Tonal Consonance versus Tonal Fusion in Polyphonic Sonorities
Author(s): David Huron
Source: Music Perception: An Interdisciplinary Journal, Vol. 9, No. 2 (Winter, 1991), pp. 135154
Published by: University of California Press
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university of California
Music Perception
Winter 1991, Vol. 9, No. 2, 135-154
Tonal Consonanceversus Tonal Fusion
in PolyphonicSonorities
DAVID HURON
ConradGrebelCollege, Universityof Waterloo
An analysis of a sample of polyphonic keyboard works by J. S. Bach
shows that the prevalence of different vertical intervals is directly correlated with their degree of tonal consonance. A major exception to this
pattern arises with respect to those intervals that contribute to tonal
fusion. The prevalence of the latter intervals is negatively correlated with
the degree to which each interval promotes tonal fusion. Bach's avoidance of tonally fused intervals is consistent with the objective of maintaining the perceptual independence of the contrapuntal voices. In summary, two factors appear to account for much of Bach's choice of
vertical intervals: the pursuit of tonal consonance and the avoidance of
tonal fusion.
Introduction
Stumpf(1890) summarizedcenturiesof observationsregardingthe tendency for some sound combinationsto cohere into a single sound image
through a process of Tonverschmelzungor tonal fusion. Tonal fusion
ariseswhen the auditorysysteminterpretscertainfrequencycombinations
as comprisingpartials of a single complex tone (DeWitt &cCrowder,
1987). Normally,suchpartialsoccurin a harmonicrelationshipwherethe
frequenciesare relatedby simpleintegerratios. Tonal fusion occurs both
in the caseof puretones, and also whereconcurrentcomplextones contain
coincidentor complementarypartials- consistent with the possible existence of a single complex tone.
In the constructionof musicalworks, one could imaginetonal fusion
to be eithera welcomeor unwelcomeperceptualphenomenon- depending
on the musicalobjective.In some cases, the musicalgoal may be to create
an integratedor consolidatedsoundimage.In suchcasesa composermight
purposelyarrangevertical sonorities so as to enhance tonal fusion. In
particular,a successionof such tonally fused sonoritieswould virtually
ensure the perceptionof a single auditory stream (Bregman,1990). A
Requests for reprints may be sent to David Huron, Conrad Grebel College, University
of Waterloo, Waterloo, Ontario, Canada N2L 3G1.
135
136
David Huron
numberof examplesof (evidentlyintentional)tonal fusion can be found
in Westernorchestralliterature,includingthe beginningof Aaron Copland's Billy the Kid ballet suite and a well-known passage in Maurice
Ravel'sBolero.In othercases,tonal fusionmay be contraryto the musical
goal. Insofaras an objectiveof polyphonicmusic is to maintainthe perceptualindependenceof the contrapuntalvoices, one might expect polyphonic composers to avoid circumstancesthat could lead to the inadvertentperceptualintegrationof the parts.A varietyof both "horizontal"
and "vertical"factors have been identifiedas contributingto auditory
streaming(Bregman,1990). Horizontalfactors that enhancethe perception of independentstreamsincludethe maintenanceof close pitch proximitywithinthe voices (Dowling,1967) andthe avoidanceof part-crossing
(Huron, 1991). Verticalfactorthat encouragestreamsegregationinclude
asynchronoustone onsets (Bregman,1990) and the avoidanceof tonal
fusion (McAdams,1982, 1984). Thus in polyphonicmusic, theremay be
good cause to avoid sonorities that promote tonal fusion.
The pitch intervalthat most encouragestonal fusion is the aptlynamed
unison. The second most fused intervalis the octave (1:2), and the third
most fused intervalis the perfectfifth (2:3) (DeWitt &CCrowder,1987;
Stumpf,1890). Thereis no generalagreementin the literatureconcerning
the rank orderingof subsequentintervals.Some commentatorsconsider
the perfectfourth (3:4) to be the next most fused interval,whereasothers
have suggestedthe double octave (1:4). Experimentaldata collected by
DeWitt and Crowder(pp. 77, 78) paradoxicallysuggeststhat majorsevenths are more prone to tonal fusion than are perfect fourths.
Were it the case that fused intervalsare avoided in polyphonicmusic
in order to maintainvoice independence,one might expect the degreeof
avoidanceof a givenintervalto be proportionalto the strengthwith which
that intervalpromotestonal fusion. Eliminatingfrom considerationthose
intervalswhose rankingis contentious,one might predict that the least
common verticalintervalto be found in polyphonicmusic would be the
unison, followed by the octave, followed by the perfect fifth.
Juxtaposedagainstsuch a predictionis the recognitionthat the choice
of harmonic intervals1in a musical work may be motivated by other
factorsapartfrom the desireto maintainthe perceptualsegregationof the
voices. Specifically,one might supposethat a composeralso endeavorsto
choose intervalsthat conform to some harmonicplan and that preserve
a certaindegreeof tonal consonance.Like theoriesof tonal fusion, traditionaltheoriesof consonancehave also reliedon the conceptof simple
1. The term "harmonic interval" is used here in contrast to "melodic interval." By
harmonic interval, music theorists mean the pitch distance between two concurrent pitches.
The term is not intended to connote a simple frequency ratio such as might arise from
the harmonic series.
Tonal ConsonanceversusTonal Fusion
137
integerfrequencyratios- the simplerratios being the most euphonious.
If traditionaltheoriesof consonanceweretrue,and composersendeavored
to maximizethe degreeof consonancein their works, then such an objective would appear to be in direct contradictionto the objective of
reducingthe amountof tonal fusion. One would not know, for example,
whether a (hypothetical)absenceof perfect intervalswould indicatethe
composer'sdesire to avoid tonal fusion, or whether it might be symptomatic of the composer'sdesire to maintaina certain degree of dissonance.
In a well-knownpaper by Plomp and Levelt(1965), it was shown that
the conceptof simplefrequencyratiosalonedoes not providea satisfactory
account of perceivedconsonance. Plomp and Levelt recalled work by
Kaestner(1909), showing that the perceivedconsonanceof intervalsdependson the spectralcontentof the participatingtones and on theirpitch
distance.2The influenceof timbreon tonal consonancehad beenpredicted
as earlyas 1863 by von Helmholtz.Plompand Leveltinvokedthe concept
of the criticalbandto explainvariousexperimentalresultsthat contradict
the "simpleratio"theoryof consonance.They also showedthat the degree
of consonancefor variousintervalsdependson the absolutepitch register
of the interval.Forexample,the intervalof a majorthirdwill sound much
more dissonantin the bass registerthan it will in the trebleregister.This
phenomenon appears to arise from changes of bandwidth for critical
bands with respectto log-frequency.Hence transposinga musicalwork
to a differentregisterwill alter the perceivedconsonanceof its vertical
intervals.
Figure1 plots threesets of publisheddata showing the degreeof tonal
consonance(i.e., pleasingnessor euphoniousness)as a function of semitone distancebetweentwo tones. The dotted and dashedlines reproduce
data from Kaestner(1909) as given in Plomp and Levelt (1965). The
dotted line representsdata for intervalsconsistingof pure tones, whereas
the dashedline representsKaestner'sdata for intervalsconsistingof complex tones.
The solid line in Figure1 reproducesdata assembledby Kameokaand
Kuriyagawa(1969b) for complex tones consisting of eight spectrally
shaped harmonics- where the second and third harmonicsdisplay the
2. It is well known in studiesof perceivedconsonancethat trainedmusiciansrespond
differentlyfrom nonmusicians(see for example,Guernsey,1928). Specifically,musicians
on the one hand,and "conor "euphoniousness"
maydistinguishbetween"pleasantness"
sonance"on the other. Van de Geer, Levelt,and Plomp (1962) have shown that naive
subjectsmakeno suchdistinctionand havearguedthat musicians'responsesaremediated
by an a prioritheoreticalunderstanding.FollowingPlompand Levelt,in this paperwe
will avoid the ambiguousterm "consonance"and use the more circumspectterm "tonal
consonance"-which Plompand Leveltdefineas the degreeof perceivedpleasantnessor
euphoniousnessof a static pitch interval.
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David Huron
Fig. 1. Perceivedtonal consonancefor chromaticintervalsup to the octave. Dotted line:
pure tone data from Kaestner(1909); dashed line: complex tone data from Kaestner
(1909); solid line: complex tone data from Kameokaand Kuriyagawa(1969b), missing
data: M3 and m7.
greatestamplitude.The effectof spectralcontenton tonal consonancehas
been explored in some detail by Kameoka and Kuriyagawa(1969a,
1969b) and by Vos (1986). Kameokaand Kuriyagawaconfirmedthat
pitchregisteris importantin the perceptionof tonal consonance.Theyalso
proposedthat consonanceis influencedby sound level and spectralmasking.3 The differencebetween Kaestner'sdata for complex tones and the
Kameokaand Kuriyagawadatacan be attributedto differencesof spectral
content in the participatingtones and to the fact that the intervalslie in
differentpitch regions. In the case of the Kaestnerdata, the lower tone
for all intervalshas a frequencyof 320 Hz (E4),whereasthe lower tone
for the Kameokaand Kuriyagawadata has a frequencyof 440 Hz (A4).
In interpretingFigure1, note that few data have been assembledconcerningthe perceptionof tonal consonancefor intervalslargerthan the
octave. Some supraoctavedata assembledby Kameokaand Kuriyagawa
3. Vos (1986) has identifiedseveraldeficienciesin the theoryof consonanceproposed
by Plomp and Levelt (1965) and in the theory proposedby Kameokaand Kuriyagawa
(1969a).In particular,Vos has demonstratedthat, at leastin the caseof musiciansubjects,
perceivedconsonance("purity"in Vos's terminology)dependson intervalsize in addition
to beatsandroughness.In thispaper,the intentis to comparecomposers'practicesdirectly
to empiricalrecordsof listeners'judgments-and so sidestepthe theoreticalmodels and
the associatedcontroversy.Vos's own carefullycollecteddatapertainonly to the intervals
of the perfectfifth and majorthird, and so cannot be used in this study.
Tonal Consonance versus Tonal Fusion
139
(1969a) cast considerabledoubt on the traditionalmusical penchantto
equatecompoundintervalsthat are octave-equivalent(as, for example,in
the case of the major tenth and major third).
Hypotheses
Let us suppose that a polyphonic composer'schoice of harmonicinterval is motivatedprimarilyby two concerns:(1) the need to avoid inadvertenttonal fusion of the voices and (2) the need to maintaina certain
degreeof intervoicetonal consonance.Stumpftells us whichintervalsmust
be avoided in order to pursue the first goal (i.e., avoid perfect consonances),whereasPlompand Levelttell us which intervalsto avoid in order
to pursuethe second goal (i.e., avoid the dissonances:m2, M2, m7, M7,
etc.).
If the first goal is pursuedby composers,we ought to see evidenceof
attemptsto minimizeoccurrencesof unisons,octaves, fifths,and perhaps
fourths.Moreover,we ought to see evidenceof a rank orderingof these
intervalssuch that perfectunisons should be the most avoided, followed
by perfectoctaves,followed by perfectfifths.If the secondgoal is pursued
by composers,we ought to see evidenceof attemptsto minimizeoccurrencesof dissonantintervals.Moreover,we oughtto see evidenceof a rank
orderingof the various intervalsaccordingto their degreeof tonal consonance. Expressedmore formally, we might identify two specific hypotheses. Hypothesis 1: The frequencyof occurrenceof an interval is
positivelycorrelatedwith its degreeof tonal consonance.Hypothesis2:
In polyphonic music, the frequencyof occurrenceof an intervalis negatively correlatedwith the degree to which it promotes tonal fusion.
Method
In order to test these hypotheses, a study was initiated to compare actual polyphonic
practice in the spelling of harmonic intervals with extant data pertaining to tonal consonance and tonal fusion. Specifically, the experimental data collected by Kaestner (1909)
and by Kameoka and Kuriyagawa (1969b) can be used as independent templates for tonal
consonance- against which intervallic practice can be correlated. Similarly, intervallic
practice also can be correlated with published data pertaining to tonal fusion collected
by Stumpf (1890) and by DeWitt and Crowder (1987).
Sample
As noted above, the phenomenonof tonal fusion may be sought or
avoideddependingon the musicalgoal. The genreof music dubbed"polyphony"is appropriatefor our study becausepolyphoniccomposersex-
140
David Huron
plicitly construct multiple concurrentmusical lines/parts/voices/streams
whose perceptualindependenceis deemed important.Thus one might
assume that there exists in polyphonic music a compositionalintent to
preservestreamsegregationbetweenthe voices- an intentionthatmaynot
be presentin othertypesof music.In orderto test our hypotheses,an initial
sample of polyphonic keyboard works by Johann SebastianBach was
selected- specifically,Bach's 15 two-part keyboard Inventions. Subsequent experimentalcontrols warrantedswitchingthe analyticsampleto
Bach's 15 three-partSinfonias.
Measurement Issues
In examiningharmonicintervals,a numberof measurementissuesarise.
In brief, these issues include the mannerin which intervalinstancesare
determined,the questionof enharmonicequivalence,the influenceof interval context, the issue of intervaltuning, and the problem of timbre.
A distinctioncan be madebetweentwo approachesto the measurement
of pitch intervals:the figurai method and the time-base method. The
figurai method determinesthe type of interval for each novel vertical
sonority- that is, a new intervalis deemedto occur each time a new note
is articulatedin eitherone of the voices formingthe interval.The time-base
methoddeterminesthe pitch distancebetweentwo voices at regularmetric
divisions- such as everysixteenthduration.The metricdivisionused for
the time-basemethodcan be definedas equivalentto the shortestduration
foundin the work. Theprevalenceof a givenintervalcan thenbe expressed
as a percentageof the total numberof intervalsidentifiedusing the timebase method. To the extent that intervalswith long durationsare perceptually more salient than intervalswith short durations (i.e., agogic
accent),the time-baseapproachis arguablythe preferredmethodfor measuringintervals.The latter approachwill be used throughoutthis article.
A second measurementissue ariseswith respectto enharmonicinterval
spellings. Musical notation permits the same (equally tempered)pitch
distanceto be spelledusingseveralaliases- as in the caseof the diminished
fifth and augmentedfourth. It is likely that harmoniccontexts dispose
listenersto perceiveone enharmonic"meaning"in preferenceto anothereven though the pitch distancesmay be identical (Cazden,1980; Krumhansl, 1979). Suggestiveresultshave come from Shackford(1961, 1962a,
1962b), who measuredintervalsizes as performedby three professional
string quartets.Shackford(1961, p. 201) found, for example, that augmented fourths are typicallyperformed18 cents wider than diminished
fifths. To the extent that performancepractice is shaped by perceptual
goals, this differencebetweentwo types of tritonesuggeststhat harmonic
Tonal Consonance versus Tonal Fusion
141
context may be perceptuallyinfluential.It is possible that the tonal consonance for a given fixed pitch distance might be rated differentlyin
differentharmonicsettings.
Unfortunately,systematicpsychoacousticdata concerningenharmonic
context have not been collected. Publishedperceptualstudies of tonal
consonance- suchas Kaestner(1909), Guthrieand Morrill(1928), Plomp
and Levelt (1965), and Vos (1986)- have disregardedthe possible effect
of harmonic context on perceivedeuphoniousness.Without such data
againstwhich to comparemusicalpractice,thereis little reasonto collect
enharmonicallydifferentiateddata. Hence, in this study, pitch distances
are measuredin semitoneswithout regardto intervalspelling. For convenience, interval sizes will be referredto by standarddiatonic terms;
thus, we will use the label "minorseventh"in preferenceto an interval
of 10 semitones.Nevertheless,it should be rememberedthat the actual
notational spellingsmay differ:what we will call a minor seventh may
be renderedas an augmentedsixth or a doubly diminishedoctave, and
so on.
A thirdmeasurementissue ariseswith respectto tuning:is it important
to establishthe precisetuningsystemunderwhichthe sampledworkswere
composed?The debate here focuses predominantlyon the contrast between just intonation and equal temperament.For perfect unisons and
octaves, equally temperedintervalsare identical to those for just intonation. In the case of successivelymore dissonant intervals,equal temperamenttuningdivergesmore and more from just intonationso that for
a minor second the differenceis about 12 cents. Stumpf claimed that
tuning differencesdo not especially affect judgments of consonance,
whereas more recent data suggest that tuning differencesare not insignificant.
With regardto tonal fusion, tuning differencesappearto be more directly influential.DeWitt and Crowder(1987) showed that tonal fusion
is slightlymorepronouncedin just intonationthan in equal temperament
tuning.However,the degreeof tonal fusionfor differentintervaltypeswas
found to correlateclosely acrossthe two tuningsystems(p. 77). The rank
orderingof intervalsin promotingtonal fusion remainsthe same in both
systems.The differencesbetweenequal temperamentand just intonation
notwithstanding,given the small size of the effect of tuning on the rank
orderingof intervalsaccordingto tonal consonanceand tonal fusion, it
is reasonableto proceedwithout attemptingto controlfor tuning- either
in the sampledworks or in the tonal fusion and tonal consonancedata.
A fourthmeasurementissueconcernsthe timbreof the originalsampled
works. Becausespectralcontentinfluencestonal consonance,it is difficult
to relate the sampledworks to any of the publisheddata pertainingto
tonal consonance. Ideally, we would like to be able to sample the in-
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David Huron
strumentaitimbresused (or imagined?)by Bachin the compositionof the
selected works. Using these timbres,we could then collect independent
datameasuringthe perceivedconsonancefor intervalsof varioussizes and
in differentpitch rangesor tessituras.It would then be possible to relate
the intervallicpracticesin the sampledworks to the instrumentaltimbres
used. Althoughthis approachhas considerablemerit, it was discounted
as impractical.In pursuingthis study, we will accept the confounding
effect of timbre, and presume that either one or both of the Kaestner
(1909) and Kameokaand Kuriyagawa(1969b) data will provide an adequatetemplateagainstwhich Bach'sintervalpracticesmay be correlated.
Becausemusic normallyconsistsof complex tones, the tonal consonance
data for pure tones will not be used in the analysis.
Preliminary Analysis
Using the time-basemethod of measurement,harmonicintervaldata
were collected for the 15 two-part Inventions.Figure2 shows the prevalenceof variousverticalintervalsin the Inventions.The dataaredisplayed
accordingto threecategoriesof intervalsdistinguishedin traditionalmusic
theory:perfectconsonances(PI, P4, P5, P8, etc.), imperfectconsonances
(m3, M3, m6, M6, etc), and dissonances(m2, M2, TT, m7, M7, etc.).
The bell-shapedcontourin these data at once revealsthat the frequencies
of occurrencefor variousintervalsare confoundedby between-voicepitch
proximity and/or pitch range (tessitura)of the individualvoices. Bach
seems disposedto keep the two voices separatedby an intervalof about
a tenth.
Contraryto our two hypotheses,we must entertainthe possibilitythat
the predominanceof various intervalsmight arise from the composer's
preferencefor maintainingan optimumbetween-voicepitch distance- or
by a preferencefor certaintessiturasfor the two voices. In the case of the
two-part Inventions,about three-quartersof the data lie beyond the interval of an octave. As noted earlier,detailedpublisheddata concerning
both tonal consonanceand tonal fusion pertainonly to intervalsup to an
octave; hence only the within-octaveintervaldata given in Figure2 can
be used in our analysis.FromFigure2, it is clear that the dozen smallest
intervalsare locatedon the risingslope of the distribution.This skews the
data toward the largerintervals,and so biases the sample.It is tempting
to amalgamateoctave-equivalentdata- however, as we have noted, octave equivalenceof compound intervalsis not a valid assumption.
Anotherproblemwith the data fromthe two-partInventionsis that the
averagelower pitch in each intervalis somewhatlow comparedwith the
Tonal ConsonanceversusTonal Fusion
143
Fig. 2. Harmonicintervalprevalence:two-partInventions.
lower pitch in Kaestner's interval study and is quite low compared with
the lower pitch in Kameoka and Kuriyagawa. The average pitch for the
bass voice of the two-part Inventions lies a little more than 9 semitones
below Kaestner's lower pitch (E4) and more than 14 semitones below the
lower pitch of Kameoka and Kuriyagawa (A4). It is difficult to estimate
the magnitude of the confounding effect of interval tessitura. Certainly,
it would be reassuring to find interval data that more closely match the
pitch region of at least one of our two sets of tonal consonance data.
In light of these problems, a more favorable sample of musical works
was sought. A better sample would exhibit a higher average pitch in the
lower voice and a closer proximity between the two voices- such that the
majority of harmonic intervals would lie within the interval of an octave.
A somewhat improved sample is provided by the upper two voices of
Bach's three-part Sinfonias - an interval distribution for which is given in
Figure 3. Figure 3 reveals that more than 85% of the intervals formed by
the middle and upper voices are an octave or smaller in size. In addition,
the mean pitch of the middle voice of the three-part Sinfonias is slightly
higher. The average pitch of the middle voice lies less than 5 semitones
away from Kaestner's lower pitch - although it remains almost 10 semi-
144
David Huron
Fig. 3. Harmonieintervalprevalence:three-partSinfonias(soprano-midvoices).
tones away from the lower pitch in Kameokaand Kuriyagawa.There
remainsome difficultieswith thesedata, but let'sproceedin any eventwith
a preliminarycomparisonof Bach'sintervalpracticeand researchon tonal
fusion and tonal consonance.
Interval Prevalence in the Upper Voices of Bach's Sinfonias
Figure4 overlaysthe firstoctave intervaldistributionfor the uppertwo
voices of the three-partSinfoniasalong with Kaestner'smeasurementsof
consonancefor complex tones. Broadlyspeaking,there is a good correlation betweenthe Bachdata and Kaestner'sdata (r = .76); howeverthere
is no correlationto the data of Kameokaand Kuriyagawa(r = -.04). The
discrepanciesbetween the Bach data and Kaestnerdata are evident in
Figure4. Specifically,the unisonand octave intervalsare substantiallyless
prevalentin the Bachdata than would be suggestedby theirrelativetonal
consonance.In addition,the majorthirdoccursless than expected,while
occurrencesof the minor third appear somewhat raised.
The suppressionof unisons and octaves is predicted by Hypothesis
2- namely, avoiding intervalsthat promote tonal fusion. If we compare
the prevalenceof the perfect intervals(PI, P4, P5, &cP8) with the data
Tonal Consonance versus Tonal Fusion
145
Fig. 4. Comparison of tonal consonance for complex tones (line) from Kaestner (1909)
with interval prevalence (bars) in the upper two voices of Bach's three-part Sinfonias.
concerningthe dispositionof these intervalsto promotetonal fusion, the
correlationsare -0.75 for the Stumpfdata and -0.80 for the DeWittand
Crowderdata. These resultssupportHypothesis2- that is, that Bach is
endeavoringto avoidtonal fusion.If we now eliminatethe perfectintervals
from consideration,we might recalculatethe correlationsfor tonal consonance.Excludingthe perfectintervals,the correlationsbetweeninterval
prevalenceand the data on tonal consonancerise considerably:r = .92
in the case of Kaestnerand r = .64 in the case of Kameokaand Kuriyagawa. The latter resultsare consistentwith both Hypotheses1 and 2. In
short, it appearsthat Bachendeavorsto promotetonal consonancewhile
concurrentlyavoiding tonal fusion.
These resultsare highly suggestive.Nevertheless,it can be arguedthat
the relativepaucity of octaves and unisons can be attributedto the fact
that these intervals are located at the lower and upper regions of the
intervaldistribution- where frequenciesnaturallydecline.A more robust
demonstrationof our hypotheseswould removethe confoundingartifacts
of the intervaldistribution.Clearly,we need to addressmore directlythe
problem of how to eliminatethe effect of intervoicepitch proximity.
Controlling for Voice Proximity: Prevalence versus Preference
In orderto addressthis problem,Bach'sintervaldata ought to be recast
such that the effects of intervoicepitch proximityare strictlycontrolled.
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David Huron
Specifically,a method is needed whereby the actual distributionof intervalscan be contrastedwith a distributionthatmightbe expectedto arise
by chance.
In Huron(1991) an auto-phasemethodwas describedthat is well suited
to the tasks of controllingthe effects of pitch proximity.An auto-phase
can be likenedto an autocorrelation.The methodcan be conceivedof by
using the following metaphor.A two-partwork may be imaginedto be
notatedon a singlelong stripof paper.The beginningand end of this strip
areconnectedtogetherto forma loop. The two partsare cut apartto form
two independentbut parallelloops in the mannerof a circularslide rule.
One of the voices can be shifted with respect to the other through a
completecircleof 360 degrees,but only when the partsare alignedat zero
degreesdoes their relationshipcorrespondto the originalmusicalscore.
The proportionsof variousharmonicintervalscan be measuredfor each
novel configurationas the parts are shiftedwith respectto each other by
a fixed metric division (such as a sixteenth duration).The intervaldistributionsfor all of the non-zero-degreearrangementsof a work can be
amalgamatedinto a singlecontrolleddistributionagainstwhich the actual
distributionintervalsin the work can be compared.This may be referred
to as a pitch-proximity-controlled
distribution.
The advantageof this methodis that each rearrangement
preservesthe
identical pitch distributionsfor the two voices, the same within-voice
melodiccontouring,durations,and within-voicerhythmicstructure.Thus
this method allows us to factor out the mean pitch proximity in the
distributionof various harmonicintervals.4
In light of this new analysismethod,our two initialhypothesescan be
reformulatedas follows. Hypothesis3: Comparedwith a pitch-proximitycontrolled distributionof intervals, the frequencyof occurrenceof an
intervalis positivelycorrelatedwith its degreeof tonal consonance.Hy4. A slight complication arises from the fact that, in imitative polyphony, entries tend
to occur at the intervals of the fourth, fifth, or octave. This means that at certain angles
in the autophase, the parts will be shifted so that two previously asynchronous entries are
now aligned. At this angle, the tally of various harmonic intervals will include a plethora
of fourths (fifths, or octaves) due to the parallel melodic contour shared by the parts. In
contrasting these ostensibly controlled results with the original interval distribution, the
original score will appear to be comparatively devoid of concurrent fourths (fifths, or
octaves) thus confounding our results. In order to determine the magnitude of this confound, an analysis of three works was done in which the autophase values for angles
displaying synchronized entries were excluded. The corresponding Z-scores for the perfect
consonants did rise slightly (by about 3%) when the synchronized entries were omitted.
The reason that the change is so small is that, in the autophase method, the original
distribution is compared with several hundred controlled distributions. Each Z-score is
calculated with respect to a distribution containing several hundred phase-shifted values.
In short, values arising when voice-entries are in-phase tend to be swamped by data for
all other phase values. Thus, the confounding effect of synchronized entries was deemed
to be insignificant.
Tonal Consonance versus Tonal Fusion
147
distributionof
pothesis 4: Comparedwith a pitch-proximity-controlled
of
of
an
interval
occurrence
the
(in
intervals, frequency
polyphonicworks)
is negativelycorrelatedwith the degreeto which that intervalpromotes
tonal fusion.
The autophasemethod does not eliminatethe effect of the tessituraof
the intervals:the data remainconfoundedby the absolute-pitchplacement
of the intervals.Nevertheless,in eliminatingthe effect of intervoicepitch
proximity,one of the major confounds (which earlierled us to dismiss
our initialrepertoiresample)has been removed.For this reason,we need
no longer restrictthe analysisto the upper two voices of the three-part
Sinfonias.As long as we bear in mind the continuedconfoundingeffect
of intervaltessitura,we can profitablyexpandour analysisto includethe
entire analytic sample of 30 polyphonicworks.
Using this approach, we calculated auto-phase functions for the 15
two-part Inventions and for all three voice-pairingsin the Sinfonias:
soprano-mid,bass-mid, and soprano-bass.An aggregate(controlled)distributionwas generatedfor each intervalin each voice-pairingfor each
work. These distributions were subsequently standardized, and the
Z-score was determinedfor the actual occurrenceof various intervalsin
the given voice-pairingfor individualworks. AverageZ-scoreswere then
calculatedfor each intervalfor all voice-pairsin the sampledrepertoire.
A positive Z-score indicatesthat the intervalis promotedor encouraged
by the composer,whereas a negativeZ-score indicatesthat the interval
is avoidedor suppressed.Z-scorevaluesnearzeroindicatethatthe interval
is neithermore common nor less common than would be expected in a
chance juxtapositionof voices.
Plottingthe mean Z-scores for the differentintervalsprovidesa good
contrastto the intervalhistogramsof Figures2 and 3. It is temptingto
assume that a frequencydistributionof intervals (e.g., Figures2 & 3)
representsBach'sintervallic"preferences."However,it is a versionof the
NaturalistFallacyto assumethat the most commonis the most preferred.
(One'sfavoritefood is not necessarilythe food that one eats most often.)
By contrast,Z-scores indicatehow a given intervalfares with respectto
a chancedistribution.Intervalsexhibitinghigh positive Z-scores may be
deemed "preferred"in the sense that they are "sought-after"-and this
fact is independentof the actual prevalenceof the interval.
Figure5 provides four Z-score plots for all of the voice pairingsexaminedin this study. Figure5a shows the mean harmonicinterval"preferences"in Bach'stwo-partInventions,whereas Figures5b-d show the
mean soprano-mid, bass-mid, and soprano-bass interval"preferences"
for the three-partSinfonias.Having determinedthe intervalZ-scoresfor
all of the sampledmusic, we can now test our hypothesesby correlating
the first octave of these scores with the comparisondata for tonal consonance and tonal fusion.
148
David Huron
Fig. 5. Harmonieintervalpreference,(a) Two-partInventions,(b) three-partSinfonias
(soprano-midvoices), (c) three-partSinfonias(bass-midvoices), (d) three-partSinfonias
(bass-sopranovoices).
Tonal Consonance versus Tonal Fusion
Fig. 5. continued
149
150
David Huron
Interval Prevalence and Preference in Bach's
Inventions and Sinfonias
A synopsisof the analyticresultsis presentedin Tables 1 and 2. Table
1 tabulatesthe correlationcoefficientsrelatingBach'sintervallicpractice
to publisheddata for tonal consonance;Table2 tabulatesthe correlations
pertainingto tonal fusion research.Insofaras Bach endeavorsto pursue
consonant intervals,the coefficientvalues in Table 1 ought to be predominantlypositive. Conversely,to the extent that Bach avoids intervals
thatpromotetonal fusion,the valuesin Table2 oughtto be predominantly
negative.Both tablesprovideseparatecorrelationsfor each voice-pairing
in each of the repertoiresstudied.In the case of Table 1, upper-rowvalues
indicatethe correlationswith Kaestner'sdata (1909), whereaslower-row
valuesindicatethe correlationswith data from Kameokaand Kuriyagawa
(1969b). Values in parenthesesgive correlationcoefficientscalculatedby
using all of the interval data. Table 1 values not in parenthesesgive
correlationcoefficientsfor all intervalsexceptthe perfectintervals:PI, P4,
P5, and P8.
In Table2, the upper-rowvaluesin eachgroupindicatethe correlations
with data from Stumpf(1890), whereasthe lower-rowvaluesindicatethe
correlationswith data from DeWitt and Crowder(1987). Both Tables 1
and 2 provide separatecolumns for analysesof intervalprevalence(i.e.,
interval frequency)and pitch proximity-controlled interval preference
(i.e., intervalZ-score).
In interpretingthe results of these two tables, two points are appropriate. First, we should give greatest credenceto the pitch proximitycontrolleddata (i.e., interval"preference").
Second,in the caseof the tonal
consonancecorrelations(Table 1), we ought to give greatercredenceto
the data for the soprano/mid-voicepair in the three-partSinfoniasbecausethe pitch region of the intervalsmore nearly approximatesthat
of the independentdata- especiallythe data from Kaestner.The mean
lower pitchesin each of the othervoice-pairingsin the musicalsampleare
significantlylower than the comparisondata, and so other voice-pair
comparisonsare likely to be less reliable.
Hypothesis1 would predictpredominantlypositive values for r in column 1 of Table 1, consistent with the pursuit of consonant intervals.
Hypothesis2 would predictpredominantlynegativevaluesfor r in column
1 of Table 2, consistentwith the avoidanceof tonal fusion. Hypothesis
3 would predictpredominantlypositivevaluesfor r in column3 of Table
1, consistentwith the pitch proximity-controlledpursuitof consonance.
Hypothesis4 would predictpredominantlynegativevaluesfor r in column
2 of Table2, consistentwith the pitch proximity-controlledavoidanceof
tonal fusion. Hypotheses1 and 2 would concurrentlypredictlargevalues
Tonal Consonance versus Tonal Fusion
151
TABLE 1
Tonal Consonance/Harmonic Interval Correlation Results
IntervalPrevalence
IntervalPreference
PerfectIntervals
PerfectIntervals
(Included) Excluded (Included) Excluded
Inventions
Bass-soprano
( + 0.49)
( + 0.23)
+0.57
+0.78
( + 0.63)
(-0.09)
+0.81
+0.67
Bass-mid
( + 0.73)
( + 0.34)
( + 0.40)
( + 0.41)
( + 0.76)
(-0.04)
+0.87
+0.78
+0.45
+0.75
+0.92
+0.64
( + 0.82)
( + 0.09)
( + 0.56)
( + 0.11)
( + 0.71)
(-0.11)
+0.97
+0.75
+0.58
+0.41
+0.95
+0.76
Sinfonias
Bass-soprano
Mid-soprano
+0.72
Correlation means
(HI)
Hypothesis
(H1+H2)
+0.74
(H3 + H4)
(H3)
note. Upper values: correlations with Kaestner (1909); lower values: correlations with
Kameoka and Kuriyagawa (1969b).
TABLE 2
Tonal Fusion/Harmonic Interval Correlation Results
IntervalPrevalence
IntervalPreference
Inventions
Bass-soprano
+0.05
-0.33
-0.72
-0.52
Bass-mid
-0.11
-0.48
+0.21
-0.19
-0.75
-0.80
-0.50
-0.63
+0.11
+0.10
-0.90
-0.73
-0.30
-0.47
Sinfonias
Bass-soprano
Mid-soprano
Correlation means
note. Upper values: correlations with Stumpf (1890); lower values: correlations with
DeWitt and Crowder (1987).
for r in column2 of Table 1, and that these valuesshould be greaterthan
the correspondingvalues for column 1; the data are consistentwith the
mutualpursuitof tonal consonancewhile avoidingtonal fusion. Hypotheses 3 and 4 would concurrentlypredict large values for r in column 4
152
David Huron
of Table 1, and that thesevaluesshouldbe greaterthan the corresponding
values for column 3. Finally,we would expect that the tonal consonance
correlations(Table1) for the mid-sopranovoice-pairwould be the highest
of all the table values, because this voice-pair correspondsbest to the
intervaltessituraof the independentdata.
As can be seen, the analysisresultsare consistentwith all of the predictions arising from the hypotheses. Comparedwith a random distribution, the most avoided intervalis the unison, followed by the octave,
followed by the perfectfifth. Bach'sevidentsuppressionof these intervals
suggeststhat he does indeedchoose intervalsin inverseproportionto the
degreeto whichtheypromotetonal fusion.At the sametime,Bachchooses
intervalsin a manner consistent with the pursuit of tonal consonance.
Moreover,the data are most consistent,not with one or anotherof the
hypotheses, but with the concurrentpursuit of tonal consonance and
avoidanceof tonal fusion.
Multiple Regression Analysis
A multiple regressionanalysis would seem to provide the most appropriatetype of statisticaltest, given the natureof this study. Multiple
regressionenables us to identifythe degreeto which two or more independentmeasuresareableto predictthe behaviorof a dependentmeasure.
In orderto pursuesuch an analysis,data must be availablefor all of the
independentmeasures,plus the dependentmeasure.Unfortunately,between the tonal consonanceand tonal fusion measuresthere are a considerablenumberof missingobservations.This meansthat morethan half
of the intervalclassesstudiedmust be omitted from a multipleregression
analysis- castingsome doubt about the meritof such an approach.Nevertheless,there is some utility in such an analysis because it can help
establishthe relativeimportanceof tonal fusion and tonal consonanceas
compositionalgoals.
Four sets of analyseswere carriedout using each of the four combinations of tonal fusion and tonal consonance data (Kaestner/Stumpf,
Kaestner/DeWitt
&cCrowder,Kameoka&cKuriyagawa/Stumpf,
Kameoka
&CKuriyagawa/DeWitt&cCrowder).The intervalprevalenceand preference data for all voice pairs in the Inventionsand Sinfoniasprovided
the dependentvariables,resultingin 32 individualanalyses.For most of
these analyses,data pertainingto only six intervalscould be used; hence
the probabilityof achievingstatisticallysignificantresults are reduced.
Althoughmany analyseshoverednear the 95% confidencelevel, only
two analyseswere found to give statisticallysignificantresults.Both analyses pertainedto the soprano-mid-voicecombinationin the three-part
Tonal Consonance versus Tonal Fusion
153
Sinfonias.In using the Kaestnerand Stumpfdata to predictintervalprevalence,R2was determinedto be .8933 (F = 16.74; p = .0114). In this case
the Stumpftonal fusion data were found to account for 56.4% of the
observedvariance,whereasthe Kaestnerconsonancedata were found to
account for a further33.0% of the variance.In using the Kaestnerand
DeWitt and Crowderdata to predict intervalprevalence,R2 was determined to be .8720 (F = 10.22; p = .0458). In this latter analysis, the
DeWittand Crowdertonal fusion data were found to accountfor 41.3%
of the observed variance,whereas the Kaestnerconsonance data were
found to account for a further45.9% of the variance.In general,these
resultsimply that in Bach's choice of harmonicintervals,the avoidance
of tonal fusion is about equallyimportantto pursuingtonal consonance.
Conclusion
An analysisof 30 polyphonickeyboardworksbyJ. S. Bachsuggeststhat
the choice of harmonicintervalsis governedby two predominantgoals:
(1) the pursuitof tonal consonanceand (2) the avoidanceof tonal fusion.
Specifically,the prevalenceof intervals(other than perfectconsonances)
is correlatedwith their degree of tonal consonance. In the case of the
perfectconsonances,the prevalenceof an intervalis inverselycorrelated
with the interval'spropensityto promotetonal fusion. Bachendeavorsto
minimizethe occurrenceof those intervalsthat most promotetonal fusion
while concurrentlypursuingtonal consonance.
Theseresultsreinforcethe view that Stumpfwas mistakenin regarding
the phenomenonof tonal fusion as the source or cause of tonal consonance.As Bregman(1990, p. 508) has noted, Stumpffailedto distinguish
properlybetween"heardas one" and "heardas smooth."It would appear
that in his polyphonic compositions,Bach attemptedto produce music
that is "heardas smooth" without being "heardas one."
The avoidanceof tonal fusion is in accordwith other polyphonicpractices used by Bach: Bach avoids inner voice entries (Huron &cFantini,
1989) andalso avoidspart-crossing(Huron,1991). Together,theseresults
suggesta significantagreementbetween compositionalpracticesin polyphonic music and empiricalresearchconcerningthe segregationof auditory streams.5
5. This research was supported in part through funds provided by the Social Sciences
and Humanities Research Council of Canada.
154
David Huron
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