Mechanisms of absolute pitch: II. Pitch shift and perfect touch

The previous article in this series reviewed the historical and modern academic literature concerning the distinctive characteristics of individual musical notes and keys. It stressed Bachem’s definition of tone chroma (TC) as the quality that allows notes/keys to be identified instantly and accurately by musicians possessing the type of absolute pitch (AP) that Bachem described as genuine. TC qualities were shown to vary in the same systematic manner as the second-order acoustical beats predicted to accumulate during the tuning of instruments to equal temperament. This article offers further evidence for the connection between TC and acoustical sensitivity, as indicated by an examination of paracusis musicalis (PM), the shifting of the pitch sense with age to a level sharper or flatter than its original level. It is also noted that AP judgments have been shown to be based on kinaesthetic and tactile sensations, which perform the same cueing functions as auditory TC, and that types of AP judgment may, therefore, exist not typically identified as absolute: for example, an absolute or perfect touch capacity observed in keyboard players. Evidence of this capacity supports the theory of instrument-specific AP.

Absolute pitch (AP), the ability to make speedy and accurate identifications of musical notes and keys, has been a contentious topic since its introduction into the academic literature by Stumpf (1883). As indicated in Part I of this article (Baggaley & Thurlow, 2022), a growing consensus of opinion indicates that AP judgments are made possible by comparing incoming pitch stimuli with a cortical template of pitch qualities: "representations of pitch that are more stable and accurate than previously recognised" (Levitin, 2019, p. 372). Deutsch et al. (1986) noted that non-AP possessors' pitch identifications can display high levels of accuracy, which suggest that AP "may frequently be present in partial form" (p. 1351). This conclusion is corroborated by the finding that non-AP possessors can instinctively sing a melody in its original key, individually as well as in ensembles, without first being given a reference note or key-the Levitin effect as discussed by Levitin (1994Levitin ( , 2007Levitin ( , 2019 and Frieler et al. (2013). Baggaley (1972) found related evidence in an analysis of note/key brightness ratings. Comparing ratings of key brightness by 15 AP possessors with those of 28 self-described non-AP musicians, Baggaley noted that the ratings followed a systematic trend around the musical circle of fifths, exhibiting the same high/low points in both groups, and that an identical trend can be predicted in the acoustical beats that arise in the traditional method of deliberately mistuning the fifth intervals on musical instruments to equal temperament. Plomp (1967) characterized this type of acoustical beating as second order to distinguish it from the rapid and usually less discernible first-order beats that arise between notes separated by less consonant intervals. The systematic correspondence between these trends suggests that second-order acoustical beats are a basis for the tone chroma (TC) sensations proposed by Bachem (1937Bachem ( , 1948Bachem ( , 1950Bachem ( , 1955 as the basis for genuine AP, as opposed to the quasi AP that depends on the use of relative pitch, and also for types of synaesthesia (the association of intersensory qualities) involving tone-color associations. Part I of this article (Baggaley & Thurlow, 2022) has given a detailed description of this tuning process and of the tuner's perception of the beats, and concludes that the distinctive tonality characteristics generated by the traditional tuning process may be recognized by AP and non-AP claimants alike.
Although TC qualities and genuine AP judgments do not seem to have been related to acoustical beating rates by other writers in the modern era, the hypothesis was hinted at by earlier scholars. Galilei (1638) noted the varying consonance qualities of different notes sounded simultaneously and which "strike the ear with a certain regularity" (p. 104). Rameau (1722) and D'Alembert (1752) suggested that variable beating sensations originate in the harmonic overtones of notes and keys; and the idea that these variations can be due to instrumental tuning error was supported in various forms by German Renaissance theorists including Mattheson (1719), Kirnberger (1771), Lingke (1779), Heinse (1795Heinse ( -1796, and Kollmann (1796). As noted in the preceding article in this series (Baggaley & Thurlow, 2022), de Laborde (1780) linked tuning error and key characteristics explicitly.
The fact that a systematic relationship between second-order beats and TC variations is observed when they are plotted on the musical circle of fifths indicates that historical claims to circle of fifths synaesthesia, as made by the composer Scriabin (Myers, 1914), are not fancifully contrived, as skeptics have suggested (Cytowic & Eagleman, 2011;Dann, 1998). Even among scholars who do not deny that AP and synaesthesia are genuine phenomena, however, there is a tendency to dismiss them as irrelevant to the musical experience (Roederer, 2008) and, as demonstrated in Part I of this article, to overlook historical conclusions in the literature that do not support this view. Indeed, in many cases, the modern literature fails to acknowledge historical conclusions in general (Baggaley, 2016;Baggaley & Thurlow, 2022). In the musical perception literature, Révész (1954) argued that "(a)bsolute pitch has just as little to do with the aesthetic experience of music as has colour memory with the aesthetic evaluation of painting" (p. 106). Roederer, in his detailed account of the psychophysics of music (2008), argues similarly that "(s)econd-order beats of musical consonances of pure tones do not play an important role in music (mainly because pure tones do not)" (p. 49). These views may well be accurate in relation to the musical experience of individuals who do not sense TC variations, and for whom the aesthetics of music are defined by other dimensions such as melody, harmony, and timbre; but for the minority who are sensitive to note and key characteristics, the properties of these pitch dimensions are likely to be an essential aspect of the esthetic experience.
A departure from the research tradition that has commonly failed to take account of the outlying experiences of AP possessors has recently been made by Carden and Cline (2019) and Marvin et al. (2020). Both teams have stressed the importance of studying introspective reports from AP possessors themselves, an approach they describe as commonly neglected. AP possessors "might be encouraged to share, document and review coping mechanisms . . . To help encourage respect for those with AP, possessors should not be reduced to the role of human tuning forks for choirs and orchestras" (Carden & Cline, 2019, p. 898). In paying attention to AP possessors' comments, these researchers report depths of experience not commonly revealed by AP researchers, for example, the nature of their synaesthetic descriptions of pitch, the decline in AP accuracy when playing unfamiliar instruments, and the confusion that AP possessors experience when music is presented to them based on unfamiliar pitch standards (e.g., Baroque pitch in which A = 415 cps). This article seeks to build upon the example of Carden and Cline, and Marvin et al., by considering rarely cited comments in the historical literature in the light of introspective observations offered by the current authors, by online bloggers, and in a detailed case study reported by Sacks (2007).

When the night sky turns yellow
A topic of particular concern to AP possessors, although one that is more apparent in their blogging contributions than in the academic literature, is their impression that they are losing their ability with advancing age. One online discussion (Reddit.com, 2012) provided them with a rare opportunity to record introspective descriptions of this loss that have potential implications for AP research. Their debate indicates that an awareness of TC enables AP possessors to appreciate how composers have used particular keys to complement moods simultaneously expressed via other musical dimensions. Emotional reactions to works by composers such as Scriabin (Timmers et al., 2006) are particularly likely to be influenced by key characteristics owing to the intense synaesthetic qualities that the composers perceived in them. As one contributor to the online debate puts it, the increasing inability with age to hear these tonal characteristics obscures "memories, associations and deep, deep feelings." The sense that one's AP is deteriorating over time is described as paracusis musicalis (PM), and occasionally as paracusis qualitatis. It has been mentioned in the academic literature, albeit often briefly, by Bachem (1948Bachem ( , 1950, Vernon (1977), Ward (1999), Athos et al. (2007), Deutsch (2013), Levitin (2019), and Baggaley and James (2020); and it has been discussed in detail as cochlear amusia in the non-academic literature by Oliver Sacks (2007). From approximately 50 years of age onwards, gradual changes in the pitch sensitivity of the basilar membrane in the cochlea can cause the perceived pitch of music to sharpen or flatten. The incidence of this phenomenon in the general and musical populations and the typical age at which it begins are unknown, for only AP possessors are usually able to recognize that their pitch perceptions are changing over time; but the onset of cochlear deterioration and its effects on frequency perception in the general population have been well documented (Howarth & Shone, 2006). Introspective PM reports cited by the above researchers indicate that if the pitch shift is less than a semitone, AP possessors may still be able to reconcile the changed pitches with those that they used to hear; but as the difference between the original pitch and the altered pitch approaches a semitone, the TC of the altered pitch is replaced by that of the tonality originally identified with that pitch. With a semitone shift, works written in C major can take on the quality of B major or D flat/C sharp major, and efforts that the composer may have made to use the original key to communicate a specific mood are frustrated. Even if the composer did not have key characteristics in mind when writing the work, the AP possessor with PM will still be unable to experience synaesthetic qualities that have become associated with the keys in the general musical culture-as, for example, the association of F major with pastoral music typified by Beethoven's sixth symphony. The confusion commonly experienced by AP possessors in dealing with out-of-tune and unfamiliar pitch levels in this way is examined by Rogenmoser et al. (2021) and is emotionally expressed in musicians' introspective comments such as One time I had to transcribe this one piece by Beethoven and . . . then it dawned on me that it was in Baroque tuning and my heart just started racing really really fast and I started crying . . . (Marvin et al., 2020, p. 815).
PM was linked in detail to the deterioration of TC perception by Bachem (1948). In a case study of a wartime shell-shock injury, he observed that outer ear damage can cause changes to both frequency volume and frequency-chroma relationship. Vernon (1977) also attributed his own PM to the flawed perception of incoming pitches by the outer ear leading to their incorrect identification in the auditory cortex. Athos et al. (2007) reported that PM shifts are usually in the sharp direction and suggested that they may be due to inconsistencies of international concert pitch tuning. The relatively sparse literature on PM contains reporting inconsistencies, however. For example, conflicting reports of the sharp/flat direction in which the pitch sensitivity of musicians is reported to have shifted may be due to the different tasks and conditions used in the measurement of their AP skills. A pitch recognition task, for example, can cause the incoming stimulus to be judged as sharp while a pitch recall task is likely to evoke the memory of the original pitch encoded in the cortex, causing the individual to sing or hum it at a flatter level than the incoming pitch. Similarly, the use of pure tones in a pitch recognition task is likely to yield less accurate and speedy judgments than when instrumental notes are presented (Baggaley, 1974;Steblin, 1987), owing to the varying rates of acoustical beating between instrumental overtones that are suggested as the basis for TC qualities in the current pair of articles.
By the time an AP possessor reaches around 70 years of age, it is possible though not inevitable that PM will have caused subjective pitches to have shifted from their original levels by as much as three semitones. If the shift is in the sharp direction, Beethoven's sixth symphony will seem to have changed from a mid-bright F major chroma to a darker A flat major. In listening to a work written to express nocturnal moods (e.g., Beethoven's Moonlight Sonata in C sharp minor), a sharpening by three semitones will cause its quality to change from a dark chroma to the brighter quality commonly associated with E minor. The TC of the musical night sky is likely to have shifted to a chroma more suggestive of daylight (e.g., yellow), in conflict with the nocturnal mood expressed in other dimensions of the music.
Although these notes about PM were initially based solely on the co-author JB's personal experience of it, he has since noted that the detailed case study of cochlear amusia by Sacks agrees with it in detail. Jacob L., a musician in his late 60s, reported to Sacks that his pitch sense had suddenly become "terribly sharp" by a quarter of a tone in the lower frequency range and by a semitone or more in the upper range (Sacks, 2007, pp. 140-151). If such pitch shifts are found to be common among AP possessors, they could account for and justify Bachem's tolerance of semitone errors in his 1937 definition of genuine AP. A year after the condition's initial emergence in Jacob L.'s pitch sensitivity, the sharpening effect had increased to three semitones. This made it difficult for him to continue playing for he could not hear the notes he was supposed to play "as clearly as he heard them in his mind's ear" (Sacks, 2007, p. 147). Jacob described his resulting musical experience as a constant conflict between the original version of each note and its distorted version, which "might interweave and alternate with each other like a moiré pattern or the two aspects of an ambiguous figure" (Sacks, 2007, p. 147). This description of the pitch shift suggests that the basilar membrane was now transmitting each incoming frequency at a sharper level than was originally associated with it, and that the TC originally associated with the sharper frequency was now being activated rather than the TC relating to the original frequency. This, it is suggested, creates conflict between the internal, "cortical representation" (Sacks, 2007, p. 144) of a note or key and the observed one. The persistence of this conflict over time in cases such as that of Jacob L. implies that internal TC representations can remain distinctly etched on the cortex despite the conflicting TCs activated by the distorted incoming frequencies. Examination of the sense of AP loss due to PM, with particular reference to the differences in pitch distortion in the higher and lower pitch ranges, may help to increase our understanding of pitch perception in the same way as studies of color blindness aided Ladd-Franklin (1929) in the development of color vision theory. Jacob L.'s description of his distinctive original TCs also raises the question of whether he had totally lost his AP since developing PM or had actually retained it intact despite the conflict between his original TCs and the distorted TCs of his PM condition. While Sacks and others (e.g., Hedger et al., 2013Hedger et al., , 2018 are justified in describing such conditions as imperfect pitch, it can nonetheless be argued that instantaneous and accurate pitch judgments in terms of original TCs indicate that genuine AP can remain perfect, so to speak, even when their frames of reference are confused by an aging cochlea. It is also debatable whether we can accurately describe any kind of pitch judgment as absolute if it can be shown to be enabled by reference to a cortical template of pitch qualities as accepted by researchers from Bachem (1937) to Levitin (2019). It is not proposed, however, that representations of TC underlying AP perception derive exclusively from acoustical variations of the type discussed in Part I of this article (Baggaley & Thurlow, 2022). The next section illustrates how musical pitch identification skill can attain levels of immediacy and accuracy comparable with genuine AP by means of sensitivity to kinaesthetic and tactile types of TC.

Absolute (perfect) touch
"You must get to the point that you can hear music from the page," the composer Robert Schumann taught his students (Reich, 1967, p. 402). The process whereby the so-called inner ear experiences the sound of the notes without them being physically played on an instrument has been labeled as notational audiation (Gordon, 1975), and has been shown to be enabled by kinaesthetic and tactile sensations associated with the auditory experience (Letailleur et al., 2020;Zatorre & Beckett, 1989;Zatorre et al., 2007). Brodsky et al. (2008) have concluded that this process involves subvocalizations of the sounds suggested by the notes on the page, a silent "singing to oneself " (p. 442). In addition, their study observed that participants claiming to have AP were significantly faster in the prescribed sight-reading task, which suggests that subvocalization may not be the only strategy that AP possessors use in sight-reading. Indeed, Rogenmoser et al. (2015) have confirmed that multiple sensory modes, including verbal, auditory, and sensorimotor information, can influence AP judgments. The synaesthesia literature also indicates that a trigger stimulus such as a musical note can induce qualities in any of the senses if the individual identifies it with a logically related set of qualities (serial, ordinal, interval, or nominal), as in number-form associations (Calkins, 1893;Simner & Holenstein, 2007); and a review of the synaesthesia literature by Baggaley and James (2020, pp. 23-58) has identified studies of 23 such variants including those involving gustatory, olfactory, and tactile sensations. Similar inductive processes are evident in the serial dimensions generated by hyper-systematizing reasoning in autism and savantism (Baron-Cohen, 2006;Baron-Cohen et al., 2009). Applying this multisensory rationale to the analysis of AP sensitivity suggests that, with the appropriate background experience, AP-type judgments can be triggered by multi-sensory qualities via a process not usually identified as absolute.
The in-depth keyboard experience of pianists and organists, for example, is potentially capable of generating instant and accurate tactile responses to music: a skill that may be described as absolute or perfect touch (PT). Reymore and Hansen (2020) have developed a related, instrument-specific hypothesis with regard to the identification of pitch "chromas" by two professional oboe players who did not previously regard themselves as AP possessors. In a pitch identification task, one of the oboists gave more accurate responses than the other, and described herself as "using kinesthetic imagery of the note that she thought she heard, replaying the note while imagining the fingering to see if it felt like the best fit to the sound" (post-experimental outcomes section). Reymore and Hansen conclude that this oboist has a previously unacknowledged ability that they describe as instrumentspecific AP (ISAP)-a notion that recalls the hypothesis proposed in Part I of this article (Baggaley & Thurlow, 2022) regarding the nature of AP and TC in individuals who have/have not experienced music played on equal temperament instruments. The idea that pitch identification skills may fulfill the requirements of speed and accuracy of genuine AP on one or more musical instruments though not on others is not found to be universal in the Reymore and Hansen study, for only one of their two oboists provided data supporting the ISAP conclusion. Their conclusions nonetheless generate useful hypotheses for future study. "(W)hile not providing any complete, formal test of our theory, (the results) serve as methodological guidelines for how this can be achieved in the future" (Reymore & Hansen, 2020, Discussion section).
Notwithstanding the current lack of empirical evidence for instrument-specific AP, the notion is convincingly corroborated by introspective comments in the historical literature. The physicist Max Planck, for example, described how in childhood his pitch sensations were "completely confused" when he played on instruments that were unfamiliar to him. I had to break off after the first few notes because I . . . did not possess the skill to make a hurried double transposition: once in my head in a lower key and then again on the keyboard back in the earlier key. (Planck, 1893, pp. 428-429) Similar confusion is reported in a case study by Lebeau et al. (2021) of the difficulty experienced by a professional violinist with AP in switching between pitch standards from A = 440 concert pitch to Baroque A = 415. Over time, both Planck and the violinist became accustomed to these changes, and Steblin (1987) indicates the modern relevance of these reports with regard to the lability or adjustment ability whereby some AP possessors tolerate a rather wide band of frequencies, usually about a semitone in width, as the acceptable pitch for a particular note . . . The adoption of low pitch (a 1 = 415) by "Baroque" orchestras today requires similar adjustment skills. (p. 149) The fluctuation of AP judgments in this way has become a recent topic for discussion leading to the conclusion that absolute and "perfect" pitch judgments are often neither absolute nor perfect (Hedger et al., 2013(Hedger et al., , 2018Levitin & Rogers, 2005).
The following introspective analysis of PT experiences by one of the current authors (AT) also supports the ISAP hypothesis and may help to bolster the likelihood of its academic acceptance. When playing keyboard music in C major, for example, the hands are placed exclusively on the white keys close to the player and toward the front of the keys. The white keys are all on the same plane and of identical width, so without the presence of one of more black notes to provide individuality the associated hand shapes are less distinctive and memorable. As the key signatures begin to introduce sharps or flats, the fingers adopt other positions in relation to one another, and to facilitate the playing of the occasional black notes, the hands instinctively move further away from the body. Some fingers also need to be raised above the level of the plane to play the black notes. As the number of accidentals in the key signature continues to increase, the hands, again instinctively, position themselves at an even higher level over the console so as to provide the fingers and thumbs with optimal access to them. Beyond the key signatures of E flat major and F sharp major, the position of the hands moves further away from the body and onto the higher plane, the fingers now positioned above the black keys rather than the white ones. At this point, some fingers need to be lowered below the new plane to play white notes, no longer on the front of the key but on the narrower part of its surface that is positioned between the black notes. In this position, the sides of the player's fingers will sometimes experience the tactile sensation of physical contact with the sides of the black note. Following intensive repetition, the key signatures and the hand and finger shapes for the chords they contain gain a distinct and individual feel, which instantly defines and dictates the hands' positions.
During his years of teaching the piano and organ, the co-author AT has observed that his students are usually, though not invariably, more successful, in transposition exercises if the material they are required to transpose is already well known to them, such as a favorite hymn tune. The transposition task is easier if the students have a mental picture already of the sound they are trying to recreate at the new pitch. So, when asking them to transpose something not already known to them, AT encourages his students, in the time-honored manner of Robert Schumann (see above), to start by simultaneously hearing and feeling the notes they see on the page. They can then use their musical instinct to translate the mental image into the different tactile shapes associated with the new key signature to recreate the music at the new position on the keyboard. This method is simpler than the more cumbersome approach of reading the notation in its printed key, mentally moving that notation to the notation required by the higher or lower pitch, then playing those notes from the mental image of the music in its new key. On looking at a musical score, the student senses the physical feeling of playing it, placing the hands in the right positions from one moment to the next just as the AP possessor's performance is prompted by the sounds of the music. On hearing music, the student may not be able to reproduce it instantly in the correct key but can nonetheless be expected to do so on reading it. The resulting skill is thus comparable with notational audiation as defined by Gordon (1975) and Brodsky et al. (2008). In the current context, it would more fittingly be described as notational tactility, and its most rapid and accurate levels as absolute or PT based on a tactile type of TC. Introspective analyses of the manner in which the skill can be taught, as in the musical transposition context, are not merely of interest from the pedagogical point of view, but illustrate the interplay of auditory, kinaesthetic, and tactile factors in skilled musical performance and the ways in which they vary between performers.
AT's approach to teaching transposition, described above, is similar to one used by advanced driving instructors in training students to give (while driving) an out-loud commentary on what they are seeing on the road ahead and their instant reactions to it. The musical equivalent of this practice asks students to identify out loud whether, for example, the next chord that they have to play is major, minor, diminished, or a seventh, whether it is in root position or first/second inversion, whether a cadence to be played is perfect, plagal, or interrupted, and so on. If the auditory detail is correct in the mind, the chances are that the fingers will recreate it correctly. Since losing one's bearings is one of the largest traps in transposition, the students can also be taught to identify common link notes between adjacent chords, whether in the same part and at the same pitch or in a different part and at a different octave. If these notes are the same in the printed key they will also be identical in the transposed key, and will provide a fixed visual point of reference between the two chords that can also be felt in the hands. AT regards it as likely that keyboard players have a more heightened tactile feeling for harmony than those who play single-note instruments; for keyboard players feel chords and harmonies under their hands and fingers, habitually anticipating the tactile qualities of the hands' positions when reading a sheet of music. In learning new music, AT also attempts to sense the fingerings that the composers used when writing their works at the keyboard, and he believes that these tactile sensations can form an essential part of the music's personality. Based on their differing skills and musical backgrounds, individual students perfect their own versions of these tactile processes, with strategies such as subvocalization being used by some of them though not all. AT, for example, does not possess AP, which perhaps enhances his tactile and visual sensations.

Conclusion
The phenomenon of pitch shift or PM and the concept of PT each shed light on the nature of AP. The existence of multiple TC dimensions is indicated involving auditory and tactile sensations. Evidence for the auditory type of TC is derived from the PM condition caused by the deteriorating basilar membrane and the pitch shift that results from it. The confusion exhibited by AP possessors whose pitch sense has shifted in a sharp or flat direction points to the lingering existence of an earlier TC memory in conflict with the current one. Barely considered in the modern AP literature, PM merits detailed investigation. The scarcity of current PM studies might suggest that locating AP possessors with PM for such research would be even more difficult than tracing AP participants in general, although it is possible that this is not so in view of the conclusions by researchers including Deutsch et al. (1986) and Levitin (2019) that AP is less extraordinary than is generally supposed, and by Howarth and Shone (2006) that PM is also relatively common in musical and non-musical populations. An appropriate technique for use in such studies would be to ask participants to sing a familiar tune and to note if they do so in its customary key (the Levitin effect). Surveys, interview studies, and pitch recognition and recall tests designed to avoid pitch-naming skills may also confirm that acoustical, tactile, and kinaesthetic bases for AP-type judgments are also relatively common in non-AP possessors, and operate in all musicians singly or in combination and at varying levels.
The kinaesthetic sensations evoked in PT may thus be described as the tactile equivalent of auditory TC, and commonly follow a logical sequence identified with, for example, the number of accidentals in the key being played and/or the necessary distance of the hands from the body. Using a combination of aural, visual, subvocal, and tactile memories, the player learns to perform sounds represented on sheet music instantly and accurately by converting the sight-read notes and keys into auditory and/or tactile memories in the same way as AP possessors evoke TC memories in recognizing and recalling notes and keys. The accurate and instant execution of keyboard performance triggered exclusively by tactile sensations is thus a process equivalent to genuine AP (PT). This view is consistent with the "sensorimotor information about the feeling of performing" conclusion discussed by Zbikowski (2010, pp. 32-38) and by Curwen (2020) in relation to synaesthetic associations. The same rationale might also be applied to percussionists' ability to anticipate rhythmic changes instantaneously (perfect rhythm?), and to the rapid embouchure changes required in playing wind instruments (perfect lip?). In other words, any musical performance skill involving instant and accurate physical adaptability may be an AP equivalent based on comparisons of incoming stimuli with a template of qualities stored in logical sequence in the cortex. This possibility adds support to the view that absolute judgments of pitch and multisensory types of TC are more common than is generally supposed. Related conclusions have been drawn by Reymore and Hansen (2020) regarding ISAP.
Future researchers will be able to examine themselves for PM using simple cellphone applications and by checking whether notes in a musical performance seem out of tune with one another. For example, if high notes on the piano, violin, flute, or piccolo sound sharper or flatter than notes in the lower ranges, as in the case study by Sacks (2007), it may not be that the instrument is poorly tuned but that the listener has different PM shifts in the higher and lower ranges, generating acoustical conflicts that cause a well-tuned Steinway grand to sound like a jangle piano. Disparities between pitch shifts on the same instrument may even cause notes tuned to perfect consonance to have different acoustical beating qualities owing to conflicts between (a) the note being tuned, (b) the frequencies in its harmonic series, and (c) other equally tempered notes on the instrument. It is only too easy to mistakenly blame these effects on the instrument, performer, or tuner as in the popular phrase "Don't shoot the pianist" and in Tom Waits (1976) lyrics "The piano has been drinking (not me)." If instruments can seem to be mistuned by a semitone or more in its high registers, it is more likely to be the listener's ear that has become mistuned.
AP is not always perfect, as analysts from Planck (1893) to Hedger et al. (2013Hedger et al. ( , 2018 have indicated. Nor can it strictly speaking be described as absolute when it depends on relative comparisons of incoming pitches with an internal TC template. The fallibility of AP, however, notably in PM, provides clues to the processes underlying it. With a tightening of research procedures, the study of these anomalies may play a decisive role in musical perception and synaesthesia research as analyses of color blindness have helped in the development of color vision theory (Ladd-Franklin, 1929). In the process, it is to be hoped that the solid conclusions of eminent scholars of up to three centuries ago, and of 20th-century musicians such as Scriabin and Bachem, will no longer be overlooked by researchers who do not experience AP and synaesthetic sensations for themselves.