406 Richard Parncutt, Sabrina Sattmann, Andreas Gaich, & Annemarie Seither-Preisler

TONE PROFILES OF I S O L AT E D M U S I C A L C H O R D S : P SYC H OAC O U S T I C

V E R SU S C O G N I T I V E M O D E L S

R I C HA R D PA R N C U T T , S A B R I NA S AT T M A N N , structures depends on our auditory experience, which

A N D R E A S G A I C H , & A N N E M A R I E S E I T H E R -P R E I S L E R in turn depends on how musical structures developed in
University of Graz, Graz, Austria the past (Lynch, Eilers, Oller, & Urbano, 1990). The
familiar pitch-time patterns of major-minor tonality
W E INVESTIGATED PERCEPTION OF VIRTUAL emerged during a long period of musical development
pitches at missing fundamentals (MFs) in musical chords (Dahlhaus, 1968/1990). The structure of early Western
of three chromas (simultaneous trichords). Tone profiles polyphony was governed by a combination of explicit
for major, minor, diminished, augmented, suspended, compositional rules and implicit perceptual principles
and four other trichords of octave-complex tones were (e.g., Eberlein, 1994; Huron, 2001).
determined. In Experiment 1, 40 musicians rated how To understand how major-minor tonality works, we
well a tone went with a preceding chord; in Experiment need to understand both the psychological and the his-
2, whether the tone was in the chord. Mean ratings for torical (here: psychohistoric) origins of Western tonal
nine non-chord tones were compared with predictions of pitch structures. Overarching questions include: Why are
four models: MFs, diatonicity, 5th-interval relations, and some pitch combinations more consonant or common
tones that complete familiar tetrachords (e.g., 7th (prevalent) than others? How did the perception of con-
chords). Profiles were accounted for by all four models sonance and dissonance develop historically? Why have
in Experiment 1, and two (MFs, 5th relations) in Exper- major and minor triads played such a central role in
iment 2. Overall, effect size was largest for MFs. In Exper- Western polyphony since the 14th century?1 Why did
iment 3, listeners heard a chord and chose a matching major-minor tonality emerge in the 16th-17th centuries
tone from 12 possibilities. Profile peaks were predicted and why did it come to dominate most (Western) music?
by pitch models (usually, the lower tone of a perfect 5th). Questions of this kind involve both humanities and
Participants who more likely attended to MFs in isolated sciences. From a scientific perspective, major-minor
harmonic complex tones (fundamental listeners) were tonality emerged historically under the constant influ-
not more sensitive to MFs in chords, suggesting their ence of universal principles of perception and cognition
responses instead depended on statistical properties of (Eberlein, 1994; Parncutt, 2011a). Scientists in disci-
familiar music. We propose a speculative, psychohistoric plines such as acoustics, psychology, biology, mathemat-
explanation: MFs influenced the historical development ics, and computing attempt to reduce major-minor
of musical structure, which in turn influenced the per- tonality to underlying psychophysical and neurocogni-
ception of enculturated modern listeners. tive principles (Bharucha, 1984, 1987; Deutsch & Feroe,
1981; Krumhansl, 1990; Tillmann, Bharucha, & Bigand,
Received: June 24, 2016, accepted February 3, 2019. 2000; Trainor & Trehub, 1994). Humanities scholars in
disciplines such as music theory, music analysis, music
Key words: harmony, tonality, pitch, missing funda-
history, and music sociology tend instead to regard
mental, chord root
major-minor tonality as a complex, partly arbitrary out-
come of historical, social, cultural, and political interac-
tions (Lowinsky, 1954; Norton, 1984). Humanities

M
AJOR- MINOR TONALIT Y IS THE MOST
familiar system of structuring pitch in West-
ern tonal music. Pitches and pitch patterns 1
In musical set theory, a set of three different chromas (octave-
are perceived in relation to a major or minor scale, generalized pitches) is a trichord (Rahn, 1980). The term triad tends to
a major or minor triad (the tonic chord), or a tone (the be reserved for major, minor, diminished, and augmented triads; that is,
tonic) (Cuddy & Badertscher, 1987; Krumhansl, 1990). for musically familiar or basic trichords. If some tones in a trichord are
doubled (played in more than one octave register), the chord comprises
The musical surface can often be reduced to, or is per- three chromas but more than three tones. A tetrachord is a set of four
ceived as, familiar harmonic progressions (Holleran, chromas. By polyphony we mean music comprising several partly inde-
Jones, & Butler, 1995). How we perceive musical pendent voices (rather than voices that move in parallel).

Music Perception, VOLUM E 36, ISSU E 4, PP. 406–430, IS S N 0730-7829, EL ECTR ONI C ISSN 1533-8312. © 2019 B Y THE R E GE N TS OF THE UN IV E RS I T Y O F CA LI FOR NIA A LL
R IG HTS RE S E RV E D . PLEASE DIR ECT ALL REQ UEST S F OR PER MISSION T O PHOT O COPY OR R EPRO DUC E A RTI CLE CONT ENT T HRO UGH T HE UNI VER S IT Y OF CALI FO RNIA PR ESS ’ S
R EPR INT S AN D P E R M I S S I O NS W E B PAG E , HT T P S :// W W W. UCPR ESS . E D U / JOU RNA LS / R E P RI NTS - PERMISSI ONS . DOI: https://doi.org/10.1525/ M P.2019.36.4.406
 Tone Profiles of Musical Chords 407

scholars and scientists ask similar questions about musi- commonality, intended to account for harmonic rela-
cal structure, but adopt different approaches. Psycholo- tionships perceived between successive chords such as
gists may ask about perception and cognition today but CEG (with MFs at D, F, and A) and DFA (with MFs at G
ignore the past, despite evidence that perception gener- and B ). A theory of harmony based on the perception
ally depends on culture and history (Nisbett, Peng, of harmonic patterns among the partials of complex
Choi, & Norenzayan, 2001). Historians and music the- sounds is promising, considering the biological impor-
orists may study the history of music and music theory tance of voiced speech sounds (Bowling & Purves, 2015;
but ignore empirical psychology. Bowling, Purves, & Gill, 2017). It is also possible to
Humanities scholars and scientists (in both ‘‘psycho- predict interesting structural aspects of tonality using
acoustic’’ and ‘‘cognitive’’ approaches) agree that pitches a pitch-commonality model that considers only spectral
in tonal musical contexts vary in importance, but use pitch and ignores harmonic pitch patterns and virtual
different words to describe such variations (e.g., stabil- pitch (Milne, Laney, & Sharp, 2015).
ity, salience, hierarchy). Krumhansl (1990) presented Parncutt (1989) adapted the pitch model of Terhardt
diverse experimental data that provided: et al. (1982) for music-theoretical purposes, assigning all
input frequencies and output pitches to 12 equally spaced
a quantitative measure of the hierarchical ordering categories per octave across the range of hearing (120
imposed on the individual tones in tonal contexts. In categories altogether). For a given input sound, repre-
music-theoretical terms, the rating might be identi- sented as a sum of pure tones with different frequencies
fied with the relative stability or structural signifi- and amplitudes, the model estimated the perceptual
cance of tones as they function within tonal salience of each audible partial, looked for harmonic pat-
contexts. It will be argued that this hierarchy is, in terns among these spectral pitches (i.e., among the per-
some sense, basic to the structuring of music itself ceived pitches of individual audible partials)3 and on that
and also to the psychological response to music. basis estimated the perceptual salience of virtual pitches.
This identification of a music-theoretical construct Relevant predictions are shown in Table 1. The pre-
and a pattern of psychological data, then, represents dictions were made using the model of Parncutt (1989),
a point of contact between the structure contained based on simple assumptions about mutual masking
within the music and described by music theory, and and harmonic pattern recognition among simultaneous
the listener’s response to that structure. (Krumhansl, partials. The free parameter settings in the model were
1990, p. 16) kM ¼ 18, kT ¼ 3, and kS ¼ 0.5. Parameter kM is the
gradient of the masking pattern of a pure tone in dB per
What is the origin of these differences between more critical band; if it is high, there is less masking and the
and less stable tones? From a psychoacoustic viewpoint, partials are more clearly audible. Parameter kT is a mea-
tones with the same amplitude and waveform, when sure of how analytically tones are perceived; if it is high,
presented simultaneously, can differ in perceptual one is more likely to experience spectral than virtual
salience for two reasons. First, masking among nearby pitches. Parameter kS measures the tendency to hear
partials tends to make inner voices in a musical texture
less salient than outer voices. Second, complex tone
perception tends to increase the salience of tones that
correspond to periodicities or fundamentals (Moore, 3
A spectral pitch is the pitch of a pure tone—whether heard in
2003; Terhardt, Stoll, & Seewann, 1982). isolation or as part of a complex sound, as a partial. Like any other
pitch, spectral pitch is fundamentally subjective and experiential in
In addition, missing fundamentals (MFs) may be per-
nature, because empirical pitch judgments are always mediated by the
ceived at pitches not corresponding to chord tones (that listener’s consciousness (Terhardt, 1998). In a common procedure for
is, at non-chord pitches corresponding to non-chord pitch judgment, a listener hears a complex sound and a pure tone in
tones).2 For example, Parncutt (1988) predicted that alternation and adjusts the frequency of the pure tone until the two
an A-minor triad (ACE) evokes weak pitches at D and sounds have the same pitch. The physiological correlates of spectral
pitch in the peripheral auditory system are complex; both spectral and
F, and Parncutt (1989) incorporated the predicted
virtual pitch depend in general on a mixture of temporal and spectral
salience of MFs at non-chord pitches in a model of pitch information and processes (Moore, 2003). If we ignore physiology and
consider only the relationship between spectral pitches and partial
2
A pitch at an MF is always a virtual pitch, but not all virtual pitches frequencies, the relationship is still complex: spectral pitches and
are at MFs. The pitch at or near the fundamental of a HCT, in which the corresponding spectral frequencies differ from each other depending
fundamental is present and audible, is usually virtual. But if the spectral on the sound levels of the partials and the degree to which they mask
pitch of the lowest partial is more salient than the coinciding virtual pitch, each other (pitch shifts). If a partial is completely masked, its spectral
as in some high-pitched musical sounds, the main pitch is spectral. pitch ceases to exist.
408 Richard Parncutt, Sabrina Sattmann, Andreas Gaich, & Annemarie Seither-Preisler

TABLE 1. Predicted Chroma-salience Profiles of Four Common Chords Constructed From Two Different Kinds of Tone According to Parncutt
(1989)

Chord Tone type C C / D  D D / E  E F F / G  G G / A  A A / B  B

Major HCT 0.87 0.01 0.19 0.03 0.66 0.09 0.01 0.64 0.05 0.15 0.03 0.12
(e.g., CEG) OCT 1.62 0.00 0.05 0.05 0.55 0.17 0.01 0.58 0.06 0.16 0.01 0.00
Minor HCT 0.81 0.01 0.09 0.65 0.03 0.19 0.02 0.74 0.17 0.03 0.21 0.05
(e.g., CE G) OCT 1.34 0.01 0.02 1.09 0.00 0.32 0.00 0.79 0.32 0.02 0.01 0.06
Suspended HCT 0.88 0.03 0.17 0.07 0.03 0.73 0.00 0.66 0.04 0.07 0.14 0.02
(e.g., CFG) OCT 1.34 0.05 0.02 0.11 0.00 1.35 0.00 0.72 0.07 0.01 0.16 0.00
Diminished HCT 0.56 0.11 0.08 0.67 0.04 0.13 0.66 0.09 0.28 0.04 0.26 0.18
(e.g., CE G ) OCT 1.08 0.02 0.23 0.93 0.02 0.31 1.11 0.00 0.71 0.00 0.02 0.49
Note: The OCTs have flat spectra; the amplitude envelopes of the HCTs are musically typical as specified in Parncutt (1989). The chords from HCTs are in close root position in
middle register and the lowest tone is always C4 (e.g., C4E4G4). The numbers in the table are predicted chroma salience, calculated by summing virtual pitch salience across 10
octave registers, for both OCT- and HCT-chords. For example, the value at C is the sum of calculated virtual pitch salience at C0, C1, C2, . . . and C9. Virtual pitch salience is
normalized such that the sum of all calculated values across all 120 pitches (12/octave x 10 octaves) equals the chord’s calculated multiplicity—the predicted number of
simultaneously perceived pitches, which for example is 2.86 for the major triad of HCTs and 3.27 for the major triad of OCTs. Numbers in bold indicate chord tones; italics
indicate the main MFs.

multiple tones; if it is high, more tones are perceived comparison to more universal or innate aspects of pitch
simultaneously. perception, which we will call ‘‘nature.’’ The distinction is
The model predicts features of tone profiles of isolated problematic: virtual pitch perception according to Ter-
chords constructed from either octave-complex tones hardt et al. (1982) is universally learned from voiced
(OCTs; cf. empirical data of Parncutt, 1993) or sounds in speech and hence a form of ‘‘nurture,’’ but the
harmonic-complex tones (HCTs; cf. Reichweger, 2010; relationship between spectral and virtual pitches is often
Thompson & Parncutt, 1997). 4 Such profiles may treated as ‘‘nature’’ because of its quasi-universality. Note
depend on experience of statistical distributions in music also that the dichotomy between ‘‘psychoacoustic’’ (or
(Pearce & Wiggins, 2012)—which chromas5 typically ‘‘sensory’’) and ‘‘cognitive’’ approaches to musical pitch
precede or follow given chords in musical scores or per- structures is not the same as the nature-nurture distinc-
formances. In the following, we will refer to musical tion; both approaches involve both psychophysics (rela-
learning processes of this kind as ‘‘nurture’’ by tionships between physical and experiential parameters)
and cognition (information processing) (Parncutt, 1989).
4
In music theory, a ‘‘chord’’ is often a familiar triad or seventh chord, or The current study aimed to evaluate the relative impor-
a sonority constructed according to the principle of stacked thirds. But the tance of nature and nurture in the perception of non-
word ‘‘chord’’ may also refer to any simultaneity of any tones from the chord tones by comparing predictions of simple models
chromatic scale, which is how we use the word in this paper. Our definition with empirical data. We measured the chroma salience
is consistent with polyphonic musical practice since the Middle Ages, in
which almost all possible pitch-class sets were used (Parncutt et al., 2018).
profiles of diverse musical chords in common use, com-
It is also consistent with the idea of additive harmony in early modernism paring the results of different empirical methods and the
(Blättler, 2017) and the jazz-theory concept of bitonal chords that combine predictions of different predictive models. A chroma
lower and upper structures (Pease & Pullig, 2001). Alternative terms for salience profile is a vector of 12 numbers, each represent-
‘‘chord’’ in this sense include ‘‘simultaneity’’ or ‘‘sonority.’’ An OCT is ing the perceptual salience of a chroma. A chroma
a complex tone whose partials are spaced at octave intervals across the
audible spectrum. Shepard tones are OCTs whose amplitude envelope is
salience profile of a chord is a representation of the
bell-shaped. In this study, the amplitude envelope of OCTs was flat before pitches perceived in the chord and their relative saliences.
amplification; low and high frequencies were instead attenuated by a mix- In Experiment 1, listeners heard a chord followed by
ture of acoustical phenomena (frequency response of sound card and a tone and rated how well the tone went with the
headphones) and psychoacoustical phenomena (auditory threshold, chord—similar to experiments reported in Krumhansl
curves of equal loudness, and masking).
5
We distinguish between pitch and chroma. Pitch is the perceived
(1990), but without a preceding tonal context. From
height of a tone on a one-dimensional scale from low to high. Chroma a musical perspective, this method is suitable for testing
is octave-generalized, musically categorized pitch. There are 12 chromas: theories of chord-scale compatibility or mappings (Biles,
C, C /D , D, etc. Each chroma can be realized in different octave registers. 2003). In Experiment 2, listeners were asked whether the
A chroma is also a psychological category: in a musical context based on tone was in the chord. In Experiment 3, they could hear
the chromatic scale, pitches lying within roughly a quartertone of a chro-
ma’s centre pitch are perceived as belonging to that chroma (cf. Burns &
the chord and any of 12 chromatic tones by clicking on
Ward, 1978). A ‘‘chord chroma’’ is a chroma corresponding to one of the an interface, and chose the chord’s clearest or main
chord’s notes; other chromas are ‘‘non-chord tones.’’ pitch. In Experiments 1 and 2, our analysis focused on
 Tone Profiles of Musical Chords 409

non-chord tones (i.e., the nine chromas that did not pitch distance between test and probe is held approxi-
correspond to chord tones), whereas in Experiment 3, mately constant.
we focused on the three chord tones. By comparing
results using different methods, we aimed to shed light
PREDICTORS AND MODELS
on the psychological nature and musical function of
In the present study, we measured chroma-salience
chords and hence on the major-minor tonal system.
profiles for several musically typical chords and com-
Like Krumhansl (1990), our study was confined to
pared them with four predictors: one ‘‘nature’’ predictor
chords constructed from OCTs. This strategy allowed
(MFs) and three ‘‘nurture’’ predictors (diatonicity, 5th
us to eliminate confounds of pitch register and chord
relations, and completion tones). The models are intro-
voicing and made it possible to demonstrate or falsify
duced here and later operationalized in Table 5 and
the psychological reality of weaker pitches6 (less salient
accompanying text.
chromas) by statistical comparison of ratings across
a finite number of pitches. For example, the strategy Missing fundamentals. In the spectrum of a typical
enabled us to ask whether chromas F and A (predicted musical chord, patterns of spectral pitches (audible par-
MFs) are evoked by the chord CEG, by comparing rat- tials) may correspond to incomplete harmonic series. In
ings for those two tones with seven other non-chord general, pitch may be perceived at the fundamentals of
tones (C /D , D, etc.). such patterns (Ritsma, 1967). Empirical research on
Analogous experiments using chords of HCTs (Reich- categorical perception of partial frequencies within
weger, 2010) suffered from a confound: the greater was complex tones (Moore, Peters, & Glasberg, 1985) sug-
the overall perceived pitch distance between test sound gests that these harmonic patterns need not be exact and
and probe tone, the lower was the goodness-of-fit rating. can be mistuned by a few tens of cents (Terhardt et al.,
In practice, this confound can be reduced, but not elim- 1982). The inherently approximate nature of pitch
inated, by careful choice of test sounds and probe tones. intervals in memory, both in this case and in music,
In general, when a listener is asked to compare a test undermines the Pythagorean concept of musical inter-
sound with a probe tone (e.g., for goodness of fit), results vals as frequency ratios (Parncutt & Hair, 2018).
depend on both pitch commonality and pitch distance Predictions about pitch perception in musical chords
(Parncutt, 1989). The distinction is assumed to depend should therefore be almost independent of variations
on categorical pitch perception (Burns & Ward, 1978): among theoretical tuning systems such as Pythagorean
pitch commonality is higher when a greater number of and Pure/Just (Barbour, 1951).
(more salient) successive pitches are perceived to lie in Diatonic and chromatic representations of the har-
the same pitch categories, whereas pitch distance is monic series are provided for reference in Table 2. In
higher when the distance between (more salient) succes- a first approximation, consider only harmonics that are
sive pitches in different categories is higher. octave-equivalent to the fundamental (harmonic num-
Parncutt (1989) demonstrated the psychological real- bers 1, 2, 4, 8, etc.). In this case, and considering only
ity of selected non-chord pitches in chords of HCTs, but fundamental frequencies of chord tones, the major triad
those pitches were octave-equivalent to chord tones: for CEG may evoke MFs at A, F and D. An MF is predicted
example, the chord C4E4G4 was shown to evoke pitches at A, for example, because E corresponds to the 3rd
at C3, G3, C5, and G5. To demonstrate the psycholog- harmonic and G to the 7th harmonic of A. In other
ical reality of non-chord chromas, one would have to words, the MFs are subharmonics of chord tones. Sim-
average over all inversions of each chord as well as ilarly, the minor triad ACE evokes MFs at D and F. In
musically typical spectral envelopes, spacings, and regis- general, the larger the number of coinciding subharmo-
tral distributions, greatly increasing the number of nics at a given pitch, and the lower the harmonic num-
experimental trials. These methodological difficulties bers involved, the more salient is the predicted pitch
evaporate when both test sound and probe tone are (Terhardt et al., 1982).
constructed from OCTs, such that the overall perceived Note that the predictions of spectral and temporal
models of pitch perception are similar to each other due
to the mathematical equivalence of time- and
6
A ‘‘pitch’’ is a subjective experience. This definition applies equally to frequency-domain representations; our aim here is not
spectral and virtual pitch. An experimental psychoacoustic paradigm to compare pitch models or approaches with each other,
explores quantitative relationships between experiential parameters
such as pitch, timbre and loudness on the one hand, and physical
but to test predictions that are common to different
parameters such as the frequencies, amplitudes, and relative phases of models. Note also that the tendency to perceive pitch
partials within complex tones on the other. at the MF in isolated tones is subject to large individual
410 Richard Parncutt, Sabrina Sattmann, Andreas Gaich, & Annemarie Seither-Preisler

TABLE 2. The Harmonic Series in Diatonic and Chromatic Representations Relative to an Arbitrary Fundamental Frequency of 110 Hz (the
Musical Note A2)

Harmonic no. 1 2 3 4 5 6 7 8 9 10
Frequencies (relative to 110 Hz) 110 220 330 440 550 660 770 880 990 1100
Note names (relative to A2) A2 A3 E4 A4 C 5 E5 G5 A5 B5 C 6
Simple diatonic intervals P1 P1 P5 P1 M3 P5 m7 P1 M2 M3
Chromatic intervals in semitones 0 0 7 0 4 7 10 0 2 4
Note: The numbers following the note names in row 3 are octave registers (middle C is the lowest tone of register 4). Abbreviations for diatonic intervals row 4 are P ¼ perfect,
M ¼ major, m ¼ minor, 1 ¼ unison, 3 ¼ third, 5 ¼ fifth, 7 ¼ seventh. The intervals in the last two lines are relative to the lowest tone or fundamental. They are octave
generalized; that is, expressed as simple rather than compound intervals (using modulo 12 arithmetic for chromatic intervals). Note that the tuning of the harmonic series
differs from 12-tone equal temperament and piano keyboards; the largest deviation occurs at the 7th harmonic, where a ratio of 4:7 is 31 cents smaller than an equally tempered
m7 interval (10 semitones).

differences (Schneider et al., 2005; Seither-Preisler, Completion tones. Any chord of 3 chromas (trichord)
Johnson, Seither, & Lütkenhöner, 2008): fundamental can be transformed into a chord of 4 chromas (tetra-
listeners are more likely to hear pitches at MFs, whereas chord) by adding one of the 9 other chromas. For exam-
spectral listeners are less likely to do so. ple, a diminished triad can be turned into a dominant,
diminished, or half-diminished 7th chord, or six other
Diatonicity. The standard diatonic scale or set corre- less familiar tone combinations, by adding a fourth
sponds to the white keys on the modern piano and tone. The relative prevalence (frequency of occurrence)
comprises seven chromas (C, D, E, F, G, A, and B), in of those 9 tetrachords in real music depends mainly on
any transposition and any musically typical or accept- their consonance (Parncutt & Hair, 2011). If a tetrachord
able tuning. It represents a closed segment of the circle of is prevalent and hence familiar, each of its four triadic
5ths, all tones being related to at least one other tone by subsets may be perceived as the tetrachord with a miss-
a P4 or P5 interval. Most types of chords in the major- ing element—just as a familiar pattern can be recog-
minor system can be played (in transposition) in this nized when elements are missing (Gestalt principle of
scale. Sometimes the same type of chord can be played closure). A listener may therefore expect to hear the
at more than one position within the scale. For instance, missing tone. Our completion tone predictor again par-
in the key of C major, there are major triads on C, F, and tially overlaps with the other predictors, because com-
G, and minor triads on D, E and A. Diatonic scales are pletion tones often lie at P4/P5 intervals from chord
highly familiar to Western listeners (Deutsch & Feroe, tones, create a harmonic pattern corresponding to part
1981), having dominated Western music since ancient of the harmonic series, or create part of a diatonic scale.
times (Gauldin, 1983). If a chord is a subset of the stan- To understand better how these four predictors over-
dard diatonic scale, listeners may expect to hear other lap, consider a simple example. Parncutt (1993, Exper-
tones in the same scale in the music that follows. iment 2) presented chords of octave-complex tones
(OCTs) followed by single octave-complex probe tones.
5th relations. On the circle of 5ths, chromatic pitches are When the chord was a C-major triad (CEG), partici-
assigned to a circular representation similar to a clock pants rated the probe tone F as sounding or fitting
face: C at 12 o’clock, G at 1 o’clock, D at 2 o’clock, and significantly better than F . Of the four proposed mod-
so on. There is convergent evidence in the literature for els, three might explain this result. In the MF model, F is
the psychological reality of this construct (Bharucha, an MF in a C-major triad but F  is not. Regarding dia-
1987; Krumhansl, 1990, 1991; Lerdahl, 1988; Shepard, tonicity, F is diatonic in two scales to which the triad
1982; Thompson & Cuddy, 1989). In a tonal musical belongs (C major and F major), whereas F  is diatonic in
context, when any tone is heard, other tones at P4 or only one (G major). Regarding 5th relations, F lies a 5th
P5 intervals above or below it may be expected (cogni- away from a chord tone (C) but F  does not. Only the
tively activated or facilitated). This idea is plausible fourth model (completion tones) cannot explain
given the central role of diatonic scales based on P8 and this particular result: neither CEFG nor CEF G is a par-
P5 intervals in Western tonal music. But this observa- ticularly familiar tetrachord, although both do occur
tion also means our diatonicity and 5th-relatedness pre- occasionally in tonal music. Thus, we are unable to
dictors overlap. They also overlap with the MF separate the theories on the basis of this example.
predictor, because many MFs lie at P5 intervals below Three of the models—diatonicity, 5th relations, and
chord tones. completion tones—presumably involve musical
 Tone Profiles of Musical Chords 411

learning (‘‘nurture’’); that is, experience of pitch-time students of a musical instrument, composition, or con-
patterns in music and their statistical regularities (Parn- ducting (University of Music and Performing Arts
cutt, Reisinger, Fuchs, & Kaiser, 2018). If the tone F Graz), 6 electrical and audio engineering students (Graz
often follows a C-major chord (or more generally, if University of Technology), 7 students in other disci-
a major triad is often followed by a tone a P4 above the plines, and 3 non-students. The age range was 18 to
root7) in the music to which we have been exposed, we 57 years (M ¼ 25, SD ¼ 7).
are likely to indicate in an experiment that F is a good All participants were musicians with over 6 years
continuation to C major—regardless of whether F is experience of regular instrumental playing and/or sing-
also a MF. Here and elsewhere, the prevalence of musi- ing (mean number of years of playing ¼ 14, SD ¼ 5;
cal patterns (such as any major triad followed by a tone mean singing ¼ 7, SD ¼ 7). Most (78%) indicated that
a P4 above the root) can be estimated by statistical they were playing an instrument regularly; of the sing-
analysis of representative databases of musical scores; ers, most (63%) indicated they were singing regularly.
such analysis can then predict note-by-note expectan- The main instrument of 7 participants was the piano, of
cies (cf. Pearce & Wiggins, 2012). 7 the guitar, of 5 the flute, and of 3 each the trumpet and
With the contrasting—but also overlapping—predic- the violin. As a second instrument, the piano was men-
tions of these four models in mind, the experiments tioned most often (13 times). Most participants (26)
were designed to answer questions such as: Do funda- named classical music as their main performance genre;
mental listeners rate MFs in musical chords higher than further genres (in descending order) were traditional
spectral listeners? Are the tone profiles more peaked, music, jazz, pop, and rock. Two participants gave sing-
and the ratings higher on average, for more familiar ing as their main instrument and 31 reported they were
chords? Does the highest peak correspond to the root? singing in a choir.
Are there peaks among the nine non-chord tones? If so, Nonmusicians were not included because of the dif-
which model or models predict them? ficulty of the experimental task. A previous, comparable
study (Reichweger, 2010) had found that nonmusicians
were often unable to differentiate between chord tones
Method
and non-chord tones in familiar musical triads.
OVERVIEW
STIMULI
Each participant participated in one preliminary exper-
All sounds were electronically synthesized. They were
iment and three main experiments in a single session in
either pure tones, HCTs with MFs, OCTs, or chords
a quiet room at the University of Graz. Each experiment
of OCTs.
comprised a series of trials whose order was random
In each trial of the preliminary experiment (AAT),
and different for each participant. Each began with a few
participants heard two successive HCTs with MFs, in
practice trials. The order of the three main experiments
which MFs and spectral components moved in opposite
was random for each participant. The preliminary
direction: if the spectral components of the second
experiment was the Auditory Ambiguity Test (AAT)
sound were higher, the MF was lower, and vice versa.
of Seither-Preisler et al. (2007), which aimed to separate
The task was to indicate whether the second tone was
participants into fundamental and spectral listeners.
higher or lower than the first. Each tone comprised
After completing the experiments, participants com-
successive harmonics spanning one octave. In some
mented briefly on their experience of participating, their
tone pairs, the tone with the higher F0 had harmonic
strategies, and the chords that they recognized. Statisti-
numbers 2-4 and the tone with the lower F0 had har-
cal tests were performed in SPSS 22.
monics 5-10. In others, the higher tone had harmonics
3-6 and the lower had 7-14; in still others, 4-8 and 9-18.
PARTICIPANTS The gradient of the spectral envelope was always -6 dB
Forty people (26 females) participated in all experi- per octave—comparable with a bowed string (e.g., vio-
ments: 16 musicology students (University of Graz), 8 lin). Each HCT had the same SPL before amplification
and realization; overall loudness was adjusted to be
7
In a psychological approach, the root is a reference chroma, relative comfortable. The frequency of the missing F0 was
to which other chord chromas are often or usually perceived. Major and between 100 and 400 Hz. The interval between the MFs
minor triads appear most often in root position (i.e., with the root in the
bass). Chord roots are often ambiguous, but the assumed root usually
was familiar from Western music (major second or M2,
corresponds to the lower tone of a P5 interval between chord chromas (or major third or M3, perfect 5th or P5, major 6th, or M6).
the upper tone of a P4). Both tone durations and the silent gaps between them
412 Richard Parncutt, Sabrina Sattmann, Andreas Gaich, & Annemarie Seither-Preisler

were 500 ms. Test presentation and response analysis always moved in opposite directions. In 10 unambigu-
were performed by an automatized Visual Basic script. ous control trials, both the MF and the spectral envelope
In the three main experiments, in each trial partici- moved in the same direction, to test the participants’
pants heard a chord of OCTs followed by a single OCT reliability. Subjects with two or more incorrectly classi-
or pure tone. The OCTs corresponded to the 12 steps of fied control trials were discarded. Participants were not
the equally tempered chromatic scale with A4 ¼ 440 Hz informed about the structure of the sounds; they simply
and unstretched octaves (2:1). Each OCT comprised 10 indicated whether the pitch rose or fell. Participants
partials of equal amplitude (before amplification); the who commented that they were sometimes unsure of
frequency range was from C1 (32.7 Hz) to B10 (15800 the direction of motion were asked to give spontaneous
Hz). The relative phase of the partials was randomized: ‘‘gut reactions.’’ Each participant was given a score
each was phase-shifted by a random angle between - between 0 and 100, representing the number of trials
180 and þ180 before superposition, to eliminate the in which her or his response corresponded to the move-
possibility that phase relationships might affect the rel- ment of the MF. Consistently spectral listeners had
ative salience of evoked pitches. The amplitude of each scores near 0, while consistently fundamental listeners
chord was adjusted to -0.5 dB relative to full scale. To had scores near 100.
avoid clicks, amplitude-linear ramps were applied to the
start (10 ms) and end (30 ms) of each sound. RESULTS
For the first 20 participants in the three main experi- The mean score was 81.8 (SD ¼ 20)—comparable with
ments, the chord of OCTs had a duration of 300 ms, the the score of 81.6 for professional musicians of Seither-
silence between the chord and the probe tone was 300 Preisler et al. (2008), who also found mean scores of
ms, and the probe tone was an OCT whose amplitude 45.9 for nonmusicians and 61.6 for amateur musicians.
was the same as that of the chord and whose duration According to the original criteria of the AAT (score of 0-
was 300 ms. For the second group of 20 participants 50, spectral listener; 50-100, fundamental listener;
(numbered 21 to 40), the chord of OCTs had a duration Seither-Preisler et al., 2007), most of our participants
of 100 ms, the silence was 300 ms as before, and the were fundamental listeners. We divided our 40 partici-
probe tone was a pure tone whose pitch was randomly pants into two equal groups relative to the median value
distributed over a two-octave range from F4 (349 Hz) of 90.5 (the distribution was not bimodal). The cutoff
to E6 (1320 Hz). The amplitude of the pure tone was value was arbitrary; in general, it depends on the acous-
the same as that of the corresponding partial within the tic parameters of the stimuli such that HCTs with higher
chord, and its duration was 200 ms. These changes harmonic numbers and fewer successive harmonics
were intended to be exploratory; we did not indepen- increase the likelihood of spectral responses (Preisler,
dently manipulate duration and probe tone type (OCT 1993). The 20 participants with lower AAT scores were
versus pure). labeled relatively spectral listeners (mean score: 66.5);
the 20 with higher scores were relatively fundamental
EQUIPMENT listeners (mean score: 97.1). The difference between the
Sounds were presented using a DELL Optiplex 960 two means was significant (p < .001; we used the Mann-
computer with Intel1 Core™ 2 Duo CPU E8400 @ 3 GHz, Whitney U-test, since the distributions were skewed).
Windows 7 64-Bit operating system, 8GB RAM, and We assume that our relatively fundamental listeners
ADI 1984A High Definition Audio Onboard Graphic consistently heard the pitch of an isolated HCT at the
Card. The sound signal was amplified using Samson MF, whereas our relatively spectral listeners sometimes
C-control mixer-amplifier and sent to Beyerdynamics responded to spectral and sometimes to virtual pitch.
DT-100 closed headphones. ‘‘Octave’’ open-source soft-
ware was used both to synthesize the sounds and run the Experiment 1
experiment.
PROCEDURE
Auditory Ambiguity Test (AAT) In each trial, a chord of OCTs and a probe tone were
heard in succession. Participants were asked to rate how
PROCEDURE well the tone went with the chord on a 7-point scale
This preliminary test comprised 110 trials, in a different from very badly to very good (Wie gut passt der Ton zum
random order for each participant. In each trial, two Akkord? 1 ¼ sehr schlecht, 7 ¼ sehr gut). The experi-
successive HCTs with MFs were presented. In 100 menter explained that the task was comparable with the
ambiguous trials the spectral envelope and the MF question ‘‘How well do two colors go with each other?’’
 Tone Profiles of Musical Chords 413

and compared that question with how well the colors

went with each other in a picture hanging on the wall.
The experimenter encouraged participants to respond
to each trial quickly and spontaneously, and clarified
that we valued their opinion and there were no clearly
defined right or wrong answers. Some participants FIGURE 1. Nineteen Tn-types of cardinality 3 according to Rahn (1980),
asked whether they should judge whether the tone in close position with C4 in the lowest voice. Trichords with familiar
would go together with the chord if it sounded simul- music-theoretical names have been labeled: sus ¼ suspended triad
taneously (even though it was presented successively); (here, a G chord with suspended 4th, also called Gsus4), dim ¼
the experimenter avoided giving a clear answer to this diminished, min ¼ minor, maj ¼ major, aug ¼ augmented.

question and instead asked participants to respond

spontaneously on the basis of the sound, without think- it divides the octave, and hence the chroma circle, into
ing about music theory. If a participant interested in the three equal intervals of four semitones.
aesthetics of new or atonal music claimed that every
tone might go equally well with every chord, the exper- HYPOTHESES
imenter repeated the instruction to respond to the On the basis of previous research, we made the follow-
sound itself as experienced. ing predictions.
In a series of 108 trials, 9 chords and 12 chromatic
H1. Relatively fundamental listeners rate MFs in
tones were presented in all combinations.8 Relative to
musical chords higher than relatively spectral
an arbitrary reference chroma (0), the chords were 015 listeners.
(e.g., CD F), 025 (CDF), 027 (sus ¼ suspended 4th H2. The main peaks in the profile for each chord cor-
chord), 035 (e.g., CE F), 036 (dim ¼ diminished triad), respond to the 3 chord tones. Thus, empirical
037 (min ¼ minor), 045 (e.g., CEF), 047 (maj ¼ major), profiles correlate with stimulus profiles, with 1 for
and 048 (aug ¼ augmented). The arbitrary reference each chord tone and 0 for the other 9 chromas.
was randomly transposed around the chroma circle in H3. The difference between chord tones and non-
each trial. The chords had been selected from all 19 chord tones is larger for more familiar chords.
possible Tn-types of cardinality 3 (Forte, 1973; Rahn, If so, a possible explanation is that Western lis-
1980), as illustrated in Figure 1. The first eight selected teners have more practice making such distinc-
chords were the most common trichords in Renaissance tions in such chords.
polyphony according to a database analysis (Parncutt H4. The mean rating over all 12 probe tones is higher
et al., 2018; see Figure 2).9 The selection reflected our for more familiar or consonant chords. Goodness
goal to understand the historical emergence of major- of fit can be regarded as a horizontal (successive)
minor tonality: we wanted to focus on pitch perception consonance judgment; listeners often cannot sep-
in chords that played an important role in that historical arate different kinds of consonance (Parncutt &
development. The last chord was a special case from Hair, 2011).
which unusual results might be expected, and it H5. Among the 3 chord tones, the highest peak cor-
increased the diversity and dissonance of the sounds responds to the conventional root. For most
to which participants were exposed. The augmented chords, that was the upper tone of a P4 interval.
triad 048 is symmetrical and hence tonally ambiguous: H6. Ratings vary among the 9 chromas not corre-
sponding to chord tones, enabling a comparison
8
At first sight, this seems like a two-way repeated measures design with of the four listed models (MFs, diatonicity, 5th
independent variables Chord (9 levels) and Chroma (12 levels). It is not,
because a main effect of the circular (modulo 12) variable Chroma would
relations, completion tones).
be meaningless: each chord has a different profile and is randomly trans-
posed relative to the others. In this simple, non-standard design, each chord
RESULTS
is statistically independent of the other chords, so regardless of the number
of tested chords there is no need to correct for multiple comparisons. We made a preliminary, exploratory test of whether the
9
Chord 015 (e.g., E-F-A) combines a consonant P4/P5 interval (5 data might depend on chord duration or tone type by
semitones) with a dissonant m2 (1 semitone) and an intermediate M3 comparing the results of the first group of 20 partici-
(4 semitones). In modern (tonal) terminology, it can be a minor triad (on pants (who heard longer chords and OCT probes) with
D) with suspended or passing M9 (on E). In early music, it can be a M3
interval (FA) with a passing M7 (E), or a P5 interval (AE) with a passing
those of the second group of 20 (shorter chords and
m6 (F). In Figure 2, this dissonant tone combination is almost always pure probes). Although our experiment resembles
prepared (i.e., the tones do not begin simultaneously). a two-way repeated measures design with independent
414 Richard Parncutt, Sabrina Sattmann, Andreas Gaich, & Annemarie Seither-Preisler

FIGURE 2. Distribution of chords corresponding to 19 Tn types of cardinality 3 (trichords) in a database of musical scores of unaccompanied choral
polyphony from the 13th, 14th, 15th, and 16th centuries (Parncutt et al., 2018). A new “chord” was identified at every onset in any voice. White bars:
Prepared chords, in which one or more notes are held from the previous chord. Black bars: Unprepared chords, in which all onsets are simultaneous.

variables Chord (9 levels) and Chroma (12 levels), it is TABLE 3. Repeated-measures Analyses of Variance of Results of
not; each chord is statistically independent of the other Experiment 1 for Each Chord Separately with Factors Group (2) and
chords, as each chord has a different profile and is ran- Chroma (12)

domly transposed relative to the others. Therefore, Interaction between

repeated-measures analyses of variance (ANOVAs) Chord Main effect of Group Group and Chroma
were calculated for each chord separately with factors
Group (2) and Chroma (12); results are shown in 015 F(1, 38) ¼ 0.13, p ¼ .91, F(11, 418) ¼ 0.98, p ¼ .47,
Table 3. Given our stance on the statistical indepen- Z2 ¼ .000 Z2 ¼ .025
045 F(1, 38) ¼ 0.01, p ¼ .93, F(11, 418) ¼ 1.10, p ¼ .36,
dence of chords from one another we did not consider Z2 ¼ .000 Z2 ¼ .028
corrections for multiple comparisons necessary. There 025 F(1, 38) ¼ 0.02, p ¼ .89, F(11, 418) ¼ 1.26, p ¼ .24,
was no main effect of Group and no interaction between Z2 ¼ .001 Z2 ¼ .032
Chroma and Group for any of the chords, which justi- 035 F(1, 38) ¼ 0.05, p ¼ .83, F(11, 418) ¼ 0.68, p ¼ .76,
fied averaging the results of the two groups in later Z2 ¼ .001 Z2 ¼ .017
analyses. We did not systematically investigate the effect 027 F(1, 38) ¼ 1.77, p ¼.19, F(11, 418) ¼ 0.54, p ¼ .87,
of chord duration or tone type. Z2 ¼ .045 Z2 ¼ .014
Results are presented in Figure 3. The bottom right 036 F(1, 38) ¼ 0.75, p ¼ .39, F(11, 418) ¼ 0.67, p ¼ .77,
Z2 ¼ .019 Z2 ¼ .017
panel of Figure 3 is a special case. The augmented triad
037 F(1, 38) ¼ 0.49, p ¼ .49, F(11, 418) ¼ 1.07, p ¼ .39,
048 (e.g., CEG ) is symmetrical: the intervals between Z2 ¼ .013 Z2 ¼ .027
adjacent tones are all 4 semitones. For this reason, and 047 F(1, 38) ¼ 0.75, p ¼ .39, F(11, 418) ¼ 0.93, p ¼ .51,
because the chord was constructed from OCTs, the Z2 ¼ .019 Z2 ¼ .024
stimuli presented to the participants for probe pitches 048 F(1, 38) ¼ 0.55, p ¼ .46, F(11, 418) ¼ 1.13, p ¼ .34,
0 to 3 were physically identical (because transposed Z2 ¼ .014 Z2 ¼ .029
around the chroma circle) to the stimuli presented for Note: Group 1 was the first group of 20 participants (with longer stimulus durations)
probe pitches 4 to 7, and for probe pitches 8 to 11. and Group 2 was the second group (with shorter durations).
 Tone Profiles of Musical Chords 415

FIGURE 3. Results of Experiment 1. Points are mean listener ratings over 40 participants. Error bars are 95% confidence intervals. Open circles are
chord tones; filled circles are non-chord tones. Tones predicted to have higher salience are marked with letters: M: missing fundamental, D: diatonic
tone, C: completion tone. The headings “3-4A” and so on are labels for Tn-types according to Rahn (1980); “015” means 0, 1, and 5 semitones relative to
an arbitrary reference pitch.

Results were therefore averaged across these three TABLE 4. Repeated-measures Analyses of Variance of Results of
groups of trials. The error bars are smaller than for the Experiment 1 for Each Chord Separately with Factors Group (2) and
other chords because they represent the means of 120 Chroma (12)
rather than 40 data per point. Interaction between
H1 was not confirmed: relatively fundamental listen- Chord Main effect of Group Group and Chroma
ers did not generally rate MFs higher than relatively
spectral listeners. For each chord, results were subjected 015 F(1, 38) ¼ 0.35, p ¼ .56, F(8, 304) ¼ 1.53, p ¼ .15,
to a repeated-measures ANOVA with factors Listener Z2 ¼ .009 Z2 ¼ .039
045 F(1, 38) ¼ 0.39, p ¼ .53, F(8, 304) ¼ 0.39, p ¼ .93,
Type (2 levels: relatively fundamental, relatively spectral)
Z2 ¼ .010 Z2 ¼ .010
and Chroma (9 levels; the three chord tones were omit- 025 F(1, 38) ¼ 2.41, p ¼ .13, F(8, 304) ¼ 1.13, p ¼ .34,
ted). This was possible given that the fundamental and Z2 ¼ .060 Z2 ¼ .029
spectral listener groups did not differ in variance accord- 035 F(1, 38) ¼ 0.34, p ¼ .57, F(8, 304) ¼ 0.86, p ¼ .55,
ing to the Levene test. Results are presented in Table 4. Z2 ¼ .009 Z2 ¼ .022
The effect of Listener Type was significant for only one 027 F(1, 38) ¼ 0.87, p ¼ .36, F(8, 304) ¼ 1.11, p ¼ .35,
chord: 037 (minor). The interaction between Listener Z2 ¼ .022 Z2 ¼ .028
036 F(1, 38) ¼ 0.09, p ¼ .77, F(8, 304) ¼ 0.76, p ¼ .64,
Type and Chroma was significant for only two chords:
Z2 ¼ .002 Z2 ¼ .020
047 (major) and 048 (augmented). Given the lack of any 037 F(1, 38) ¼ 5.20, p < .05, F(8, 304) ¼ 0.89, p ¼ .53,
consistent, significant main or secondary effect of Lis- Z2 ¼ .120 Z2 ¼ .023
tener Type, we averaged over all listeners in Figure 3. 047 F(1, 38) ¼ 0.41, p ¼ .52, F(8, 304) ¼ 2.14, p < .05,
H2 was confirmed: the main peaks in each chord Z2 ¼ .011 Z2 ¼ .053
profile corresponded to chord tones. We performed 048 F(1, 38) ¼ 0.08, p ¼ .78, F(8, 304) ¼ 2.51, p < .05,
an ANOVA with factors Chord (9) and Tone (2). Tone Z2 ¼ .002 Z2 ¼ .062
was set to 1 for the three chord tones and 0 for the nine Note: Group 1 was relatively fundamental listeners and Group 2 was relatively
non-chord tones in each chord. The difference in mean spectral listeners.
416 Richard Parncutt, Sabrina Sattmann, Andreas Gaich, & Annemarie Seither-Preisler

rating between chord and non-chord tones was com- there was a ceiling effect, such that all three tones were
pared across 9 chords. There was a main effect of Tone: heard to go (very) well with the preceding chord.
responses for chord tones were higher than for non- H6 was partially confirmed. A one-way ANOVA with
chord tones (H2), F(1, 39) ¼ 85.72, p < .001, Z2 ¼ .69. 9 levels yielded a main effect of Chroma among non-
H3 was also confirmed: participants could more eas- chord tones for chords 015, 036, 037, 047. For 035 we
ily distinguish chord tones from non-chord tones in observed a trend.
more familiar (consonant) chords. In the same ANOVA
• For 015, F(8, 312) ¼ 2.25, p < .05, Z2 ¼ .05
with factors Chord and Tone, the effect of Tone was
• For 036, F(8, 312) ¼ 2.65, p < .01, Z2 ¼ .06
greater for chord 047 (maj) than all other chords except
• For 037, F(8, 312) ¼ 2.27, p < .05, Z2 ¼ .06
chords 027 (sus) and 037 (min), F(8, 312) ¼ 5.85, p <
• For 047 F(8, 312) ¼ 2.59, p < .05, Z2 ¼ .06
.001, Z2¼ .13.
• For 035, F(8, 312) ¼ 1.79, p < .08, Z2 ¼ .04
H4 was not confirmed: ratings were not generally
higher for more familiar or consonant chords. An There was no significant effect of Chroma among
ANOVA with two factors, Chord (9 levels) and Chroma non-chord tones for chords 025, 027, 045, or 048.
(12), revealed a main effect of Chord, F(8, 312) ¼ 5.37, p
• 025: F(8, 312) ¼ 1.09, p ¼ .37, Z2 ¼ .03
< .001, Z2 ¼ .12, but the mean rating for a chord did not
• 027: F(8, 312) ¼ 1.24, p ¼ .28, Z2 ¼ .03
depend in a clear way on its familiarity/consonance. In
• 045: F(8, 312) ¼ 1.32, p ¼ .23, Z2 ¼ .03
order of mean response (from highest to lowest), the
• 048: F(8, 312) ¼ 1.39, p ¼ .20, Z2 ¼ .03
chords were 036, 025, 037, 048, 015, 035, 047, 027, 045.
H5 was partially confirmed: the profile peak some- Although only half of the chords produced a significant
times corresponded to the music-theoretic root. For effect of Chroma for non-chord tones, we proceeded to
each chord separately, a one-way repeated-measures investigate predicted differences among non-chord tones
ANOVA was applied to the ratings for the three in all nine chords, looking for higher mean goodness-of-
chord-tones, ignoring ratings at non-chord pitches. fit ratings at predictions of four theories: MFs, diatonic
There was a significant main effect of Chroma for four tones, 5th-related tones, and completion tones.
of the nine chords: 015, 027, 037, and 047.
• For 015, the ANOVA yielded F(2, 78) ¼ 3.16, p < MODELS
.05, Z2¼ .08, but Bonferroni-adjusted post hoc Predictions of the four models for non-chord tones are
analysis produced no significant differences. summarized in Table 5. For each chord and each of the
• For 027, an ANOVA with Greenhouse-Geisser four theories, the first chroma in the list is the one most
correction yielded F(1.58, 61.68) ¼ 4.77, p < .05, strongly predicted by the theory. For example, the
Z2 ¼ .11, and a Bonferroni-adjusted post hoc strongest non-chord tone in chord 015 according to the
analysis revealed a significant difference (p < diatonic predictor is 3 (i.e., 3 semitones or a m3 above
.01) between tone 0 and tone 2, tone 0 receiving the 0 in 015). The other pitches are listed in descending
higher ratings; 1.05, 95%-CI(0.22, 1.88). order of predicted strength. If two or more pitches are
• For 037, F(2, 78) ¼ 6.08, p < .01, Z2 ¼ .14, a Bon- predicted to have equal strength, they are listed in rising
ferroni-adjusted post hoc analysis produced a sig- numerical order. This criterion was applied in the same
nificant difference (p < .01) between tone 0 and way to all four theoretical predictions. Exact procedures
tone 3, tone 0 receiving higher ratings; 1.10, 95%- for the four predictive models were as follows.
CI(0.43, 1.77).
MFs. Predictions for MFs were chroma-salience profiles
• For 047, an ANOVA with a Greenhouse-Geisser
according to Parncutt (1988) with the following root-
correction yielded F(1.62, 63.31) ¼ 10.97, p < .001,
support weights: 10 for the P1/P8 interval, 5 for P5, 3 for
Z2 ¼ .22, and a Bonferroni-adjusted post hoc
M3, 2 for m7, and 1 for M2/M9 (see Parncutt, 2009,
analysis revealed a significant difference (p <
Appendix). The results of these calculations are pre-
.01) between tone 0 and tone 4, tone 0 receiving
sented in Table 6a, in which the two main MFs for each
higher ratings; 1.25, 95%-CI(0.40, 2.10), tone 7
chord are also marked. Table 6b presents predictions of
higher than tone 4; 1.13, 95%-CI(0.35, 1.90).
a similar algorithm that additionally accounts for mask-
While all observed differences were broadly consistent ing among partials (Parncutt, 1993). Nearby partials
with both typical music-theoretic positions and the pre- mask each other, and the smaller the interval, the higher
dictions of Parncutt (1988) and Parncutt (1993), many the degree of masking. In the chord 015, for example,
predicted differences did not reach significance. Perhaps tones 0 and 1 mask each other, which reduces their
 Tone Profiles of Musical Chords 417

TABLE 5. Predicted Profile Peaks for Nine Non-chord Tones (in Semitones Above 0 in the Chord Label) for Each Chord in the Experiments
According to Four Different Theories, Starting with the Highest Predicted Peak in Each Case

Chord MFs Diatonic tones 5th-related tones Completion tones

015 10, 6, 3, 8, 9, 2, 4, 7, 11 3, 8, 10, 6, 7, 2, 4, 9, 11 6, 7, 8, 10, 2, 3, 4, 9, 11 8, 10, 9, 3, 7, 6, 2, 4, 11
025 10, 7, 1, 8, 4, 3, 6, 9, 11 7, 9, 10, 3, 4, 8, 11, 1, 6 7, 9, 10, 1, 3, 4, 6, 8, 11 9, 8, 10, 7, 3, 4, 1, 6, 11
027 5, 10, 3, 8, 4, 9, 1, 6, 11 5, 9, 4, 10, 3, 11, 6, 8, 1 5, 9, 1, 3, 4, 6, 8, 10, 11 5, 4, 10, 9, 3, 6, 1, 8, 11
035 8, 10, 1, 11, 2, 7, 4, 6, 9 10, 7, 8, 1, 2, 6, 9, 4, 11 10, 7, 8, 1, 2, 4, 6, 9, 11 9, 8, 7, 10, 2, 1, 4, 6, 11
036 8, 11, 5, 2, 1, 4, 7, 9, 10 1, 5, 8, 10, 2, 4, 7, 9, 11 1, 5, 7, 8, 10, 11, 2, 4, 9 8, 10, 9, 11, 1, 7, 2, 4, 5
037 5, 8, 11, 2, 9, 1, 4, 6, 10 5, 10, 2, 8, 1, 9, 4, 6, 11 2, 5, 8, 10, 1, 4, 6, 9, 11 10, 9, 8, 2, 5, 11, 1, 4, 6
045 10, 9, 1, 2, 8, 6, 3, 7, 11 2, 7, 9, 10, 11, 1, 3, 6, 8 7, 9, 10, 11, 1, 2, 3, 6, 8 9, 8, 7, 10, 2, 11, 1, 3, 6
047 9, 5, 2, 3, 8, 6, 1, 10, 11 2, 9, 5, 11, 6, 10, 1, 3, 8 2, 5, 9, 11, 1, 3, 6, 8, 10 10, 9, 11, 2, 5, 1, 3, 6, 8
048 1, 5, 9, 2, 6, 10, 3, 7, 11 1, 2, 3, 5, 6, 7, 9, 10, 11 1, 3, 5, 7, 9, 11, 2, 6, 10 2, 6, 10, 3, 7, 11, 1, 5, 9

TABLE 6. Predicted Chroma-salience Profiles of 9 Tested Chords

A)
Chord in semitones Chord name 0 1 2 3 4 5 6 7 8 9 10 11
015 – 10 13 2 3 0 15 5 2 3 3 6 1
045 – 13 3 3 1 10 15 2 2 3 5 6 0
025 – 11 3 12 1 2 15 0 7 3 0 9 0
035 – 10 4 2 11 0 17 0 2 8 0 6 3
027 sus 16 0 12 3 2 6 0 15 3 2 4 0
036 dim 10 1 5 10 1 7 10 0 10 0 1 8
037 min 15 1 2 13 0 8 0 10 8 2 1 3
047 maj 18 0 3 3 10 6 2 10 3 7 1 0
048 aug 13 5 3 0 13 5 3 0 13 5 3 0
Note: Top row: interval in semitones relative to reference pitch 0. Body of table: Weight W according to Parncutt (1988, Equation 1, p. 77) with root-support weights 10 (for the
P1/P8 interval), 5 (for P5), 3 (for M3), 2 (for m7), and 1 (for M2/M9). Chord tones are underlined, predicted roots are bold, and predicted MFs are italic.

B)
Chord in semitones Chord name 0 1 2 3 4 5 6 7 8 9 10 11
015 – 22 35 4 9 0 55 11 9 7 6 24 2
045 – 52 7 11 2 22 46 4 5 14 11 16 0
025 – 33 11 34 4 6 51 0 21 9 0 29 0
035 – 35 12 7 30 0 52 0 6 24 0 18 8
027 sus 49 0 33 11 6 18 0 51 8 7 11 0
036 dim 40 4 20 37 4 28 40 0 39 0 4 31
037 min 49 3 6 41 0 26 0 34 25 7 3 9
047 maj 54 0 9 9 28 18 6 29 9 20 3 0
048 aug 45 17 10 0 45 17 10 0 45 17 10 0
Note: Top row: interval in semitones relative to reference pitch 0. Body of table: Audibility A according to Parncutt (1993, Equation 5, p. 45) with the same root-support weights
as for part A.

salience relative to tone 5. Comparing parts a and b of CDF) is part of two major scales: 0, 1, 3, 5, 6, 8, 10 (D 
Table 6, we see that masking has little effect on the rank major) and 0, 1, 3, 5, 7, 8, 10 (A  major). We therefore
order of predictions. expect chromas 3, 6, 7, 8, and 10 to receive higher mean
ratings following 015 than other non-chord tones. Of
Diatonic tones. The predictions of this model were cal-
these, 3, 8, and 10 belong to both scales, so ratings for
culated in two steps. First, we listed all major scales to
these tones are predicted to lie between those for chord
which each chord belonged. Second, we counted how
tones (0, 1, 5) and other diatonic tones (6, 7).
many of these scales each chroma belonged to, and
weighted it with this number. Chromas that belong to 5th-related tones. The first predictions listed in the
the same diatonic scale(s) as a chord are marked in Figure fourth column of Table 5 are tones that are 5th-related
3 as ‘‘diatonic tones.’’ For example, the chord 015 (e.g., to at least one chord tone. For each chord, between 2 and
418 Richard Parncutt, Sabrina Sattmann, Andreas Gaich, & Annemarie Seither-Preisler

6 pitches were predicted by this method. If a tone was for other non-chord tones. The interaction was not
5th-related to two chord tones, it was placed first in the significant.
list. For example, in chord 025, tone 7 is 5th-related to For 5th-related tones, the two repeated-measures fac-
both tone 0 and tone 2. Finally, the remaining tones were tors were Chord and 5th Relation, the latter set to 1 for
inserted in numerical order. 5th-related tones and 0 for other tones. There were signif-
icant main effects of Chord, F(8, 312) ¼ 5.66, p < .001, Z2 ¼
Completion tones. For each trichord in Experiment 1,
.13, and 5th Relation, F(1, 39) ¼ 12.76, p < .01, Z2 ¼ .25);
we listed the main tetrachords of which it could be part.
5th-related tones (mean ¼ 4.07) were rated higher than
Consulting data on the prevalence of tetrachords in our
other non-chord tones (mean ¼ 3.75); however, there was
database (cf. Parncutt et al. 2018), we identified the
no interaction between Chord and 5th Relation.
non-chord-chromas that would create a familiar tetra-
For completion tones, the factors were Chord and
chord if added to the trichord. For example, if chro-
Completion (completion tones versus other non-chord
matic tone 10 is added to chord 015, it becomes (0, 1, 5,
tones). There were main effects of Chord, F(8, 312) ¼ 6.45,
10), which is a minor triad on 10 (10, 1, 5) with an
p < .001, Z2 ¼ .14, and Completion, F(1, 39) ¼ 8.16,
added major 9th (0). Put another way, chord 015 has
p < .01, Z2 ¼ .17; completion tones (mean ¼ 4.03) were
a missing root at 10. If chromatic tone 8 is added to 015,
rated higher than the other non-chord tones (mean ¼
the result is a major 7th chord relative to root 1. To
3.85). However, there was no interaction between Chord
create the lists in Table 5, we subjectively estimated the
and Completion.
consonance or prevalence in mainstream tonal music of
In summary, Experiment 1 provided tentative evi-
the harmonic (simultaneous) tetrachord created by
dence in favor of all four listed theories, but the effect
adding each possible completion tone to each trichord
size was higher for MFs (Z2 ¼ .33) than for the other
played on the piano. In the process we referred to our
models (.11, .25, and .17 respectively), suggesting MFs
tetrachord prevalence data, but did not use it system-
were responsible for most of the variance.
atically, because our participants were most familiar
with pop/jazz styles of the 20th century, for which we
have no comparable data. Instead, we drew subjectively Experiment 2
on our experience as musicians and theorists. We
assume that other music theorists will generate similar Experiment 2 was a repeat of Experiment 1 with just one
data and that any individual differences will not bias change in the empirical method. The question that par-
our final conclusions. Tetrachords that include two or ticipants answered in each trial was: ‘‘Is the tone in the
more semitone intervals (e.g., 015 plus 2, 4, or 11) were chord?’’ (Ist der Ton im Akkord?); and the rating scale
always placed last in the list—in rising numerical order, was labeled 1 ¼ definitely not and 7 ¼ definitely. The
as before. new question focused the attention of participants on
the chords themselves, rather than on the contexts in
which the chords occur in music. The question also
TESTING MODEL PREDICTIONS
corresponded more directly to the idea of MFs,
We tested the predictions for MFs in Table 5 by running
which—if they exist psychologically—should be per-
an ANOVA with two repeated-measures factors: Chord
ceived as physically real tones. We therefore expected
(9 levels) and MF strength (2), the latter being 1 for the
a higher rate of ‘‘errors’’ (participants mistakenly indi-
first two MFs and 0 for other pitches. There were main
cating that a tone is in a chord) when a chord is followed
effects of Chord, F(8, 312) ¼ 6.04, p < .001, Z2 ¼ .13, and
by an MF than when it is followed by another non-
MF Strength, F(1, 39) ¼ 19.37, p < .001, Z2 ¼ .33, con-
chord tone. All other aspects of Experiment 2 were
firming that the two predicted MFs (mean rating ¼ 4.17)
identical to Experiment 1, including initial hypotheses.
were rated higher than other seven non-chord tones
Because Experiments 1 and 2 are similar, we use a lower
(mean ¼ 3.80). The interaction was not significant.
p value for significance (.025) for the results section of
To test the predictions for diatonic tones in the table,
Experiment 2 than if the two experiments were consid-
the factors were Chord and Diatonicity (non-chord dia-
ered independent (where p would be .05) Detailed
tonic chromas versus other non-chord tones). The com-
results are shown in Figure 4.
parison was limited to eight chords (the non-diatonic
augmented triad was omitted). Both main effects were
significant: Chord, F(7, 273) ¼ 5.78, p < .001, Z2 ¼ .13, ADDITIONAL HYPOTHESES
and Diatonicity, F(1, 39) ¼ 4.79, p < .05, Z2 ¼ .11, with When comparing the results of Experiments 1 and 2,
means of 3.95 for non-chord diatonic chromas and 3.76 the following additional hypotheses were tested.
 Tone Profiles of Musical Chords 419

FIGURE 4. Results of Experiment 2. The symbols and letters have the same meaning as in Figure 3.

H7: Mean ratings for Experiments 1 and 2 over 108 Z2 ¼ .25).10 The interaction was not significant. We
trials correlate with each other because the task is so then ran an ANOVA with factors Chord (9) and Diato-
similar. Confirmed: r ¼ .79, p < .001. There was no nicity (2 levels: diatonic chromas versus other non-
interaction between Trial Number (a combination of chord tones) and Greenhouse-Geisser correction. Only
Chord and Chroma) and Experiment, nor was there an the main effect of Chord was significant, F(5.3, 259.7) ¼
interaction between Experiment and Chord or Chroma. 3.54, p < .01, Z2 ¼ .08. An ANOVA with Chord (9) and
H8: The overall mean result for Experiment 2 is lower Completion (2 levels, completion tones versus other
than for Experiment 1 because listeners are more likely non-chord tones) with Greenhouse-Geisser correction
to think a chromatic tone goes with a 3-tone chord than revealed that only the main effect of Chord was signif-
is part of it. Confirmed: a 3-way ANOVA with factors icant, F(1, 39) ¼ 19.37, p < .001, Z2 ¼ .33. Finally, an
Chroma (12 levels), Chord (9), and Experiment (2) ANOVA with Chord (9) and 5th Relation (2 levels:
revealed a main effect of Experiment, F(1, 39) ¼ 10.88, 5th-related chromas versus other non-chord tones) and
p < .01, Z2 ¼ .22. The overall mean rating for Experi- Greenhouse-Geisser correction produced main effects
ment 1 was 4.18; for Experiment 2, 3.98. of chord, F(6.1, 238.9) ¼ 3.95, p < .01, Z2 ¼ .09) and
H9: The MF predictor is more successful than the 5th-relatedness, F(1, 39) ¼ 7.70, p < .01, Z2 ¼ .17, and
other predictors in Experiment 2, because the question a significant interaction, F(6.0, 235.3) ¼ 2.89, p < .05, Z2 ¼
posed to participants in that experiment focused their .07. Both significance level and effect size were higher
attention on the chords themselves rather than the for MFs, F(1, 39) ¼ 13.26, p < .01, Z2 ¼ .25, than for 5th-
contexts in which they appeared (either for all partici- related tones, F(1, 39) ¼ 7.70, p < .05, Z2 ¼ .17, consistent
pants or only for fundamental listeners). To test H9, we with H9.
first considered MFs, running an ANOVA with two Summarizing the comparison of Experiments 1 and 2:
factors, Chord (9 levels) and MF-Strength (2). There In Experiment 1, participants were asked to rate how
were two main effects: Chord, in which some chords
attracted higher mean ratings than others, F(8, 312) ¼ 10
Because Experiments 1 and 2 are similar, the p value for significance
3.27, p < .01, Z2 ¼ .08, and MF-Strength, in which MFs is lower for the results section of Experiment 2. It lies between .05 (if the
(mean ¼ 3.89) were rated higher than other non-chord two experiments are regarded as independent) and .025 (if they are
tones (mean ¼ 3.56), F (1, 39) ¼ 13.26, p < .01, regarded as identical).
420 Richard Parncutt, Sabrina Sattmann, Andreas Gaich, & Annemarie Seither-Preisler

FIGURE 5. Comparison of four theories. Mean listener ratings for the first two pitches predicted by each model from among the 9 non-chord tones,
over all chords. The higher the mean rating, the more accurate the prediction of the corresponding model. Baseline is the mean of all 9 non-chord tones
for all chords. Error bars are 95% confidence intervals.

well the probe tone went with the preceding chord. That than baseline.11 In Experiment 2, the MF theory appeared
instruction logically included all four predictors: MFs, to account for the data best, followed by completion tones.
diatonic tones, 5th-related tones, and completion tones, This is evidence that MFs in musical chords have psycho-
as reflected in the results. In Experiment 2, participants logical reality—at least when those chords are constructed
were asked if the probe tone was in the chord, which from OCTs. The importance of completion tones in the
logically included only MFs—perceived as if they are data reflects the importance of chord extensions in music
physically present. In both experiments, MFs and 5th- theory—the idea that 7th chords are constructed by add-
related tones were rated higher than other non-chord ing 7ths to triads, and consequently that Western tonal
tones, but diatonic and completion tones were only music is based on triads (Childs, 1998).
rated higher in Experiment 1, consistent with the dif- The surprisingly poor performance of the diatonicity
ferent instruction. model in Experiment 2 can be explained as follows.
Trials in which a chord was followed by a diatonic tone
were more familiar because such combinations happen
A Comparison of Four Theories
more often in music. That made it easier for musically
trained participants to recognize that the tone was not
Figure 5 presents an alternative comparison of the pre-
part of the chord, increasing the number of negative
dictions of four theories for Experiments 1 and 2. For
responses. If, for example, a listener hears the chord
each chord, we considered the two most likely MFs,
015 followed by the diatonic tone 3 (a normal and
diatonic tones, 5th-related tones, and completion tones
familiar succession in tonal music), the familiar diatonic
according to the models, among the nine non-chord
relationship helps her or him realize that the tone is not
tones in each case, and considered the mean ratings
part of the chord.
given those two non-chord tones by all participants. For
each theory, we predicted that mean ratings for those
Results for Fundamental Versus
two chromas would exceed mean ratings for all nine
Spectral Listeners
non-chord tones. If the mean ratings for chromas pre-
dicted by a given theory A were higher than those pre-
The results presented above are averaged over relatively
dicted by theory B, we would then have more
spectral and relatively fundamental listeners as
confidence in theory A. The comparison is problematic,
because the predictors overlap: one and the same tone 11
The models overlap, so the data do not satisfy the independence
could be predicted by more than one theory. condition for ANOVA. To avoid overstating the finding, we apply the
The results as shown in the figure suggest that in Exper- approximate rule that two means are different if confidence intervals do
iment 1 all models except diatonicity performed better not overlap, or overlap only slightly (Goldstein & Healy, 1995).
 Tone Profiles of Musical Chords 421

determined by AAT. We averaged over the two groups Chord and Chroma, F(88, 334) ¼ 5.14, p < .001, Z2 ¼ .12,
because previous analyses had revealed no main effect but this time no interaction with Listener Type.
of listener type; however, group differences were found Relatively fundamental listeners were predicted to hear
in several other analyses. MFs more clearly or more often. To test this idea, we
We predicted a larger difference between ratings of performed an ANOVA that was restricted to non-chord
different listener types in Experiment 2, because the tones (9 per chord x 9 chords), for each experiment
question posed in that experiment (‘‘Is the tone in the separately. Independent variables were chord and tone
chord’’) focused the listener’s attention on tones in type (repeated measures) and listener type (between).
the chord itself, whereas the question asked in Experi- Here, tone type had two levels: MFs and other tones.
ment 1 (‘‘Does the tone go with the chord’’) referred to We expected an interaction between listener type and
musical context. Results contradicted this prediction. tone type, but found one neither for Experiment 1 nor
For Experiment 1, a 3-way ANOVA with factors Chord for Experiment 2. We also conducted similar analyses in
(9 levels), Chroma (12), and Listener Type (2) with which the two levels of tone type were defined differ-
Greenhouse-Geisser correction showed significant main ently: completion tones versus other non-chord tones,
effects of Chord, F(8, 304) ¼ 5.29, p < .001, Z2 ¼ .12, and diatonic tones versus other non-chord tones, and 5th-
Chroma, F(7.5, 283.8) ¼ 19.41, p < .001, Z2 ¼ .34, but not related tones versus other non-chord tones. Again, no
Listener Type. However, there were significant interac- two-way interactions between listener type and tone type
tions between Chroma and Listener Type, F(11) ¼ 2.8, were found.
p < .01, Z2 ¼ .07, and between Chord and Chroma, F(88, In sum, we found no clear, consistent, or theoretically
2
3344) ¼ 4.22, p < .001, Z ¼ .10. When the same ANOVA explicable differences between the results of relatively
was performed for Experiment 2, there were significant spectral and relatively fundamental listeners. A possible
main effects of Chord, F(5.4, 205.9) ¼ 2.4, p < .05, Z2 ¼ explanation is that fundamental listeners were perceiv-
.06, and Chroma, F(6.3, 239.6) ¼ 16.02, p < .001, Z2 ¼ .30, ing musical chords primarily on the basis of musical
but not of Listener Type; there was also a significant experience, rather than hearing MFs directly as we had
interaction between Chord and Chroma, F(88, 334) ¼ hypothesized. This hypothesis is consistent with the
5.14, p < .001, Z2 ¼ .12, but this time no interaction strong dependency of spectral versus fundamental lis-
with Listener Type. tening on stimulus exposure reported by Seither-
When results for individual chords were analyzed Preisler et al., (2008). Evidently neither fundamental
separately, there was sometimes an interaction between nor spectral listeners are capable of focusing attention
Chroma (12 levels) and Listener Type (2). In Experi- on MFs in musical chords.
ment 1, we found this interaction for four of nine The finding that MFs accounted for non-chord tone
chords: 035, 037, 047, and 048. For Experiment 2, we profiles in both experiments, but especially in Experi-
found this interaction for three of nine chords: 025, 047, ment 2, combined with the observed lack of any
and 048. However, we could not attach a particular consistent significant difference between the results of
meaning to the chords for which this difference was spectral and fundamental listeners, can now be explained
found and those for which it was not found. differently. A psychohistoric explanation involves two
We also conducted an ANOVA in which independent stages. In the first, MFs influenced how often corre-
variables were tone type (with two levels: chord tone sponding tones appeared immediately before and after
versus non-chord tone) and listener type. The interac- given chords in music from previous centuries. In the
tion for Experiment 1 was significant, F(1) ¼ 5.43, p < second stage, those statistical regularities influenced the
.05, Z2 ¼ .13; relatively fundamental listeners rated perception of all listeners—both fundamental and spec-
chord tones higher than relatively spectral listeners, by tral. The first stage involved intuitive (subconscious)
comparison to non-chord tones. This contradicted our perception, in connection with composition and impro-
hypothesis, according to which relatively fundamental visation. These processes are always to some extent cre-
listeners would more likely hear certain non-chord ative and experimental (even if contemporary theorists
tones (MFs), thereby reducing the difference between did not use terminology of that kind), otherwise musical
chord tones and non-chord tones. styles would not have changed historically. The second
When the same ANOVA was performed for Experi- stage involved codified compositional conventions that
ment 2, there were significant main effects of Chord, were presented repeatedly to listeners, causing them to
F(5.4, 205.9) ¼ 2.4, p < .05, Z2 ¼ .06, and Chroma, F(6.3, be enculturated by these statistical regularities. A single-
2
239.6) ¼ 16.02, p < .001, Z ¼ .30, but not of Listener stage process is also possible, in which some modern
Type; there was also a significant interaction between listeners perceive MFs directly.
422 Richard Parncutt, Sabrina Sattmann, Andreas Gaich, & Annemarie Seither-Preisler

Experiment 3

The main aim of Experiment 3 was to test the psycho-

logical reality of chord roots by testing whether listeners
spontaneously perceived the roots of diverse musical
chords. A secondary aim was to test the models from
the previous experiments on contrasting data, and to
explore the effect of task on empirically determined
chroma-salience profiles.
Music theorists first started to conceptualize chord
roots and chordal invertibility (inversions relative to
roots) in the early 17th century (Parncutt, 2011a; Rivera,
1984). This new development in the history of ideas was
a response to two centuries of compositional practice in
which most three- and four-voice chords had corre-
sponded to what were later called major and minor triads
in root position (Parncutt et al., 2018). To understand
this historic process, we must consider both culture-
specific ideas and perceptual universals (Eberlein, 1994). FIGURE 6. Screenshot for Experiment 3. Participants clicked on the 12
The method for this experiment was inspired by Ter- unlabeled chroma buttons to hear the individual tones. Akkord ¼ chord,
hardt’s (1972) standard procedure for determining the Weiter ¼ next trial, Beenden ¼ stop.
pitch of any short sound (described in Terhardt, 1998,
pp. 312–313; see also Terhardt & Grubert, 1987). In that RESULTS
procedure, a test sound and a pure reference tone are The results of Experiment 3 for all 40 participants are
heard in alternation. A listener adjusts the frequency of presented in Figure 7. From visual inspection, the chord
the pure tone until the two sounds have the same pitch. profiles are much more peaked than for Experiments 1
The SPL of the pure tone is held constant (e.g., 40 or 60 and 2. In most cases, participants clearly differentiated
dB SPL). The procedure produces valid, reliable, quan- between chord tones and non-chord tones. Unlike in
titative pitch estimates. For Experiment 3, we reduced Experiments 1 and 2, they also differentiated among
the number of response possibilities by defining 12 chord tones. As before, the differentiation was more
response categories in advance. Rather than allowing difficult for chords such as 036 and 048, presumably
participants to continuously adjust the frequency of the because they were more dissonant or less familiar.
reference tone, we asked them to choose a pitch from Among chord tones, the most commonly matched
a set of possibilities. chroma corresponded to the music-theoretical root in
all cases where it could clearly be defined as the higher
METHOD tone of a P4 interval. This definition yielded clear pre-
Participants saw an interface with 12 buttons in a circu- dictions for 7 of the 9 chords, the exceptions being 036
lar arrangement (the chroma circle, illustrated in Fig- and 048. Considering each chord in Figure 7 in turn, the
ure 6). At the start of each trial, they clicked on a central Wilcoxon Test (without adjusting for multiple compar-
button and heard a chord. They then focused their isons) yielded the following results. The p value for
attention on the first pitch that they heard in the chord significance is .05/3 ¼ .017, because in each chord we
and found it on a circular display of 12 tones. The chord made 3 comparisons (all 3 intervals between all 3 tones).
could be heard one or two times in each trial (mean:
1.8), and the tones could be heard as often as needed. In • For chord 015, tone 5 was rated higher than tone
each trial, the chord was randomly transposed around 0 (p ¼ .003)
the chroma circle, but the frequencies of the tones in the • For 025, 5 > 0 (p ¼ .004), 5 > 2 (p < .001)
circular interface remained constant: C was always at • For 027 (suspended 4th chord), 0 > 7 (p < .001),
the 12 o’clock position, E  at 3 o’clock, F  at 6 o’clock. 0 > 2 (p < .001), and 7 > 2 (p < .001)
Nine chords were presented six times each, making 54 • For 035, 5 > 0 (p < .001) and 5 > 3 (p < .001)
trials in all. The order of trials was random and different • For 036, 0 > 3 (p ¼ .005)
for each participant. The chords were physically identi- • For 037 (minor), 0 > 7 (p < .001), 0 > 3 (p < .001),
cal to those used in Experiments 1 and 2. and 7 > 3 (p ¼ .008)
 Tone Profiles of Musical Chords 423

150 3-4A (015) 3-7 (025) 3-9 (027)

100

50

0
0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11
Number of times selected

150 3-7B (035) 3-10 (036) 3-11A (037)

100

50

0
0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11

150 3-4B (045) 3-11B (047) 3-12 (048)

100

50

0
0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11

Pitch class of probe tone relative to chord

FIGURE 7. Results of Experiment 3, in which listeners actively selected the best-matching tone from 12 possibilities.

• For 045, 5 > 0 (p < .001), and 4 > 0 (p ¼ .002) chord becomes 037, the minor triad. The relatively low
• For 047, 0 > 7 (p < .001) and 0 > 4 (p < .001) mean rating for the chord’s 3rd (the tone 3 in 036) can
be explained by masking.
The only chord for which findings contradicted pre-
dictions was the diminished triad (036). The root of the
diminished triad CE G  is often considered to be A ; the Qualitative Data
triad may function as an incomplete dominant 7th on A 
in the key of D . Parncutt (1988) similarly predicted that After each experiment, participants were asked to com-
036 has a strong MF at 8. The results of Experiments 1 ment briefly on any aspect of their experience, including
and 2 were consistent with this prediction, but Experi- how they felt about the task (Wie ist es dir bei dieser
ment 3 contradicted it—perhaps because participants Aufgabe gegangen?) and what strategies they used. We
could play the target chord twice, focusing their atten- conducted 120 short interviews (3 experiments x 40
tion on physically present tones (analytic listening). The participants). Their comments suggested that most par-
result corresponded instead to the music-theoretic prin- ticipants did not recognize the chord that they heard in
ciple of stacked 3rds (tertian harmony; Rameau, 1721), each trial. Those who did, did not identify the interval
which dominates theoretical treatises on both classical between a reference chroma in the chord (such as the
and jazz harmony (e.g., Rawlins & Bahha, 2005). A pos- root) and the probe tone.
sible psychoacoustic explanation: the diminished triad Of the 40 participants, 32 were asked which chords
may be perceived as a mistuned major or minor triad. they recognized (the first 8 were not asked this ques-
That is feasible given that the partials of a HCT can be tion). Most replied that they heard major, minor, dimin-
mistuned relative to a harmonic series by as much as ished, and augmented chords (which were indeed 4 of
a semitone and still be perceived as part of the pattern the 9 chords), and nine replied that they heard dimin-
(Moore et al., 1985). If tone 0 in the diminished triad 036 ished and augmented more often than major and minor
is lowered by semitone, the chord becomes 047, the (in fact, each of these chords was presented equally
major triad; and if tone 6 is raised by a semitone, the often). Ten participants reported hearing 7th chords,
424 Richard Parncutt, Sabrina Sattmann, Andreas Gaich, & Annemarie Seither-Preisler

although there were none in the experiment; one par- five vectors were predictions of five models. The first
ticipant reported hearing only 7th chords throughout model was a simple stimulus model corresponding to
the entire experiment. These responses are consistent music notation, in which the presence of a tone is indi-
with the psychological reality of MFs at non-notated cated by 1 and the absence of a tone by 0; chord 015 was
chromas. represented by the vector 110001000000. The second
Those 32 participants were also asked to list the kinds model was the octave-generalized model of harmonic
of chord that they had heard and estimate the percent- pitch-pattern recognition by Parncutt (1988), as shown
age of each chord in the experiments. Of these, most in Table 6a. The third model was similar to the second
immediately objected they could only guess the answer, but also considered masking between nearby partials
but only one refused to try. Major triads were men- (Parncutt, 1993), see Table 6b. The last two predictors
tioned by 27 participants, minor triads by 25, dimin- are explained below.
ished triads by 17, augmented triads by 17, dominant Table 7 presents correlation coefficients, calculated by
7th chords (which never occurred) by 14, suspended 4th comparing two vectors of 108 numbers (9 chords x 12
triads by 2, and half-diminished 7th chords (which chromas; two-tailed significance tests). Within these
never occurred) by 2. Percentage estimates were gener- vectors, subvectors for each chord (groups of 12 values)
ally inaccurate. One participant reported hearing 5% had been converted to z-scores (with mean ¼ 0 and
major, 5% minor, 45% diminished, and 45% aug- standard deviation ¼ 1) before correlation. We first
mented, while another reported 60% major and 40% considered Pearson’s (linear) correlations (Table 7a),
minor chords. These findings are consistent with our since we were primarily concerned with how well each
assumption that most or all participants were unable to chroma is implied by or goes with the chord (conceived
recognize chord-tone relationships and were therefore of as a real number) rather than the rank order of the
unable to respond on the basis of music-theoretic chromas. We were also concerned to optimize the mod-
knowledge. els by improving the Pearson correlations. Spearman
Although we did not ask how difficult the experi- (rank) correlations are also shown, because the data are
ments were, out of all 40 participants, Experiment 1 was not normally distributed. Each correlation coefficient
spontaneously described as difficult by 11, Experiment 2 has specific advantages and disadvantages (Hauke &
(‘‘Is the tone in the chord?’’) by 26, and Experiment 3 by Kossowski, 2011).
13. Five found it difficult to concentrate, and another All correlations in Table 7 would have been highly
five found the experiments tiring or exhausting. Of the significant (p < .01) if considered alone. However, our
20 participants who heard the shorter 100-ms test primary interest was to compare coefficients with each
sounds, 12 complained they were too short. Regarding other and draw general, tentative conclusions from
timbre, 10 participants said that the sounds reminded those comparisons.
them of the organ, 6 of the piano, and 5 participants Consider first the Pearson correlations. Results of
complained the timbre was unpleasant. Experiment 1 correlated well with results of Experiment
We also conducted short interviews following AAT. 2, but less well with Experiment 3, in which participants
Six participants reported hearing tones go up and down actively chose the best-fitting tone and the profiles well
simultaneously and chose the movement that sounded predicted conventional chord roots. The experimental
more important. results also correlated well with the first three models
(stimulus model, Pa (88), Pa (93)). For Experiment 1,
Correlation Analyses the pitch models correlated better than the stimulus
model as expected—but not for Experiments 2 and 3,
To further test the psychological reality of MFs at non- suggesting that the pitch models could be improved by
chord tones, we correlated various results and predictors combining them with the stimulus model.
with each other. We first created a matrix of 8 vectors of We therefore created a linear combination of each
108 values each (12 values for each chord times 9 pitch model and the stimulus model (Pa (88)’, Pa
chords). Note that all 12 chromas are included in this (93)’). Relative to Table 6, a constant value of 20 was
analysis—both the three chord tones and the nine non- added to the predicted weight of the three chromas
chord tones. Each vector represents either experimental corresponding to chord tones. In chord 047, for exam-
data or theoretical predictions according to different ple, the models output profiles of 12 values; 20 was
models. added to the predicted salience of chromas 0, 4, and
The first three vectors were ratings from Experiments 7. By trial and error, we found that the value 20 roughly
1, 2, and 3, averaged over all 40 participants. The next maximized the Pearson correlations between model
 Tone Profiles of Musical Chords 425

TABLE 7. Correlation Coefficients, Calculated by Comparing Two Vectors of 108 Numbers (9 chords x 12 chromatic pitches).

A) Pearson
Data Model
Expt 1 Expt 2 Expt 3 Stimulus Pa (88) Pa (93) Pa (88)’ Pa (93)’
Data Expt 1 1 .85 .77 .81 .83 .84 .85 .86
Expt 2 .85 1 .79 .85 .82 .81 .87 .86
Expt 3 .77 .79 1 .82 .79 .79 .84 .83
Model Stimulus .81 .85 .82 1 .86 .84 .96 .93
Pa (88) .83 .82 .79 .86 1 .99 .97 .98
Pa (93) .84 .81 .79 .84 .99 1 .95 .98
Pa (88)’ .85 .87 .84 .96 .97 .95 1 .99
Pa (93)’ .86 .86 .83 .93 .98 .98 .99 1
Note: Pa (88)’ is a linear combination of Pa (88) and the stimulus model; Pa (93)’ similarly. All correlations are p < .01 (two-tailed comparisons).

B) Spearman
Data Model
Expt 1 Expt 2 Expt 3 Stimulus Pa (88) Pa (93) Pa (88)’ Pa (93)’
Data Expt 1 1 .73 .62 .72 .69 .69 .69 .69
Expt 2 .73 1 .57 .74 .65 .65 .65 .66
Expt 3 .62 .57 1 .74 .54 .53 .55 .54
Model Stimulus .72 .74 .74 1 .74 .73 .75 .75
Pa (88) .69 .65 .54 .74 1 .98 1.00 .99
Pa (93) .69 .65 .53 .73 .98 1 .98 1.00
Pa (88)’ .69 .65 .55 .75 1.00 .98 1 .99
Pa (93)’ .69 .66 .54 .75 .99 1.0 .99 1
Note: Pa (88)’ is a linear combination of Pa (88) and the stimulus model; Pa (93)’ similarly. All correlations are p < .01 (two-tailed comparisons).

predictions and empirical data for the three experi- previously defined). The tones of compatible scales can
ments. This result can be explained if some participants be MFs, diatonic tones, 5th-related tones, or completion
intuitively recognized some of the chords, deducing tones, or a combination of these.
which pitches were chord tones based on musical expe- Experiment 2 demonstrated that although MFs are
rience or music theory. The success of this combined not consciously perceived at non-chord tones in musical
model suggests that the salience of chord tones was chords, listeners’ perceptions are influenced by them.
underestimated in the original models, relative to other Historically, they could be an important factor influenc-
chromas. ing variations in salience of non-chord tones. If so,
The Spearman correlations in Table 7b confirm that a systematic consideration of such MFs belongs to the
all correlations are significant at the p < .01 level, but foundations of Western music theory.
they do not reflect the superior performance of the Results of Experiment 3 were consistent with pre-
adjusted models Pa (88)’ and Pa (93)’. The better per- dicted chord roots according to virtual pitch theory.
formance of the stimulus model by comparison to other Whereas some participants may have responded on the
models confirms that participants were generally able to basis of music-theoretic knowledge, our qualitative data
distinguish chord tones from non-chord tones. suggest that individual chords were seldom correctly
recognized, reducing the chance that results were arti-
General Discussion facts of music-theoretic knowledge. Results were con-
sistent with both Terhardt’s (1974, 1982) claim that
Experiment 1 shed light on the perceptual and cognitive chord roots are virtual pitches and Thomson’s contrast-
foundations of chord-scale compatibility in music the- ing (1993) claim that they are cultural phenomena.
ory. A comparison of results with predictions of differ- Taken together, the results of Experiments 1, 2, and 3
ent models suggests that the scales with which a chord is suggest that the models of Parncutt (1988, 1993) cor-
compatible depend on both ‘‘nature’’ and ‘‘nurture’’ (as rectly identify the main MFs in musical chords, but
426 Richard Parncutt, Sabrina Sattmann, Andreas Gaich, & Annemarie Seither-Preisler

overestimate their perceptual salience. One possible A psychohistoric approach would ideally acknowl-
explanation is that the MFs of chords in music result edge the role and relevance of the history of musical
from incomplete, approximate harmonic series of structure, the history of music perception, the history
slightly asynchronous partials; the models ignore the of music theoretic ideas, and the historical, social, and
mistuning and asynchrony. A second explanation is that ideological context dependency of music perception
musicians learn to ignore MFs during musical training (Cazden, 1945). The points in this list are causally inter-
and practice, including ear-training courses. connected (confounded) and hence resistant to empir-
For methodological reasons, the chords in our experi- ical scientific investigation. A psychohistoric approach
ments were built from OCTs and not from HCTs. These also acknowledges and addresses the ‘‘two cultures’’
chords, like those presented by Krumhansl (1990), problem of Snow (1959) by introducing humanities
sounded similar to chords played on a church organ, issues into scientific discourse and vice versa. It has the
suggesting that our participants perceived them as if potential to reconcile persistent contradictions between
they comprised HCTs. If so, we can imagine a two- scientific approaches such as Krumhansl (1990) and
stage cognitive process: first, chord recognition based Terhardt (1974), while at the same time acknowledging
on similarity, and second, access to perceptual (non- the positive contribution of both, and on that basis pro-
linguistic) ‘‘knowledge’’ about the chord as it occurs in vide a new foundation for a comprehensive psycholog-
music, such as profiles of prevalence of preceding and ically founded theory of major-minor tonality.
following tones. If we tried to understand historic music perception
Comparing the results of the pretest (AAT) and the from a purely psychoacoustic viewpoint, we might pre-
main experiments, the data suggest that our participants dict that, since the advent of counterpoint, European
sometimes directly perceived tone sensations at MFs listeners have been influenced by weak MFs at non-
and at other times responded according to statistical chord tones—that is, pitches that are not octave-
distributions in music to which they had been exposed. equivalent to chord tones (e.g., the tone A in the chord
Results of Experiment 2, in which participants were CEG). In temporal theories of pitch perception, these
asked if the probe tone was ‘‘in the chord,’’ were better tone sensations or tonal implications correspond to
accounted for by a theory of MFs than results of Exper- approximately periodic patterns in the waveform after
iment 1, in which participants were asked if the tone auditory filtering. In spectral approaches, they corre-
‘‘went with the chord.’’ Results of Experiment 2 also spond to fundamental frequencies of incomplete,
suggest that variations in the salience of non-chord approximately harmonic patterns of audible partials.
tones were better accounted for by a psychoacoustic In a psychohistoric paradigm, this subconscious, his-
theory of MF perception than by three competing the- torically undocumented aspect of musical pitch percep-
ories based on experience of tonal music: diatonicity, tion influenced the statistical probability of certain
5th relations, and completion tones (tones that com- tones preceding and following certain chords. For
plete a familiar, more complex chord). The lack of a con- example, the probability that the tone A would precede
sistent significant difference between relatively or follow the chord CEG, regardless of context, was
fundamental and relatively spectral listeners in Experi- boosted because A was weakly implied as an MF. Statis-
ment 2 suggests in addition that participants were not tical regularities of that kind were then internalized by
directly perceiving tones at non-chord-chromas; Western listeners (cf. Tillmann et al., 2000).
instead, they may have been imagining pitches that Our findings are limited to relationships between
often occur before and after those chords in music— chromas and do not consider octave register. Future
because in the past MFs were sometimes perceived in work may return to this issue, following Parncutt
those chords at those pitches. (1989) and Terhardt et al. (1982). Consider for example
On this basis, we propose a speculative psychohistoric the A-minor triad ACE, in close position with A in the
explanation for the observed variations in perceptual bass (e.g., A3C4E4). The tone A usually has audible
salience of non-chord tones. A psychohistoric account harmonics at chromas A, E, C , G, and B; C, at C, G,
considers both the acoustics of musical sounds and his- E, B , and D; and E, at E, B, G , D, and F . The triad’s
toric changes in their perception. By contrast, psycho- spectrum has (approximate) fundamental frequencies at
acoustic theories of pitch perception focus on physical 6 or more chromas: physically present at A, C, and E,
properties of the real-time stimulus such as periodicity and missing (MFs) at D, F, and B. Ignoring voicing and
or harmonicity. A psychoacoustic approach usually does register, the MF at D is associated with partials at D, A,
not consider the situation in which a sound is perceived F , C, and E; the MF at F, with C, A, and G; and the MF
or the (musical) experience of the listener. at B, with B, F , A, and C . If register is taken into
 Tone Profiles of Musical Chords 427

account, specific pitches are predicted to be more salient differences due to Adorno’s listener typologies and the
than others, depending on the chord’s voicing. The the- increasing diversity of modern musical styles and musi-
ory could be tested by manipulating the amplitude of cal audiences (Lilienfeld, 1987). Issues of this kind can
selected harmonics of selected MFs, testing whether the be clarified by combining psychological and music-
salience of those MFs changed according to predictions. theoretical approaches.
We have not considered non-human pitch perception Our findings have additional applications in music
or neural substrates of pitch perception. That non- analysis and composition. Pitch salience could be
human animals perceive MFs (e.g., Heffner & Whitfield, notated in musical scores as notehead size (Parncutt,
1976) is unsurprising given the ecological and social 2011b); non-notated pitches might be gray instead of
significance of fundamental frequency in conspecific black. In algorithmic composition, Ferguson and Parn-
vocalizations (e.g., Biben, Symmes, & Bernhards, cutt (2004) applied the pitch algorithm of Parncutt
1989) and the susceptibility of the fundamental to (1989) to composition in a relatively complex and dis-
masking in noisy environments (Sinnott, Stebbins, & sonant style; future work may generate more consonant,
Moody, 1975). Prior to the present study, we know of accessible music, and revisit the question of ‘‘new tonal-
no evidence for the perception of MFs at non-chord ities.’’ In computer-based expressive performance,
tones within musical chords in either human or non- musical expression (including timing and dynamics)
human subjects. It is difficult enough to demonstrate depends on harmonic accent, which in turn involves
MF perception within musical chords with musically both vertical dissonance and horizontal harmonic rela-
trained listeners; nonmusicians were excluded from our tionships (Bisesi & Parncutt, 2011); a better understand-
experiments because the task was too difficult. Nor are ing of MFs in chords could improve algorithms to
there published empirical studies on neural mechan- predict harmonic accent, leading to more convincing
isms underlying individual pitches perceived within artificial performances.
musical chords or implied by musical chords. Studies Music-psychological studies of pitch perception and
such as Maess, Koelsch, Gunter, and Friederici (2001) cognition tacitly assume a one-to-one correspondence
and Patel, Gibson, Ratner, Besson, and Holcomb (1998) between notated and perceived pitches. Our findings
considered music-syntactic relationships and incongru- undermine this assumption. Aspects of Krumhansl’s
ities, but not individual pitches. Although each chord in (1990) cognitive structures may be explicable by varia-
our experiments was presented in isolation, post- tions in pitch salience and MFs. These include the tone
experiment interviews suggested that chords were per- profiles of musical keys (tonal hierarchies; Parncutt,
ceived as musical entities, implying that their perception 1989, 2011a) and tone profiles of chord progressions
was affected by musical experience—an aspect that non- (Huron & Parncutt, 1993; Parncutt & Bregman, 2000).
humans are unlikely to be sensitive to and mechanistic Melodies in major and minor keys may be perceived as
temporal models of pitch perception are unlikely to prolongations of tonic triads (Parncutt, 2014; cf. Forte &
account for. Gilbert, 1982; Schenker, 1906/1954). In a psychohistoric
Our findings may inspire new approaches to analysis approach, ratings of chords (harmonic functions) rela-
and composition. 20th-century music theorists repeat- tive to tonal contexts depend on the prevalence of sim-
edly addressed issues of pitch salience: Schoenberg and ilar chord progressions in music, which in turn depend
followers such as Webern or Boulez tried to abandon on pitch commonality and preferences for root progres-
syntactic relations and hierarchical distinctions between sions such as falling 5ths (Parncutt, 1989, 2005).
musical tones, making them compositionally less
important. Discussion about the artistic virtues and per- Conclusion
ceptibility of such procedures is ongoing; no matter how
hard a composer tries to avoid hierarchical cognitive We measured tone profiles for a relatively large number
structures, the listener will still construct them in an of musically representative, isolated musical chords,
attempt to make sense of the music (Dibben, 1994, using contrasting empirical methods and a large num-
1999; Imberty, 1993). From a psychological viewpoint, ber of participants. Our results may represent the best
it is practically impossible to achieve atonality, since in existing body of data for the testing of explanatory
passages regarded as ‘‘atonal’’ some tones or chromas psychoacoustic, cognitive, and music-theoretic models.
generally sound more important than others. Even if we We then demonstrated that models of MFs, diatonicity,
tried to equalize tone saliences for the average listener, 5th relations, and completion tones can account in part
applying algorithmic models to real-time measurement for tone profiles of typical musical chords (Experiment
and adjustment, there would still be individual 1), but that MFs and 5th relations dominate when the
428 Richard Parncutt, Sabrina Sattmann, Andreas Gaich, & Annemarie Seither-Preisler

listener’s attention is more focused on tones in the were exposed, and hence their real-time music percep-
chord itself (Experiment 2). We also presented what tion. This ‘‘psychohistoric’’ approach addresses a long-
is presumably the most conclusive evidence to date for standing issue about the status of virtual pitches in tonal
the psychological reality of chord roots (Experiment 3). music and suggests that the contrasting approaches of
In three different approaches to analysis of data from Krumhansl and Terhardt may be complementary rather
Experiments 1 and 2 (ANOVA, comparison of mean than contradictory.
ratings at predicted chromas, and correlation analysis),
we presented convergent evidence that peaks in the Author Note
tone profiles of musical chords are significantly influ-
enced by MFs. We thank Erica Bisesi, Andreas Fuchs, Fabio Kaiser,
An analysis of individual differences (fundamental Daniel Reisinger, and Craig Sapp for computing support
versus spectral listeners) suggested, however, that indi- and assistance with music database analyses.
vidual listeners do not perceive these MFs directly. Parts of this paper were presented at the Ninth Trien-
A possible explanation involves the well-documented nial Conference of the European Society for the Cogni-
sensitivity of listeners to statistical distributions in the tive Sciences of Music, Manchester, August 17-22, 2015.
music to which they are exposed. We speculate that MFs Correspondence concerning this article should be
may have been perceived by past listeners, which influ- addressed to Richard Parncutt, Centre for Systematic
enced statistical pitch distributions of past music, which Musicology, University of Graz, Merangasse 70, 8010
in turn influenced the music to which our participants Graz, Austria, E-mail: parncutt@uni-graz.at

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