Redefining Pitch Proximity: Tessitura and Mobility as Constraints on Melodic Intervals
Author(s): Paul Von Hippel
Source: Music Perception: An Interdisciplinary Journal, Vol. 17, No. 3 (Spring, 2000), pp. 315327
Published by: University of California Press
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university of California
Music Perception
Spring 2000, Vol. 17, No. 3, 315-327
RedefiningPitch Proximity:Tessituraand Mobility as
Constraintson Melodic Intervals
PAUL VON HIPPEL
Ohio State University
In descriptionsof melodicstructure,pitchproximityis usuallydefinedas
the tendencyfor smallpitchintervalsto outnumberlargeones. Thisdefinitionis validas far as it goes;however,an alternativedefinitionis preferable.The alternativedefinespitchproximityin termsof two moreba- a constrainton tessitura(or pitch distribution)and a
sic constraints
constrainton mobility(or freedomof motion). This new definitionoffersseveraladvantages.Whereasthe usualdefinitionpredictsonly interval size, the new definitionpredictsintervaldirectionas well. The usual
definitionpredictssmallintervalsgenerally,whereasthe new definition
predictscontext-sensitivevariationsin intervalsize. Finally,if the new
definitionis given the first few notes in a melody,it can assigna probability to each of the pitches that could occur next. In sum, the new
definitionoffersa morepreciseanddetaileddescriptionof melodicstructure.
ReceivedFebruary
29, 1999.
22, 1999;acceptedforpublicationSeptember
the simplestpropertyof melodic structureis the rule known as
Perhaps
pitch proximity.This rule has traditionallybeen definedas a tendency
for smallpitchintervalsto outnumberlargeones. A preponderanceof small
intervalshas beenmeasuredin a wide varietyof musicalcultures,including
indigenousmelodiesfrom Europe,Africa,North America,and the Caribbean (Dowling, 1967; Ortmann,1926; Watt, 1924; Zipf, 1949).1A classic
demonstrationby Watt (1924) is displayedin Figure1.
Althoughthe traditionaldefinitionis valid as far as it goes, I will argue
that an alternativedefinitionis preferable.This alternativedefines pitch
proximityas the resultof two more basic constraints.The first constraint
affects a melody'sdistributionof pitch heights- in a word, its tessitura.
1.
Dowling'sdemonstrationreliedon data reportedby Fucks(1962) and Merriam
(1964).Thedatareportedby Merriam,in turn,was originallycollectedbyMerriam,Whinery
and Fred(1956).
Addresscorrespondenceto Paul von Hippel, School of Music, Ohio State University,
1866 CollegeRoad, Columbus,OH 43210. (e-mail:von-hippel.l@ohio-state.edu)
315
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316
Paulvon Hippel
Fig. 1. In an analysisof 56 Liederby Schubert,Watt(1924) foundthat smallmelodicintervalsgreatlyoutnumberlargeones. Similarresults,less fullytabulated,wereobtainedfor the
songs of two Native Americantribes- the Ojibway and the Lakota.
The second constraintaffects a melody'sfreedomof movementwithin its
tessitura- in a word, its mobility.These two constraints,on tessituraand
mobility,limit a melody'schoice of intervals.
At first glance, the proposedchange of definitionmay appeartrivial.I
will demonstrate,however,that the new definitionoffersimportantadvantages overthe old one. Whereasthe old definitioncan predictonly the sizes
of melodic intervals,the new definitionpredictstheir directionsas well.
The old definitionpredictssmall intervalsregardlessof context, whereas
the new definitionpredictssystematicvariationsin intervalsize. Finally,
the new definitionfits a statisticalformalismthat, given the first several
notes in a melody,assigns a probabilityto each of the pitches that could
occurnext. This formalismandits corollarieswill be developedin the pages
that follow.
Constraint 1: Tessitura
To begin, we can observeinformallythat everymelody is limitedin its
range of pitches, and that most melodiesseem to favor the centerof their
range.This preponderanceof moderatepitch heightscould have a variety
of causes. On many instruments,pitches of moderateheight are easierto
play.Evenwhen extremepitchescan be playedeasily,they tend to produce
less definitepitch perceptions(Terhardt,Stoll, & Seewann,1982). Moreover, because melodies move primarilyby small interval,melodies must
generallypass throughthe center of their range in order to reachthe ex-
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Redefining Pitch Proximity
317
tremes.Whateverthe cause,theredoes seem to be a tendencyfor melodies
to use the fringesof their range sparingly.It is for this reason that musicians use the word "tessitura,"which "differsfrom range in that it does
not take into accounta few isolatedpitchesof extraordinarilyhigh or low
pitch" (Apel, 1969, p. 839).
To make these observationsmore concrete,tests were conductedon a
diversesamplecomprising176 indigenousfolk songs from four different
continents:30 songs from a Chinesecollection (Chung-kuoyin yiieh yen
chiu so [ChineseMusic ResearchInstitute],1959), 24 songs from a South
Africancollection (Makeba, 1971), 42 songs chosen at random from a
Native AmericanOjibway collection (Densmore,1910, 1913; computer
data files availablein von Hippel, 1998), and 80 from a Europeancollection (Schaffrath,1995)- 4 songs chosen at random from each of 20 regions of Europe.One of the Europeanfolk songs is notated in Figure2,
and a tally of the song'spitch heightsis displayedas a bar graphin Figure
3. The bargraph'smoundedshapeshows clearlythat most of this melody's
pitchesoccurtowardthe middleof its range.Moreprecisely,the pitchheights
in this melody- like those in the other sampledmelodies- approximatea
normalor Gaussiandistribution.2
It is evidentfromFigure3 that the size of melodicintervalsis constrained
by the pitch distribution.Giventhe pitchesused in this melody,the largest
possibleintervalis 14 semitones- and that intervalcan be formedonly by
straddlingtwo rarepitchesat opposite extremesof the range.In contrast,
therearemanyways to formsmallerintervalsusingmoreabundantpitches
nearthe middleof the range.To be more precise,we mightpoint out that,
givena normaldistributionof pitchheights,abouttwo thirdsof a melody's
Fig. 2. A German folk song ("Lasset uns schlingen dem Friihling Blumlein") from the Essen
Folksong Collection (Schaffrath, 1995).
As a simple test of normality, we can calculate skew and kurtosis values for each
2.
of the 176 sampled melodies. Values of 0 would be consistent with a normal distribution,
whereas a consistent tendency toward positive or negative values would indicate a departure from normality. According to a sign test, the skews of these melodies did not tend
significantly toward positive or negative values (75 positive, 100 negative; %2=3.2, p = .07).
The kurtoses did exhibit a significant tendency toward negative values (23 positive, 153
negative; x2 = 95, p < .0001), but the median kurtosis (-.7) did not seem large enough to
warrant adjustments.
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318
Paulvon Hippel
Fig. 3. The pitch heightstalliedhere come from the melody in Figure2. The distribution
exhibitsan approximatelynormalshapein whichmost pitchesoccurnearthe centerof the
melody'srange.The standarddeviationof this distributionis an index of tessitura.
pitcheswill be within a standarddeviationof the meanpitch.The standard
deviation of a melody'spitch heights- 3.5 semitones for the melody in
Figure2- can thereforeserveas a usefulindex of tessitura.Medianvalues
for the tessituraindex are 3.1 semitonesfor the Europeanfolk songs, 3.8
for the Chinese,4.0 for the South African,and 4.7 for the Ojibway.(The
medianrangesare roughlyfour times as large:12 semitonesfor the European folk songs, 15 for the Chinese,17 for the SouthAfrican,and 16.5 for
the Ojibway.)
The effect of tessituraon intervalsize can be measuredby transforming
eachof the sampledmelodiesinto a "scrambledtwin." A scrambledtwin is
producedby simplyreorderinga melody'spitchesat random.The resulting
twin is identicalto the originalmelodyin its pitchdistribution,but random
in its pitch order- random,that is, in its intervals.The intervalspresentin
a scrambledtwin arethe exclusiveresultof the pitchdistribution- a result,
that is, of the scale, the relativeprevalenceof differentscale degrees,and
the tessitura.If intervalsize is constrainedby tessitura,we would expect
the scrambledtwins to exhibit a preponderanceof small intervals.
For the purpose of this comparison,an intervalcan be defined as the
semitonedistancebetweenany consecutivepitchesthat are not separated
by a phraseboundaryor rest. To preservethe numberof intervalspresent
in the originalmelodies,scrambledtwins can be constructedin such a way
that the positions of rests and phrase boundariesare fixed, even as the
positions of pitches are randomized.Scrambledtwins of this type were
constructedfor each of the 176 sampledmelodies.Talliesof noncompound
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RedefiningPitchProximity
319
Fig. 4. Noncompoundintervalsizes in folk songs from four differentcontinentsare comparedwith intervalsizes in correspondingsets of "scrambledtwins."Eachscrambledtwin
was composedby rearrangingthe pitchesof the correspondingfolk song in a randomorder.
The prevalenceof smallintervalsin the scrambledtwinscan resultonly fromconstraintson
tessitura.The greaterprevalenceof smallintervalsin the originalfolk songs can be attributed to constraintson mobility.
intervalsin the originalmelodiesand their scrambledtwins are displayed
in Figure4.
To begin by examining specific intervals, the distributions for the
scrambledtwins displaysharppeaks at the consonantintervalsof 5 and 7
semitones(the perfectfourthand perfectfifth), as well as a deep troughat
the dissonantintervalof 1 semitone (the minor second). These peaks and
troughsare probablyan artifactof scale structure.Both the pitch content
of common scales and the relativeprevalenceof certainscale degreestend
to maximizethe possibilityof consonantintervalsand minimizethe possibilityof dissonantintervals(Huron, 1994; Smith, 1997).
Settingasidethesedetailedobservations,Figure4 exhibitsa coarsergeneral
pattern:In all foursetsof scrambledtwins,smallintervalsgreatlyoutnumber
largeones.Thispatternis consistentwith the simpleclaimthatthe constraint
on tessitura,by itself,would tendto producean excessof smallintervals.
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Paul von Hippel
320
Constraint 2: Mobility
Althoughsmall intervalsare prevalentin the scrambledtwins, they are
considerablymore prevalentin the originalmelodies. The comparisonis
summarizedin Table 1, which shows that the mean intervalsize for each
repertoireis one to four semitonessmallerthan the mean intervalsize for
that repertoire'sscrambledtwins. Whenthe 176 originalmelodiesareconsideredindividually,all but 5 of them have a smallermean intervalsize
than the correspondingscrambledtwin. Theseresultsshow that, when the
pitchdistributionis controlled,melodicintervalsremainconsiderablysmaller
than would be expectedby chance.
This simple analysiscorroboratesthe intuitionthat intervalsize is subject to a secondconstraintbeyondthat of tessitura.This secondconstraint
limitsthe freedomwith which a melodycan move throughits tessitura- it
is, in other words, a constrainton mobility.The effect of mobility constraintscan be describedby the observationthat, even allowing for the
compact distributionof a melody'spitch heights, each pitch tends to be
close to the pitch before.More formally,we would expect most melodies
to have a positivecorrelationbetweenthe heightsof consecutivepitchesthat is, a positive lag-one autocorrelation. The strength of this
autocorrelationmay serveas an index of constraintson mobility.
For all but one of the 176 melodies in our sample, the lag-one
autocorrelationbetweenconsecutivepitch heightsis positive.When pitch
heightis measuredin semitones,the medianvalue of this autocorrelationis
.46 for the Chinesefolk songs, .59 for the Europeanfolk songs, .59 for the
SouthAfricanfolk songs, and .84 for the Ojibwayfolk songs. Notice that
the Ojibwayfolk songs have the most constrainedmobility;this explains
how they can have the widest tessituraof the four repertoires(medianindex, 4.7 semitones), yet still have the smallest mean interval size (1.6
semitones).
Table 1
Mean Interval Sizes in Four Folk Song Repertoires and in Their
Scrambled Twins
Meanintervalsize (semitones)
in ScrambledTwins
FolkSongRepertoire in OriginalMelodies
Chinese
European
South African
Ojibway
23
2.2
2.3
1.6
3^8
3.7
4.3
5.5
MelodiesWithSmaller
MeanIntervalSizesThan
TheirScrambledTwins
26
79
24
42
out
out
out
out
of
of
of
of
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30
80
24
42
Redefining Pitch Proximity
321
To illustratethis index of mobility,consideragain the melody displayed
in Figure1. In Figure5, we have plotted each pitch in this melody against
the pitchthat succeedsit. The stronglag-oneautocorrelationbetweensuccessivepitchheights(.70) indicatesthat, evencomparedwith the restof the
Europeansample,the mobilityof this folk song is highlyconstrained.
Forecasting Pitch Heights
Tests of the four sampledrepertoiresindicatethat the autocorrelation
between successive pitch heights is approximatelylinear. That is, on a
scatterplotlike Figure5, the points fit a straightline ratherthan a higher
ordercurve.3The existenceof such a linear autocorrelationsuggeststhat
the height of any pitch can be predicted,with some error,by a first-order
autoregressionformula.In particular,if the first severalnotes in a melody
of ru
(?!, ..., PiA)havea meanof m semitonesand a lag-oneautocorrelation
the most likelyheightfor the next pitch(P,)will fit the followingequation:
P.-m=r1(PiA-m)
(1)
where,in practice,the predictedpitch height (P.)would be roundedto the
nearestdiscretepitch.
Put into words, this equationmay be interpretedin the followingterms.
- m > 0), the next pitch tends to be someAftera high pitch (i.e., when Piml
mean
to
the
what lower (i.e., closer
by a factorof rx).Similarly,aftera low
pitch,the next pitch tendsto be somewhathigher.Both of these tendencies
are strongerwhen the antecedentpitch is extreme(i.e., P,_xis distantfrom
m) or when the constrainton mobilityis weak (i.e., rxis small).
These phenomena,familiarto statisticiansas "regressiontoward the
mean," are straightforwardresults of the constrainton tessitura.Simply
put, after an extreme pitch- or, for that matter,before it- most of the
availablepitches are quite a bit closer to the middle of the tessitura.For
this reason,upwardintervals,especiallylargeones, tend to begin low and
end high,whereasdownwardintervals,especiallylargeones, tend to begin
high and end low. These tendenciesare illustratedin the two panels of
Figure6, which shows that for the Germanfolk song in Figure1- as for all
of the 176 sampledmelodies- intervals(as measuredin semitones)exhibit
As an informal test of linearity (Darlington, 1990), we examined residual
3.
scatterplots for eight randomly selected melodies (two from each of the four repertoires);
these scatterplots exhibited no obvious signs of curvilinearity. As a more formal test
(Darlington, 1990), we tried to fit quadratic curves to the scatterplots for each of the 176
melodies. In general, these curves explained only marginally more of the variance than
could be explained by a straight line (median increment in R2= .01). It therefore seems that
the correlation between successive pitch heights could be reasonably characterized as linear.
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322
Paulvon Hippel
Fig. 5. The pitch successionsplotted here come from the melody in Figure3; the area of
each point is proportionalto the numberof timesthat the correspondingpitchsuccession
occurs in the melody.Each of the melody'spitchestends to be followed by a pitch that,
consideringthe melody'stessitura,is nearby.Accordingly,the lag-oneautocorrelationbetween successivepitchheights(hererx= .70) can serveas an index of the melody'smobility.
(Note. Musiciansoften use the terms"antecedent"and "consequent"in a specializedsense
relatingto musicalrhetoric.In this paper,however,the terms simply referto successive
pitches.)
a negativecorrelationwith the heightof theirstartingpitch, and a positive
correlationwith the heightof theirendingpitch.
The tendenciesillustratedin Figure6 have a numberof straightforward
corollaries.One corollaryis that the peak pitch in a phrase- which is, by
definition,high- tends to be approachedand left by skip. A secondcorollaryis that aftera skip- whichtendsto landon an extremepitch- a melody
is likelyto retreatby changingdirection.Althoughboth of thesetendencies
have been claimed as independentprinciplesof melodic structure(Eitan,
1993, 1997; Meyer,1956; Nanino & Nanino, ca. 1600; Narmour,1990;
Toch, 1948), recentanalysessuggestthat they are mereside effectsof constraintson tessitura(Huron, 1999; von Hippel & Huron, 1999).
Assigning Probabilities to Pitches
A testof the scatterplotin Figure5- andsimilartestsforthe othersampled
melodies- indicatesthat the variancesaroundthe autoregressionline are
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324
Paul von Hippel
roughly homogeneous. That is, regardless of the antecedent pitch height,
the heights of consequent pitches are about equally variable.4 Given this
property, our autoregression formula can do more than simply predict the
pitch that is most likely to occur next- the formula can also assign a probability to the subsequent occurrence of any given pitch. In particular, if the
first several notes of a melody (Pu ..., PM) fit a normal distribution with
mean of m, a standard deviation of s, and an autocorrelation of ru the
probabilities for the next note will fit a normal distribution. The mean of
this probability distribution, as indicated above in equation (1), will be P,=
fi(Pi-i- m) semitones, and the standard deviation will be Vl - rx2semitones
(Brockwell &cDavis, 1996). Note that the width of the probability distribution (Vl-ra2) depends on both tessitura and mobility; in particular,pitches
are predicted with greater certainty when the tessitura is narrow (s is small)
or the mobility is highly constrained (rxis large).
In practice, this continuous probability distribution will be broken into
discrete bins corresponding to the available pitch classes. An example is
displayed in Figure 7. Here the melody from Figure 2 has been interrupted
at the end of its sixth measure, and a probability distribution has been
generated for the following pitch. Because the mobility is constrained, the
peak of the probability distribution is near the most recent pitch; moreover, because the most recent pitch is high in the tessitura, the peak of the
probability distribution is somewhat lower.
For melodies with a downward trend- for example, most Ojibway melodies (Densmore, 1913) - the next note will usually be lower than our formula predicts. But for a melody that maintains a consistent tessitura, the
formula's predictions should be roughly correct- at least with respect to
pitch height. In Figure 7, for example, it is encouraging to observe that the
Fig. 7. Here the melody from Figure 2 has been interrupted at the end of its sixth measure,
and our definition of pitch proximity has been used to generate a probability distribution
for the next pitch. Because the mobility is constrained, the most likely pitches are predicted
to be close to the most recent pitch. Moreover, because the most recent pitch is high in the
tessitura, the most likely pitches are predicted to be somewhat lower.
A simple test for homogeneity of variance is the rank correlation between the
4.
predictions of the regression equation and the absolute values of the residuals (Madansky,
1988). For the 176 sampled melodies, the median value of this rank correlation is minuscule (r = -.05), indicating that the variances are close to homogeneous.
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Redefining Pitch Proximity
325
second-highestprobability(18%) is assignedto the pitch A, which does in
fact occurnext in the melody.
Like prior definitionsof pitch proximity,the new definition takes no
account for scale structure.It assigns probabilitiesbased entirelyon the
heightsof pitches,regardlessof wherethose pitchesfit in a melody'sscale.
In Figure7, for example,the definitionassignsthe highestprobability(19%)
to the pitch At, which does not fit the fragment'sD-majorscale. If I were
proposinga completemodel of melodic structure,ratherthan an isolated
definitionfor pitch proximity,I would probablycombinethe bell curvein
Figure7 with a seconddistributionrepresentingthe prevalenceof different
scale degrees.The key-profiledistributionsexploredby Krumhansl(1990)
and her colleaguesmight be suitablefor this purpose.
Conclusion
To review,researchershave traditionallydefined pitch proximity as a
simpleabundanceof smallpitch intervals.I advocatereplacingthis definition with a more refined alternative.Statistically,the proposed revision
definespitch proximityas a positivelag-one autocorrelationbetweensuccessivepitch heightsdrawnfrom a quasi-normaldistribution.More informally,the new definitionis based on the premisethat intervalsare constrainedboth by a melody'stessituraand by its mobility.
This new definitionoffers severaladvantagesover the old one.
1. Whereasthe old definitioncan predictonly the sizes of melodic
intervals,the new definitioncan predicttheir directionsas well.
In general,the new definitionpredictsthat intervalswill retreat
from the extremesof the tessituraand approachthe middle:intervalsthat start on high pitches will proceed downward, and
intervalsthat start on low pitcheswill proceedupward.
2. Whereasthe old definitionpredictssmall intervalsregardlessof
context, the new definitionpredictsthat intervalsizes will vary
dependingon the melody's position in its tessitura. Near the
middleof a melody'stessitura,the new definitionpredictssmall
intervals,but nearthe extremes,the new definitionpredictsrelatively largeintervals.
3. The new definitionsubsumestwo tendenciesobservedby music
theorists:thetendencyfor a melodicpeakto be approachedby skip
andthetendencyfor a skipto be followedby a changeof direction.
Althoughearlierwork has presentedthesetendenciesas independent rulesof melodicstructure,underthe new definitionthey are
seenas emergingfrombasicconstraintson mobilityandtessitura.
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326
Paulvon Hippel
4. The new definitionfits a statisticalformalismthat, giventhe first
severalnotes in a melody, assigns a probabilityto each of the
pitchesthat could occur next. More specifically,the new definition predictsthat the probabilitiesfor the next pitch will fit a
normaldistributioncenteredbetweenthe most recentpitch and
the mean pitch of the melody.Both the peak and the width of
this distributionare defined by statisticalmeasuresof tessitura
and mobility.
In sum, comparedwith the old definitionof pitch proximity,the predictions associatedwith the new definitionaremorepreciseanddetailed,more
sensitiveto melodic context, and broaderin their implications.In other
words, the new definitionenablesa sharperand deeperdescriptionof melodic structure.
Although this definitionof pitch proximitywas developedto describe
melodic structure,it might plausiblydescribemelodicpsychologyas well.
A rule of pitch proximityis often used to account for such cognitivephenomenaas melodicexpectations,melodicpreferences,andauditorystreaming (Bregman,1990; Carlsen, 1981; Dowling, 1967). In these cognitive
applications,just as in descriptionsof musical structure,pitch proximity
has traditionallybeen defined simply in terms of intervalsize. In light of
the presentresults,however,thereis good reasonto try an alternativedefinition.Perhapsmelodiccognition,likemelodicstructure,couldbe described
more preciselyin termsof constraintson tessituraand mobility.5
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