RHYTHM TRANSCRIPTION OF POLYPHONIC MIDI PERFORMANCES BASED ON A MERGED-OUTPUT HMM FOR MULTIPLE VOICES

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1 Proceedigs SMC , Hamburg, Germay RHYTHM TRANSCRIPTION OF POLYPHONIC MIDI PERFORMANCES BASED ON A MERGED-OUTPUT HMM FOR MULTIPLE VOICES Eita Nakamura Kyoto Uiversity eakamura@sap.ist.i.kyoto-u.ac.p Kazuyoshi Yoshii Kyoto Uiversity yoshii@kuis.kyoto-u.ac.p Shigeki Sagayama Meii Uiversity sagayama@meii.ac.p ABSTRACT This paper presets a statistical method of rhythm trascriptio that estimates the quatised duratios (ote values) of the musical otes i a polyphoic MIDI performace (e.g. piao) sigal. Hidde Markov models (HMMs) have bee used i rhythm trascriptio to combie a model for music scores ad a model describig the temporal fluctuatios i music performaces. However, whe applied to polyphoic music, covetioal HMMs have a problem that they are based o represetatio of polyphoic scores as liear sequeces of chords ad thus caot properly describe the structure of multiple voices. We propose a statistical model i which each voice is described with a HMM ad polyphoic performaces are described as merged outputs from multiple HMMs, based o the framework of merged-output HMM. We develop a rhythm-trascriptio algorithm based o this model usig a efficiet Viterbi algorithm. Evaluatio results showed that the proposed model outperformed previously studied HMMs for rhythm trascriptio of polyrhythmic performaces.. INTRODUCTION Music trascriptio is a fudametal problem i music iformatio processig, requirig the extractio of pitch ad rhythm iformatio from music audio sigals. There have bee may studies o covertig a music audio sigal ito a piao-roll represetatio based o acoustic modellig of musical soud [, ]. To obtai a music score, we must recogise quatised ote legths (or ote values) of the musical otes i piao rolls. For this purpose, may studies have bee devoted to solvig the problem of covertig MIDI performaces to music scores, which is called rhythm trascriptio or quatisatio [ ]. I accordace with the geeral tred, statistical modellig has bee gatherig attetio recetly i this field. Hidde Markov models (HMMs) [] are the most popular models used i recet studies o rhythm trascriptio [5 ]. Ideed a moophoic score, whe represeted as a series of musical otes, ca aturally be described with a Markov model. I additio, temporal fluctuatios i performaces ca be described by a cotiuous-space HMM Copyright: c 6 Eita Nakamura et al. This is a ope-access article distributed uder the terms of the Creative Commos Attributio. Uported Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial author ad source are credited. with a latet variable correspodig to time-varyig tempos [, 4, 5]. Whe HMMs are used for modellig polyphoic music, we immediately face the problem of score represetatio. A polyphoic score has multilayer structure, where cocurretly soudig otes are grouped ito several streams or, i music termiology, voices. A covetioal way is to represet a polyphoic score as a liear sequece of chords [7]. However, this represetatio may ot retai sequetial regularities withi voices, such as those i polyrhythmic scores. Furthermore, properties of music performace, like the pheomeo of loose sychroy betwee voices [7, 8], caot be captured without explicitly modellig the multiple-voice structure. The purpose of this paper is to costruct a statistical model for rhythm trascriptio that ca describe the multiple-voice structure of polyphoic music scores ad performaces. We costruct a model that describes polyphoic performaces as merged outputs from multiple compoet HMMs, each of which describes the geerative process of music scores ad performaces of oe voice. Our model is based o the merged-output HMM [9, ], which has bee developed to describe, i a evet-drive maer, symbolic data of polyphoic music. We derive a efficiet iferece algorithm that ca simultaeously separate performed otes ito voices ad estimate their ote values. The proposed model is compared with previously studied HMM-based models by evaluatig the accuracy of rhythm trascriptio for piao performaces. A complete model descriptio ad exteded evaluatio results will be preseted i our forthcomig paper []. The mai cotributio of this study is the costructio of a rhythm-trascriptio algorithm that ca explicitly hadle multiple voices with guarateed optimality. A statistical model with multiple-voice structure based o twodimesioal probabilistic cotext-free grammar (PCFG) models has bee studied [, ], but the algorithms developed i those studies had to use provided voice iformatio or a pruig techique that would sacrifice optimality.. RELATED WORK I this sectio, we review previous HMM-based models for rhythm trascriptio ad discuss the problem of polyphoic extesios. I this paper, a voice meas a uit stream of musical otes that ca cotai chords. 8

2 Proceedigs SMC , Hamburg, Germay & b b b 4? b b b 4 Figure. Two differet represetatios of a music score i previously proposed HMMs.. HMM-Based Models for Moophoic Music HMMs for rhythm trascriptio usually cosist of two compoet models; a score model describig the probability of a score ad a performace model describig the probability of a performace give a score. HMMs i previous studies [5 ] ca be classified ito two groups accordig to the way the score model describes the sequece of otes. I oe class of HMMs for rhythm trascriptio, which we call ote HMMs, a score is represeted as a sequece of ote values ad described with a Markov model (Fig. ) [5, 6]. To describe the temporal fluctuatios i performaces, oe itroduces a latet variable correspodig to a (local) tempo that is also described with a Markov model. A observed duratio is described as a product of the ote value ad the tempo that is exposed to oise of oset times. I aother class of HMMs, which we call metrical HMMs, a differet descriptio is used for the score model [8 ]. Istead of a Markov model of ote values, a Markov process o a grid space represetig beat positios of a uit iterval, such as a bar, is cosidered (Fig. ). The ote values are give as differeces betwee successive beat positios. The same performace model as i ote HMMs ca be used. Icorporatio of the metre structure is a advatage of metrical HMMs.. Polyphoic Extesios There are two directios of polyphoic extesios: usig a simplified represetatio of polyphoic scores or usig a exteded model describig multiple voices. The first directio is based o a fact that ay polyphoic score ca be represeted as a sequece of chords or, more precisely, ote clusters cosistig of oe or more otes as far as oly osets are cocered. For ote HMMs, chordal otes ca be represeted as self-trasitios i the score model (Fig. ) ad their iter-oset itervals (IOIs) ca be described with a probability distributio with a peak at zero [7]. Similar extesios are possible for metrical HMMs. For the secod directio, a PCFG model has bee exteded to describe the multiple-voice structure of scores []. I additio to the divisios of a time iterval, duplicatios of itervals ito two voices are cosidered. Ufortuately, a tractable iferece algorithm could ot be obtaied for this model, ad the correct voice iformatio had Figure. A polyrhythmic passage (Chopi s Fataisie Impromptu) represeted as a sequece of chords. s / 4 x x x x 4 x 5 x 6 x 7 4 Figure. A schematic illustratio of the merged-output HMM. The symbols ad represet auxiliary states to defie the iitial trasitios. to be provided for evaluatios. Takamue et al. state that this problem is solved usig the geeralised LR parser []. Although detailed explaatios are lackig, their method uses pruig ad its optimality is ot guarateed. Although the above two descriptios of polyphoic scores are both logically possible, there are istaces i which models based o the simplified represetatio caot describe the ature of polyphoic music well. First, complex polyphoic scores such as polyrhythmic scores are forced to have urealistically small probabilities. This is because such scores cosist of rare rhythms i the simplified represetatio eve if the compoet voices have commo rhythms (Fig. ). Secod, the pheomeo of loose sychroy betwee voices (e.g. two hads i piao performaces [7]), called voice asychroy, caot be described. Ideed, the importace of icorporatig the multiple-voice structure i describig polyphoic music is well-established i studies o score-performace matchig [7, 8]. The situatio calls for a similar treatmet of multiple voices for polyphoic rhythm trascriptio.. Merged-Output HMM Recetly merged-output HMM has bee proposed as a HMM-based model for describig symbolic sigals of polyphoic music with multiple voices. I the model, each voice is described with a HMM ad the total sigal is represeted as merged outputs from these HMMs (Fig. ). The merged-output HMM ca be see as a variat of factorial HMM []. To appropriately describe the ature of symbolic sigals ad capture sequetial regularities withi each voice, oly oe of the compoet HMMs is ivolved with each output i a merged-output HMM, whereas all compoet HMMs cotribute to every output i a stadard 9

3 Proceedigs SMC , Hamburg, Germay factorial HMM. Basic iferece algorithms for mergedoutput HMMs have bee provided i our previous studies [9, ].. PROPOSED MODEL We preset a HMM-based model for rhythm trascriptio that describes polyphoic performaces with multiplevoice structure. Give a polyphoic MIDI performace sigal, the model ca simultaeously separate performed otes ito voices ad estimate their ote values. To costruct a model based o a previously studied HMM [7] ad apply the framework of merged-output HMM [9, ], we address the followig issues: () pitches should be explicitly modelled to appropriately describe voices; () tempos of multiple voices should be boud to assure loose sychroy betwee voices. After explaiig the ote HMM i detail i Sec.., a model satisfyig these requiremets is preseted i Sec.., ad a sketch of iferece algorithm is give i Sec... A music score is specified by multiple sequeces, correspodig to voices, of pitches ad ote values. Sice polyrhythm ad voice asychroy typically ivolve two voices, we formulate the model with two voices idexed by a variable s =,. A MIDI performace sigal is specified by a sequece of pitches ad oset times.. Model for Each Voice For each voice we first costruct a model based o the oe preseted i a previous study [7]. Let N s be the umber of score otes i voice s ad let r (s) deote the ote value of the -th ote. The ote values = (r (s) ) Ns = are geerated by a Markov chai with the probability give as Cat( (s) ), () ii Cat( (s) ) ( =,...,N s ), () where Cat deotes the categorical distributio, (s) ( (s) ii,r ) r is the iitial probability, ad (s) ii = =( (s),r) r is the (statioary) trasitio probability. Chordal otes are represeted as self-trasitios of ote values (Fig. ). The probability values are to be leared from music data. To describe the temporal fluctuatios, we itroduce a tempo variable, deoted by v (s), that describes the local tempo for the -th ote. To represet the variatio of tempos, we put a Gaussia Markov process o the logarithm of the tempo variables as l v (s) l v (s) N(l v(s), v ), () where N deotes the ormal distributio. If the ( )-th ad -th otes belog to a chord, their IOI approximately obeys a expoetial distributio [5] ad the probability of the oset time of the -th ote, deoted by t (s), is the give as t (s) t (s) Exp( ), (4) where Exp deotes the expoetial distributio ad is the scale parameter. Otherwise, t (s) t (s) has a duratio correspodig to ote value ad the probability is described with a ormal distributio as t (s) t (s),v(s),r(s) N(t(s) + r(s) v(s) ; t ). (5) The measured values of the parameters are t =. s ad =. s [5] (the value of v will be explaied later). Remarks should be made here: First, the umber of observed osets must be N s + so that there are N s IOIs correspodig to N s score otes. Secod, we do ot put a distributio o the oset time of the first ote t (s) because we formulate the model to be ivariat uder time traslatios ad this value would ot affect ay results of iferece. We will use the otatio v (s) =(v (s) ) Ns = ad t (s) =(t (s) ) Ns+ =. Fially we describe the geeratio of pitches = ( ) Ns+ = as a Markov chai (we itroduce a auxiliary symbol for later coveiece). The probabilities are p(s) where (s) (s) Cat( (s) ), (6) =( (s) Cat( (s) ) ( =,...,N s +), (7) = ( (s),p) p is the iitial probability, ad,p) p is the (statioary) trasitio probability. These parameters are to be leared from music data. The above model ca be summarised as a autoregressive HMM, which we call a voice HMM, with hidde states (, v (s) ) ad outputs (, t (s) ). Although so far the probabilities of pitches are idepedet of other variables, they will be sigificat oce multiple voice HMMs are merged ad the posterior probabilities are iferred.. Model for Multiple Voices We combie the multiple voice HMMs i Sec.. usig the framework of merged-output HMMs [9]. Simply speakig, the sequece of merged outputs is obtaied by gatherig the outputs of the voice HMMs ad sortig them accordig to oset times. To derive iferece algorithms that are computatioally tractable, however, we should formulate a model that outputs otes icremetally i the order of observatios. This ca be doe by itroducig stochastic variables s =(s ) N+ =, which idicate that the -th observed ote belogs to voice s, with the followig probability: s Ber(, ), (8) where Ber is the Beroulli distributio. s represets how likely the -th ote is geerated from the HMM of voice s ad, to improve the results of voice separatio, we put o the parameter coditioal depedece o the lowest ad highest pitches of simultaeously soudig otes. If voice s is chose, the the HMM of voice s outputs a ote, ad the hidde state of the other voice HMM is uchaged. Such a model ca be described with a HMM with a state space labelled by k = (s,p (),r (),t (),p (),r (),t (),v ). Here we have a sigle tempo variable v that is shared by the two voices i order to assure loose sychroy betwee them. P (k k ), 4

4 Proceedigs SMC , Hamburg, Germay for s P (v v h, is give as s () r r() )A (s) r(s) p () where we have defied A (s) r(s) = (s) (,t (s) (s),r(s) p() (,t (s),t(s) (t () t () )+($ ),t(s) ; v ),p(s) ; v ) i, (9) P (t (s) t (s),v, ) () ad deotes Kroecker s delta for discrete variables ad Dirac s delta fuctio for cotiuous variables. The probability P (v v ) is defied i Eq. (), ad P (t (s) t (s),v,r (s) ) is defied i Eqs. (4) ad (5). For ote values the iitial probability is give as Cat( (s) ii ), ad for pitches the iitial probability is set as i Eq. (6). The first oset times t () ad t () do ot have distributios, as explaied i Sec.., ad we practically set t () = t () = t where t is the first observed oset time. Fially the output of the model is give as p =, t = t (s), () ad thus the complete-data probability is writte as P (k, p, t) = Y P (k k ) p N = N + N deotes the total umber of score otes, ad the followig otatios will be used: v = (v ) N =, p = (p ) N+ =, t = (t ) N+ =, ad k = (t t (s) ). () (k ) N+ =. Note that whereas p ad t are observed quatities, p (), p (), t (), t () are ot because we caot directly observe the voice iformatio ecrypted i s.. Iferece Algorithm Rhythm trascriptio based o the proposed model ca be performed by estimatig the most probable hidde state sequece ˆk give the observatios (p, t). Oce ˆk is obtaied, we ca extract the voice iformatio ŝ ad the ote values ˆr () ad ˆr (). These are the result of voice separatio ad rhythm trascriptio. The maximisatio of the probability P (k p, t) ca be i priciple doe with the Viterbi algorithm []. However, due to the complexity of our model, we eed refiemets to the stadard Viterbi algorithm to derive a computatioally tractable algorithm. First, sice the state space of the merged-output HMM i Sec.. ivolve both discrete ad cotiuous variables, a exact iferece is ot computatioally tractable. To solve this problem, we discretise the tempo variable i a rage that is commo i music practice. Other cotiuous variables t, t (), ad t () ca take oly values of observed oset times ad thus ca, i effect, be treated as discrete variables. Secod, it appears that a Viterbi algorithm derived i the way proposed i [9] has rather large computatioal cost for the preset model ad i practice difficult to execute. The large computatioal cost derives from the fact that we eed to model pitches ad oset times for the voice HMMs. This problem ca be reduced by otig that the pitch ad oset time are observed quatities ad ca be represeted by a variable describig the historical iformatio of voices associated to otes, as suggested i []. Extedig the formalism of itroducig a latet variable to describe this iformatio, we ca derive a efficiet algorithm. Details will be give i our forthcomig paper []. We have cofirmed that this algorithm ca be executed i a stadard moder computer eviromet with a practical time (withi a few hours for a performace with hudreds of otes). 4. Setup 4. EVALUATION We evaluated the proposed model by comparig the accuracy of its rhythm trascriptio with that of previously studied models based o HMMs. Two data sets of MIDI recordigs of classical piao pieces were used. Oe ( polyrhythmic data set) cosisted of 8 performaces of 5 (excerpts of) pieces that cotaied agaist or agaist 4 polyrhythmic passages, ad the other ( stadard polyphoy data set) cosisted of performaces of pieces that did ot cotai polyrhythmic passages. Pieces by various composers, ragig from J. S. Bach to Debussy, were chose ad the players were also various: Some of the performaces were take from the PEDB database [], a few were performaces we recorded, ad the rest was take from public domai websites. All ormal, dotted, ad triplet ote values ragig from the whole ote to the d ote were used as cadidate ote values. The trasitio ad iitial probabilities of the ote values ad pitches, ad the value of s, were leared from a data set of classical piao scores that had o overlap with the test data. For the tempo variable, we discretised v ito 5 values logarithmically equally spaced i the rage of. to.5 sec per quarter ote (correspodig to BPM ad 4 BPM). The stadard deviatio i Eq. () was set as v =.8, usig the value i [5] as a referece. For compariso, we implemeted the ote HMM [6] ad the metrical HMM [8] that is exteded to hadle polyphoy. The parameters of the score models were also traied with the same score dataset. The performace model was the same as that for the proposed model. We used as a evaluatio measure the rhythm correctio ratio, i.e., the ratio of the smallest umber of edit operatios eeded to correct the estimated result to the umber of otes i the data. I additio to ote-wise correctio (shift operatio), the scalig operatio applied for a subsequece of ote values was icluded. This is because there is arbitrariess i choosig the uit of ote values: For example, a quarter ote played i a tempo of 6 BPM has the same duratio as a half ote played i a tempo of BPM. The smallest umber of ecessary edit operatios N e ca be calculated by a dyamic programmig similar to that used i computatio of the Leveshtei distace (see our forthcomig paper [] for details). The rhythm correctio ratio R is the give as R = N e /N. Whe separated voices are give, we ca apply the above editig of ote values for each voice ad the the total rhythm correctio cost is the sum of the rhythm correctio costs i all voices. 4

5 Proceedigs SMC , Hamburg, Germay Data set Model R [%] Polyrhythmic Proposed 6. ±.6 Note HMM [6] 8.9 ± 4.9 Metrical HMM [8] 4. ± 5. Stadard polyphoy Proposed 7.9 ±. Note HMM [6] 7. ±. Metrical HMM [8] 7.9 ±.4 Table. Average rhythm correctio rates R with stadard errors. Lower is better. 4. Results Results i Table show that the proposed model clearly outperformed the other models for performaces with polyphoic passages. Fig. 4 shows a example that a polyrhythmic passage is successfully trascribed with the proposed model with mior errors. We see that the proposed model correctly recogised the agaist 4 polyrhythms. O the cotrary, the Note HMM did ot recogise the polyrhythms (cf. Fig. ) ad had frequet errors i chord clusterig. For performaces i stadard polyphoy, o the other had, the ote HMM was slightly better tha the proposed model ad the metrical HMM. Presumably, the mai reaso is that the rhythmic patter i the reduced sequece of chords is ofte simpler tha that of melody/chords i each voice i the case of stadard polyphoy because of the priciple of complemetary rhythm [4]. I particular, otes/chords i a voice ca have tied ote values that are ot cotaied i our cadidate list (e.g. quarter ote + 6th ote value), which ca also appear as a result of icorrect voice separatio (Fig. 5). It is also observed that the trascriptio by the merged-output HMM ca produce desychroised cumulative ote values i differet voices. This is due to the lack of costraits to assure the matchig of these cumulative ote values ad the simplificatio of idepedet voice HMMs. Further improvemets are expected by icorporatig such costraits ad iteractios betwee voices ito the model. For the ote HMM ad the proposed model, there were grammatically wrog sequeces of ote values, for example, triplets that appear i sigle or two otes without completig a uit of beat. This ca be avoided with a refied score model with beat/bar structure [6, ]. O the other had, these grammatical errors were ot observed i the trascriptios by the metrical HMM owig to the explicitly icluded metrical structure. 5. CONCLUSION To develop a rhythm trascriptio algorithm that captures the voice structure, we costructed a stochastic model of musical score ad performace usig the framework of merged-output HMMs. The evaluatio results cofirmed that the proposed algorithm worked better for polyrhythmic performaces tha the previously proposed HMMbased algorithms. Soud files ad more examples are accessible i our demostratio web page: demo.html A importat future directio of developig advaced trascriptio techiques is to capture the phrase or motivic structure of music. Recogitio of offsets ad articulatios ad detectio of oramets are challegig problems. The treatmet of voice structure is a fudametal problem for these issues, ad the results of this study may be applicable to solvig these problems. Ackowledgmets This work is partially supported by JSPS KAKENHI Nos. 46, 645, 6889, 67, 5K654, 6H744 ad 6J5486, JST OgaCREST Proect ad Kayamori Foudatio. E. Nakamura is supported by the JSPS fellowship program. 6. REFERENCES [] A. Klapuri ad M. Davy (eds.), Sigal Processig Methods for Music Trascriptio, Spriger, 6. [] E. Beetos et al., Automatic Music Trascriptio: Challeges ad Future Directios, J. Itelliget Iformatio Systems, vol. 4, o., pp ,. [] H. Loguet-Higgis, Metal Processes: Studies i Cogitive Sciece, MIT Press, 987. [4] P. Desai ad H. Hoig, The Quatizatio of Musical Time: A Coectioist Approach, Comp. Mus. J., vol., o., pp , 989. [5] T. Otsuki et al., Musical Rhythm Recogitio Usig Hidde Markov Model (i Japaese), J. Iformatio Processig Society of Japa, vol. 4, o., pp ,. [6] H. Takeda et al., Hidde Markov Model for Automatic Trascriptio of MIDI Sigals, Proc. MMSP, pp. 48 4,. [7] H. Takeda et al., Rhythm ad Tempo Aalysis Toward Automatic Music Trascriptio, Proc. ICASSP, vol. 4, pp. 7, 7. [8] C. Raphael, Automated Rhythm Trascriptio, Proc. ISMIR, pp. 99 7,. [9] M. Hamaaka et al., A Learig-Based Quatizatio: Usupervised Estimatio of the Model Parameters, Proc. ICMC, pp. 69 7,. [] A. Cemgil ad B. Kappe, Mote Carlo Methods for Tempo Trackig ad Rhythm Quatizatio, J. Artificial Itelligece Res., vol. 8 o., pp. 45 8,. [] M. Tsuchiya et al., Probabilistic Model of Two- Dimesioal Rhythm Tree Structure Represetatio for Automatic Trascriptio of Polyphoic MIDI Sigals, Proc. APSIPA, pp. 6,. [] N. Takamue et al., Automatic Trascriptio from MIDI Sigals of Music Performace Usig - Dimesioal LR Parser (i Japaese), Tech. Rep. SIG- MUS, vol. 4-MUS-4, o. 7, pp. 6, 4. [] L. Rabier, A Tutorial o Hidde Markov Models ad Selected Applicatios i Speech Recogitio, Proc. IEEE, vol. 77, o., pp ,

6 Proceedigs SMC , Hamburg, Germay & C? C &? &? Figure 4. Trascriptio results of a polyrhythmic passage. For the result with the proposed model (merged-output HMM), the staffs idicate the estimated voices. & c? c Ó J J & & & w Figure 5. Trascriptio results of a stadard polyphoic passage. For the result with the proposed model (merged-output HMM), the staffs idicate the estimated voices. [4] C. Raphael, Automatic Segmetatio of Acoustic Musical Sigals Usig Hidde Markov Models, IEEE Tras. o PAMI, vol., o. 4, pp. 6 7, 999. [5] E. Nakamura et al., A Stochastic Temporal Model of Polyphoic MIDI Performace with Oramets, J. New Music Res., vol. 44, o. 4, pp. 87 4, 5. [6] A. Cot, A Coupled Duratio-Focused Architecture for Realtime Music to Score Aligmet, IEEE Tras. o PAMI, vol., o. 6, pp ,. [7] H. Heiik et al., Data Processig i Music Performace Research: Usig Structural Iformatio to Improve Score-Performace Matchig, Behavior Research Methods, Istrumets, & Computers, vol., o. 4, pp ,. [8] B. Gigras ad S. McAdams, Improved Score- Performace Matchig Usig Both Structural ad Temporal Iformatio from MIDI Recordigs, J. New Music Res., vol. 4, o., pp. 4 57,. [9] E. Nakamura et al., Merged-Output Hidde Markov Model for Score Followig of MIDI Performace with Oramets, Desychroized Voices, Repeats ad Skips, Proc. Joit ICMC SMC 4, pp. 85 9, 4. [] E. Nakamura et al., Merged-Output HMM for Piao Figerig of Both Hads, Proc. ISMIR, pp. 5 56, 4. [] Z. Ghahramai ad M. Jorda, Factorial Hidde Markov Models, Machie Learig, vol. 9, pp. 45 7, 997. [] M. Hashida et al., A New Music Database Describig Deviatio Iformatio of Performace Expressios, Proc. ISMIR, pp , 8. [] E. Nakamura et al., i preparatio. [4] F. Salzer ad C. Schachter, Couterpoit i Compositio: The Study of Voice Leadig, Columbia Uiversity Press,