as if the fictitious singer faced a gap between physical hearing and musical cognition, between tones and notes. The flow of the music got stuck and only a cognitive effort would make it continue.

The above speculation of Helmholtz is primarily supported by his description of an economy of expression. In the chapter on voice leading, he generalized the relationship between the listener and the flow of music as moments of stasis and expressivity:

Here we see that the infringements of harmony take on a peculiar role. They appear to the listener as deviations from the usual flow of music. Such deviations serve the aim of expression, and they can be perceived against the background of a stable equilibrium in the musical flow that is granted by the rules.

A second factor in Helmholtz's speculation on false relations can also be contributed to the interplay of rules and infringements. Given the listeners’ changing habits, the context of the rule itself was flexible. Listeners constantly readjusted their hearing and expectations to what at a given moment counted as ‘usual’ and what as an ‘effective violation’. As Helmholtz explains, this also held for false relations:

The metaphor of fluid mechanics comes full circle in this passage. We see that the ‘old ecclesiastical music’ imposed rules that modern composers could twist and bend, thereby shaping new listening habits and rules. As a physiologist wrenches a complex theory of hearing from the laws of fluid mechanics, a composer can be said to impose an original, expressive modification on the system of harmony. Undeniably, a strong notion of progress governs much of Helmholtz's thinking here, one that was quite common in his time.Footnote ³²

However, for the experimenter, progress in music comes with a specific problem: the aesthetic endeavour imposed by composers provided no stopping rule. By admitting that nothing determined whether an infringement would be considered progressive or harmful, Helmholtz deprived himself of a fixed frame. As a consequence, it became impossible to query which physiological phenomena would play into this notion of bending the rules. The repetition of an experiment ceteris paribus, that is, keeping all factors stable except the one of interest, was illusory, since the repetition of an expressive infringement in music would not necessarily produce the same effect twice, but instead lead to a habitual effect. The habit, for instance, of considering something a dissonance or consonance as such did not give access to a physiological experiment. In that particular case, Helmholtz found a way to deal with the shifting expressive aspects of dissonance and consonance in various intervals. His own theory, namely that the roughness resulting from acoustic beats in various intervals, was flexible enough to explain changing tastes and definitions across centuries.

However, if false relations pointed to a rule that could be infringed upon by the composer at any time, this meant that there was nothing to hold onto for the experimenter. Expression therefore could not only justify infringements, but it could even turn upside down the entire system of rules:

So, when music expressed effort or excitement, physicality overruled harmony. To say that the rules of music become meaningless is after all a surprising statement. By the same token, this eliminated the option to have to investigate the misstep. Without a reference point for the infringement, there was no longer any deviation to be explored and measured in the experimental investigation. As will be shown in part two below, for Helmholtz's experimental research into hearing to occur, a deviation was necessary between two states: one undistorted and the other a deviation from it.

Objects and Devices of Auditory Physiology

The English term ‘false relations’ and the German ‘Querstand’ are fully equivalent in the sense that they refer to the same infringement of a rule in voice leading. Whereas the German term perhaps fits more easily into a physiologist's account of the rules of music, the English term more explicitly addresses the purpose of the present argument. Physiology can comment on phenomena in organic life, but it cannot possibly explain why something is considered ‘false’. Yet, the study of the senses has always had an interest in ‘the oddities or departures from the common and commonplace accuracies of perception’.Footnote ³⁴ Eighteenth-century studies in perception adhered to the notion that perception relates to the perceived object as being either veridical or illusory. The moon illusion, a traditional example that goes back to Aristotle and was discussed by Kant in his Critique of Pure Reason, illustrates this quite well. The size of the moon as we perceive it may change, depending on the position of the moon and our own position in relation to the moon, but this does not entail that the actual size of the moon changes as well.Footnote ³⁵

Perceptual phenomena changed their status in early nineteenth-century physiology. If sensory perception is produced by the body in one way or the other, then to call such phenomena illusions is not justified. What the body produces is what we must accept. Johann Wolfgang von Goethe was among the first to carry out extensive experimentation on the physiology of vision – as opposed to a physics of light or a philosophy of visual perception. He spoke of a ‘subjective experiment’ (subjektiver Versuch) in which the experimenter's body was involved as the site of experimentation, as opposed to an ‘objective experiment’ (objektiver Versuch), in which the experimenter took on the role of an observer of phenomena occurring outside of their own body. In his Farbenlehre, he reported extensively on his studies of subjective phenomena in optics.Footnote ³⁶

Following Goethe, physiologists such as Jan Evangelista Purkyňe and Johannes Müller claimed that subjective phenomena in perception occurred just as regularly as any physical phenomena outside of the human body and therefore merited scientific investigation. The point of departure for nineteenth-century physiology was thus the study of these ‘subjective phenomena’, like the after-images one sees when closing the eyes after just looking into a source of light. After-images clearly demonstrated that the body itself produced phenomena. These phenomena disrupted the discourse on perception that had considered them either as physically demonstrable or as figments of the imagination.

In exactly these terms, the author of an 1805 article on the Tartini tone described the third tone that appears when two strong notes with constant pitch resound. Ulrich Vieth, translator of Thomas Young into German, doubted that the combination tone, as he called the third tone, could be explained as either purely physical or purely imaginative. On the contrary, both explanations seemed valid. Any tone thus would be called a figment of the imagination if the criterion for this was that regular motion results in the perception of a tone.Footnote ³⁷ Vieth did not consider sensory physiology suitable to find the answers to his questions – recall that his text was published five years before the first publication of Goethe's Farbenlehre. And in fact, for the study of hearing it proved difficult to claim any heard phenomenon as being subjective, yet verifiable at the same time. Johannes Müller, for instance, argued that the rattle of a stagecoach that remains in one's ears for days after travelling could serve as an example. He could not give any advice, though, on how to investigate this phenomenon in a systematic way.

Helmholtz instead scrutinized the wealth of knowledge on sound gathered through the ages in music composition. His preference for periodic sound, that is to say sounds with a clear pitch, privileged Western music for this purpose. Periodic sound was easier to reproduce and to research, and it provoked all kinds of distortions. According to Helmholtz, distortions, in particular, offered insight into the body's production of the sensation of sound and more specifically the limitations of such production. For his research into hearing, he took combination tones as his point of departure. He was able to demonstrate that the third tone occurred in the body as a distortion, which results when two vibrations are superimposed in a small-sized medium, such as in the middle ear.Footnote ³⁸

The article on combination tones perfectly presented his method of auditory physiology. It demonstrated that the body produced sound rather than just channelling vibrations to the mind. For understanding this bodily production of sound, distortion was a privileged measurable object.Footnote ³⁹ As long as the coming into being of tones themselves was difficult to determine, distortions demonstrated the limits within which tones can be heard. Distortions such as the third tone proved that the ear itself is a site where sound emerges. Distortions, as defined in this way, made hearing perceptible.

Another important aspect of Helmholtz's methodology on combination tones is that the conditions needed for third tone to emerge were dependent on the two initial tones. Not just any sound produced the third note, but a musical tone that was in line with an aesthetics of a beautiful note that was not so far away from Helmholtz's idea of the hydrodynamic equilibrium in musical harmony. So, all the phenomena involved were musical notes, including the distortion itself. As a consequence, the constellation of a third note joining two notes played simultaneously could be easily observed with regard to the resulting intervals. The third tone would either fit, or be in contradiction with, the rules of counterpoint or distort them like a note that was out of tune. In the history of music theory, this third tone even served as a starting point for constructing entire systems of harmony before and after Helmholtz; the treatises by Giuseppe Tartini and Paul Hindemith are prominent examples.Footnote ⁴⁰

Distortion in this context acquires a second meaning. The combination tones not only witness the limitations of sound emerging in the body, they also function as a potentially disturbing occurrence within the system of musical harmony. This made Helmholtz ask whether the system of rules governing music composition also had taken these occurrences into account. He designed an extended series of experiments in which he scrutinized compositions with respect to the traces of distortion within them. In so doing, he tried to confirm that the ‘best’ composers had avoided distortion by combination tones. In fact, he found a lack of distortions in Mozart's Ave verum corpus and Palestrina's Stabat mater.Footnote ⁴¹ In Palestrina, for instance, Helmholtz analysed the chords one by one, and then arranged them on a scale of ‘harmoniousness’. The resulting statistics demonstrated that Palestrina's choice of chords avoided the registers in which distortion through difference tones would have been audible. How this particular finding related to voice leading was of no concern to Helmholtz at this point in his research, though.

It is important to note that Helmholtz's work on auditory physiology stood somewhat apart from physiological experimentation more generally. As noted previously, physiology developed rapidly during the nineteenth century. A major factor in this had been the call for a new mode of experimentation. French physiologist François Magendie was among the first to urge his colleagues to adopt experimental methods in his Précis élémentaire de physiologie (1817).Footnote ⁴² According to him, physiologists had to renounce the strict coupling between physiology and post-mortem anatomy. Neither the gaze into a corpse, nor a fixed notion of what they should be looking for in it would help physicians to understand the living body. Magendie's own handbook for medical students instead dealt with experimentation extensively.Footnote ⁴³

As explained earlier in this article, physiology eventually succeeded in subjecting living organisms to experimentation, although an understanding of life itself still remained out of reach. For this, experimentation was set on a new footing. Earlier models of experimentation in physics had proposed to test a pre-formulated hypothesis under the conditions of a controlled experience in order to either confirm or contest it. Such an experimentum crucis was not feasible on living objects, however, since most factors of their functioning were unknown. Physiological experimentation instead turned towards an exploratory mode of inquiry.Footnote ⁴⁴ Physiologists provoked the emergence of new phenomena, even though no hypotheses for their explanation were available. The scrutiny of after-images mentioned earlier is an example for this. Physiologists produced extended series of such visual impressions in order to understand the patterns and laws potentially inherent in them. In modern terms, one could say that the physiologists were gathering data, before subjecting the data to a hypothesis.

For the study of hearing, the exploratory mode was problematic. Controlled experimentation imposed additional constraints. Stable reproduction of sounds was by no means easy. More importantly, it dawned to nineteenth-century researchers working in the domain of acoustics that they did not know enough about sound to tell what they heard. Even when it seemed that the same sound was reproduced, this did not guarantee that the auditory experience was the same. Helmholtz's use of music was peculiar in this respect. At first glance, he seems to continue a tradition of using music for identifying, consistently reproducing, and denoting sound. Upon a closer look, though, music appears to have a different function. A comparison with the study of vision can help to clarify this peculiarity.

Vision and Hearing

The role of music in Helmholtz's auditory physiology is different in important ways from that of the art of painting in his treatise on physiological optics. In the foreword to the book On the Sensations of Tone he states that music has a different status than the other arts, because musical harmony deals with ‘the natural power of immediate sensation’:

Painters eagerly embraced this idea of an immediacy of colour. They looked for new ways to see and depict the world and found them in what Helmholtz explained about vision, not painting.

Music, in contrast, pervades the argument of On the Sensations of Tone. Yet, the reactions of composers to his text only become noticeable in the early twentieth century.Footnote ⁴⁶ Music and visual arts differ in the way they impact and relate to Helmholtz's sensory physiology in a number of ways. The first of these concerns the stability of the objects. Seeing the same thing twice is not the same as hearing the same sound repeated. Yet experimentation needs reiteration. Acoustic experimentation had to ensure that the phenomena under investigation could be stably reproduced. Part of Helmholtz's success in putting acoustics on a new footing was founded on keeping one sound stable until his query into this sound was done, rather than rely on identical reproduction of that sound. He used the siren for this purpose, as well as electro-magnetically produced vibration in tuning forks. Although he was well aware that this excluded transient features in sound perception from the scope of his investigation, he was at pains to demonstrate that there was still enough to be discriminated by the ear and thus to be scrutinized by the physiologist.Footnote ⁴⁷

A second aspect is that anatomical knowledge about the ear, and especially the neural structures of the organ of hearing, made a leap with technical improvements in microscopy in the mid-nineteenth century. To begin with, the enterprise of presenting a new physiological theory of hearing was triggered by the findings of neurological structures in the inner ear by Marquis Alfonso Corti, who was first to apply the new techniques of microscopy in the ears of mammals. He mentioned in passing that what he saw there resembled the action of a piano, and Helmholtz famously expanded on this idea.Footnote ⁴⁸ Speculation reached its end, though, when it came to the connection between both ears. For the physiologist, it was not evident that the eyes and ears work together in any comparable manner. The optic nerves cross directly behind the eyes and before reaching the brain. Their chiastic juncture could be detected without microscopy and accordingly was known long before Helmholtz. Nothing of the same sort was known for the two ears. It took until the end of the century for microscopic neuroanatomists to discover the complex and tiny structures that connect the acoustic nerves before the brain.

Helmholtz therefore treated the ears like two independent organs. He considered the way in which auditory data from the two ears merged to be the same as the way in which we combine input from different senses. Not long before he started working on sensory physiology, astronomers had stumbled upon the phenomenon that sense data from different organs are synchronized differently in each individual. The consistent deviations among the personnel who notated the passing of stars in the telescope to the beat of a pendulum could only be explained based on each member's individual manner of relating auditory and visual perception. This individual footprint of correlating beat and vision was termed the ‘personal equation’.Footnote ⁴⁹ If the two ears had to be considered independent, then their combination was likely to result in equally individual manners of concatenating input. For Helmholtz, this was a reason to leave out binocular hearing in his auditory physiology, whereas in his physiological optics binocular vision provided an important share of the examples he discussed in the part that dealt with higher order physiological processes in vision.

This takes us to a third aspect in the comparison. Vision and hearing yield different ways of dealing with the cognitive aspects of perception. Both books follow roughly the same three-part structure, which in turn derives from Helmholtz's distinction of three levels in his concept of sensory physiology. He explains further in the preface to On the Sensations of Tone:

The third part of Helmholtz's Treatise on Physiological Optics dealt with experimenting on the ‘unconscious inferences’ we make when dealing with two sources of optical input. A simple experiment could consist in closing one eye and then comparing the way in which the impression of a shiny surface is reduced to a stark contrast in the subsequent view of that surface. This experiment points to some unconscious operation that merges both visual impressions into a third quality, namely the perception of the shiny surface. This unconsciously occurring inference explains why we see the stark contrasts on both surfaces at slightly different locations merge. The contrast is due to a reflection on that surface; it is not a feature of that surface's colour.

The way in which the experimental physiologist reached this insight consisted of a simple change in the perceptual condition. Closing the eyes alternately revealed the different location of the contrast on the perceived surface. The inference thereby became a conscious one. On a higher level of abstraction, the change in the condition could be called a ‘defamiliarization’ of vision. Various instruments were used, and sometimes explicitly devised, for experiments that forced the subject into perceptual situations, in which such unconscious inference was interrupted. The stereoscope is one example, which used a combination of mirrors to swap the images between the left and right eyes.Footnote ⁵¹ Helmholtz also used very simple ways to achieve defamiliarization, like closing one eye or bending over so as to look between one's legs.

According to Helmholtz, timbre was the equivalent to such unconscious concatenation in hearing – not in the least because humans can distinguish timbre with one ear only. Concatenation had to be understood more literally in this case. As is well known, Helmholtz assumed that the ear operates like a mathematician using Fourier analysis. The theorem proposed by Jean Baptiste Joseph Fourier in 1822 postulated that any periodic function could be analysed into simple parts of sinusoidal shape in integer ratios. Fourier himself had suggested that his theorem might prove useful for understanding the motion of a vibrating string, but had not mentioned the sense of hearing. Georg Simon Ohm later postulated that the same theorem could be applied to the tones of music and more generally to any periodic sound. In his first article, a mistake in his calculus made him infer that we hear only one note where there are several frequencies (he obtained an infinite amplitude for the fundamental frequency). Another physicist, August Seebeck, detected the mistake and flatly denied the postulate altogether. In his last paper on acoustics from 1846, Seebeck nevertheless conceded that Fourier analysis was probably the means to analyse tones and went on to speculate that this might relate to timbre, but did not delve into the matter. Ohm by then had already given up on acoustics; Seebeck died prematurely in 1849.

Helmholtz did not credit Seebeck for this insight, nor the otherwise unknown school teacher from Königsberg named Eduard Brandt, whose posthumously published paper informs the reader that he had shown his ideas on Fourier analysis and timbre to Helmholtz in the summer of 1855, shortly before the latter left Königsberg for Bonn. These ideas concerned the following postulation: if the analysis of periodic waves yielded different results for different timbres, then the distinction of timbre by the ear could, inversely, be related to Fourier analysis. Helmholtz pursued this idea, crediting Ohm for naming the insight that periodic sound could be analysed in the terms of Fourier analysis, while keeping the decisive insight into how Fourier's theorem could be linked to hearing for himself. As opposed to Brandt, Helmholtz also provided a label for the distinction. Differentiating shades of timbre from now on would refer to ‘Klangfarbe’.Footnote ⁵²

Starting from the Fall of 1855, Helmholtz worked on devising an experimentum crucis for this hypothesis: if the ear proved capable in distinguishing a minimal timbral difference between periodic sounds, then Fourier's theorem would become meaningful. Just as the analysis of different shapes of periodic waves resulted in different patterns of sinusoidal components according to Fourier's theorem, so these same patterns could be considered to designate a difference in timbre for the human ear.

Although timbre took on the role of binocular vision in the sense of showing how sensations merge into higher-level perceptions, it did not serve as the main example in the third part of the book. Instead, harmony and voice leading – or rather, the topography of periodic sound – provided the subject matter for the psychological physiology of hearing. Here, he questioned the extent to which musical systems can be reduced to Fourier analysis, providing a trigonometric space in which hearing detects affinities between the sounds of instruments.Footnote ⁵³ Many musical instruments exhibit harmonic spectra and thereby suggest that humans had reacted to this feature in sound. Based on this observation, Helmholtz again and again questioned the systems of rules that had yielded musical systems, whether in other places of the world like China and Persia or in other times like Greek Antiquity or the Italian Renaissance.Footnote ⁵⁴

Deviation as a Research Device

The experimental setups for a psychological physiology of hearing used neither defamiliarization nor did they focus on distortion. Distortion shed light on the normal functioning of the body. For this, distortion (in the sense of Helmholtz) had to be seen as a physiological feature itself. Defamiliarization, in turn, relied on everyday understandings. Against the background of unquestioned features in perception, its functioning became accessible to the experimenter.

Unfamiliar modes of hearing were difficult to achieve and to deal with, as Helmholtz knew. He kept polemicizing against his contemporaries to whom it did not occur that a discussion of locally or historically distant ways of music-making could not be carried out by playing the music in question on a piano. Helmholtz himself exploited this very difference between listening habits in his work. Whether he discussed Greek and medieval music theory, Renaissance composition, or Chinese and Arabic scales, he always emphasized that in using other instruments, the musicians did not have the habit of hearing equal temperament. Such distant instances of listening habits, therefore, were the ideal space for setting up an experiment. For this he devised a harmonium in a system of (almost) just intonation. The harmonium was less an instrument for re-enacting distant music than for producing a significant deviation from present habits.

Helmholtz's polemics against the piano as a tool for understanding distant instances of music was grounded in his empiricist stance towards perception. He argued that perception is not an innate human faculty, but it must be learned. Learning how to perceive always occurred under concrete historical and material conditions. Nineteenth-century music theorists, more often than not, had acquired their musical skills using a piano. One could not expect them on these grounds to intuit, for instance, the fine distinctions of melodic modes in Greek theory, not to speak of the latter's affective contents.

The effects of learning, though, were likely to be just as much individually determined as the concatenation of the perception from both ears. So, the problem remained just how to ask a valid question about them. The investigation of distant listening habits offered a solution. He read both music theory sources and composition as descriptions of different conditions for learning to hear. On the harmonium, he tested whether those conditions might indeed produce effects that went unnoticed on a piano, but would came to the fore when played with the stable and sharp sounds of the harmonium. These experimentally produced sounds were significantly different, yet valid cases of making sense from hearing periodic sound under conditions Helmholtz could not have invented himself. Music, in this way, became a reservoir of producing cases for creating significant deviation.