Expressive Musical Timing

Transcription

1 Axel Berndt, Tlo Hähnel Department of Smulaton and Graphcs Otto-von-Guercke Unversty of Magdeburg {aberndt Abstract. Tmng s crucal for the qualty of expressve musc performances. It s a means to medate muscal form and largely effects the emotve and motor characterzaton of a pece. Dfferent epochs of musc hstory do not only dffer n ther compostonal styles but also caused change processes n performance practse. To facltate muscologcal nvestgatons n hstorcally nformed performances and ther changes, especally n the late baroque perod, we had to create a software tool less an autonomous performance system, rather a hgh-level performance edtng system. Flexble formal models for performatve features should allow to explore the parameter spaces and to smulate and evaluate a broad varety of styles: reserved baroque, affectve romantc, unprecse begnners, and so on. Ths paper focusses on the tmng aspects of muscal performances. We dentfed four general feature classes, namely tempo, rubato, constant asynchrony and human mprecson. To explore ther degrees of freedom we ran a number of evaluatons of professonal recordngs, lve performances, and expermental recordngs. On ths bass we elaborate adequate formal models and descrbe ther mplementaton wthn a MIDI-based performance system. 1 Introducton To nvestgate dfferences and change processes of performance practce n musc hstory we developed a software tool to model dfferent performatve styles. A major desgn element for ths s human muscal tmng [10]. Performance practces have been nvestgated largely from the romantc era up to contemporary musc [16, 17, 19, 20]. However, these fndngs were hardly transferable to hstorcally nformed performance of prevous stylstc perods. Exstng performance systems turned out to be less applcable for the muscologcal task that rather necesstates a knd of a hgh-level performance edtng system than an autonomous performance system [4, 11, 21, 22]. Ths text reports of our own nvestgatons especally nto the temporal aspects of expressve musc performance and ther synthess. In a performance analyss and evaluaton phase we explored the shape and characterstcs of performatve features and how they change accordng to the structural and stylstc contexts (analyss by synthess and measurement). On ths bass we developed flexble formal models to descrbe and resynthesze these features wthn computer performances. Wth these tools t s possble to support musc producton, e.g., n sound studos, for performance synthess, flm and game scorng. It s useful to demonstrate the development of hstorcal performance practces and also to facltate further nvestgatons of musc hstory, e.g., the nterrelaton between the development of musc theory, organology and performance practce. Ths paper s structured as follows: Secton 2 wll ntroduce the basc tmng concept and correspondng defntons. In Secton 3 we revew and summarze our evaluaton of sklled muscans. Sectons 4 and 5 detal the formal models and ther mplementaton whch derve from the performance analyses. Secton 6 provdes a concluson and exposes future perspectves. 2 Tmng Features Expressve muscal tmng s a complex nterplay of multple aspects. To facltate a flexble smulaton and reasonable parameterzaton, all tmng aspects have to be separated and treated ndependently from each other. Ths clam led to a phenomenon-based concept of muscal tmng: Sngular features are ndvdually shaped dervatves of a general class. The ndvdual characterstcs are adjusted accordng to the muscal context of ther applcaton and the compostonal and performatve style. We dentfed four such feature classes: Tempo descrbes the basc beat count per tme unt (e.g., 100 beats per mnute) that establshes the muscal meter. It can reman constant, change dscretely or contnuously over tme. Contnuous tempo changes rarely feature a lnear shape. The actve work wth tempo changes s mportant for

2 the fguraton of clmaxes, melodc destnatons, and muscal phrases. Thus, t s appled over segments wth muscally medum-term and long-term extent, hence macro tmng. Rubato Muscal meter s rarely performed exactly. Lttle prolongatons and compressons are used conscously and subconscously to express fgures, motfs, metrc accentuatons, peak tones etc. Accordngly, rubato defnes muscally short-term tmng devatons that take place wthn clearly delmted frames. Any devatons also have to be compensated wthn the same tmeframe to keep up wth the basc tempo. For that reason they are called balanced devatons. These occur as sngular phenomenons but more often as repettve schemes (e.g., n the Vennese waltz). Constant Asynchrony Dfferent onsets between muscans n an ensemble do often show systematc behavour. It s orgnated n the herarchy of nstruments n the score (leadng parts are ahead) and s affected n ts extent by the overall tempo [14]. Human Imprecson Wthn the analyses devatons occur that cannot defntely be traced to a systematc reason. Even f effects of fngerng, shftng, and room acoustcs are taken nto account, a rest of mprecson remans. To determne approprate parameter settngs for partcular muscal contexts and styles we carred out a number of performance evaluatons as descrbed n the followng. 3 Analyss and Retreval The parameter spaces defned n the prevous secton were dscovered by an analyss of lve, studo, and expermental recordngs. 3.1 Methodology The studo recordngs were represented by 10 professonally produced recordngs of G. P. Telemann s Trumpet Concerto n d major (TWV 51:D7). Ths chamber concert s one of the most famous baroque concerts for trumpet and typcal n ts form [7]. After a global analyss of tempo changes we focused on the frst sx bars of the frst movement, the adago. There, the most tempo changes emerged, leadng us to assume a possble phrase-dependent performance. We also present results of rubato analyses, usng the example of three motfc groups of four sxteenth notes. The lve recordngs were made durng the 5th Internatonal Telemann Competton for hstorcal woodwnd nstruments (recorder, baroque flute and baroque oboe). The advantage was that all nterpretatons played lve and no recordngs were mproved afterwards. In addton, the recordng equpment (AKG C 1000 mcrophones and Zoom Handy H4 recorder) as well as ts poston and the room all recordngs were made n, were the same. An nternatonal jury selected 15 players for recorder, who all studed or had been studyng hstorcally nformed performance. We analyzed the compulsory pece, the sonata n f mnor for recorder and basso contnuo (from Telemann s Der getreue Musc Mester ). In the frst movement, sgned Trste, all performers played very dstnct rubat. In the expermental recordngs 10 professonal muscans for classcal musc as well as experts n hstorcally nformed performance played baroque and modern nstruments (brass, strngs and woodwnds). To get results that are free of any further contextual nfluences, they performed rtardand and accelerand on non-melodc tone repettons. Beats and onsets were detected n two steps, whch combned automatc onset detecton and human tappng. The automatc onset detecton generated a number of onset hypotheses based on algorthms descrbed by Duxbury et al [2], the power over tme of an audo sgnal 1 and changes n energy between varous frequency bands n a sequence of spectra 2 [1]. These were hypotheses for sngle onsets and also for onsets, whch are of secondary relevance (e.g., arpeggated cembalo play). By tappng, we set markers wthn the hypothess space. These markers provded clues to dentfy the most vald hypotheses whch were used for tempo and rubato analyses. 3.2 Results of Tempo Analyses Every performance of the Trumpet Concerto showed outstandng devatons n tmng to mark phrase boundares. However, we could not fnd any evdence for permanent tempo changes over a long perod of tme. To evaluate whether successve devatons n note lengths can be descrbed wth a curve, we approxmated the successve devatons wth lnear, quadratc and cubc functons n the frst sx bars of the adago. Only two performances showed a sgnfcance of α > 0.05 wth the cubc functon as the best fttng one, followed by the quadratc functon. In contrast to the phrase arch performance, whch conssts of an acceleraton to a turnng pont and a slowdown untl the end of a phrase, we could nether detect tempo changes over the whole phrase, nor an acceleraton to a turnng pont (only for one out of ten MzSpectralFlux/ - 2 -

3 lnear quadr. cubc quadr. cubc lnear nterpreter I R2 F α nterpreter III nterpreter V nterpreter II R2 F α nterpreter IV Table 1: Approxmated curves for the retard n bar sx of the adago. quadr.= quadratc functon. 5 performers (not lsted) dd not show any sgnfcant values. performers we could not exclude acceleraton lnear wth a weak correlaton coeffcent R2 = 0.12 at the sgnfcance level α > 0.05). Yet we found evdence for a gradual retard, albet lmted to the last bar of the phrase. Table 1 shows 5 of 10 performances. For nterpreter V a lnear retard could not be excluded. The nterpreters I IV played a retard, descrbng a cubc or a quadratc functon, respectvely. These fndngs match wth prevous studes of retard [5, 9]. Frst Half and Second Half Characterstcs To extract rtardand and accelerand separately, we analysed the expermental recordngs, whch had no muscal context to the greatest possble extent. Although all muscans were experts, they found these context-free scores hard to play. Some had dffcultes to count the bars, some played faster nstead of slower or had a lot of questons about the dfferences n tmng. All muscans performed addtonal devatons on metrcal accents, whch (sometmes n hgh degree) dstorted the accelerand and rtardand. In the analyss the performers showed less problems n playng a rtardando, especally the fnal retard. The standard devatons of all performances for the fnal retard (mean SD = 14% of the entre tempo change) were less than for the nner-pece retard (mean SD = 17%). Moreover, the quadratc shape of the fnal retard had a hgher degree than the nner one whch was weakly curved. However, the larger part of the tempo changes stll took place n the second half of the whole process. In contrast to the rtardand, the larger part of accelerand took place n the frst half. Both, fnal and nner accelerand showed a logarthmc shape, both wth a mean SD = 17% and a weaker curvature for the nner accelerand. Ths corresponds to the dstncton of nner and fnal rtardand (see Fgure 1). 3.3 Results of Rubato Analyses Dfferences n Devatons f na lr t ar da nd o o and er el acc ando er l nal f acce nner do an rd a t r er nn Fgure 1: Expermental performances of accenerand and rtardand: Fnal characterstcs are more curved than n a pece. contour occurrence lsll 11 contour occurrence lsll 9 Adago lsls lssl 6 4 Trste lsls lssl 3 13 slsl 3 other 6 lsss 5 other 0 P 30 P 30 Table 2: Dfferent relatons n a group of four sxteenth notes n the Trumpet Concerto (Adago) and sonata for recorder (Trste). l=long, s=short Devatons n note lengths are restrcted to the bar level or shorter. In hs poneerng work, Quantz [12] recommended to lengthen the frst and thrd note n a group of four sxteenth notes n a slow tempo. At a fast tempo only the frst of four notes has to be lengthened, but besde these rules everythng has to be played under the major prems of dversty 3. 3 The connecton between Telemann and Quantz as well as other contemporary muscans has been proven recently [8, 13, 15]. -3-

4 Durng the frst sx bars of the adago, groups of four sxteenth notes occur three tmes, so 30 examples have been avalable from the studo recordngs. As shown n Table 2, the performers played dfferent relatons of these four notes, mostly n the way long-short-long-long (lsll ), followed by long-short-long-short (lsls). Dfferences n contour as well as n markedness refer to the ntenton of varety of performance. A few exceptons gave a vew on a random error: Interpreters, who ntended to play the same rubato, match wth a mnmal accuracy of 12ms. In the expermental recordngs many performances vared to an unexpectedly small extent of less than one percent of the underlyng meter. Wthn ths amount errors of measurement and performance nose could not be clearly dfferentated. meandev at on( er r orbar s =+/ -2SD) s ce an m r o f r pe 0 l3 al dent t ymapp ng( nor ubat o) Frst-Note-Lengthenng ln. log. pow. R lssl F α R lsll F α Table 3: Best fttng curves for the rubato over four sxteenth notes n the Trste. ln.=lnear, log.= logarthmc, pow.= power functon. l =long, s=short More obvous results we found n the lve recordngs of the Trste movement. Here, two groups of four sxteenth notes occur and are performed by 15 nterpreters. As seen n Table 2, for most of the part the contour long-short-short-long (lssl ) was performed.4 A defnte result has been the lengthenng of the frst note. Compared to the mean devatons of the last three sxteenths, the frst was played from 15% up to 140% longer (wth mean x = 66% and SD = 28%). Nevertheless, the hgh varaton of the frst note shows ts outstandng poston. Ths can also be seen n Fgure 2, were the accumulated ntervals are shown. The lengthenng of the frst note results n a maxmum deflecton of the performance curve, referrng to an assumed equal play. The most frequent contours lssl and lsll n the recorder sonata (see Table 2) were approxmated separately, as demonstrated n Table 3. As the curve does not descrbe the relatve devatons, the logarthmc functon as well as the lnear and power functon ft best. Apart from the statstcal relevance of these fndngs, the logarthmc functon has to be preferred: A 4 It has to be mentoned that the cembalst decdes the length of the last sxteenth note, and therefore f t wll become long or short. The recorder only plays four sxteenth notes and ends n a pause, flled by the contnuo. Fgure 2: Rubato over four sexteenth notes n the Trste movement. lnear functon would descrbe an exact equal playng that was excluded because of the above-mentoned results. The power curve just overlad the lnear one (wth margnal devatons n a more unfavourable drecton than the mean curve), and therefore has to be excluded as well. The often found last-note-lengthenng [3, 4] depends on the mportance of the followng frst note. In the Trste the last notes have been played longer at the end of the phrase. Therefore, the last note should not be seen as a lengthened note. It rather has to be seen as an addtonal pause before the followng phrase or fgure. In the late nneteenth century, Remann [18] sowed the seed wth hs rules of agogc from whch contemporary performances developed and stll refer to, respectvely. In ths regard, we dscovered dfferences, exemplfed on hstorcally nformed performance. They concern tempo and rubato features, ther shape characterstc and ntensty. Thus, a synthetc performance must guarantee the flexblty n rubato and also tempo, partcularly the varous shapes and curvatures. In the followng we wll descrbe our approach to ths. 4 Macro Tmng For the formalzaton and mplementaton of tmng features we decded to keep the macro-mcro dstncton. Ths Secton wll focus on macro tmng and descrbe our approach to modelng expressve muscal tempo wthn a performance system. In order to support the huge amount of characterstcs, as dscovered -4-

5 prevously, we present an extended tempo representaton and correspondng mplementatonal detals for an adequate playback. Snce the MIDI standard s stll present and extensvely supported by home computer systems as well as by professonal musc and studo hard- and software we decded to base our mplementatons on the MIDI protocol, too. Therefore, we appled the MdShare engne [6], whch provdes a low-level MIDI-API and a real-tme schedulng system for MIDI events. The new tempo map formalsm, whch s descrbed below, was mplemented as an XML data structure. 4.1 Formalzaton Muscal tempo features are represented as tempo nstructons T. These are lsted n the tempo map M T n tmely ncreasng order: M T = (T 0, T 1,..., T n) Each nstructon T m (0 m n) s a 5-tuple T m = (d m, t 1m, t 2m, b m, m) wth d m provdng the tempo-ndependent date of the nstructon wthn the pece of musc. Ths can be, e.g., a MIDI tcks value, as used here, or t can follow the Bar-Beat-Unt conventon. Both are equvalent and can easly be converted usng the tcks-per-quarternote value (henceforth clcks) whch s provded by each MIDI fle header. Whle d m marks the begnnng of the tempo nstructon T m, ts range s termnated ether by d m+1 of the succeedng nstructon T m+1 or, n case of T m already beng the last nstructon n M T, by the date of the last event wthn the pece of musc. Wthn ths range, T m defnes a contnuous tempo transton from tempo t 1m to t 2m (both measured n beats per mnute). Therefore, b m provdes the muscal length of one beat n floatng pont format (e.g., quarter note: 1/4 0.25, half note: 1/2 0.5 and so forth). To model the dfferent tempo transton characterstcs, as dscovered n Secton 3, the usually lnear transton s twsted va a potental functon n the doman [0; 1] (corresponds wth the degree of tempo change). The exponent gven by m ( m R, m 0) steers the deflecton of the functon: m = 0 to gnore t 1m and run t 2m mmedately (subto) as constant tempo, 0 < m < 1 for frst half characterstcs, m = 1 to acheve the mechancal lnear behavor, m > 1 for second half characterstcs. The ntensty of the rtardando/accelerando grows the more the transton deflects from the lnear course. Extreme devatons approach a subto-lke behavour. 4.2 Implementaton The MIDI standard only defnes constant temp and dscrete,.e. stepwse, tempo changes. Seemngly contnuous transtons can be acheved by dscretzaton wth suffcently small step heghts and wth step lengths of at least the shortest nter-event dstance. However, the hgh-level representaton of tempo nstructons gets lost. The result s stll only an approxmaton and subject to more or less audble alasng effects (abrupt/bouncy tempo changes). To overcome these lmtatons, the MIDI nternal tempo map s gnored and substtuted by our new macro-tmng representaton, whch s provded by an XML fle accompanyng the MIDI data. Accordng to these nformaton, the mllseconds date of each MIDI event s determned by functon ms(d) before sendng t to the scheduler (wth d beng the MIDI tcks value to be converted): d : d d 0 ms(d) = con 1(d d m) + ms(d m 1) : t 1m = t 2m tran(d d m) + ms(d m 1) : otherwse for m as the ndex of tempo nstructon T m whch holds: d m < d d m+1 wth d m+1 as the tck date of nstructon T m+1 or of the last event f T m has no successor n the tempo map. In case of a constant tempo,.e. t 1m equals t 2m, the converson s easly done by functon con k (d l ) wth d l = d d m beng the local MIDI tcks poston wthn the range of T m and k {1; 2}: con k (d l ) = d l t km 4 b m clcks Otherwse, T m defnes a substantal tempo transton. Therefore, the converson s done by functon tran(d l ): tran(d l ) = con 1(d l ) + (con2(d l) con 1(d l )) d m+1 l ( m + 1)(d m+1 d m) m+1 The result of functon ms(d) added to the mllseconds date when the playback begn. Snce all muscal parts/midi tracks are processed ndvdually, ths date s also a bass for synchronzaton to compensate numercal mprecson. The soundng result s a proper alasng-free tempo transton. In our mplementaton we furthermore dstngush between global and local tempo maps. The global ones consttute a unversal tmng base for all tracks. Local tempo maps, by contrast, only provde the macrotmng base for one track. Local nformaton domnate the global, accordngly, the global tempo map s gnored f there s a local one

6 5 Mcro Tmng Mcro-level tmng features add fne-structured detals to the macro-tmng curve. They often show a repettve or rregular but contnual behavor. 5.1 Formalzaton Mcro-level tmng features have to be dfferentated as those wth systematc and those wth apparently random behavor. Both can change over tme; mprecse ensemble play can, e.g., be orgnated n a stuaton wthn the composton whch s techncally dffcult to realze. As a result, both feature types are to be organzed as dated entres wthn ordered lsts M S (systematc devatons) and M R (random mprecson). q q q q 0 < < 1 q q q q = 1 q q q q > 1 rubato tme 1 r 2 r 1 0 q q metrcal tme q q 1 Systematc Devatons Rubat and constant devatons are both of systematc type, hence, summarzed nto one formalsm (m s the ndex of the M S lst entry): S m = (d m, f m, r 1m, r 2m, m, c m) In correspondence to the tempo formalsm (see Secton 4) d m ndcates the tempo-ndependent date (n MIDI tcks) from when on the mcro-level devatons are to be appled. It s termnated by d m+1. The smplest knd of devaton, the constant asynchrony, s a plan delay. Its value s gven by attrbute c m n mllseconds (c m Z). All remanng attrbutes defne the rubato devaton scheme (see Fgure 3) whch s contnuously appled to all consecutve tmeframes of length f m (n tcks). One tmeframe may, e.g., cover the length of a measure, a fgure, or even less. Snce all rubato devatons have to compensate wthn the same frame, f m also defnes the meter of overall synchrony. Consequently, the rubato devaton scheme must be a mappng of metrcal tme to rubato tme n the doman [0; f m). The potental functon n [0; 1) agan turned out to provde a suffcent behavour. So the tmeframe was scaled down to ths doman. Exponent m controls whether the tme has to: be stretched n the begnnng and compressed n the end (0 < m < 1), be accelerated n the begnnng and stretched n the end ( m > 1), or reman unchanged ( m = 1). An ntal non-negatve delay can be set by r 1m. Correspondngly, r 2m sets the arrval at the end of the frame early. Both attrbutes ndcate a relatve poston wthn the tmeframe and have to hold the followng condton: 0 r 1m < r 2m 1; r 1m, r 2m R Fgure 3: Three rubato devaton schemes for a tmeframe of four quarter notes wth dfferent settngs for. Random Imprecson Up to now, the performance s perfect regardng macro and mcro tmng. Nevertheless, human muscans are rarely able to perform all nuances perfectly. Even extensvely traned professonals are subject to psychologcal and motor condtons whch ntroduce a gaussan dstrbuton nto the accuracy of mcro tmng. Frberg s KTH rule system mplements ths performance nose by two components: the muscan s motor delay and long term tempo drfts [4]. The latter,.e., long term tempo drfts, have to be classfed as macrotmng features (see Secton 4). Human mprecson causng mcro devatons, however, have to be appled and parameterzed more carefully to measure up wth the qualtatve aspects of player s apttude. It can change over tme accordng to the muscal context and techncal dffculty. Thus, we organze these devatons as dated nformaton n an ordered lst M R. An entry R m = (d m, σ m) defnes the normal dstrbuton around the exact mllseconds date of all note events n [d m; d m+1) wth a standard devaton of σ m (n mllseconds). 5.2 Implementaton To get the defnte mllseconds date for each note event, all mcro-tmng devatons have to be added to the macro tmng. The whole sequence of date transformatons s llustrated n Fgure 4. For numercal reasons, the rubato transformaton s already appled to the tempo-ndependent tck date by - 6 -

7 event date (n tcks) Rubato Transformaton Tempo Transformaton Constant Asynchrony Random Imprecson schedule date (n ms) rubato tmng (n tcks) rubato and macro tmng (n ms) stll perfect tmng (n ms) Fgure 4: The complete date converson ppelne. functon rub(d) wth d as the tck date to be transformed. [ ( ) m lp(d) rub(d) = (r 2m r 1m) + r 1m] f m + d f f m Once agan, the ndex m refers to the entry n M S wth date d m < d d m+1,.e., the last entry before d. Functon lp(d) determnes the local poston of d wthn ts tmeframe lp(d) = (d d m)modf m and d f s the date of that tmeframe d f = d lp(d). All remanng mcro-tmng features are appled after the tempo-based converson from tcks nto mllsecond dates. The constant asynchrony c m s also a mllsecond value, and s smply added. A random mprecson wth standard devaton σ m s added to the resultng precsely tmed date. Whle all prevous transformatons were keepng the order of events, ths last step runs the rsk of causng nconsstences (e.g., a note-on event delays after ts correspondng note-off). Before addng the random devaton, a consstency check has to prevent ths stuaton. Altogether, the defnte mllseconds tmng of tck date d, tmng(d), s done as follows: tmng(d) = ms(rub(d)) + c m + rand(σ m) We already dfferentated macro tmng as global or local. Lkewse, both mcro-tmng feature maps M S and M R can be defned globally for all parts or locally per part n order to get lvely performances wth muscans who play more or less well together and exact. 6 Concluson and Future Perspectves Ths paper traced our nvestgatons nto expressve muscal performances wth regard to tmng aspects. We dentfed four dfferent classes of tmng phenomenons, namely tempo, rubato, constant devatons and random mprecson. Our evaluatons have shown that these features cannot be appled statcally. In dfferent muscal contexts and performance styles they showed very dstnctve characterstcs that we wanted to ntroduce nto computer performed musc. Therefore, we developed adequate formalzatons that we mplemented wthn a performance system. However, knowledge about the tools of expressve performances does not represent knowledge about ther applcaton. As mentoned, the characterstcs of all tmng features vary accordng to the muscal context. Thus, one drecton of our future work wll be to nvestgate the assocaton of structural and performatve features and ther characterstcs. In the context of hstorcally nformed performances our models do already provde valuable quanttatve nformaton whch allow a better nsght nto performance practces, and how they changed n hstory. Moreover, tmng s only one of many aspects that expressve musc performance deals wth. Our future research wll also nclude the analyss of dynamcs,.e. loudness, and artculaton. Acknowledgement We lke to express thanks to all muscans for ther partcpaton n the recordngs and for the nsprng dalogs. References [1] S. Dxon. Onset Detecton Revsted. In Proc. of the 9th Conf. on Dgtal Audo Effects (DAFx-06), pages 18 20, Montreal, Canada, Sept [2] C. Duxbury, J. P. Bello, M. Daves, and M. Sandler. Complex Doman Onset Detecton for Muscal Sgnals. In Proc. of the 6th Conf. on Dgtal Audo Effects (DAFx-03), London, UK, Sept [3] A. Frberg. A Quanttatve Rule System for Muscal Expresson. PhD thess, Royal Insttute of Technology, Sweden, [4] A. Frberg, R. Bresn, and J. Sundberg. Overvew of the KTH Rule System for Muscal Performance. Advances n Cogntve Psychology, Specal Issue on Musc Performance, 2(2 3): , July

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