Non-Linear Phenomena in Contemporary Vocal Music

Nonlinear Phenomena in Contemporary Vocal Music

*Jurgen Neubauer, †Michael Edgerton, and ‡Hanspeter Herzel

Berlin, Germany

Complex and multiphonic voice signals of vocal improvisors are analyzedwithin the framework of nonlinear dynamics. Evidence is given that nonlinearphenomena are extensively used by performers associated with contemporarymusic. Narrow-band spectrograms of complex vocalizations are used tovisualize the appearance of nonlinear phenomena (spectral bifurcationdiagrams). Possible production mechanisms are discussed in connection withprevious research,personalperformance andpedagogical experience. Examplesfor period doubling, biphonation and irregular aperiodic phonation in vocalsonorities of contemporary vocal improvisors are given, and glottal whistleproduction encompassed with biphonation and triphonation is shown.Furthermore, coincidences of harmonics-formant matching associated withabrupt transitions to subharmonics and biphonation in the vocal output areprovided. This also shows the recurrent use of nonlinear phenomena byperformers. It is argued that mechanisms such as source-tract coupling orvocal fold desynchronization due to asymmetry are used in a reproducibleway for musical tasks.

Key Words: Contemporary vocal music—nonlinear dynamics—multiphoniccomplex sonorities—artistic reproducibility—glottal whistle.

Accepted for publication February 24, 2003.From the Institute for Theoretical Biology, Invalidenstrasse

43, D-10115 Berlin, Germany.*Electronic mail: [email protected]†Electronic mail: [email protected]‡Electronic mail: [email protected] of Voice, Vol. 18, No. 1, pp. 1–120892-1997/$30.00� 2004 The Voice Foundationdoi:10.1016/S0892-1997(03)00073-0

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I. INTRODUCTION

Composers, performers and listeners of contempo-rary classical music have long recognized the vitalityof complex multiphonic instrumental and vocal so-norities. However, the theoretical understanding ofthese complex sounds relied mainly upon the meth-ods of mechanical reproduction (i.e. fingering charts

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ith embouchure indications), while scientificuestions were largely avoided (except e.g. 1).Beginning in the early 1980’s, theories of nonline-

rity were applied to complex musical signals. Thised some to reconceptualize their understanding ofhe elements involved in the production of sound.uring the last decade, when combined with the

knowledge gained through analog/digital sound syn-thesis, a conception of a scalable parameter spaceapplied to instruments and voices was developed.

his scaling suggests that selected parameters maye systematically varied between minimal and maxi-al values to produce new complex sonorities.In this paper, complex and multiphonic voice sig-

als from vocal improvisors will be analyzed withinhe framework of nonlinear dynamics. We classifyoice samples from solo vocal improvisations aseing harmonic voice, subharmonics, biphonationtwo independent pitched melodies) or irregular ape-iodic behavior.

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NEUBAUER ET AL2

Nonlinear phenomena in voice are widely observedin newborn cries,2,3 pathological voices,4–8 animalvocalizations,9–12 and occasionally in speech.13,14

Further, it has been reported that nonlinear vocalphenomena carry functional and communicative rele-vance for animals and humans.15,16

In a previous case study, Paul Ward17 investigatedvocalizations of a teenage subject who had the abilityto produce biphonation. This subject achieved suchbehaviors through asymmetrical control of the leftand right vocal folds. Captured on high speed pho-tography and cinefluorography, the subject demon-strated the capacity to produce two different, butsimultaneous pitch movements, otherwise having acompletely normal voice. Further, she had the pro-ficiency to produce such behaviors within clearlyidentifiable musical scales and not simply as con-tour relationships.

This study is representative of a growing body ofevidence that suggests normal larynges have theability to produce nonlinear phenomena includingsubharmonics,18–21 biphonation17,22–24 and aperi-odic, irregular behavior (deterministic chaos).21,25–27

In addition, it has been documented in the musicalliterature that these and many other non-standardvocal outputs are widely used by performers ofcontemporary music.26,28–39 Here we provide aframework to characterize these frequently usednonstandard vocalizations in contemporary vocalmusic and discuss possible physiological mechanisms.

In this paper, we document the extensive use oftemporally and spectrally complex vocalizations byperformers of contemporary vocal music within theframework of nonlinear dynamics. We discuss theobserved complex temporal and spectral patternsusing the terminology of nonlinear phenomena. Mu-sically, we argue that nonlinear phenomena in thevoice can be consciously used for artistic purposes.

A. Nonlinear phenomenaThe human voice production system can be con-

sidered as multiple nonlinear coupled oscillators(e.g. left and right vocal folds, different modes ofvibrations within single folds, ventricular folds, epi-glottis, aerodynamical oscillators).40 The vocal foldsdisrupt the outward flowing airstream using a wave-like vibratory motion. This oscillation is driven by

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Journal of Voice, Vol. 18, No. 1, 2004

he airflow provided by alveolar (lung) pressure to-ether with visco-elastic forces within the larynxo produce phonation. The theoretical description oformal periodic phonation requires already a nonlin-ar theory, describing e.g. phonation onset from pre-honatory standstill as an instability of the resttate resulting in self-sustained oscillations. Further-ore, nonlinear laws govern the dependence of

ormal phonation upon the driving pressure, ratef airflow, amplitude of vocal fold oscillation, stress-train properties of vocal fold tissue and vocalold collision.

In this paper we focus on vocal behaviors ofocal improvisors that deviate from normal periodichonation. In the following discussion the termonlinear phenomena will consequently refer toubharmonics, biphonation, and irregular aperiodichonation (associated with deterministic cha-tic behavior).

. PrinciplesUnderstanding the dynamics of nonlinearly cou-

led oscillators requires an introduction to a few basiconcepts of nonlinear dynamics. A detailed intro-uction to nonlinear dynamics is beyond the scopef this paper and can be found elsewhere.7,41–43

ere we sketch only briefly the basic ideas.Oscillating systems are omnipresent in the physi-

al, biological and chemical worlds. These systemshare the common feature to be able to self-sustainperiodic behaviour. Coupling several oscillators,

ven just two, can generate highly complex temporalatterns of motion for the overall system, even moreomplex than the individual oscillators. The analysisf real-world phenomena produced by several cou-led self-sustained oscillators uses methods fromonlinear dynamics theory. Therefore, the behaviorf a real-world system is mapped into phase space.he phase space is built from the dynamical vari-bles (e.g., for a mechanical mass-spring oscillatorhese variables would be the excursion from theesting position and the velocity of the mass attachedo the spring) necessary to determine the state of theystem. At every time point, the behavior of a systemay be represented by a single phase space point.There are qualitatively different sorts of complex

ibratory patterns of nonlinearly coupled oscillators.

NONLINEAR PHENOMENA IN CONTEMPORARY VOCAL MUSIC 3

In the corresponding phase space representationsthese patterns are characterized by different geomet-rical objects, termed attractors. In the simplest casethe oscillators cease to move at all; the systemreaches a static equilibrium. This attractor is termed‘steady state’ and can be represented by a singlepoint in phase space. When the coupling betweenthe oscillators synchronizes their vibrations, theoverall behavior is a periodic, regular oscillation.Technically, such a periodic self-sustained oscilla-tion is termed ‘limit cycle’. In phase space a limitcycle attractor forms a closed loop. The natural fre-quencies of two oscillators can be quite differentfrom each other. Then, coupling can still synchron-ize or entrain these oscillations, but the resultingfrequency might be quite different from the naturalones. For instance, two oscillators with a frequencyratio of approx. 1 : 2 will be entrained to a ratio ofexactly 1 : 2. Then the overall system output willhave the frequency of the slower oscillator. Thisbehavior is termed ‘subharmonic oscillation’ or‘folded limit cycle’, which can be represented asa folded closed loop in phase space. If the couplingbetween oscillators is weak, oscillators can vibratefreely with independent frequencies not related asinteger multiples. Then, the phase space representa-tion is a ‘torus’, a two-dimensional object in phasespace, as the overall system (i.e. the superpositionof at least two oscillators) never exactly repeatsitself. Eventually, two desynchronized coupledoscillators can generate nonperiodic, irregular vibra-tions. This behavior that never repeats but that isstill bounded is termed ‘[deterministic] chaos’. Forthe sake of clarity the term ‘deterministic’ is used tostress the absence of any random influences on thesystem. In phase space, deterministic chaos formsa limited complex geometrical object that has a frac-tal, noninteger dimension.

Different sorts of vibratory patterns govern thedynamics of nonlinear coupled oscillators for con-stant control parameters of the system – for examplevocal fold tension or subglottal pressure. Above,we described the most prominent attractor typessteady state, [folded] limit cycle, torus, and [deter-ministic] chaos. Often, external control parametersvary slowly compared to the typical dynamical be-havior, e.g. vocal fold vibrations. A slowly varyingparameter can induce sudden, abrupt transitions

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etween different attractor types – a behavior termedifurcation. In this paper a few bifurcation types aref particular relevance: The transition from the resttate (steady state) to a limit cycle is termed ‘Hopfifurcation.’ An abrupt change from a limit cycleo a folded limit cycle (subharmonic oscillation) isermed ‘period doubling bifurcation’. When anotherndependent oscillation appears in a vibratingystem, this transition from a limit cycle to a toruss termed ‘secondary Hopf bifurcation.’ Often, eitherascades of period doubling bifurcation or secondaryopf bifurcations are precursors of the sudden onsetf deterministic chaos. In other words, a small con-rol parameter shift then induces an abrupt change ofhe system output to nonperiodic, irregular behavior.

. Applications to voice productionThe framework of nonlinear dynamics can be used

o classify complex vocal behavior of (contempo-ary) vocal improvisors. Steady state behaviorccurs when the vocal folds are at rest. Then asubglottal air pressure increases, a Hopf bifurcationhanges the steady state attractor into a limit cycles the vocal folds begin to vibrate in a normal,eriodic way. Often during speech and song, periodoubling bifurcations occur which can then be seens subharmonic oscillation. Subharmonics may belassified as a special type of limit cycle whichppears via transition from a periodic oscillationo an oscillation with alternating amplitudes. Lessrequent, though still seen in speech and song, arehenomena featuring two or more independentrequencies. This phonation, geometrically classi-ed as a torus in phase space, may be produced withsymmetrical vocal fold vibration and has beenermed biphonation.7,44 As mentioned above, sub-armonics and tori often are precursors of determin-stic chaos. In this case, the behavior is nonperiodic,rregular and complex, but not necessarily random.

In order to provide evidence of deterministichaos, phase space analysis is required.2,7 In earliertudies attractor dimensions and Lyapunov expo-ents have been estimated from voice signals.45–50

comprehensive phase space analysis of differentttractors of voice signals is beyond the scope ofhis paper (for attractor dimensions, Lyapunovxponents see e.g. 27,51–53. For rigorous mathe-atical and statistical tests of the presented data


NEUBAUER ET AL4

the data are available on request). Fortunately, manyconclusions about the dynamics can be drawn fromnarrow-band spectrograms. These can be interpretedas spectral bifurcation diagrams where time isconsidered a control parameter (cf. 54–56): Sub-harmonics (e.g., see Fig. 1) after period-doublingbifurcations correspond to parallel lines in betweenharmonics. These occur typically at multiples of 1/2, 1/3 or 1/4 of the original pitch, generally at n/mof the original pitch, where n and m are small inte-gers. In the case of biphonation (e.g. Fig. 2), seem-ingly independent spectral components with nosimple ratio (such as 1/2, 1/3), that are modulateddifferently or move independently, appear in thespectrum. Finally, chaos (see Fig. 3) is characterizedby broadband noise-like segments that appear viaabrupt transitions. Distinct spectral peaks embeddedin the broadband spectrum appear due to recurrentquasiperiodic behavior within the chaotic segment.Deterministic chaos is often interrupted by windowsof normal periodic phonation or subharmonicphonation.

II. MUSICAL PHENOMENA

Nonlinear phenomena have been well-docu-mented in many disciplines, but regularly ignoredby the mainstream of performers and composers ofcontemporary classical vocal music who treat the

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FIGURE 1. Period doubling cascade from a harmonic behaviorwith a pitch of about 500 Hz to subharmonics at 1/2, 1/4 or1/6 of the pitch (Sample from solo vocal improvisation investi-gazioni (diplofonie e triplofonie) by the vocal improvisor Deme-trio Stratos.65

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IGURE 2. Biphonation with two independent frequenciestwo independent pitched melodies) termed f and g and vari-us linear combinations of these frequencies (Sample fromolo vocal improvisation passagi 1,2 by the vocal improvisoremetrio Stratos.65

oice with much the same musical aesthetic andrinciples of phonation that existed during the 18thnd 19th centuries. So that while instruments havencorporated multiphonics, complex inharmonic sig-als and other transient phenomena,1,57 the voice hasemained for the most part the carrier of simpleelodic formuli.Nonlinear phenomena have been identified as im-

ortant instrumental resources for experimental newusic since the late 1950s when Bruno Bartolozzi

ttempted to systematize a framework for the pro-uction of multiphonic sonorities for woodwinds.58

hese innovations, combined with the continuingechnical development and conceptual radicalism oflectroacoustic music, suggested to many that theoice should share in the wealth of new sound andts construction. Therefore during the 1960s, com-osers such as Dieter Schnebel, Luciano Berio, Johnaton, Giacinto Scelsi, Gyorgy Ligeti, Kennethaburo, Pauline Oliveros, Sylvano Bussotti, Robertrickson and Mauricio Kagel began to explore theroduction and organization of non-standard vocalusic.59 However, most of this work did not attempt

o systematically utilize nonlinear phenomena. Thiss completely understandable, for unlike instru-

ents, the human voice cannot easily be taken apartnd put back together. Combined with the lack oftandardized fingering charts for vocal sound pro-uction within the larynx, most composers attempted


FIGURE 3. Transition from harmonic behavior via period dou-bling, period quadrupling to an irregular, noise-like segment,where the vocal tract filter is revealed by different shadings(Sample from solo vocal improvisation entre nosotros - epitafioa las ballenas by the vocal improvisor Fatima Miranda.63

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to explore performance technique and expressionthrough phonetic based articulatory procedures or toa far lesser degree through the combination ofmultiple vocal sound sources, combining primarilyharmonic with inharmonic input, or special phenom-enon, such as subharmonics or overtone singing. Inthe last two decades of the 20th century, the activesearch for expression through the vocal instrumenthas subsided and new compositions again featuremostly traditional modes of vocal performance.However, nonlinear phenomena of the voice are ex-tensivelly used by performers who are often classi-fied as vocal improvisers.

For this study, complex vocalizations from solovocal improvisations by vocal improvisors GreetjeBijma,60 Jaap Blonk,61 Anna Homler,62 Fatima Mi-randa,63 David Moss,64 and Demetrio Stratos65 werechosen. We show examples for the appearance ofnonlinear phenomena in these singers and provideideas about possible production mechanisms. Weargue that these mechanisms are used in a reproduc-ible way for musical tasks.

III. MATERIAL AND METHODS

We present narrow-band spectrograms as time-frequency plots of sound samples of vocal improvi-sors. Narrow-band spectrograms can be interpreted

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s spectral bifurcation diagrams where time is con-idered a control parameter.54–56 The samples wereecorded at a sampling rate of 44100 Hz with aesolution of 16 bits/sample. For the calculationf the spectrograms we used a window size of 2048ample points for the Fourier analysis. Thereforehe frequency resolution can be calculated as4100/2048 Hz≈21.5 Hz which corresponds to themallest frequency of the Fourier analysis. We usedHanning window for the Fourier analysis to ac-

ount for finite window size effects. The sound sig-als were centered about their mean values. Aemporal overlap of 2000 sample points was used forhe sliding window Fourier analysis. If not otherwisepecified, the dynamical range of the spectrogramsas chosen as 60 dB sound intensity level. All spec-

rograms were normalized with respect to theiraximum intensity level. A high spectral resolution

f 21.5 Hz rather than a high temporal resolutionas chosen for the calculation of the spectrograms.e assume that in vivo physiological parameters,

uch as vocal tract configuration or subglottal pres-ure change slowly over time66 which can still beesolved by the low temporal resolution of the spec-rograms.

All samples were recordings from single singersnly. The singers are vocal improvisors performingolo vocal improvisations with no processing. Inusical terms, their articulary gestures can be de-

cribed as nonturbulent open vowel-like sonorities.hese improvisors aim to reach an artisticallyleasing and highly cultured state. All recordingsonsist of the pure microphone signal without post-rocessing, neither dynamically (dynamical com-ression or enhancement) nor spectrally (filtering,itch shifting, chorusing, phasing). The sound sam-les were digitally copied from publicly availableD recordings.Our study is based on voice samples that were

hosen from 300 different samples of 7 differentingers. We chose the shown representative subset ofamples due to preliminary perceptual evaluationoncentrating on episodes with independent fre-uencies, fast transitions, pitch instabilities, whis-le like vocalizations and broadband sounds. Withespect to contemporary music, voice samples wereelected that combine artistically interesting multi-one complex vocal sonorities with dynamically in-eresting nonlinear phenomena.


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IV. RESULTS

In Fig. 1 we show an example of period doublingwith windows of inharmonicity. At 1.6 s a perioddoubling bifurcation (P2) occurs indicated by anadditional stack of harmonics interspersed be-tween the existing harmonics. At about 2.5 s a periodquadrupling bifurcation (P4) occurs, at 2.7 s a bifur-cation to a period sextupling (P6) appears – a periodsix times the fundamental pitch. After a P4 bifurca-tion (2.8 s) the system jumps via a P6 bifurcationagain (3.0 s). The fundamental frequency of about500 Hz suggests that the vocal folds of the male singervibrate in the falsetto register of a male voice. Thisfinding is consistent with our perceptual judgement.

In Fig. 2 we show an example of biphonation,the occurrence of two independent frequency con-tours that may be perceived as two different pitchsequences (melodies). At about 0.6 s and about2.0 s the time course of the two frequencies crosses,which supports our observation of independence ofthe two frequencies. In addition to the two indepen-dent frequency contours f and g, we observe fre-quency components which can be explained bylinear combinations of f and g. During the biphonicepisode the amplitudes of the combination frequen-cies are relatively small, indicating that the amountof coupling between the oscillators is low. Atabout 3.0 s a register transition to a nonbiphonicfalsetto voice occurs, revealing stronger harmoniccomponents than during the biphonic episode.

In Fig. 3 we give an example of a female voicewith a sudden transition from normal, harmonic pho-nation to one with a broadband spectrum. In thisexample, sequences of period doubling and quadru-pling bifurcations appear before the onset of irreg-ular aperiodic oscillation. Within the irregularsegment, starting at about 2.0 s, we can see residualspectral components related to the previous har-monic components. Additionally, formants at ap-proximately 1.0 kHz, 1.4 kHz and 3.0 kHz can beseen. The subharmonic components bifurcating atabout 0.5 s and approximately 1.3 s are perceivedas time varying mixtures of harmonic and inhar-monic components.

In Figures 4 and 5 we present examples in whichtime-varying formant frequencies matching harmon-ics coincide with bifurcations to subharmonic re-gimes. Fig. 4 is the visualization of an extremly high

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IGURE 4. Coincidence of formant matching harmonic asso-iated with sideband modulations of harmonic componentarrows) (dynamical range: 90 dB sound intensity level) (Samplerom solo vocal improvisation in principio by the vocal improvi-or Fatima Miranda.63

nd light fundamental frequency of a female voice atbout 1.25 kHz modulated by a tremolo frequencyf about 4 Hz. Sidebands of the second harmonicppear when the upward moving formant frequencyatches the second harmonic at about 1.0 s (arrows).he result is to reinforce the amplitude of this har-onic. The sidebands of the fundamental frequency

nd the third harmonic are also visible, but lower inntensity than the sidebands of the second harmonic.

In Fig. 5 we show an example of a formant match-ng harmonics associated with a period multiplyingifurcation. A male singer used the falsetto registero phonate at about 550 Hz. In Fig. 5, a formantrequency coincides with the the fourth harmonicround 0 s (arrow). At the same time period quintu-ling (P5) occurs. Then, from 0.5 s to 1.5 s theormant frequency moves downward and coincidesoughly with the third harmonic at about 1.5 sarrow). At this point period quintupling (P5) occurs.

In Fig. 6 we show recurrent instances of abruptransitions from regular phonation to sub-harmonicnd irregular phonation. The female voice produceshonation at nearly 900 Hz. The left part of Fig. 6egins with an irregular segment with a periodicindow followed by another irregular segment. At

bout 1.0 s an extremely high and light phonationith an increased perception of pitch sets in. Note


FIGURE 5. Recurrent source-formant interaction: Formant in-duced P5 bifurcation with five subharmonics vanishing andreappearing when a formant frequency matches the fourth orthe third harmonic, respectively (arrows). (dynamical range: 90dB sound intensity level, figure contrast and intensity manuallyadjusted to emphasize formants) (Sample from solo vocal im-provisation investigazioni (diplofonie e triplofonie) by the vocalimprovisor Demetrio Stratos.65

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that about 1.0–2.5 s the frequency component of900 Hz is an artefact of the recording environmentand is the result of excess reverberation during re-cording, and is not representative of phonation. Atabout 2.5 s a transition to a chaotic segment withresidual harmonics occurs followed by harmonicwindows with subharmonics including P6 at 3.3 s.At about 3.4 s a chaotic segment with residualharmonics starts, which is interrupted by short in-stances of subharmonic windows. In the right partof Fig. 6 the qualitative behavior is similar, althoughdetails vary. At about 0.4 s the high and light phona-tion bifurcates to a chaotic behavior. This segmentis interrupted by a subharmonic window. After thechaotic segment the voice transitions again to high,light phonation. At about 2.5 s another chaotic seg-ment starts which, again, is interrupted by a subhar-monic window at about 3.2 s.

In Fig. 7 we observe a female ‘glottal whistle’starting at a pitch of above 2.0 kHz descending toabout 1.5 kHz. After the initial ‘glottal whistle’, asecond frequency of approx. 3.5 kHz appears atabout 1.0 s and is functionally independent. At 2.0 sa third frequency shows up which vanishes atabout 2.5 s. As the two remaining frequency con-tours descend further, we observe multiple combina-tion frequencies increasing in amplitude. This

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ncrease of intensity can be related to an increasef coupling over time between the two independentrequencies. At about 5.4 s a third component ap-ears which perceptually has a vocal fry-likeharacter.

V. DISCUSSION

In this paper we have shown examples of thententional use of nonlinear phenomena in contem-orary vocal music. In multi-tone complex sonoritiesf vocal improvisors we find sub-harmonics, bipho-ation, sudden onset of irregular phonation (possiblyeterministic chaos) interspersed with more periodicindows and sudden frequency jumps (register

hanges). We argue that nonlinear phenomena, well-nown in voice pathology, play an essential rolen modern artistic vocalizations. In the followingections we will discuss bifurcation analyses, poten-ial physiological mechanisms, issues relative to re-roducibility, and their musical relevance.

. Bifurcation analysisAccording to the theory of nonlinear dynamics, all

ystems feature a limited set of qualitatively differentynamical behaviors. In this paper we document howhis applies to sound production of the voice, inarticular for solo vocal improvisations by perform-rs using their voices for artistic sound production.he attractors relevant to the physical descriptionf oscillatory systems are limit cycle (related toeriodic oscillation), folded limit cycle (related toubharmonics), torus (related to two frequency oscil-ation) and chaos (related to highly irregular, aperi-dic and noise-like behavior).Transitions between different dynamic regimes

i.e. different attractors) are predicted to occur evenor slowly varying system parameters like pitch,ormants, subglottal pressure or the varying interac-ion with supraglottal tissue structures. A compre-ensive visualization of transitions can be achievedy bifurcation diagrams42 which display differentynamical behavior depending on one or two vary-ng system parameters. Such diagrams were calcu-ated for a simplified two-mass model of vocalolds67 and continuum models.68 Furthermore, bifur-ation diagrams were measured for excised larynx


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FIGURE 6. Recurrent instances of abrupt transitions from regular phonation to subharmonics and irregular phonation (arrows)within the same vocalization sequence (Sample from solo vocal improvisation entre nosotros - epitafio a las ballenas by the vocalimprovisor Fatima Miranda.63

experiments69 and analyzed for a voice with unilat-eral paralysis.70

Here we use time-frequency plots (spectrograms)as spectral bifurcation diagrams.54–56 In contrast tobifurcation diagrams obtained from mathematicalmodels or experimental setups, where single systemparameters can be varied, we have no direct mea-surements of the varying system parameters in invivo multi-tone complex sonorities of vocal improvi-sors. Thus in spectral bifurcation diagrams, time isconsidered a “control parameter” of the system.

There are a few hints on changing parametersof pitch, formants, and subglottal pressure due todifferent shading of spectral lines and noiselike ele-ments in the spectrographic displays. Thus we canspeculate about the physiological mechanisms thatdetermine the vocal outputs.

B. Physiological mechanismsWe suggest that formant induced transitions might

play a role in the production of sub-harmonic andbiphonic sequences. In Fig. 4 we show an exampleof a formant induced transition for a harmonic pho-nation to an oscillation with sidebands. Thus asecond frequency modulates the initial harmonicoutput during formant matching which can be ob-served as sidebands in the spectrogram. With respectto nonlinear dynamics nomenclature, such transi-tions might be called secondary Hopf bifurcations.They change the behavior of the system from a limitcycle oscillation to a toroidal oscillation–a torus in

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hase space. The dominant time varying formanteems to play a crucial role in destabilizing thelottal oscillator coupled to the vocal tract. Similarly,n Fig. 5 we give an example of a formant inducedifurcation for a harmonic vocal output to subhar-onic oscillation. This conclusion is supported by

revious model studies in simplified models forource-tract interaction. Sub- and supraglottal reso-ances were found to be relevant for high-pitchedhonation and, moreover, biphonation.71

Second, it has been reported that sub- and supra-lottal oscillators (and resonance factors), such asentricular fold phonation, epiglottic and arytenoidartilage, can contribute to a variety of complexodes of phonation.72

Third, another mechanism that can induce transi-ions of dynamical behavior is vocal fold asymmetry.eft-right asymmetries and anterior-posterior asym-etries facilitate subharmonic oscillation, biphona-

ion and even chaotic vibration.17,24,69,73

Fourth, excessive subglottal pressure, increasedtiffness of the vocal fold mucosa, or reduced pre-honatory glottal shape (phonation neutral area)eading to chaotic vibrations have been identifiedn a recent study of a simplified two-mass model,ven for a symmetrical vocal fold configuration.47

xcessively high airflow with a lax laryngeal postureas been implicated in the simultaneous productionf subharmonics and deterministic chaos.74 We arguehat vocal improvisors exploring the full range ofocal abilities are able to use exceptional, but not


FIGURE 7. Female glottal whistle with biphonation and tri-phonation. Final segment with triphonation with two glottalwhistle and third, vocal fry–like oscillator (Sample from solovocal improvisation Signals by the vocal improvisor AnnaHomler.62

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pathological, methods to induce multi-tone complexvocal sonorities such as (multiple) subharmonics,biphonation and irregular aperiodic sound signals.(see Fig. 3 and 6).

Fifth, we argue, that vocal improvisors can useglottal whistles to produce complex sonorities. Theterm ‘glottal whistle’ describes vortex-generatedsound with partially adducted glottal or supraglottalconstrictions that may also feature slight vocal foldvibrations (vortex-induced fold vibrations). We sug-gest that these biomechanic or aerodynamic vi-brations may be produced at different locationssimultaneously within the glottis. The subglottalairstream as the driving force could facilitate onepossible coupling between these different oscillators.Thus, biphonation or triphonation could occur dueto several interacting oscillators: left versus rightvocal fold, anterior versus posterior parts of the vocalfolds, glottal versus supraglottal vibrating structures,or vortex-induced vibrations. Moreover, typical for-mant frequencies in the range of several kHz, thusmatching the pitch of the glottal whistle, could eitherstabilize biphonation or induce transitions to oscilla-tions with sidebands or even irregular behavior.74

The highly variable time course (melody) shown inFig. 7 suggests the lack of accurate control mecha-nisms. The melody may thus result from pressure/flow regulated behavior or from relaxation behaviorof tissue stiffness.

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. Reproducibility, intention, and controlResulting from personal performance experi-

nces, the development of special training programsf amateur and professional voices, and the acousticnalysis of hundreds of sound samples of vocal im-rovisors, we claim that singers are able to use non-inear phenomena intentionally. We argue thathe above mentioned physiological mechanismsnsure reproducibility in a musical context. Theseechanisms include vocal tract resonance tuning,

ubglottal pressure adjustments, or voluntary desyn-hronization of vibration modes using asymmetries.espite variation in minute details of bifurcations to

ubharmonics, biphonic or irregular segments (seeig. 5 and 6), the vocal sonorities can be produced

n artistically controlled methods. In contrast toathological phonation where nonlinear phenomenaccur accidentially, certain vocalists are able to con-rol the recurrent production of complex sonorities.

usically, control is important for sound output: byhanging perceptually robust variables over time,n intended affect, emotion or meaning can be pro-uced.75,76 Next, as many performance techniquesf vocal improvisors may last the duration of a per-ormer’s career or even life-span, it is obvious thathese techniques are not necessarily aberrent behav-ors to be avoided,77 but rather can be physicallynd spiritually beneficial.78,79

. Musical relevanceThe insight provided by nonlinear dynamics to

he understanding of complex musical vocalizationsan offer valuable information about an instrumentith no levers, buttons or strings – the voice. This

ype of analysis offers the identification of a phenom-non with a class and a process. For example, bipho-ation is identified as a torus that appears via aecondary Hopf bifurcation.

Historically, sonorities of vocal improvisors areescribed by a combinatorial multiphonic frame-ork which has its origin in the broad linguistic

lassification. This scheme identifies the degree ofoicing that appears within speech or song, tradition-lly identified as either voiced or unvoiced.35,80

Voiced’ refers to mostly harmonic outputs producedithin the laryngeal framework, whereas ‘unvoiced’

efers to both harmonic and inharmonic sounds pro-uced at different locations superior to the larynx


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produced by manners such as stops, fricatives, ap-proximants, whistles or buzzes. Many sonorities ofvocal improvisors feature the perception of two ormore identifiable tones, often the result of multiplesource components.30 As a result, numerous catego-ries have been developed that combine 1) voicedsound with another voiced sound (‘voiced-voiced’);2) voiced sound with an unvoiced sound (‘voiced-unvoiced’); 3) unvoiced sound with another un-voiced sound (‘unvoiced-unvoiced’).35

In this paper, we integrate this combinatorialmultiphonic framework with the theory of nonlineardynamics to describe archetypical phenomena – sub-harmonics, biphonation, and irregular aperiodic pho-nation. We give ideas for linking physiologicalmechanisms with nonlinear phenomena in a fewtypical multiphonic behaviors within the combinato-rial framework. Thus, we suggest a robust musicalterminology which merges production-related termi-nology with nonlinear dynamics.

Within the voiced-voiced category we argue thatnonlinear phenomena could occur due to glottalasymmetries (left-right or anterior-posterior), desyn-chronization of other complex glottal vibratorymodes and coupling of phonation with supraglottalvibrations (e.g. the ventricular folds). Within thevoiced-unvoiced category, which includes vocal foldoscillation with unvoiced supraglottal oscillation orfriction, we argue that all nonlinear phenomena canbe expected for coupled glottal and supraglottalunvoiced oscillation, such as glottal phonation withlip buzz. The combination of vocal fold oscillationwith frication should lead to irregular or turbulentoutputs as the aerodynamic oscillators are thoughtto have a strong effect when glottal phonation iscombined with pharyngeal frication. Within the un-voiced-unvoiced category, which includes tonguewith oral cavity frication or labial whistle with pharyn-geal articulation, we can define subgroups according tothe manner of (un)voicing, such as unvoiced oscillationwith unvoiced oscillation, unvoiced oscillation withunvoiced fricative, or unvoiced fricative with unvoicedfricative that indicate relative amounts of pitchednessversus inharmonicity. The first subcategory of ‘un-voiced oscillation with unvoiced oscillation’ is associ-ated with limit cycle oscillations, subharmonics andbiphonation. Both other subcategories are morelinked to irregular or turbulent behavior.

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One readily-available tool relevant for sound anal-sis is the spectrogram. In the context of acoustichaos spectrograms are called spectral bifurcationiagrams.54 When time is considered as a controlarameter, they are also termed ‘visible speech’.56

pectrograms may also contribute to the trainingnd treatment of desired or aberrant voice signalss windows of visualization. When combined withnformation regarding airflow, laryngeal posturingnd tension, etc., we propose to use them similar tonstrumental fingering charts (standard pedagogicalools for producing nonstandard extended complexnstrumental sonorities81). Through the identifica-ion of specific aerodynamical and physiologicalarameters, approximate scales within a parameterpace can be introduced. In this way, multiple param-ters can be continuously varied between minimalnd maximal values, such as the pitch/rhythmic axisound in common-practice music. This procedureay be referred to as scalability,35 and suggests thathat was formerly considered extraneous acoustic/hysiologic elements now attain a greater role duringhe production of multi-tone complex musicalignals.

For the clinician and performer, if (control) pa-ameters encompassing multi-tone complex pho-ation can be described with scales, then the efficacyf training and treatment might attain significantenefits. For contemporary compositional practice,t least two benefits may be proposed. First, bysing these parameters as scalar properties (betweeninimal and maximal values), artistic quality is

nhanced by offering coherence of compositionalrocedure across multiple parameters. Second, re-arding multi-tone complex sonorities and extendedonstandard gestures, the heightened emphasisf these parameters might offer exciting additionalusical outputs worth continued artistic exploration.

cknowledgments: We thank Tecumseh Fitch for stim-lating discussions and the Deutsche Forschungsge-einschaft for support.

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Non-Linear Phenomena in Contemporary Vocal Music

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