Download - Modelling and Synthesis of Violin Vibrato Tones · Modelling and Synthesis of Violin Vibrato Tones Colin Gough School of Physics and Astronomy, University of Birmingham, Birmingham

Transcript
Page 1: Modelling and Synthesis of Violin Vibrato Tones · Modelling and Synthesis of Violin Vibrato Tones Colin Gough School of Physics and Astronomy, University of Birmingham, Birmingham

Modelling and Synthesis of Violin Vibrato TonesColin Gough

School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK, [email protected]

A model for vibrato on stringed instruments is presented, based on the measured acoustic response for animpulsive force at the bridge. The model highlights the importance of both the violin's dynamic responseand that of the performance acoustic in determining the fluctuations in amplitude of any note played withvibrato. Comparison of the influence of vibrato on the recorded and synthesised tones of violins ofdifferent quality and of electric violins will be presented, in addition to considering the additionalinfluence of holding the violin and the sympathetic vibrations of unstopped strings.

1 IntroductionIt is well known that a bowed violin noteplayed with vibrato leads to both frequencyand amplitude modulation of the radiatedsound, resulting from the multi-resonantresponse of the violin body [1-3]. In thispaper, we consider how the dynamic responseof the violin and the acoustic environmentsignificantly affect the radiated sound. In arecent paper [4], such effects were shown togive sound waveforms with very large andcomplex modulations in amplitude, in whichthe time-delayed reflections from thesurrounding walls play a very significant role,as first noted by Meyer[3].

In addition, we develop a model to simulatethe effect of vibrato on the sound of a violinat any position in the performance space,derived from measurements of the soundproduced by an impulsive at the bridgerecorded at the listener s position.

Musical illustrations, downloadable from theinternet [5], demonstrate the importance ofvibrato and other fluctuations in defining thecharacteristic sound of different musicalinstruments. By inference, such fluctuationsare therefore also likely to be important in anysubjective assessment of violin tone quality.

The focus of this paper is to highlight thedynamic effects that determine thecharacteristic sound of a violin when playedwith vibrato, rather than attempting tosynthesise violin vibrato sounds with 100%authenticity.

2 Vibrato and violin tone qualityIf a violin or any other continuously soundinginstrument could be played without anyfluctuations in amplitude or frequency, thesound would be indistinguishable from that ofa characterless signal generator. This isillustrated in Figure 1 and SOUND 1 [5], bythe envelope and section of waveform of along note A (440Hz) played with vibratorecorded at 2 m distance from the violin,which is followed by a note produced bycontinuously repeating a single periodwaveform extracted at random from therecorded sound. The repeated note has thesame overall spectrum of partials and asimilar but unchanging waveform as theparent waveform, but lacks the fluctuations infrequency and amplitude and additional noise,which characterise the sound as that of aviolin.

Figure 1: Envelope and section of waveform of abowed violin vibrato note A (441Hz) and a

continuously repeated single waveform

Note the very large fluctuations in amplitudeof the recorded sound largely arising from theuse of vibrato with a typical frequency width

f/f ~ 3% and vibrato rate of ~ 5 Hz.

721

Page 2: Modelling and Synthesis of Violin Vibrato Tones · Modelling and Synthesis of Violin Vibrato Tones Colin Gough School of Physics and Astronomy, University of Birmingham, Birmingham

Forum Acusticum 2005 Budapest Mustache, Klein, Paprika

The continuously repeated waveform isexactly the kind of sound predicted by asimple physicists model , in which the violinis excited by a constant frequency sawtoothbowing force on the bridge producing a combof harmonic partials differentially excitingthe many acoustically radiating vibrationalmodes of the instrument. Since the resultingsound has none of the characteristics of thesound of a real violin, such a model alone isunlikely to provide much useful informationon what might distinguish the sound of areally fine instrument from that of a mass-produced factory instrument.

We therefore argue that it is the characteristicfluctuations in the sound of a violin thatenable the listener to identify the instrumentas a violin and, by inference, its quality also.

In this paper we focus our attention onfluctuations in frequency and amplitude fromthe use of vibrato, but other fluctuations frominherent bow noise [6] and both controlledand inadvertent fluctuations of bow pressureand velocity may also contribute to anysubjective assessment of tone quality. .

It has often been remarked that it is the initialtransient that enables the listener to identifythe sound of an instrument. Although this isundoubtedly true for the immediaterecognition of certain instruments, the soundof individual instruments can generally bedistinguished equally well when the initialtransient is removed. This is illustrated fortypical violin, flute, trumpet, oboe andsawtooth notes with the first 50 ms transientremoved, followed by the sound of the fullwaveform ([5], SOUND 2)

3 model for vibratoWe now describe a model for vibrato in thetime-domain, involving the dynamic responseof the violin and the equally important roomacoustics.

For simplicity, we assume the violin isexcited by a sawtooth driving force at thebridge generated by a simple Helmholtz waveon the bowed string. As a first approximation,

we ignore any additional ripples or noise onthe driving force produced by reflectionsbetween the bow and bridge, the finite widthof the collection of bow hairs and interactionwith the excited body modes [6].

The sawtooth waveform can be considered asa succession of Helmholtz step-functionssuperimposed on a linearly increasing force ofno acoustic significance. Each successiveHelmholtz step will excite a transientresponse of the violin involving all thecoupled modes of the instrument. In thesteady state, the resultant vibrations will bedetermined by the superposition of a sequenceof such responses.

For a sequence of step-functions Fn at times tn,the response is given by

where h(t) is the unit step-function response.If h(t) is known, the response can becomputed for a sequence of steps with aperiodically modulated spacing, to simulatethe effect of vibrato.

To synthesise a realistic waveform requires aknowledge of the step-function response h(t)for the violin. This can be derived byintegrating the measured impulse response I(t)excited by a delta-function input,

The impulse response can be measured bystriking the top of the bridge in the bowingdirection with a force hammer or small mass.

The derived step-function response differsfrom the impulse response in giving greateremphasis to the lower frequency components,with relative weightings 1/ . This isillustrated in SOUND 3 [5] by the initialacoustic tap response measured 2m awayfrom the violin, followed by the derived step-function response, the transient responsessounding rather like the tick and tock of agrandfather clock. The step-function responseis then repeated at 1, 10 and 200 Hz, the finalsound simulating the sound of a violin, but

722

Page 3: Modelling and Synthesis of Violin Vibrato Tones · Modelling and Synthesis of Violin Vibrato Tones Colin Gough School of Physics and Astronomy, University of Birmingham, Birmingham

Forum Acusticum 2005 Budapest Mustache, Klein, Paprika

without vibrato, noise or any otherfluctuations.

This simple demonstration highlights theimportance of the transient response indefining the sound of an instrument. Theresponse can be measured at any point on thebody of the instrument or in the surroundingsound field. The latter is important, if one isinterested in the quality of a violin sound. Theradiated sound is determined not only by thestructural resonances excited but also by theirradiation efficiency and directivity.

In any realistic performing situation, asopposed to playing the instrument in the openair or inside an anechoic chamber, thetransient acoustic response involves bothdirectly radiated sound and time-delayedreflections from the surrounding walls.Because of the inverse square law decrease inintensity with distance, the relativeimportance of the direct and reflected soundwill vary strongly with position from theviolin.

2 cm from by ear 2 m awayfront plate2 cm from by ear 2 m awayfront plate

Figure 2: Transient acoustic responses measured atdifferent distances from the violin body

This is illustrated in Fig.2 and in SOUND 4[5] for the sound of a violin with dampedstrings excited by a short impulse at thebridge. The impulse is first measured close tothe front plate, where the sound is dominatedby the violin rather than the room acoustic,close to the player s ear, where the roomacoustic already contributes significantly tothe transient response, and at a distance of 2m,where the transient sound is dominated byreflections from the surrounding surfaces. Thefeedback of sound from the performance

acoustic provides very important feedbackand encouragement to the player inoptimising the quality of sound produced.

At a distance from the violin, the sound of aviolin is just as strongly influenced by thedynamic response of the room acoustics as itis by the violin. This has importantimplications for any subjective inter-comparisons of violin tone quality,particularly in an over-resonant acoustic. Asour simulations show, the influence of theroom acoustics is particularly important whenthe instrument is played with vibrato (see alsoref [4]).

4 Synthesis of vibrato tones

4.1 Violin vibrato tones

The impulsive acoustic response of a numberof violins of different quality has beenmeasured at different positions from theviolin in both an anechoic space and arelatively small (~ 5x6x3 m3) furnished room.Impulse responses were excited by swinging asmall pith ball (1.3g) against the top of thebridge in the bowing direction.Measurements were made with the violin bothfreely-suspended and held by a player in thenormal way, with the strings damped andallowed to vibrate freely.

The step-function responses were computedby numerical integration of the impulseresponses. Simulated vibrato tones were thencomputed from equ.1 for varying amplitudesand rates of vibrato. To give a further senseof realism, the sawtooth waveform wasmodulated with a 25 ms time exponential riseand decay time.

Fig.3 shows the simulated sounds of thefreely suspended Vuillaume violin withdamped strings derived from tap tonesrecorded very close to the instrument and thenat a distance of 2m, first with no vibrato andthen for a vibrato with f/f = 1.5% at 3 and6Hz. The ear can follow the cyclic changes infrequency at and below 3 Hz but at typicalvibrato rates of 4-6Hz the sound simply

723

Page 4: Modelling and Synthesis of Violin Vibrato Tones · Modelling and Synthesis of Violin Vibrato Tones Colin Gough School of Physics and Astronomy, University of Birmingham, Birmingham

Forum Acusticum 2005 Budapest Mustache, Klein, Paprika

appears to pulsate, as illustrated by SOUND 5[5] for the six waveforms shown. At typicalvibrato rates, fluctuations in amplitude aremore important in the identification of thesound of the violin than those in frequency, aspreviously noted by Melody and Wakefield[7].

0 Hz 3 Hz 6 Hz

2 cm

2 m

0 Hz 3 Hz 6 Hz

2 cm

2 m

Figure 3: Simulated Vuillaume vibrato sounds fromrecorded tap tones at 2cm and 2m for a sinusoidal f/fmodulation of 1.5% and vibrato rates of 3 and 6 Hz.

The initial transient and final decay of thesimulated and all real waveforms retain manyof the spectral features of the generating step-function response. However, once established,the waveform and sound, in the absence ofvibrato, remain featureless. However, whensimulated with even small amounts of vibrato(see [4]), the envelopes acquire a complexityvery similar to that observed in real vibratotones.

It is important to note, that the amplitudefluctuations are strongly asymmetric withrespect to time. This is a characteristicsignature of the importance of dynamiceffects, which is not predicted by earliermodels for vibrato [1-3], in which dynamicprocesses were ignored.

The difference in waveforms and sounds ofthe simulated tones played with and withoutvibrato is dramatic. We argue, as others havedone before [1-3], that the complexity inenvelope not only creates a much moreinteresting sound for the listener, but alsoenables a solo instruments to be heard abovethe collective sounds of a large orchestra.Essentially, the fluctuations in amplitudeprovide a continuous sequence of transientsor time-varying fluctuations, which continue

retain the interest of the listener quite unlikethe predictable sound of a note played withoutvibrato or that of a crude electronicsynthesiser.

4.2 Electric violin tones

It is a striking fact that the sound of anelectronic violin is remarkably similar to thatof a real violin (SOUND 6 [5]). This isdespite having an almost flat frequencyresponse without the prominent spectrum ofresonances that characterise the acousticspectrum of the violin, which manyresearchers have attempted to correlate withthe defining quality of an instrument.

It is therefore interesting to see what ourmodel tells us about the sound of an electricviolin. We have therefore recorded impulsefunctions from the piezo-electric transducermounted in the purposely massive bridge of a4-string Violectra violin by David Bruce-Johnson.

7 s

no vibrato vibrato single waveformrepeated

7 s

no vibrato vibrato single waveformrepeated

Figure 4: The note A (440 Hz) of an electric violinplayed without and with vibrato and the sound of a

continuously repeated single waveform with a similarenvelope.

Figure 4 and SOUND 7 [5] compares thereproduced sounds of the note A (440Hz) onthe electric violin, first without and then withvibrato, followed by a continuously repeatedsingle period wave extracted from the vibratowaveform. As demonstrated earlier for theviolin, the continuously repeated single periodwave has none of the fluctuations that mightcharacterise the instrument as a violin, thoughwithout prior knowledge the sound of theelectric violin could easily be mistaken for areal violin.

724

Page 5: Modelling and Synthesis of Violin Vibrato Tones · Modelling and Synthesis of Violin Vibrato Tones Colin Gough School of Physics and Astronomy, University of Birmingham, Birmingham

Forum Acusticum 2005 Budapest Mustache, Klein, Paprika

Because of the absence of easily excitedstructural modes, the recorded impulsefunction decays very much more rapidly thanthat of a real violin. The waveforms generatedby the sawtooth waveform will therefore lackthe complexity generated by the much longertransients of the violin.

Violectra 0.5% vib 1% vibin room @ 6 Hz @ 6 HzViolectra 0.5% vib 1% vibin room @ 6 Hz @ 6 Hz

Figure 5: Bowed vibrato tone on an electric violin andthe same tones synthesised from the recorded tap-tone.

This is illustrated in Figure 5 and SOUND 8[ ], which compares the waveforms andreproduced sound of an electric violin playedwith vibrato with synthesised vibrato tonesgenerated from the short impulse response ofthe piezoelectric transducer. The width f/fof the vibrato for the real instrument was ~1%, while the synthesised tones are for 0.5%and 1% modulation

As anticipated, the simulated waveformsexhibit none of the complexity observed forviolin simulations, because of the very muchfaster decay of the transient response. Thecomplexity of the recorded sound waveformsarises almost entirely from the transientresponse of the small room in which therecorded sounds were replayed.

5. Further Comments

In practice, violins are played under the chinrather than being freely suspended and anyunstopped strings remain undamped, both ofwhich significantly affect the transientresponse and hence complexity of thesimulated waveforms.

Modal analysis measurements by Marshall[8]and more recently by Bissinger[9] have

demonstrated that holding the violinintroduces a significant amount of additionaldamping of the vibrational modes. Figure 6and SOUND 9 [5] illustrates the additionaldamping of the transient response when theviolin is held under the chin rather than beingfreely suspended followed by the additionalringing of the open strings when they are leftfree to vibrate. The transients were recordedclose to the top plate in an anechoicenvironment.

0 3kHz

0 0.01 0.02 0.03 0.045

4

0

104

105

50 ms traces

0

500 1000 1500 2000 2500 30000.1

1

10

100

103

1000

0.1500 1000 1500 2000 2500 3000

0.1

10

100

500 1000 1500 2000 2500 30000.1

1

10

100

0 0.01 0.02 0.03 0.04

0

freely supporteddamped strings

helddamped strings

heldfree strings

dB

0

-20

-40

-60

0 3kHz

0 3kHz

0 3kHz

0 0.01 0.02 0.03 0.045

4

0

104

105

50 ms traces

0

500 1000 1500 2000 2500 30000.1

1

10

100

103

1000

0.1500 1000 1500 2000 2500 3000

0.1

10

100

500 1000 1500 2000 2500 30000.1

1

10

100

0 0.01 0.02 0.03 0.04

0

freely supporteddamped strings

helddamped strings

heldfree strings

dB

0

-20

-40

-60

0 3kHz

0 3kHz

Figure 6: Transient response of a freely supportedVuillaume violin with damped strings, the violin when

held and held with undamped strings. Short periodFFTs delayed by successive intervals of 12.5 ms.

The short time FFTs, delayed by successiveintervals of 12.5 ms, show that the increaseddamping from holding the violin isparticularly pronounced for frequencies below~1.5kHz. This is unsurprising, because atthese frequencies the normal modes ofvibration will generally involve significantbending and flexing motions of the violinbody, which will be damped when the edgesand neck of the violin are supported by theplayer.

An impulse at the bridge will excite all thepartials of any strings left free to vibrate,which will contribute strongly to the transientresponse and radiated sound, as illustrated inFig.6 and the recorded sound. The sound ofthe unstopped strings can easily heard in thedecaying sound of any short note played on

725

Page 6: Modelling and Synthesis of Violin Vibrato Tones · Modelling and Synthesis of Violin Vibrato Tones Colin Gough School of Physics and Astronomy, University of Birmingham, Birmingham

Forum Acusticum 2005 Budapest Mustache, Klein, Paprika

the violin and must contribute significantly tothe sound and complexity of any bowed note.

Figure 7 and SOUND 10[5] illustrate theinfluence of the above effects on thesimulated sound. The simulated sounds areshown for first the freely held violin with allthe strings damped and then with the stringsleft free to vibrate. Simulations are thenillustrated for impulse functions measuredwhen the violin is held by the player in thenormal way, first with damped then withundamped strings. The simulations are for avibrato width of 1.5% vibrato at 5Hz. Thedifference in the simulated sound isparticularly marked when the strings are leftfree to vibrate.

violin freely supported held in normal way

Strings damped undamped damped undamped

violin freely supported held in normal way

Strings damped undamped damped undamped

Figure 7: Simulated waveforms for freely suspendedand conventionally played Vuillaume violin with

strings damped and left free to vibrate.

6. SummaryAlthough our model for synthesizing thesound of a violin is relatively simple, itproduces quite realistic violin vibrato soundsand serves to highlight and explain many ofthe observed features of real violinwaveforms. In particular, it emphasizes theimportance of dynamic effects in the time-domain associated with the vibrational modesof the violin, the acoustic in which the violinis played and the freely vibrating strings onthe instrument.

The importance of vibrato in enhancing thequality of the sound of the violin, to producea warmth and singing quality to the tone likethat of the singing voice, was emphasized bymany early writers, notably LeopoldMozart[10]. Whereas some modern playersmay use an excessive amount of vibrato when

playing baroque and early classical music,the complete absence of vibrato advocated bysome players and conductors appears perverse,on both aesthetic and scientific grounds. Oursimulations demonstrate that fluctuations, andespecially the use of vibrato, are importantfeatures in defining the sound of the violin,without which the sounds produced arecharacterless and lacking in interest to the ear.

References[1] H. Fletcher, L.C. Sanders, Quality of

Violin Vibrato Tones, J. Acoust. Soc. Am.41,1534-1544 (1967)

[2] M.V. Matthews and K. Kohut, ElectronicSimulation of Violin resonances, J.Acoust. Soc. Am. 53, 1620-1626 (1973)

[3] J. Meyer, Zur klanglichen Wirkung desStreicher-Vibratos, Acustica 76, 283-291(1992)

[4] C.E. Gough, Measurements Modellingand Synthesis of Violin Vibrato Sounds,Acustica 91, 229-240 (2005)

[5] Downloadable as a packaged PowerPointpresentation with linked *.WAV files atwww.cm.ph.bham.ac.uk/Gough/Forum-05

[6] M.E. McIntyre, R.T. Schumacher, J.Woodhouse, Aperiodicity in bowed-stringmotion, Acustica 49, 13-32 (1981)

[7] M. Mellody and G.H. Wakefield. Time-frequency Characteristics of ViolinVibrato: Modal Distribution Analysis andSynthesis. J. Acoust. Soc, Am. 107, 598-611 (2000)

[8] K.D. Marshall, The Musician and theVibrational Behaviour of the Violin,Catgut Acoust Soc. J. 45, 29-33 (1985)

[9] G. Bissinger, General Normal ModeAcoustics, Acta Acustica 91 (2005)

[10] Leopold Mozart, Fundamental Principlesof Violin Playing (1756) (translated E.Knocker, Oxford University Press, 1948)

726