Vol. 43 No. 2 Feb. 2002,, MIDI A Probabilistic-model-based Quantization Method for Estimating the Position of Onset Time in a Score Masatoshi Hamanaka

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1 Vol. 43 No. 2 Feb. 2002,, MIDI A Probabilistic-model-based Quantization Method for Estimating the Position of Onset Time in a Score Masatoshi Hamanaka, Masataka Goto,, Hideki Asoh and Nobuyuki Otsu, This paper describes a method for organizing onset times performed along a jam-session accompaniment into the normalized (quantized) positions in a score. The purpose of this study is to align onset times of a session recording to the quantized positions in order to store the performance data in a reusable form. Unlike most previous beat-tracking-related methods focusing on predicting or estimating beat positions, our method deals with the problem of eliminating the onset-time deviations under the condition that the beat positions are given. To quantize polyphonic MIDI recordings of jam session, we propose a method of using Hidden Markov Model for modeling onset-time transition and deviation. The main advantage of this method is to quantize a performance by using a model that is statistically learned from session recordings of the same player. Experimental results showed that the model that statistically learned MIDI recordings was effective enough to surpass the performance of semi-automatic quantization of commercial sequence software. 1. 1) Doctoral Program in Engineering, University of Tsukuba 21 Information and Human Activity, PRESTO, JST National Institute of Advanced Industrial Science and Technology (AIST)

2 Vol. 43 No quantize ) 3) 4) 8) 2) 3) 4) 8) 9) HMM 9) MIDI y P (y ) Bayes (1) P (y )P ( ) P ( y) = (1) P (y) P ( ) 4) 6) 9)

3 236 Feb Fig. 1 Forward model and inverse model in the quantization problem. ˆ (1) (2) ˆ = argmax P ( y) θ = argmax P (y )P ( ) (2) θ 2.2 P (y ) P ( ) HMM P ( ) P (y ) (1) 9) (2) (3) 2 k 1 1/480 k i /12 i HMM Start End Start End 16 4 Start End 2 Start 1 1 End Start End HMM HMM

4 Vol. 43 No Fig Overview of the quarter-note Hidden Markov Model. ii 1 12 kk a ij i j a is =0a Ei =0 i>j a ij =0 b i (k) i k Start End S HMM π i π S =1 π i (i =1, 2,..., 12, E) = 0 2 HMM HMM 3(a) HMM 3 1 HMM 4 HMM Fig. 3 A model consisting of a single HMM and a model consisting of four different HMMs. HMM 3 1 HMM HMM 4 HMM HMM 9)

5 238 Feb Start End HMM P ( y) Viterbi T y =(y 1,y 2,..., y T ) =(θ 1,θ 2,..., θ T ) y t 1 1/480 θ t 1 12 δ t (i) = max P [θ 1,θ 2,..., θ t 1 θ 1,θ 2,...,θ t 1 ; θ t = i; y 1,y 2,..., y t λ] (3) δ t (i) t i λ AmodBA B n End Start 0 δ t+1 (j)=max i [δ t (i)a ij ]b j (y t+1 mod 480) (4) A ij = a ij (n =0) n A ij = a ie a Sj a SE (n >0) Viterbi End Start (4) δ t (i) 4 Fig. 4 A trellis for finding the maximum likelihood state transition. 2 q t r tt+1 2 q t q t ( θt ) 1 q t = 480 (5) 12 r tt+1 r tt+1 2 r tt+1 (n) =(q t+1 q t )+n 480 (6) n End Start 2 ˆn y t+1 y t r tt+1 n (7) ˆn = argmin n {(y t+1 y t ) r tt+1 (n)} (7) 5 1q t =0 1q t+1 =0 (6)

6 Vol. 43 No Fig. 5 An example of state transition for a dotted half note. 6 4 HMM Fig. 6 An example of a search tree when using four HMMs. q t+1 q t 0 (7) ˆn = argmin n {(y t+1 y t ) n 480} (8) 1200 y t+1 y t 1680 ˆn End Start 4 HMM 4 HMM Start End 1 3. y P ( ) a ij P (y ) b j (k) (1) a 0 σ y a σ = =480 3 (2) 3 ABC MIDI y h 12 1 = (3) a h Twelve Tone System Cakewalk Pro Audio

7 240 Feb y a ij b j (k) b j (k) 51 =480 7 (a) σ =201 =480 b 5 (k) (b) C 2 C2 b 5 (k) b 5 (k) (a) b 5 (k) (b) 5 C C2 4 HMM a ij C HMM 4 HMM HMM 1 4 AB 2 C () = () () (9) Fig. 7 7 b 5 (k) b 5 (k) obtained from artificial data and humanperformance data. Fig. 8 8 C2 a ij a ij obtained from C2-performance data. Table Percentages of four kinds of beat included in correct data. A B C σ =10 σ =20 σ =30 A1 A2 B1 B2 C1 C % 38.8% 38.8% 21.4% 5.5% 6.5% 2.7% 37.2% 0.0% % 37.8% 37.8% 54.2% 68.7% 58.2% 85.8% 42.1% 39.7% 6.9% 6.9% 6.9% 6.8% 10.1% 11.9% 6.4% 5.9% 26.7% 16.4% 16.4% 16.4% 17.6% 15.7% 23.3% 5.1% 14.8% 33.8%

8 Vol. 43 No Table 2 Comparison between a commercial sequence software and our method (a percentage of correct onsets). A B C σ =10 σ =20 σ =30 A1 A2 B1 B2 C1 C % 58.2% 62.0% 67.6% 85.6% 79.4% 88.6% 57.0% 97.7% % 67.9% 65.9% 54.5% 37.3% 36.8% 34.7% 70.7% 45.5% % 70.7% 60.8% 57.7% 48.4% 57.8% 51.3% 56.1% 82.5% 100% 99.7% 96.3% 74.9% 66.0% 63.3% 48.1% 81.0% 77.6% 1 HMM 99.6% 95.9% 86.5% 75.9% 84.8% 80.0% 90.5% 85.1% 95.0% 4 HMM 99.5% 95.9% 86.5% 82.3% 89.8% 81.4% 92.8% 85.5% 95.7% () = () () (10) (1) A1 C1 16 C (2) σ = %σ =30 96% A1 C1 8 3 (3) HMM HMM 2 HMM HMM 5 HMM b j (k) 1 HMM b j (k) σ =10 1 b j (k) HMM 100%

9 242 Feb HMM 4 HMM Table 3 Comparison between the one-hmm model and the four-hmm model (a percentage of correct beats). A B C σ =10 σ =20 σ =30 A1 A2 B1 B2 C1 C2 1 HMM 98.6% 93.8% 81.3% 79.0% 85.4% 84.2% 91.9% 83.9% 94.9% 4 HMM 98.6% 93.8% 81.3% 84.9% 89.2% 87.5% 93.5% 84.7% 97.3% Table 4 4 Quantization results by using the model parameters for other performances. A B C σ =10 σ =20 σ =30 A1 A2 B1 B2 C1 C % 85.3% 79.4% 91.6% 55.5% 77.9% % 79.0% 59.2% 73.5% 76.9% 93.1% % 53.4% 60.8% 55.2% 82.5% 86.1% 1 HMM 4 HMM 4 HMM b 1 (k) 4 HMM 1 HMM HMM 4 HMM 1 3 b 1 (k) b 1 (k) 9 C HMM 1 HMM 4 HMM 9 Fig. 9 An example of quantization results. (4) HMM HMM σ = HMM 1 1 AB 6.4% C 20%2 AB 2 C AB

10 Vol. 43 No AB 1 C HMM HMM 4 HMM 32 HMM 1) 2000-MUS-34, Vol.2000, No.19, pp (2000). 2) Desain, P. and Honing, H.: The Quantization of Music Time: A Connectionist Approach, Computer Music Journal, Vol.13, pp (1989). 3) Vol.5, No.1, pp (1990). 4) AI SIG-Challenge , pp.7 14 (1998). 5) Goto, M. and Muraoka, Y.: An Audio-based Real-time Beat Tracking System and Its Applications, Proc. ICMC, pp (1998). 6) bit pp (1998). 7) Dannenberg, R. and Mont-Reynaud, B.: Following an Improvisation in Real Time, Proc. ICMC, pp (1987). 8) Allen, P. and Dannenberg, R.: Tracking Musical Beats in Real Time, Proc. ICMC, pp (1990). 9) 99-MUS-33, Vol.99, No.106, pp (1999). ( ) ( ) SCI5th World Multiconference on Systemics, Cybernetics and Informaticsin Art

11 244 Feb jus 10 UNIX 1993 NICOGRAPH 93 CG WISS IEEEICMAISCA

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