Key Distances Shorten Perceived Time of Real Music 613 · layer, Berio superposed his own ‘‘in-Contemporary-style’’ layer by inserting additional musical elements or structures

IN TERK E Y DISTA NCES ALSO SHORTEN SUBJ ECTI VE TI M E REPRO DUCTIO NS IN

REA L MODULATI NG TONAL MUSIC

ER ICO ARTIOL I FIRMIN O &JO SE LINO OLIVE IRA BUEN O

University of Sao Paulo at Ribeirao Preto, Ribeirao Preto,Sao Paulo, Brazil

REAL AND LONG MUSIC STIMULI ARE SELDOM USED

in music cognition and subjective time literatures. Wefound longer time reproductions for Berio’s Symphonyfor Eight Voices and Orchestra than for Mahler’s Sym-phony No. 2. Berio recomposed Mahler’s symphony,inserting complexity in all its elements. The contextual-change model explains these data, predicting longer timereproduction for a greater amount of information. There-after, we found synthetic tonal modulations eliciting timereproductions in an inverse function of interkey dis-tances, with major impact for sudden ones, an oppositeresult from contextual change prediction. Alternatively,we proposed the Expected Development Fraction (EDF)Model, claiming disproportion between expected andperceived times. In this study, we manipulated interkeydistance using the real piece Inspiraçao by Brazilian com-poser Garoto. Results confirmed the EDF Model predic-tion. Formatting the EDF Model’s key spatial-axisaccording to adapted key profile measures of Krumhansl(1990) allowed close correspondence between simula-tions and data.

Received: September 5, 2014, accepted August 10, 2015.

Key words: interkey distance, time reproduction, EDFModel, Krumhansl’s modeling, memory

W HY DO PEOPLE TEND TO PERCEIVE SOME

pieces of music as being shorter and othersas being longer than their physical duration?

In the experimental psychology of time, the response tothe demand, ‘‘how long did a piece of music last?’’ iscalled time estimation behavior. Cognitive processesrelated to memory (e.g., Block & Reed, 1978; Ornstein,1969), expectation (e.g., Boltz, 1989; Jones & Boltz,1989), and attention (e.g., Gibbon, Church & Meck,1984; Zakay & Block, 1996) are typically proposed toexplain a more general underlying process called

subjective time (e.g., Allan, 1979; Block, 1990; Fraisse,1982; 1984; Grondin, 2010; Jones, 1990). Such a timeprocess conceptually relates to the global temporalproperty of a music piece’s total duration and not totime components such as rhythm, tempo, or pulses.The perception of the ‘‘components’’ level or micro-structure of musical time has been intensively studied(e.g., Acevedo, Temperley, & Pfordresher, 2014; Cao,Lotstein & Johnson-Laird, 2014; Droit-Volet, Ramos,Bueno, & Bigand, 2013; Honin, 2013; London, 2012;Prince & Schmuckler, 2014). However, relatively fewstudies address the perception of the full durationallevel or macrostructure of musical time (e.g., Clarke& Krumhansl, 1990; Kellaris & Kent, 1992; North &Hargreaves, 1999; Phillips & Cross, 2011).

Firmino and collaborators claim that there is at leastone noticeable musical factor capable of shortening thesubjective time: the pitch structure of tonal modulationor key change in Western tonal music. Firmino andBueno (2008) presented to participants 20 s-long mod-ulating chord sequences varied by the interkey distance(e.g., close, CF, versus distant, CG�) and by the time andpathway whereby the key change itself occurs (e.g., sud-den, CG �, versus gradual, CE �G �). Participants wereunexpectedly requested to reproduce the duration of themusic only after listening to the music stimulus; that is,under a retrospective paradigm. A time reproduction taskrequires participants to produce a silent time intervalthat should correspond to the total duration of the justpresented music stimulus by pressing a computer stop-watch button (for reviews on subjective time methods,see Grondin, 2008; Zakay, 1990). As a result, Firminoand Bueno (2008) found that an increase of the interkeydistance implied a decrease of time reproduction andthat a sudden modulation shortened the time muchmore.

Such findings were compared with the presupposi-tions of the most influential time model related to pro-cesses under the retrospective paradigm: the contextualchange model by Block and Reed (1978; see also Block &Zakay, 2008). This memory-based model claims that theremembered duration implies cognitive reconstructionof the changes involving the stimulus (e.g., environmen-tal features and the stimulus content itself). Thus, more

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Key Distances Shorten Perceived Time of Real Music 613

retrieved contextual changes elicit longer time estima-tions. In time research, the concept of ‘‘changes’’ relatesto psychological information pertinent to the durationalevent or episode, which can be cognitively bounded,grouped, and structured; thus, one can also talk aboutthe amount and level of complexity of information(Fraisse, 1984; Jones & Boltz, 1989; Ornstein, 1969;see also Moles, 1958/2001). Because tonal modulationhad never been tested before in time research, Firminoand Bueno (2008) had to enlarge the principles of thecontextual change model by supposing that tonal mod-ulations with farther interkey distances might implymore contextual changes than closer ones, as music-theoretical common sense might suggest (Lerdahl,2001; Piston & DeVoto, 1941/1987; Schoenberg, 1922/1978). Thus, the consequent expected positive relation-ship between tonal distance and subjective time wascontradicted by Firmino and Bueno’s (2008) data.

Alternatively, Firmino and Bueno (2008) proposedthe Expected Development Fraction (EDF) Model, whichis based not only on memory but also on expectation.The EDF Model states that when a modulating piece ofmusic ends, the listener holds its duration; that is, theperceived musical time. He/she also expects such a musicpiece to continue – that is, the expected musical time orexpected development – due to the effect of modulation.This expectation for the music to continue is greaterwhen the destination key is more distant from the orig-inal key and when the modulating pathway is quicker.This implies that the expected musical time should havebeen longer than the perceived musical time in the lis-tener’s mind. Because there is then a disproportion – orfraction – between perceived and expected musicaltimes, the listened music seems to have been cut shortbefore its ‘‘ideal’’ ending. Because the listened musicseems to be conclusively shorter, the listener henceshortens its duration through the time reproductionbehavior – that is, the reproduced musical time – whenrequired to do such a time task.

Because there is an inevitable response delay betweenthe music stimulus ending and the time reproductionbeginning due to the experimenter’s verbal instructionon the time task, the EDF Model assumes the mediationof memory processes for the storage and maintenanceof the perceived and expected musical times describedabove. Firmino and Bueno (2008) claim that the per-ceived musical time is part of low-order cognitive rep-resentation or a musical implicit working memory(IWM) that tracks the immediate acoustic-musical reg-ularities of the music stimulus. (For studies on behav-ioral and neural bases of musical working memory, see,e.g., Schulze & Tillmann, 2013; Schulze, Dowling, &

Tillmann, 2012; Schulze, Zysset, Mueller, Friederici, &Koelsch, 2011; for a theoretical approach, see Berz,1995.) Conversely, the expected musical time is part ofhigh-order tonality cognition or musical semantic mem-ory (SM) in Western people’s minds. (For studies onbehavioral and neural bases of musical semantic mem-ory, see, e.g., Groussard, Rauchs, et al., 2010; Groussard,Viader, et al., 2010). Such memory is assumed to beimplicitly learned by long-term passive exposure todaily tonal music regularities (Collins, Tillmann, Barrett,Delbe, & Janata, 2014; Tillmann, Bharucha, & Bigand,2000). In these conditions, both IWM and SM arealways implicitly expressed because both modulatingmusic stimulus and time reproduction response arenonverbal or nondeclarative (for reviews on memorysystems, see Schacter & Tulving, 1994; Tulving & Craik,2005). In the EDF Model, for convenience, these mem-ory processes are conceptually grouped into a broadercognitive structure called key memory/expectation sys-tem (KMES). If a time reproduction is opportunelyrequired, another device called the temporal compara-tive reproducer (TCR) takes the perceived and expectedmusical times held in KMES, sets the comparisonbetween them, summarizes the comparison with a frac-tion, applies the fraction back to the perceived musicaltime (thus causing an impression of music shortness),and expresses such shortness through time reproduc-tion. (For a complete explanation of all of these pro-cesses, see Firmino & Bueno, 2008.)

Firmino, Bueno, and Bigand (2009) also observed thistime-shortening effect for reverse modulations. Theyshowed that even a distant intermediate key (e.g., CG�C)is able to elicit a greater shortening of time reproduc-tions than is a close one (e.g., CFC). The EDF Modelcould also account for such findings.

In this study, we aim to verify whether the inversefunction between interkey distance and subjective timewould also work for real music pieces filling longerdurations. The stimuli of Firmino and Bueno (2008)and of Firmino et al. (2009) were chord sequences with29 chords lasting 20 s, which were digitally synthesizedwith Shepard tones (Shepard, 1964; see also Krum-hansl, Bharucha, & Kessler, 1982). Although sucha duration is relatively long when compared with thosegenerally used by the literature, it may still be consid-ered short when compared with the widespread typicalpractice of tonal modulations in real music pieces.Furthermore, the sound spectra of traditional musicalinstruments commonly used by musicians are muchdenser, richer, and more complex than are the soundspectra of Shepard tones (see Campbell & Greated,2001).

614 Erico Artioli Firmino & Jose Lino Oliveira Bueno

This study thus used digital studio recordings of thereal music piece Inspiraçao (Inspiration) for classicalguitar by Brazilian composer Garoto. These recordingsincluded interkey distances unfolded by tonal modula-tions, properly played on the guitar by a highly trainedguitarist, filling a duration of 90 s (1 min and 30 s). Thepiece Inspiraçao originally possesses a reverse modula-tion beginning in the key of A, modulating to the key ofB�, and returning to the key of A. Thus, its composi-tional structure was modified to create three versions: 1)nonmodulating control, keeping only A; 2) close modu-lation, changing from A to B (that is, two fifths apart onthe circle of fifths); and 3) distant modulation, changingfrom A to B� (that is, five fifths apart on the circle offifths). These traditional music theory interkey dis-tances (from A to B, close, and from A to B�, distant)are reaffirmed by Lerdahl’s (2001, p. 69) music-theoretical computations; that is, in a hierarchical scaleoriented to A major, with bounds in A ¼ 0 and E� ¼ 30,B ¼ 14 and B� ¼ 23. Moreover, A and B share five scaletones, whereas A and B� share only two; scale overlap iswidely accepted as a measure or at least a correlate ofinterkey distance, as discussed by Lerdahl (2001),Krumhansl (1990), and others.

In a prior exploratory study, we had already used realand long music stimuli. Bueno, Firmino, and Engel-mann (2002) used the first 90 s of the third movementof Symphony No. 2 by Gustav Mahler and the first 90 sof the third movement of the Symphony for EightVoices and Orchestra by Luciano Berio. Onto a recom-position of Mahler’s ‘‘in-Romantic-style’’ symphonylayer, Berio superposed his own ‘‘in-Contemporary-style’’ layer by inserting additional musical elementsor structures (e.g., pitch, rhythm, timbre, dynamics,words, phonemes, and music and written quotations).As a result, Berio’s symphony became richer and morecomplex in psychological information than was Mah-ler’s, as usually understood in the time research area.Participants were required to reproduce prospectivelythe music stimulus durations; that is, they were warnedabout the time task before listening to the music stim-uli. Outcomes were that Mahler’s and Berio’s excerptswere under and overestimated, respectively. Consider-ing this very long duration to be estimated, one couldadmit some expressive intervenience of memory pro-cesses even under the prospective paradigm. Thus, byovergeneralizing the contextual change model again,one should expect that an effortful reconstruction ofthe listening moment of Berio’s symphony during thetime task might have elicited longer time reproduc-tions when compared with that of Mahler’s (see alsoBlock & Zakay, 2006).

A distinct model for time estimation that substantiallyemphasizes the role of multiple structures of temporalevents has been proposed by Jones, Boltz, and collabora-tors. Jones and Boltz (1989) claimed that events withlow or high temporal coherence afford analytic orfuture-oriented dynamic attending modes, respectively(see also Boltz, 1989; Drake, Jones & Baruch, 2000;Large & Jones, 1999). For less coherent events, theyexplain, ‘‘to estimate the duration of these events, peoplewill be biased by their attention to local details and willjudge events filled with more items to be longer’’ (Jones& Boltz, 1989, p. 461). Once more considering the stim-uli of Bueno et al. (2002) as examples, Berio’s symphonyunfolding apparently low coherence might have beentracked by the analytic attentional mode, and Mahler’ssymphony unfolding apparently high coherence mighthave been tracked by the future-oriented attentionalmode, thus justifying the longer time reproductions forthe former. For highly coherent events, Jones and Boltz(1989) further derived the expectancy/contrast model,predicting that a perceived event occurring at theexpected instant elicits almost accurate time estimation,whereas occurring earlier or later than the expectedinstant elicits under or overestimations, respectively.This model received empirical support from Boltz(1989), who tested the metric position of ending tonesin melodies, and from Schmuckler and Boltz (1994),who tested the metric position of tonally expected/unexpected chords in chord sequences.

Another study that used real and long music stimuliin time research is Ziv and Omer (2010). They pre-sented to participants a 135 s long (2 min and 15 s)tonal piece, Johann Sebastian Bach’s Fugue in C sharpmajor from Well-Tempered Clavier (Book I), and a 134 slong (2 min and 14 s) atonal piece, Arnold Schoenberg’sMusette from Suite for Clavier Op. 25 from Piano Worksfor Two Hands. The authors found that ‘‘in the prospec-tive paradigm, the tonal musical piece was estimated asshorter than the atonal piece, whereas the inverse wasfound in the retrospective paradigm’’ (Ziv & Omer,2010, p. 190). However, the time estimation task of thisstudy consisted of a verbal report; that is, participantswrote their answers (i.e., time numbers) on a question-naire. Given that this is a different methodology of timeresearch from the time reproduction task used by Fir-mino and collaborators, it seems inappropriate to com-pare the inferred processes of Ziv and Omer (2010).

Interkey distances and tonal modulations have exten-sively been studied through the probe tone method.This technique was first introduced by Krumhansl andShepard (1979) and consists of a tonal context presen-tation (e.g., ascending major scale) followed by a probe


tone randomly selected from the 12 chromatic tone set.Participants are asked to judge how well the probe fitswith the context. In several experiments over the years,the participants’ tonal ratings have mirrored the tonalhierarchy proposed by Western traditional music theory(e.g., Lerdahl, 2001; Schoenberg, 1922/1978; for reviews,see Krumhansl, 1990, 2000, 2004; Krumhansl & Cuddy,2010). Krumhansl and Kessler (1982) obtained stablemajor and minor key rating profiles for several contexts.They then computed statistical correlations for eachpair of profiles to bring forth degrees of similarities asmeasures of interkey distances. For better visualization,these correlations were submitted to nonmetric multi-dimensional scaling (Kruskal, 1964). The solutionrevealed a four-dimensional representation that placedthe spatial coordinates for all 24 major and minor keysonto two crossed circles (a torus), one emphasizing thecircle of fifths and the other emphasizing the relativeand parallel relationships. By combining the probe tonemethod with such a key toroidal representation, Krum-hansl and Kessler (1982) also expanded the investiga-tion to tonal modulation effects.

Addressing dynamic perception of keys through‘‘approximately real’’ music, Toiviainen and Krumhansl(2003) have investigated real-time responses elicited bythe organ music piece Duetto BWV 805 by J. S. Bach,which was played by a computer interface via musicalinstrument digital interface (MIDI) information, patch-ing a church organ timbre from the Unity DS-1 1.2software library. A concurrent probe-tone was soundedcontinuously with the music, whereby listeners had tojudge how well the probe fit with the music at each pointin time. By also exploring a self-organizing map (SOM;Kohonen, 1997), the authors showed that data projec-tions onto such a map depicted changes either in theperceived keys or in their strengths. The patterningretained a close resemblance to the first torus of Krum-hansl and Kessler (1982). Several other efforts in mod-eling tonality sense through artificial neural networksor other types of computer implementations have beenmade (e.g., Bharucha, 1987; Collins et al., 2014; Temperley,1999; Tillmann et al., 2000; for a recent review, seeTemperley, 2013).

The behavioral research on the psychological realityof tonality not only has confirmed statements and intui-tions from music theory but also has found neural cor-relates for tonality perception. For instance, throughfunctional magnetic resonance imaging, Janata et al.(2002) found a circumscribed area in the rostromedialprefrontal cortex, tracking activation for the cognitivespatial dimension of tonality in highly trained musi-cians. In this area, distinctly activated voxels were

evoked in straight correspondence to each key froma presented melody continuously modulating to all24 major and minor keys (for further specific commentson this study, see Zatorre & Krumhansl, 2002). Like-wise, through event-related brain potential imaging,Koelsch, Gunter, Schroger, and Friederici (2003) foundthat even nonmusicians listening to modulating chordsequences are able to exhibit ‘‘an early right anteriornegativity reflecting the processing of violation of musi-cal regularities and a late frontal negativity taken toreflect processes of harmonic integration’’ (p. 1149).They also found that ‘‘modulations elicited a tonic neg-ative potential suggested to reflect cognitive processescharacteristic for the processing of tonal modulations,namely, the restructuring of the ‘hierarchy of harmonicstability’ (which specifies musical expectations), pre-sumably entailing working memory operations’’(Koelsch et al., 2003, p. 1149).

Vuvan and Schmucker (2011) have applied the probetone method to auditory imagery. Participants success-fully imaged both major and minor tonal hierarchies, asindicated by the probe tone ratings. This demonstratedthat auditory imagery functions comparably to auditoryperception. The imaged minor tonal hierarchies wereless robust than the imaged major ones, showing thatminor tonalities are less psychologically stable thanmajor ones, again a similar pattern to the perceptionof these pitch structures.

Although Firmino and Bueno (2008) used a very dif-ferent experimental method from the probe tone tech-nique, it makes sense to posit that tonality inductionmight have occurred in the course of listening to themodulating chord sequence similarly to what happenedin Toiviainen and Krumhansl (2003). Moreover, thetonality imagery might have occurred in the course ofretrospective time reproduction, as found by Vuvan andSchmucker (2011).

Because the EDF Model’s hypothesis of supporting aninverse function between interkey distance and subjec-tive time has intuitively seemed to the present authorsto prevail even for real music, the production of simula-tions following the stimuli of the present study andother similar stimuli containing other interkey distancesbecame the second major goal of this study. Such simu-lations simply aim to show the heuristic scope of theEDF Model. Thus, because the key profiles of Krum-hansl and Kessler (1982) have intensively been demon-strated as a hierarchy-of-stability process underlyingperception, cognition, and even imagery, our EDFModel should also pertinently incorporate the correla-tions between those key profiles as measures of interkeydistances along the EDF Model’s key spatial-axis (see


explanation below). These correlations are listed onpage 38 (Table 2.4.) of the now classic book CognitiveFoundations of Musical Pitch by Krumhansl (1990),and, as described above, were the ‘‘seeds’’ for the gen-eration of the cognitive torus of keys by Krumhansl andKessler (1982). The original EDF Model of Firmino andBueno (2008) predicts time reproductions triggered bytonal modulations unfolding suddenly or gradually,through forward or reverse pathways, embracing twoor more keys, and filling any time interval. However,the original EDF Model was elaborated emphasizingmajor keys because Firmino and Bueno (2008) mostlyfocused on its capabilities of representation and mech-anism. Therefore, adding Krumhansl’s measures to theEDF Model’s logical architecture not only may offer anadditionally very solid empirical and theoretical base tothis model but also will allow the processing of all 24major and minor keys.

Expected Development Fraction (EDF) Model:Including Krumhansl’s Key Profile Correlations

First, we provide a descriptive summary of the EDFModel’s operations (for a full specification, see Firmino& Bueno, 2008). While listening to modulating music,the listener’s key memory/expectation system (KMES)is induced. Within KMES, the implicit working memory(IWM) device works like a ‘‘tracker’’ or ‘‘parser’’ for theinterkey distance and specific pathway of modulation,thus making and holding the spatial-temporal tonalroute with its corresponding perceived musical timeTIWM. The TIWM equals the music stimulus’ total dura-tion. As another consequence, the semantic memory(SM) device projects and holds the expected musicaltime TSM. The length of such expected developmentdepends on how far the interkey distance is. The com-parison of IWM’s and SM’s musical times is done by thetemporal comparative reproducer (TCR) device duringthe accomplishment of the time task. Raw inputs to themodel are the physical total duration and the beginningand ending positions and instants of key-permanenceand key-change of the modulating tonal stimulus. Ren-dered outputs from the model are the cognitive repre-sentations in IWM and SM and the simulated timereproduction behavior.

In IWM, the spatial-temporal tonal route implies theconjunction of the spatial S and temporal T dimensionsof tonality being thus conventionally assigned asnumerically identical; that is, S � T, measured by unitsof tonal cognition interval (tci) and seconds (s), respec-tively. (For a study on the interaction between musicalpitch and time dimensions, see, e.g., Prince, Schmuckler,

& Thompson, 2009.) These pitch and time dimensionsare respectively referred to as the key spatial-axis andkey temporal-axis in the EDF Model. Figure 1 depicts anexample of a spatial-temporal tonal route for a reversemodulation involving the two types of functions used inthe EDF Model: constant, for permanence or tonal rest;and parabolic, for modulation itself or tonal motion. Ifmodulation goes forward (e.g., from C to D) andreverses backward (e.g., from D back to C), then theconcavity of the parabola is oriented upward and down-ward, respectively.

The expectation (E) feeling is simulated by vectorsthat express the tangential relationship between thetraversed tonal-space and elapsed tonal-time; that is,E ¼ S / T. This feeling corresponds to the kinematicprocess of tonal-velocity, meaning that the experienceof waiting for the arrival of a destination-key is exactlythe same experience as proceeding through key motion.(For a description of similar musical inertia processes,see Larson, 2012; Larson & VanHandel, 2005.) In Figure1, the home vector Ho (between t0 and t1) indicates nullexpectation because there is no modulation in this phaseof original-key establishment, featuring nonvariable Salong T. Both the local vectors Lod (original-to-destination, between t1 and t2) and Ldo (destination-to-original, between t3 and t4) indicate the maximumexpectation reached at the end of the modulations thattransit back and forth, respectively. In these modulationphases, S varies along T because the vectors Lod and Ldo

secantly follow their respective upward and downwardparabolas. The enlarging of the angles a and � allowsvisualization of the increases of expectation. The amor-tized local vectors Lod’ and Ldo’ indicate the minimumexpectation reached at the end of the establishments ofthe destination and returned-original keys, respectively.In these permanence phases, the vectors Lod’ and Ldo’follow in parallel the respective straight lines between t2

FIGURE 1. Spatial-temporal tonal route and expectation vectors from

the EDF Model for an example of reverse tonal modulation: S and T are

the key spatial and temporal axes, po and pd are original and destination

key positions, Ho and Hd are home vectors, Lod and Ldo are local vectors,

Lod’ and Ldo’ are amortized local vectors, Godo is the global vector, and R

and M are rest and motion phases.


and t3 and between t4 and t5. The narrowing of theangles a and � to the angles a’ and �’ allows visualiza-tion of the decreases of expectation. Although S doesnot vary along T in t2-t3 and t4-t5 rest moments, theseamortized vectors continue to simulate the decreases ofexpectations. Hence, IWM not only tracks the entire keypathway of a given modulating music but also tracks thedynamics of rising and falling of expectation evoked ineach instant.

If any type or any number of modulations occurs ina music piece, there will always be a certain overallquantity of expectation for the music to continue at themusic final instant. This quantity depends mainly onhow far in interkey distance and how fast in pathwaythe modulations are unfolded. Thus, in SM, the contentof the resultant expectation in the IWM is exactly theexpected musical time TSM. The TSM is a Gestalt-basedresultant projection of three expectation modes respon-sible for simulating local, global, and developmentalexpectation effects. That is, the local Eloc refers to theinterference of the penultimate key upon the final key,the global Eglo refers to the interference of the first keyupon the final key, and the developmental Edev refers tothe accumulation of local effects throughout the mod-ulating music. The contents of these expectation modesoccur in the form of virtual times Tloc, Tglo, and Tdev intothe listener’s mind, respectively. They are computed bythe linear additions of their specifically involved vectors:in Figure 1, Tloc¼ |Hd|þ |Ldo’|, Tglo¼ |Godo| (Godo is theglobal vector), and Tdev¼ |Ho|þ |Lod’|þ |Ldo’|. The TSM

is in turn computed by the geometric mean of thesetimes, that is, TSM ¼ (Tloc . Tglo . Tdev)1/3. These expec-tation modes also allow explaining effects from modu-lating music unfolding more than two keys.

Finally, when the participant is requested to repro-duce the music time, he/she must subsequently callupon KMES’s contents through TCR. Then, TCR veri-fies how short the perceived musical time TIWM in IWMis in relation to the expected musical time TSM in SM,through the expected development fraction Fr; that is,Fr ¼ TIWM / TSM, as a percentage. This disproportionFr is applied back to TIWM to express such shortnessthrough the time reproduction behavior Test, that is,Test ¼ Fr . TIWM.

Now, let us borrow the correlations between key pro-files from Krumhansl (1990, p. 38, Table 2.4), which aresuitable measures of interkey distances to pertinentlyformat the key spatial-axis of the EDF Model. WhileKrumhansl reports correlations in relation to C major,in our case the EDF Model’s key spatial-axis is centeredon A major, which is the main key of the piece Inspir-açao used in this study. Likewise, the tested duration of90 s is taken as the range of the axis. Because Krum-hansl’s correlations fall into the range of 1 (for the key ofC as the best correlation or closest distance) to�.683 (forthe key of F� as the worst correlation or farthest distance),they had to be transformed mathematically. Meaning,they had to be rescaled to the range of 90 to 0, invertedto the range of 0 to 90, and key-relabeled from the tonal-ity of C major to the tonality of A major. The inversionwas computed by subtracting each rescaled value from90. Hence, Krumhansl’s original correlations, C ¼ 1.000,a¼ .651, F¼ .591, G¼ .591, e¼ .536, c¼ .511, g¼ .241,d ¼ .237, f ¼ .215, B� ¼ .040, D ¼ .040, E� ¼ �.105,A ¼ �.105, b ¼ �.158, A � ¼ �.185, E ¼ �.185,c � ¼ �.298, f � ¼ �.369, b � ¼ �.402, D � ¼ �.500,B ¼ �.500, g� ¼ �.508, e� ¼ �.654, and G� ¼ �.683,were transformed into new values, A ¼ 0.000, f� ¼18.663, D ¼ 21.872, E ¼ 21.872, c � ¼ 24.813, a ¼26.150, e ¼ 40.588, b ¼ 40.802, d ¼ 41.979, G ¼ 51.337,B ¼ 51.337, C ¼ 59.091, F� ¼ 59.091, g� ¼ 61.925, F ¼63.369, C�¼ 63.369, a�¼ 69.412, d�¼ 73.209, g¼ 74.973,B�¼ 80.214, G�¼ 80.214, f¼ 80.642, c¼ 88.449, and E�¼90.000. The resultant key spatial-axis of the EDF Model isgraphically depicted in Figure 2.

Note that Krumhansl’s correlations formatting theEDF Model’s key spatial-axis are originally linear orone-dimensional measures that were thereafter submit-ted to nonmetric multidimensional scaling, resulting inother measures as coordinates of a four-dimensionaltoroidal figure. Therefore, by borrowing the former cor-relations and not the latter coordinates, the EDF Modelcan only track from modulating music the linear inter-key distances and the linear modulating pathways butnot the full geometric perspective of tonality.

The next section describes our experiment, whichobserved the effects of tonal modulations to close anddistant keys unfolded by 90 s long real music stimuli

FIGURE 2. Key spatial-axis of the EDF Model formatted by the correlations between key profiles, which were adapted from Krumhansl (1990).


upon retrospective time reproductions from nonmusi-cians. Thereafter, simulations from the EDF Model arepresented with respect to the tested stimuli and othersto be discussed.

Method

PARTICIPANTS

Sixty-one Brazilian undergraduates (female ¼ 34) fromthe University of Sao Paulo at Ribeirao Preto (USP/RP),whose ages ranged from 18 to 30 years (M ¼ 22.87;SD ¼ 3.86), voluntarily participated in the experiment.They all reported having normal hearing and no musictraining, that is, they had never systematically practiceda musical instrument or singing.

The highly trained guitarist Gustavo Silveira Costa(Professor from the Department of Music of USP/RP)kindly performed the music pieces that were recordedand tested experimentally.

APPARATUS

Stimulus preparation occurred inside a soundprooflaboratory/music studio at the Center for Experimen-tal Aesthetics of USP/RP. The music pieces wererecorded with a Luna/M-Audio condenser micro-phone, Firewire 1814/M-Audio interface, Pro ToolsM-Powered 8.0/Digidesign software, and TD85/Kossclosed headphone, all connected to an iMac/Applecomputer. The performer played a classical guitarcrafted by hand by Brazilian luthier Sergio Barbosain 2006, with Augustine Regals nylon strings. A sinu-soidal feedback-beep (pitch A, 440 Hz, 70 dB SPL, 0.05s) was synthesized through the Csound 4.19 softwareto be used as a sonic warning response to the timereproduction key pressing.

The experiment was conducted in a soundproofroom with suitable lighting. The equipment com-prised the following: an IBM/PC notebook contain-ing the program WaveSurfer (implemented in VisualBasic 6.0 for Windows) used to present the stimuliand feedback-beeps and to store the participant’stime reproductions; a Leadership USB-computer-keypad with the ‘‘enter’’ key covered by a blue labelwith the word ‘‘play’’ for stimulus presentation, andwith the ‘‘þ’’ and ‘‘-’’ keys covered by a green labelwith the word ‘‘start’’ and a red label with the word‘‘end’’ for the beginning and ending of time repro-duction, respectively (the other keys and the surfaceof the keypad were painted black); and a J55i/JBLclosed on-ear headphone for listening to stimuli andfeedback-beeps.

MATERIALS

Stimuli were modifications of the tonal structure of themusic piece Inspiraçao for classical guitar by Braziliancomposer Garoto (Annibal Augusto Sardinha 1915-1955; for a biography, see Mello, 2013). The adoptedscore edition was made by Brazilian guitarist Paulo Bel-linati (1991). The analysis presented here convention-ally complies with the harmonic-tonal rules of Westerntraditional music (see, e.g., Piston & DeVoto, 1941/1987;Schoenberg, 1922/1978). The piece Inspiraçao originallypossesses a distant reverse tonal modulation that beginsat A, passes through B�, and then returns to A. It pos-sesses 2/4 meter and a predominant rhythmic figurationof sixteenths featuring the compositional style of a slowBrazilian chorinho. The indicated tempo beat rate is58-60 bpm on the score, but the piece also has a fewindications of rubato. The adopted tempo was a beatrate of 60 bpm with a steadied metrical grid withoutrubatos, which standardized the total duration of 90 scommon to all versions. The first forty-four measureswere taken for experimental usage, and they unfoldedthe following sections (see score of the excerpt inFigure 3): establishment of original key of A (measures1-34), modulation to destination key of B� (measures35-36), establishment of destination key of B� (measures37-42), and reverse modulation to original key of A(measures 43-44). Measures 1-34 in A were kept intactin all of the versions. However, the remaining measures35-44 were modified according to the specific version:(1) nonmodulating control version AA, featuring perma-nence on A, an interkey distance of zero fifths apart onthe circle of fifths; (2) close modulation version AB,featuring a key change from A to B, an interkey distanceof two fifths apart on the circle of fifths; and (3) distantmodulation version AB�, featuring a key change from Ato B�, an interkey distance of five fifths apart on thecircle of fifths (see Figure 4). An extra measure 45 wasadded for tonal and phrase resolutions at the tonicchord according to the specific version.

In version AA, along measures 35-36, the chords [B7/F�]–[F7] of harmonic functions A:[V4

3/V]–[Aug6/V]and original durations [half note]–[half note] werereplaced by the new chords [B7/F�–F7]–[E7] of har-monic functions A:[V4

3/V–Aug6/V]–[V7] and dura-tions [quarter note–quarter note]–[half note] (Aug6means augmented sixth chord throughout this text).Along measures 37-42, the original excerpt in B� wasreplaced by the repetition of its corresponding excerptin A along measures 1-6. Along measures 43-44, thechords [G�7(�5)]–[F7(�5)–E7(13)] of harmonic func-tions B�:[Aug6/V]–[V7–A: V7] and original durations[half note]–[quarter note–quarter note] were replaced


FIGURE 3. The opening 44 measures of the music piece Inspiracao, with harmonic analysis of the passage used for construction of the experimental

versions along measures 35-44 bounded by brackets. The instants in which modulation and establishment of destination key occur are indicated within

rectangles.


FIGURE 4. Measures 35-45 relative to the music piece Inspiracao, modified to create three stimuli manipulated according to interkey distance: zero in a),

two in b), and five c) fifths apart on the circle of fifths.


by the new chords [B7/F�]–[E7(�5)] of harmonic func-tions A:[V4

3/V]–[V7] and durations [half note]–[halfnote]. Finally, the additional measure 45 contained thechord Amaj7 of harmonic function A:[I] and duration[half note].

In version AB, along measures 35-36, the chords [B7/F�]–[F7] of harmonic functions A:[V4

3/V]–[Aug6/V]and original durations [half note]–[half note] werereplaced by the new chords [B7/F�–E�7(b5)]–[F�7] ofharmonic functions A:[V4

3/V–VII7/V/V/V]–[V7/V/V]and durations [quarter note–quarter note]–[half note].Along measures 37-42, the original excerpt in the keyof B� was replaced by its entire transposition to the keyof B. Along measures 43-44, the chords [G �7( �5)]–[F7(�5)–E7(13)] of harmonic functions B�:[Aug6/V]–[V7–A: V7] and original durations [half note]–[quarternote–quarter note] were replaced by the new chords[G7(�5)]–[F�7(�5)] of harmonic functions B:[Aug6/V]–[V7] and durations [half note]–[half note]. Finally, theadditional measure 45 contained the chord Bmaj7 ofharmonic function B:[I] and duration [half note].

In version AB�, along measures 35-36, the chords [B7/F�]–[F7] of harmonic functions A:[V4

3/V]–[Aug6/V]and original durations [half note]–[half note] wereentirely preserved. Along measures 37-42, the originalexcept in B� was entirely preserved. Along measures43-44, the chords [G �7(�5)]–[F7(�5)–E7(13)] of har-monic functions B�:[Aug6/V]–[V7–A: V7] and originaldurations [half note]–[quarter note–quarter note] werereplaced by the new chords [G�7(�5)]–[F7(�5)] of har-monic functions B�:[Aug6/V]–[V7] and durations [halfnote]–[half note]. Finally, the additional measure 45contained the chord B�maj7 of harmonic function B�:[I]and duration [half note].

PROCEDURE

Before music stimulus recordings, the performer guitar-ist received the following instructions written down ona sheet of paper (in Portuguese language): ‘‘(1) You willplay three versions of the music piece Inspiraçao bycomposer Garoto: one that holds the key of A frombeginning to ending; another that modulates fromA to B; and another that modulates from A to B�. (2)To establish the total duration of each one of the threeversions at 90 s, their tempos were set to 60 isochronousbeats (quarter notes) per minute. (3) For this reason, inthe course of each recording, you will listen to metro-nomic clicks through the headphones, so that you canreceive further support for maintenance of tempo at60 bpm. (4) The development of the guitar fingering,musical dynamics, and timbre should be freely chosenby you, but it is recommended that these choices

conform to each of the three modulating versions, a pos-ture which can yield different performances amongthem. (5) Each of the three versions will be recordeda number of times until you deem it satisfactory orsufficient for your final choice of the best performance.(6) After recording, you will listen to the recordingsagain so that you can choose one unique performancedeemed satisfactory or sufficient for each one of thethree versions of the music piece Inspiraçao.’’

In the experimental circumstance, each participantlistened to only one stimulus through headphones.Then, he/she reproduced its duration retrospectively(unexpectedly). The participants subsequently pressedthe ‘‘start’’ and ‘‘end’’ keys on the (stopwatch) computer-keypad, indicating through beeps the beginning andending of a silent time interval that should correspondto the just heard target music duration. Accordingly,their oral instructions (in Portuguese language) werethe following: (1) Before listening: ‘‘You have to pressthe play key to listen to a music piece through the head-phones. When the music finishes, you take the head-phones off, and I will give you the next instructions. Putthe headphones on in a comfortable way and you canstart whenever you want.’’ (2) After listening: ‘‘Press thestart key and then let the time pass by. When you thinkthat the time passing by is the same as the time of thelistened music, press the end key. Put the headphoneson and begin whenever you want.’’ Response delaysbetween stimulus ending and time beginning averaged28.38 s. The interkey distance defined the between-group variable.

After the time reproduction task, participants submit-ted to a short debrief concerning the strategy they used toreproduce the music duration and concerning their cul-tural and musical backgrounds and health conditions.

Results

Figure 5 plots means (M) and standard deviations (SD)of time reproductions relative to interkey distances dueto the three 90 s-long modulating and nonmodulatingversions of the music piece Inspiraçao. Interkey dis-tances are oriented to the key of A. The original keyof A, close destination key of B, and distant destinationkey of B� are, respectively, zero, two, and five fifthsapart from the original key of A on the circle of fifths,and they situate at the positions 0.000, 51.337, and80.214 tci in the EDF Model’s key spatial axis. Timereproduction means due to the versions AA, AB, andAB� are 119.23 s (SD ¼ 20.43 s), 106.61 s (SD ¼ 21.27s), and 89.62 s (SD ¼ 18.29 s), respectively. The mod-ulating versions AB and AB � elicited shorter time


reproductions than did the nonmodulating controlAA. The distant modulating version AB � elicitedshorter time reproductions than did the close modu-lating version AB. These effects were confirmedby one-way ANOVA and post hoc Holm-Sidak test,F(2, 58) ¼ 10.85, p < .05, MSE ¼ 4386.79.

Simulations

Figure 6 depicts the spatial-temporal tonal routes andexpectation vectors evoked from the EDF Model for thethree music stimuli of this study.

The route and vectors of the nonmodulating controlversion AA (Figure 6 in a)) proceed through the keytemporal-axis, keeping A’s key-position 0 tci until theend at instant 90 s, exhibiting permanence on thisunique key and hence no evocation of expectation forfuture keys. The nonexpected musical time is computed

by the geometrical mean of the home vector HA-AA

(as local effect), the global vector GAA which is equalto HA-AA (as global effect), and the developmental vectorwhich is also equal to HA-AA (as developmental effect);that is, TSM-AA¼ (|HA-AA| . |GAA| . |HA-AA|)1/3 ¼ (90 x 90x 90)1/3¼ 90 s. The fraction is then computed by FrAA¼TIWM-AA / TSM-AA¼ 90 / 90¼ 1. This value indicates thatthe perceived musical time seems to be 100% equal to theexpected musical time for version AA. Hence, perceivedmusical time is kept invariable through the reproducedmusical time TestAA ¼ FrAA. TIWM-AA ¼ 1 x 90 ¼ 90 s.

The route of the close modulation version AB (Figure6 in b)) proceeds through the key temporal-axis untilinstant 69 s, keeping A’s key-position 0 tci, shiftsupward by parabolic function until instant 73, reachingB’s key-position 51.337 tci, and then follows in parallelalong the key temporal-axis, again keeping such posi-tion until the end at instant 90 s. The increase of localvector LAB while enlarging angle � exhibits the fading inof expectation in this phase of key-change motionbetween instants 69 and 73 s. The decrease of amortizedlocal vector LAB’ while narrowing angle � to �’ exhibitsthe fading out of expectation in this phase of keeping-key rest between instants 73 and 90 s. The expectedmusical time is computed by the geometrical mean ofhome vector HA-AB plus amortized local vector LAB’ (aslocal effect), global vector GAB (as global effect), anddevelopmental vectors HA-AB and LAB’ taken again (asdevelopmental effect); that is, TSM-AB ¼ [(|HA-AB| þ|LAB’|) . (|GAB|) . (|HA-AB| þ |LAB’|)]1/3 ¼ (124.466 x103.612 x 124.466)1/3 ¼ 117.086 s. The fraction is thencomputed by FrAB¼ TIWM-AB / TSM-AB¼ 90 / 117.086¼0.768. This value indicates that the perceived musicaltime seems to be 76.8% shorter than the expected musi-cal time for version AB. Hence, perceived musical time isproportionally shortened through the reproduced musi-cal time TestAB ¼ FrAB . TIWM-AB ¼ 0.768 x 90 ¼ 69.12 s.

The route of the distant modulation version ABb(Figure 6 in c)) proceeds through the key temporal-axis until instant 69 s, keeping A’s key-position 0 tci,shifts upward by parabolic function until instant 73,reaching Bb’s key-position 80.214 tci, and then followsin parallel along the key temporal-axis, again keepingsuch position until the end at instant 90 s. The increaseof local vector LABb while enlarging angle � exhibits thefading in of expectation in this phase of key-changemotion between instants 69 and 73 s. The decrease ofamortized local vector LABb’ while narrowing angle � to �’exhibits the fading out of expectation in this phase ofkeeping-key rest between instants 73 and 90 s. Theexpected musical time is computed by the geometricalmean of home vector HA-ABb plus amortized local vector

FIGURE 5. Participants’ means (M, diamond dots) and standard

deviations (SD, vertical bars) and the EDF Model‘s simulations

(triangle dots) of time reproductions concerning interkey distances in

three 90 s versions of the music piece Inspiracao: AA, AB, and AB �,respectively, are situated at zero, two, and five fifths on the circle of

fifths and at 0.000, 51.337, and 80.214 tci in the EDF Model’s key

spatial-axis. Linear regressions for data, LR(data), and the EDF

model’s simulations, LR(EDF), with continuous and dashed lines,

respectively, are also plotted.


LABb’ (as local effect), global vector GABb (as global effect),and developmental vectors HA-ABb and LABb’ taken again(as developmental effect), that is, TSM-ABb¼ [(|HA-ABb|þ|LABb’|) . (|GABb|) . (|HA-ABb| þ |LABb’|)]1/3 ¼ (151.917 x120.558 x 151.917)1/3 ¼ 140.649 s. The fraction isthen computed by FrABb ¼ TIWM-ABb / TSM-ABb ¼ 90/ 140.649 ¼ 0.639. This value indicates that the per-ceived musical time seems to be 63.9% shorter than theexpected musical time for version AB �. Hence, per-ceived musical time is proportionally shortenedthrough the reproduced musical time TestABb ¼ FrABb

. TIWM-ABb ¼ 0.639 x 90 ¼ 57.51 s.Figure 5 also plots the EDF Model’s simulated time

reproductions Test together with participants’ performedtime reproductions. Linear regressions for both EDFmodel’s simulations (dashed straight line) and data (con-tinuous straight line) were also computed and plotted.Their respective slopes �.40 and �.35 are not signifi-cantly different to a high degree (p ¼ .99, t-value ¼0.0097, degree of freedom ¼ 2; see Cohen, Cohen, West,& Alken, 2003, regarding calculation of significance ofthe difference between two slopes). Because the lines arethen effectively parallel, it is possible to affirm that theEDF Model’s linear ‘‘behavior’’ proportionally mirroredthe data’s linear ‘‘behavior.’’ Hence, the EDF Model cap-tured the inverse function between interkey distances andtime reproductions for real modulating tonal music, par-ticularly while incorporating the measures from Krum-hansl (1990) along the EDF Model’s key spatial-axis.However, the model did not capture the participants’

time over-reproductions; that is, the time reproductionsabove the stimulus’ physical duration of 90 s. The differ-ence between the mean of data values 105.15 s and themean of the EDF Model values 72.26 s is 32.90 s. Thisover-reproduction effect in relative contrast with the pre-dictions of the EDF Model is explained in the Discussionsection.

Advancing one step further, Figure 7 plots the predic-tions of the EDF Model concerning cognitive interkey

FIGURE 6. Spatial-temporal tonal routes and expectation vectors from the EDF Model concerning the three stimuli of this work. Only the involved key

positions are presented along key spatial-axis. Only the instants at which key-permanence and key-change start and finish are presented along the key

temporal-axis.

FIGURE 7. Predictions of the EDF Model for several versions of the

music piece Inspiracao, exploring tonal modulations to all 24 major

and minor keys. Values for interkey distances and time estimations

are progressively arrayed concerning keys. Key labels are oriented to

the tonality of A major. The crossing pattern between the lines

highlights the inverse function between interkey distance and

subjective time.


distance and time reproduction behavior for all 24 majorand minor keys taken as destination keys of several pos-sible versions of the music piece Inspiraçao along thelines of this study. In such a stimulus set, the total dura-tion and the instants of permanence and key change arethe same as the tested versions. As a result, the dashedline with squares for distance and the continuous linewith triangles for time appear as a crossing pattern, a pic-ture that again highlights their inversional relationship.

Discussion

The present data confirm the principle of the ExpectedDevelopment Fraction (EDF) Model from Firmino andBueno (2008) that the greater the interkey distances intonal modulations, the shorter the time reproductions,with nonmusician participants and even for real andlong (90 s) music stimuli. Briefly, the EDF Model’s prin-ciple has been defined as follows: when perceiving mod-ulating music, listeners expect a temporal developmentsuitable to the traversed interkey distance; that is, thefarther the distance, the longer the development. Suchexpected development seems to be longer than theduration of perceived modulating music. The dispro-portion between expected and perceived durations isapplied back to perceived duration, leading to timeshortening. The present data therefore extend our pre-vious results, which were obtained with artificial (She-pard tones) and shorter (20 s) chord sequences.Firmino and Bueno (2008) and Firmino et al. (2009)showed this same inverse function between interkeydistance and time reproduction, the former testingsudden/gradual forward modulation and the lattertesting reverse modulation.

Additionally, after formatting the EDF Model’s keyspatial-axis with adapted measures of interkey distancesfrom correlations between key profiles of Krumhansl(1990; see also Krumhansl & Kessler, 1982; Toiviainen& Krumhansl, 2003), such a model not only renderssimulated time reproductions in parallel with the presentdata (see Figure 5) but also extends the spatial-temporalcognitive processing of modulations to all 24 major andminor keys (see Figure 7).

To the best of our knowledge, the present study is thefirst to manipulate real and long music stimuli comply-ing with a rigorously systematic methodology. A highlytrained guitarist played versions of the modulatingmusic piece Inspiraçao by Brazilian composer Garotoon a legitimate classical guitar. The interkey distancewas specifically varied, whereas other music composi-tional factors such as melody, rhythm, tempo, register,and timbre were kept constant. Only harmony and

contour were slightly modified by respective replace-ments and transpositions in the last few measures tosettle the different tonal modulations. Time reproduc-tion was adopted as a pertinent time estimation methodbecause it substantially imitates the inseparable tempo-ral nature of music, when precisely registered bycomputer.

Time models exclusively based on memory — predict-ing an increasing function between amounts of informa-tion or complexity and subjective time such as thecontextual change model by Block and Reed (1978) —do not account for these findings. By assuming that theincrease of interkey distance implies a greater amount ofinformation/complexity of the modulation’s composi-tional structure — inducing a greater amount of memoryresources — one might expect a lengthening of time asa consequence, which would yield an opposite effect fromthat indicated by our data (but see Bueno et al., 2002).Nevertheless, this ‘‘deadlock’’ does not mean that mem-ory processes are not involved. Actually, there are fivemajor reasons for assuming memory implication in thepresent experimental circumstance: 1) the tested musicpieces were real examples carrying relatively rich com-plexity in terms of sound spectra and compositionalstructure; 2) the tested music piece’s duration was long;3) participants were warned concerning the time taskonly after the music piece presentation; 4) there wereconsiderable response delays between stimulus endingand time task beginning (i.e., average 28.38 s); and 5)participants had to reproduce the stimulus duration byusing a stopwatch computer-keypad.

For a similar condition, Firmino and Bueno (2008)have proposed a musical implicit working memory(IWM, as a device of the EDF Model) because differentinterkey distances exposed by music stimuli might beheld for a while to bring forth different time reproduc-tions in a systematic manner. Such memory seems to betemporary because it is supposedly held active duringthe experiment while it is useful for the demanded task.Such memory expresses itself implicitly because boththe stimulus – having no sung lyrics – and the timeestimation response – being a silent reproduction – arenonverbal. This implicit property contradicts traditionalmodels of working memory such as Baddeley’s (2012)multicomponent model (see also Baddeley & Hitch,1974). This model prescribes that all acoustic stimuliare processed by a phonological loop device verbally(or explicitly). Along the same lines as Firmino andBueno (2008), Koelsch et al. (2003) claim engagementof musical working memory for modulating chordsequences as indicated by event-related brain potentials.In doing so, they emphasize the role of an inherent time


in the processing: ‘‘because time is involved in therestructuring of the tonal hierarchy, working memoryoperations are most presumably strongly entailed in thisprocess’’ (Koelsch et al., 2003, p. 1155; see also Berz, 1995;Schulze & Tillmann, 2013; Schulze et al., 2011, 2012).

The experimental circumstance of the present studyposits a challenge to the traditional working memoryconcept because music durations, response delays, andtime reproductions appear to clearly extrapolate the typ-ical time span of such a working memory to a greatextent (e.g., Baddeley, 2012; see also Barrouillet & Camos,2012). Although extremely controversial in the literature,general common sense seems to suggest a somewhat con-ventional time span with approximately 2 to 10 s asa sliding perceptual window. Fraisse (1984) has proposeda ‘‘psychological present’’ averaging approximately 2 to 3s, with an upper limit of 5 s. By contrast, our modulatingstimuli kept the original key during 68 s, modulatedduring 4 s, and kept the destination key during 18 s inthe course of a 90 s total duration (see Figures 3 and 4).Accordingly, such a short time span of traditional work-ing memory might miss the modulating process andmight consequently imply rapid forgetting of musicaltraces. Moreover, almost 30 s of subsequent responsedelay plus approximately 100 s of time reproductionmight extinguish the few remaining traces completely.In the face of these considerations, a type of workingmemory carrying a relatively long-term manifestationshould be admitted, one still held active while being use-ful, but with a greater time span to help explain thepresent data. Ericsson and Kintsch (1995) have intro-duced such a long-term working memory:

We propose that a general account of workingmemory has to include another mechanism based onskilled use of storage in long-term memory (LTM)that we refer to as long-term working memory (LT-WM) in addition to the temporary storage of infor-mation that we refer to as short-term workingmemory (ST-WM). Information in LT-WM is storedin stable form, but reliable access to it may bemaintained only temporarily by means of retrievalcues in ST-WM. Hence LT-WM is distinguished fromST-WM by the durability of the storage it providesand the need for sufficient retrieval cues in attentionfor access to information in LTM. (p. 211)

These authors assert that such LT-WM accounts forthe large demands on working memory for text com-prehension and expert performance such as mental cal-culation, medical diagnosis, and chess: ‘‘Our proposalfor LT-WM simply argues that subjects can acquire

domain-specific memory skills that allow them toacquire LT-WM and thus extend their working memoryfor a particular activity’’ (Ericsson and Kintsch, 1995,pp. 213-214; see also Ericsson & Lehmann, 1996; Cau-chard, Cane, & Weger, 2012; Hu & Erickson, 2012).Even musically untrained Western people seem to pos-sess somewhat skilled cognitive knowledge concerningtonality and key change (e.g., Collins et al., 2014;Koelsch et al., 2003; Krumhansl, 1990), which in otherwords is a domain-specific memory skill for music. Ourparticipants might similarly have accessed such tonalityknowledge while facing retrieval cues concerning keysand key-change when they approximately imaged tonal-ity during the time reproduction task. (For a study ontonality imagery, see Vuvan & Schmucker, 2011; fora study on the relationship between musical imageryand working memory, see Kalakoski, 2001.) In fact,from debriefing after the experiment, participantsreported roughly trying ‘‘to sing into the head’’ or ‘‘torepeat the listened music piece,’’ to accomplish the timetask. Furthermore, some participants shyly attempted tohum or even sing aloud while reproducing the time.

In turn, it is also pertinent to consider the IWM pro-posed by Firmino and Bueno (2008) as possessinga long-term attribute similar to the LT-WM posited byEricsson and Kintsch (1995), allowing the EDF Modelto easily address situations in which there are long stim-uli, long response delays, and long time reproductions.Thus, the theoretical formulation of the role of IWMinto the EDF Model should be slightly modified as fol-lows: the key change pathway tracked by IWM frommodulating music affords the needed retrieval cues forthe evocation of the expected development projectionsin the EDF Model’s musical semantic memory (SM)device. When required by a time task, the temporalcomparative reproducer (TCR) device applies the dis-proportion between IWM’s and SM’s durations back tothe IWM duration, which leads to time shortening.Such estimated time is held available by IWM until theend of the experimental circumstance while it is useful;that is, until the end of reproduction. Because SM refersto high-order cognitive structures related to spatial andtemporal dimensions of tonality, SM surely stands asa memory type belonging to the long-term class. SMis also supposed to be a final state of an incidentallyimplicit learning process in Western adult people’sminds. (For details, see Firmino & Bueno, 2008; forstudies on musical semantic memory, see Groussard,Rauchs, et al., 2010; Groussard, Viader, et al., 2010; fora study on tonality learning, see Tillmann et al. 2000;and for a study on subjective time learning, see Boltz,Kupperman, & Dunne, 1998.) Clarke and Krumhansl


(1990) have also suggested similar large-scale memoryfor time perception of long music stimulus.

In this current state of elaboration, the EDF Modeldoes not capture the aspect of time reproductions thatwere greater than the stimulus’ physical duration (seeFigure 5). Thus, it makes sense to suppose that otherpsychological factors must be acting together. Indeed,this time over-reproduction effect may hypotheticallybe explained by continuing to consider the workingmemory processes discussed above. The post-experiment debriefs reveal that participants retrievednot only musical information (reporting they ‘‘tried tosing within their heads’’) but also nonmusical verbal-visual information (reporting they ‘‘tried to rememberwhat they thought or saw’’) to accomplish the time task.That is, although participants largely paid attention toand held the musical information, they also randomlypaid attention to and held other minor nonmusicalinformation surrounding the music stimulus. It is pos-sible to suppose then that there is a sum of the two typesof retrieved information coming from the nonverbaland nonvisual long-term IWM in the EDF Model andfrom the verbal and visual short-term multicomponentmodel for the working memory of Baddeley (2012) dis-cussed above. Such information fades (or is forgotten)slowly in the former and rapidly in the latter. Thus, ifparticipants are required to perform the time task earlier,they will be under a strong effect of the two memories,implying behavioral expression through time over-reproductions. Otherwise, if participants are required toperform the time task later, they will be under a continu-ing moderately strong effect of IWM, but a weak effect ofthe multicomponent model, implying behavioral expres-sion through time under-reproductions (that is, shorterthan the physical stimulus’ duration). The present experi-ment’s response delay, averaging 28.38 s, seems relativelyshort (or early to reproduce), implying time over-reproductions by the participants. Conversely, our previ-ous experiment’s response delay, averaging 71.93 s, seemsrelatively long (or late to reproduce), implying timeunder-reproductions by the participants (see Firmino &Bueno, 2008, p. 282). This hypothesis should be tested inanother study opportunely and more systematically.

Finally, although the expectancy/contrast model byJones and Boltz (1989) might present some resemblancesto some principles of the EDF Model, Jones, Boltz, andcollaborators have not theoretically or empiricallyexpanded their time model from the processing oftones-in-melodies (Boltz, 1989) or chords-in-chord-sequences (Schmuckler & Boltz, 1994) to the moreabstract cognitive level of tonal modulation. Indeed, theirmodel is heavily based on the multi-temporal structuresof the stimulus and not on its pitch structure. Hence,

such an overview suggests that it would be somewhatunlikely for this model also to fit the present data directly.

To summarize, by manipulating interkey distances inreal and long modulating tonal music pieces, we havefound again an inverse function between interkey dis-tance and subjective time. Our EDF Model – first elab-orated from data related to synthetic modulating chordsequences in Firmino and Bueno (2008) – presentsa suitable explanation by proposing cognitive processesbased on the disproportion between perceived andexpected musical time, which are held within a musicalimplicit working memory and a musical semantic mem-ory, respectively. By contrast, proposals from currenttime models such as contextual change and expec-tancy/contrast models reveal themselves as insufficientto account for such data. In addition, formatting theEDF Model’s key spatial-axis according to adapted keyprofile measures of Krumhansl (1990) allowed closecorrespondence between the present simulations anddata. Following this same line of research, future workshould investigate the effects of modulating unaccom-panied tonal melodies, which might reveal that evenmelodies may evoke tonality-timing properties, and thematter of previous stages of learning, which mightreveal how one would acquire such interwoven knowl-edge of tonality and timing.

Author Note

Erico Artioli Firmino received a scholarship from Fun-daçao de Amparo a Pesquisa do Estado de Sao Paulo(FAPESP) while a Postdoctoral Research Fellow at Uni-versity of Sao Paulo at Ribeirao Preto (USP/RP).

Jose Lino Oliveira Bueno received a Research Fellow-ship from Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico (CNPq) and Research Grantsfrom CNPq and FAPESP while a Full Professor of Psy-chobiology at USP/RP.

The authors thank the guitarist Gustavo SilveiraCosta, Professor of the Department of Music at USP/RP, for kindly playing and recording the music excerptsof the present work. The authors also thank Joao LuisSegala Borin for technical support and Roberta de Souzafor statistical support. The authors also thank DavidTemperley and the two reviewers whose critical com-ments brought a noteworthy improvement of the writ-ing of this study.

Correspondence concerning this article should beaddressed to Erico Artioli Firmino, Center for Experimen-tal Aesthetics, Department of Psychology, Faculty of Phi-losophy, Sciences, and Letters, University of Sao Paulo, Av.dos Bandeirantes, 3900 – Sala 2 Anexo Bl 3, 14040-901,Ribeirao Preto/SP, Brazil. E-mail: [email protected]


References

ACEVEDO, S., TEMPERLEY, D., & PFORDRESHER, P. Q. (2014).Effects of metrical encoding on melody recognition. MusicPerception, 31, 372-386.

ALLAN, L. G. (1979). The perception of time. Perception andPsychophysics, 26, 340-354.

BADDELEY, A. (2012). Working memory, theories models andcontroversy. Annual Review of Psychology, 63, 1-29.

BADDELEY, A., & HITCH, G. J. (1974). Working memory. In G. A.Bower (Ed.), Recent advances in learning and motivation(Vol. 8, pp. 47-90). New York: Academic Press.

BARROUILLET, P., & CAMOS, V. (2012). As time goes by: Temporalconstraints in working memory. Current Directions inPsychological Science, 21, 413-419.

BELLINATI, P. (1991). The great guitarists of Brazil: The work ofGaroto (Vol. 1). San Francisco, CA: Guitar Solo Publications.

BERZ, W. L. (1995). Working memory in music: A theoreticalmodel. Music Perception, 12, 353-364.

BHARUCHA, J. J. (1987). Music cognition and perceptual facili-tation: A connectionist framework. Music Perception, 5, 1-30.

BLOCK, R. A. (1990). Cognitive models of psychological time.Hillsdale, NJ: Erlbaum.

BLOCK, R. A., & REED, M. (1978). Remembered duration:Evidence for a contextual-change hypothesis. Journal ofExperimental Psychology: Human Learning and Memory, 4,656-665.

BLOCK, R. A., & ZAKAY, D. (2006). Prospective rememberinginvolves time estimation and memory processes. In J.Glicksohn & M. S. Myslobodsky (Eds.), Timing the future: Thecase for a time-based prospective memory (pp. 25-49). London:World Scientific.

BLOCK, R. A., & ZAKAY, D. (2008). Timing and remembering thepast, the present, and the future. In S. Grondin (Ed.),Psychology of time (pp. 367-394). Bingley, England: Emerald.

BOLTZ, M. (1989). Time judgments of musical endings: Effects ofexpectancies on the ‘‘filled interval effect.’’ Perception andPsychophysics, 46, 409-418.

BOLTZ, M., KUPPERMAN, C., & DUNNE, J. (1998). The role oflearning in remembered duration. Memory and Cognition, 26,903-921.

BUENO, J. L. O., FIRMINO, E. A., & ENGELMANN, A. (2002).Influence of generalized complexity of a musical event on sub-jective time estimation. Perceptual and Motor Skills, 94, 541-547.

CAMPBELL, M., & GREATED, C. (2001). The musicians guide toacoustics. Oxford, UK & New York: Oxford University Press.

CAO, E., LOTSTEIN, M., & JOHNSON-LAIRD, P. N. (2014).Similarity and families of musical rhythms. Music Perception,31, 444-469.

CAUCHARD, F., CANE, J. E., & WEGER, U. W. (2012). Influence ofbackground speech and music in interrupted reading: An eye-tracking study. Applied Cognitive Psychology, 26, 381-390.

CLARKE, E. F., & KRUMHANSL, C. L. (1990). Perceiving musicaltime. Music Perception, 7, 213-252.

COHEN, J., COHEN, P., WEST, S. G., & ALKEN, L. S. (2003).Applied multiple regression/correlation analysis for the behav-ioral sciences (3rd ed.). Mahwah, NJ: Lawrence EarlbaumAssociates.

COLLINS, T., TILLMANN, B., BARRETT, F. S., DELBE, C., & JANATA,P. (2014). A combined model of sensory and cognitive repre-sentations underlying tonal expectations in music: From audiosignals to behavior. Psychological Review, 121, 33-65.

DRAKE, C., JONES, M. R., BARUCH, C. (2000). The developmentof rhythmic attending in auditory sequences: Attunement,referent period, focal attending, Cognition, 77, 251-288.

DROIT-VOLET, S., RAMOS, D, BUENO, J. L. O., & BIGAND, E.(2013). Music, emotion, and time perception: The influence ofsubjective emotional valence and arousal? Frontiers inPsychology, 4, 1-12.

ERICSSON, K. A., & KINTSCH, W. (1995). Long-term workingmemory. Psychological Review, 102, 211-245.

ERICSSON, K. A., & LEHMANN, A. C. (1996). Expert and excep-tional performance: Evidence of maximal adaptation to taskconstraints. Annual Review of Psychology, 47, 273-305.

FIRMINO, E. A., & BUENO, J. L. O. (2008). Tonal modulation andsubjective time. Journal of New Music Research, 37, 275-297.

FIRMINO, E. A., BUENO, J. L. O., & BIGAND, E. (2009). Travellingthrough pitch space speeds up musical time. Music Perception,26, 205-209.

FRAISSE, P. (1982). Rhythm and tempo. In D. Deutsch (Ed.), Thepsychology of music (1st ed., pp. 149-180). New York: AcademicPress.

FRAISSE, P. (1984). Perception and estimation of time. AnnualReview of Psychology, 35, 1-36.

GIBBON, J., CHURCH, R. M., & MECK, W. H. (1984). Scalartiming in memory. Annals of the New York Academy ofSciences, 423, 52-77.

GRONDIN, S. (2008). Method for studying psychological time. InS. Grondin (Ed.), Psychology of time (pp. 51-74). Bingley, UK:Emerald Group.

GRONDIN, S. (2010). Timing and time perception: A review ofrecent behavioral and neuroscience findings and theoreticaldirections. Attention, Perception, and Psychophysics, 72,561-582.

GROUSSARD, M., RAUCHS, G., LANDEAU, F., VIADER, F.,DESGRANGES, B., EUSTACHE, F., & PLATEL, H. (2010). Theneural substrates of musical memory revealed by fMRI andtwo semantic tasks. NeuroImage, 53, 1301-1309.

GROUSSARD, M., VIADER, F., HUBERT, V., LANDEAU, B., ABBAS,A., DESGRANGES, B., EUSTACHE, F., & PLATEL, H. (2010).Musical and verbal semantic memory: Two distinct neuralnetworks? NeuroImage, 49, 2764-2773.


HONIN, H. (2013). Structure and interpretation of rhythm inmusic. In D. Deutsch (Ed.), The psychology of music (3rd ed.,pp. 369-404). New York: Academic Press.

HU, Y., & ERICSSON, K. A. (2012). Memorization and recall ofvery long lists accounted for within the long-term workingmemory framework. Cognitive Psychology, 64, 235-266.

JANATA, P., BIRK, J. L., HORN, J. D. V., LEMAN, M., TILLMANN,B., & BHARUCHA, J. J. (2002). The cortical topography oftonal structures underlying western music. Science, 298,2167-2170.

JONES, M. R. (1990). Musical events and models of musical time.In R. A. Block (Ed.), Cognitive models of psychological time(pp. 207-240). Hillsdale, NJ: Erlbaum.

JONES, M. R., & BOLTZ, M. (1989). Dynamic attending andresponses to time. Psychological Review, 96, 459-491.

KALAKOSKI, V. (2001). Musical imagery and working memory.In R. I. Godoy, & H. Jorgensen (Eds.), Musical imagery(pp. 43-55). Lisse: Swet and Zeitlinger.

KELLARIS, J. J., & KENT, R. J. (1992). The influence of music onconsumers’ temporal perceptions: Does time fly when you’rehaving fun? Journal of Consumer Psychology, 1, 365-376.

KOELSCH, S., GUNTER, T., SCHROGER, E., & FRIEDERICI, A. D.(2003). Processing tonal modulations: An ERP study. Journalof Cognitive Neuroscience, 15, 1149-1159.

KOHONEN, T. (1997). Self-organizing maps. Berlin: Springer.KRUMHANSL, C. L. (1990). Cognitive foundations of musical pitch.

New York: Oxford University Press.KRUMHANSL, C. L. (2000). Rhythm and pitch in music cognition.

Psychological Bulletin, 126, 159-179.KRUMHANSL, C. L. (2004). The cognition of tonality: As we know

it today. Journal of New Music Research, 33, 253-268.KRUMHANSL, C. L., & CUDDY, L. L. (2010). A theory of tonal

hierarchies in music. In M. R. Jones (Ed.), Springer handbookof auditory research (pp. 51-87). New York: Springer.

KRUMHANSL, C. L., BHARUCHA, J. J., & KESSLER, E. J. (1982).Perceived harmonic structure of chords in three relatedmusical keys. Journal of Experimental Psychology: HumanPerception and Performance, 8, 24-36.

KRUMHANSL, C. L., & KESSLER, E. J. (1982). Tracing the dynamicchanges in perceived tonal organization in a spatial map ofmusical keys. Psychological Review, 89, 334-368.

KRUMHANSL, C. L., & SHEPARD, R. (1979). Quantification of thehierarchy of tonal functions within a diatonic context. Journalof Experimental Psychology: Human Perception andPerformance, 5, 579-594.

KRUSKAL, J. B. (1964). Nonmetric multidimensional scaling:A numerical method. Psychometrika, 29, 115-129.

LARGE, E. W., JONES, M. R. (1999). The dynamic of attending:how people track time-varying events. Psychological Review,106, 119-159.

LARSON, S. (2012). Musical forces: Motion, metaphor, andmeaning in music. Bloomington, IN: Indiana University Press.

LARSON, S., & VANHANDEL, L. (2005). Measuring musical forces.Music Perception, 25, 119-136.

LERDAHL, F. (2001). Tonal pitch space. New York: OxfordUniversity Press.

LONDON, J. (2012). Hearing in time: Psychological aspects ofmusical meter. Oxford, UK & New York: Oxford UniversityPress.

MELLO, J. (2013). Gente Humilde: Vida e musica de Garoto[Humble people: Life and music of Garoto]. Sao Paulo, SP:Ediçoes SESC/SP.

MOLES, A. (2001). Theorie de l‘information et perception esthe-tique [Information theory and esthetic perception]. Paris:Flammarion. (Originally work published 1958)

NORTH, A. C., & HARGREAVES, D. J. (1999). Can music movepeople? The effects of music complexity and silence on waitingtime. Environment and Behavior, 31, 136-149.

ORNSTEIN, R. (1969). On the experience of time. Harmondsworth,UK: Penguin.

PHILLIPS, M., & CROSS, I. (2011). About musical time: Effect ofage, enjoyment, and practical musical experience on retro-spective estimate of elapsed duration during music listening.In A. Vatakis, A. Esposito, A., M. Giagkou, F. Cummins, &G. Papadelis (Eds.), Multidisciplinary aspects of time and timeperception (pp. 125-136). Berlin: Springer.

PISTON, W., & DEVOTO, M. (1987). Harmony (5th ed.). NewYork: Norton & Company, 1987. (Original work published1941)

PRINCE, J. B., & SCHMUCKLER, M. A. (2014). The tonal-metrichierarchy: A corpus analysis. Music Perception, 31, 254-270.

PRINCE, J. B., SCHMUCKLER, M. A., & THOMPSON, W. F. (2009).The effect of task and pitch structure on pitch-time interac-tions in music. Memory and Cognition, 37, 368-381.

SCHACTER, D., & TULVING, E. (1994). Memory systems.Cambridge, MA: MIT Press.

SCHMUCKLER, M., & BOLTZ, M. (1994). Harmonic and rhythmicinfluences on musical expectation. Perception andPsychophysics, 56, 313-325.

SCHOENBERG, A. (1978). Theory of harmony (R. E. Carter,Trans.). Berkeley, CA: University of California Press. (Originalwork published 1922)

SCHULZE, K., & TILLMANN, B. (2013). Working memory forpitch, timbre, and words. Memory, 21, 377-395.

SCHULZE, K., DOWLING, W. J., & TILLMANN, B. (2012).Working memory for tonal and atonal sequences duringa forward and a backward recognition task. MusicPerception, 29, 255-267.

SCHULZE, K., ZYSSET, S., MUELLER, K., FRIEDERICI, A. D., &KOELSCH, S. (2011). Neuroarchitecture of verbal and tonalworking memory in nonmusicians and musicians. HumanBrain Mapping, 32, 771-783.

SHEPARD, R. (1964). Circularity in judgements of relative pitch.Journal of the Acoustical Society of America, 36, 2346-2353.


TEMPELEY, D. (1999). What’s key for key? The Krumhansl-Schmuckler key-finding algorithm reconsidered. MusicPerception, 17, 65-100.

TEMPERLEY, D. (2013). Computational models of music cogni-tion. In D. Deutsch (Ed.), The psychology of music (3rd ed., pp.327-368). New York: Academic Press.

TILLMANN, B., BHARUCHA, J. J., & BIGAND, E. (2000). Implicitlearning of tonality: A self-organizing approach. PsychologicalReview, 107, 885-913.

TOIVIAINEN, P., & KRUMHANSL, C. L. (2003). Measuring andmodeling real-time responses to music: Tonality induction.Perception, 32, 741-766.

TULVING, E., & CRAIK, F. I. M. (2005). The Oxford handbook ofmemory. Oxford, UK: Oxford University Press.

VUVAN, D. T., & SCHMUCKER, M. A. (2011). Tonal hierarchyrepresentations in auditory imagery. Memory and Cognition,39, 477-490.

ZAKAY, D. (1990). The evasive art of subjective time measurement:Some methodological dilemmas. In R. A. Block (Ed.), Cognitivemodels of psychological time (pp. 59-84). Hillsdale, NJ: Erlbaum.

ZAKAY, D., & BLOCK, R. A. (1996). The role of attention in timeestimation processes. In M. A. Pastor (Ed.), Time, internalclocks, and movement (pp. 143-164). Amsterdam: Elsevier.

ZATORRE, R. J., & KRUMHANSL, C. L. (2002). Mental models andmusical minds. Science, 298, 2138-2139.

ZIV, N., & OMER, E. (2010). Music and time: The effect ofexperimental paradigm, musical structure and subjective eva-luations on time estimation. Psychology of Music, 39, 182-195.


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Key Distances Shorten Perceived Time of Real Music 613 · layer, Berio superposed his own ‘‘in-Contemporary-style’’ layer by inserting additional musical elements or structures