A Vocabulary-Free Infinity-Gram Model for Chord Progression Analysis

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Ἰοκάστη Αντωνόπουλος
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1 1 1 n n n n n n A Vocabulary-Free Infinity-Gram Model for Chord Progression Analysis Kazuyoshi Yoshii 1 and Masataka Goto 1 This paper presents a novel nonparametric Bayesian n-gram model as a statistical language model for symbolic chord sequences. Standard n-gram models based on heuristic smoothing have three fundamental problems that they have no theoretical foundation, that the value of n is fixed uniquely, and that a vocabulary of chord types is defined in an arbitrary way. To solve these problems, we propose a vocabulary-free infinity-gram model. It accepts any combinations of notes as chord types and allows each chord appearing in a sequence to have an unbounded and variable-length context. Our experiments showed that the perplexity obtained by the proposed model is significantly lower than that obtained by the state-of-the-art models. 1 (AIST) 1. 1) 2) 3) 6) 5),6) 7) 9) n 1),2),4),6) 9) n n n n 1 n 1 n 1 n n n 0 n 10) n 1) n 2) n 3) ( major, minor, augmented, diminished, seventh, ninth,, ) 3) 1 c 2011 Information Processing Society of Japan

2 1 G:7 G:7 P() P( φ) P( G:7) P( G:7) n n n n n 11) 14) 1 n 1) 2) 3) 2 3 n Major maj Minor min Diminished dim Augmented aug Major Seventh maj Minor Seventh min Seventh Dim. Seventh dim Half Dim. Seventh hdim Min. (Maj. Seventh) minmaj Major Sixth maj Minor Sixth min Ninth Major Ninth maj Minor Ninth min Suspended Second sus Suspended Fourth sus Harte 5) ),4) C# Db 1 C major Gb diminished seventh F#:dim7 N 205 ( ) 1 12 {C,C#,D,D#,E,F,F#,G,G#,A,A#,B} 2 c 2011 Information Processing Society of Japan

3 2 2.2 n 12 C major C,E,G C: D major D: D: C major F C: N n 2 n 3 n 3.1 W V w W n 1 u W n 1 X M x 1 x M x m W (1 m M) 1 n n x m n 1 P (x m x 1 x m 1 )=P (x m x m (n 1) x m 1 ) (1) X u w P u (w X) X u w c uw P u (w X) Pu ML (w X) = c uw (2) c u ( ) c u = w c uw n uw X c uw = Kneser-Ney n Kneser-Ney (KN) 15) n 10) c uw 1 d u P u (w X) d u u Interpolated KN (I-KN) w n n 1 P IKN u (w X) = max(c uw d u, 0) c u + d u t u c u P IKN π(u)(w X) (3) t u X u W π(u) u. P IKN π(u) (w) w n 1 (3) π(u) u n w 1/V d 0,,d n 1 15) 3.3 Pitman-Yor Teh 11) Pitman-Yor (Hierarchical Pitman-Yor Language Model: HPYLM) n 3 c 2011 Information Processing Society of Japan

4 I-KN HPYLM HPYLM I-KN Pitman-Yor Pitman-Yor (PY) 16) PY (DP) DP W PY W n d θ G 0 PY G PY(d, θ, G 0 ) (4) d θ G 0 PY G θ G 0 G HPYLM PY W G φ φ 0 G φ (w) w 1 u G u G φ G φ G u G φ PY G u PY(d 1,θ 1,G φ ) d 1 θ 1 1 n 1 n G u G π(u) PY G u PY(d u,θ u,g π(u) ) (5) d u θ u u n 1 G π(u) n 1 G π(u) d π(u) θ π(u) n 2 G π(π(u)) PY G φ G φ PY(d 0,θ 0,G 0 ) (6) G 0, 3 φ D:min D:min G:7 Pitman-Yor CRF: G 0 (w) =1/V. G 0 n HPYLM n 1 3 n = HPYLM X X x m (Chinese Restaurant Franchise: CRF) 11) CRF X M W V 4 c 2011 Information Processing Society of Japan

5 x 1 x M m 1 x 1 x m 1 m x m x m n 1 u = x m (n 1) x m 1 3 u w t uw t u w k (1 k t u ) c uwk k w c uwk =0 u x m (i) c uwk d u k (1 k t u ) (ii) d u t u + θ u k = t u +1(i) x m k w x m m w c uwk (ii) k w π(u) π(u) x m w u k w x m w x m m w t uw c uw k π(u) π(π(u)) φ φ G 0 S u w Pu HPY (w S) = c uw d u t uw c u + θ u + d u t u + θ u P c u + θ π(u)(w S) HPY (7) u (7) P HPY π(u) (w S) (7) u π(u) c uwk =0 t uw =0 M X (7) (3) θ u =0 t uw =1 HPYLM I-KN m =1: M xm n 1 4 i = 1: m =1: M xm xm n 1 Pitman-Yor u w P u (w X) X S (7) P (S X) Pu HPY (w S) Pu HPY (w X) = Pu HPY (w S)P (S X) (8) S (MCMC) Pu HPY (w X) 1 L Pu HPY (w S l ) (9) L i=l L p(s X) l 4 CRF (Exchangeability) S p(s X) CRF x m = w x m w w (7) 5 c 2011 Information Processing Society of Japan

6 (MCMC) L d 0,,d n 1 θ 0,,θ n 1 0 <d u < 1 0 <θ u 11) 3.4 Pitman-Yor HPYLM n 1 n 1 12) Pitman-Yor (Variable-order Pitman-Yor Language Model: VPYLM) VPYLM x m n z m z m z m n n VPYLM x m n z m n (Stick-Breaking Process: SBP) 17) z m SBP x m φ x m 1 x m 2 i 1(1 i ) u i 1 η ui 1 1 η ui 1 n 1 z m = n n 1 P u (n η) =η un 1 (1 η ui 1 ) (10) i=1 n u η u 0 <η u < 1 α β p(η) = Beta(η u α, β) (11) u tree VPYLM HPYLM X m x m z m (10) z m x m x m CRF HPYLM n 1 VPYLM c uwk S η u n 1 w Pu VPY (w, n S, η) =Pu VPY (w n, S)P u (n η) (12) Pu VPY (w n, S) HPYLM (7) P u (n η) (10) (12) n η Pu VPY (w S) = Pu VPY (w n, S)P u (n S) (13) n n P u (n S)= Pu (n η)p(η S)dη (10) (11) n 1 a un 1 + α b ui 1 + β P u (n S) = (14) a un 1 + b un 1 + α + β a i=1 ui 1 + b ui 1 + α + β a ui 1 b ui 1 S u i 1 6 c 2011 Information Processing Society of Japan

7 3.4.3 u w 4 HPYLM x m z m x m x m = w z m = n P u (n S,w) P u (w, n S) =P VPY u (w n, S)P u (n S) (15) 3.5 Pitman-Yor n 13),14) G 0 Pitman-Yor (Nested Pitman-Yor Language Model: NPYLM) G 0 (w) =1/V V G 0 (w) 0 VPYLM VPYLM CRF w G 0 (w) 2 4. NPYLM 14) 4.1 n NPYLM NPYLM G 0 VPYLM G V = n G w w 0 :w 1 w 12 w 0 w =N w 0 =N w 0 13 G 0 (w) =p(w π, τ )=π w0 12 i=1 τ w i i (1 τ i ) 1 w i (16) π = {π C,π C#,,π B,π N } N τ = {τ 1,,τ 12 } w =N G 0 (w)=π N π τ p(π, τ )=Dir(π a 0 ) 12 i=1 Beta(τ i b 0,c 0 ) (17) 13 a 0, b 0 c VPYLM u w G 0 (w) (16) π τ S π τ (16) (17) 7 c 2011 Information Processing Society of Japan

8 p(π, τ S) =Dir(π a 0 + n) 12 i=1 Beta(τ i b 0 + n i,c 0 + n i ) (18) 13 n n v (v {C, C#,, B, N}) n v φ v w (w 0 = v) n i i w (w i =1) n i i w (w i =0) VPYLM φ n v n i n i n v n i n i π τ (18) 5. n Harte 5) ,761 VPYLM n Good-Turing (GT) Witten-Bell (WB) I-WB KN I-KN Modified KN (MKN) I-MKN HPYLM VPYLM GT WB 10) MKN KN (3) d u c uw 10) I- n 7 The SRI Language Modeling Toolkit (SRILM) 18) n GT WB I-WB KN I-KN MKN I-MKN HPYLM VPYLM (±0.03) (±0.10) n: (±0.18) 13.4 (±0.33) (±0.25) (±0.23) n: MAP (±0.23) 12.9 (±0.35) (±0.23) (±0.22) n: ( ) (±0.23) 11.9 (±0.22) (±0.23) HPYLM VPYLM 4 VF- VF-VPYLM VF-HPYLM n VPYLM n n n HPYLM VPYLM VPYLM HPYLM VPYLM I-KN HPYLM HPYLM I-KN I-MKN HPYLM HPYLM 11) 8 c 2011 Information Processing Society of Japan

9 Intro Verse A 5 n = n = Verse B Chorus E:min LetItBe n Let It Be VPYLM n 5 VF-VPYLM n n 1 u w P u (w, n X)= P S u(w, n S)P (S X) C:7 F:7 C:7 seventh 4 VF-VPYLM HPYLM VPYLM 4 3 (n 3 ) C:7 F:7 C: B:maj C: F:min E:min C: D:min7 E:min B:maj D:min7 G:sus D:min n GT WB I-WB KN I-KN MKN I-MKN n HPYLM VF-HPYLM (±0.29) 16.5 (±0.60) 6. n VPYLM VF-VPYLM 15.8 (±0.29) 14.6 (±0.55) n n n major minor (1967 ) c 2011 Information Processing Society of Japan

10 NPYLM (B) ) Ogihara,M. and Li,T.: N-gram Chord Profiles For Composer Style Representation, in 9th International Conference of Music Information Retrieval (ISMIR), pp (2008). 2) Pérez-Sancho, C., Rizo, D. and nesta, J. M. I.: Genre Classification Using Chords and Stochastic Language Models, Connection Science, Vol.21, No. 2-3, pp (2009). 3) Paiement,J.-F., Eck,D. and Bengio,S.: A Probabilistic Model for Chord Progressions, in 6th International Conference of Music Information Retrieval (ISMIR), pp (2005). 4) Scholz, R., Vincent, E. and Bimbot, F.: Robust Modeling of Musical Chord Sequences Using Probabilistic N-grams, in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp (2009). 5) Harte, C., Sandler, M., Abdallah, S. and Gómez,E.: Symbolic Representation of Musical Chords: A Proposed Syntax for Text Annotations, in 6th International Conference of Music Information Retrieval (ISMIR), pp (2005). 6),,,, No MUS-85, pp. 1 8, (2010). 7) Cheng,H.-T., Yang,Y.-H., Lin,Y.-C., Liao,I.-B. and Chen,H.H.: Automatic Chord Recognition for Music Classification and Retrieval, in IEEE International Conference on Multimedia & Expo (ICME), pp (2008). 8),,,,,,, Vol.52, No.4, pp (2011). 9) Khadkevich, M. and Omologo, M.: Use of Hidden Markov Models and Factored Language Models for Automatic Chord Recognition, in 10th International Conference of Music Information Retrieval (ISMIR), pp (2009). 10) Chen,S. and Goodman,J.: An Empirical Study of Smoothing Techniques for Language Modeling, Technical Report TR-10-98, Computer Science Group, Harvard University (1998). 11) Teh,Y.W.: A Bayesian Interpretation of Interpolated Kneser-Ney, Technical Report TRA2/06, NUS School of Computing (2006). 12), Pitman-Yor n-gram,, Vol.48, No.12, pp (2007). 13),,,, No NL-190, p. 49, (2009), 14) Mochihashi,D., Yamada,T. and Ueda,N.: Bayesian Unsupervised Word Segmentation with Nested Pitman-Yor Language Modeling, in ACL-IJCNLP, pp (2009). 15) Kneser, R. and Ney, H.: Improved Backing-Off for M-gram Language Modeling, in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Vol.1, pp (1995). 16) Pitman, J. and Yor, M.: The Two-Parameter Poisson-Dirichlet Distribution Derived from a Stable Subordinator, Annals of Probability, Vol.25, No.2, pp (1997). 17) Sethuraman, J.: A Constructive Definition of Dirichlet Priors, Statistica Sinica, Vol.4, pp (1994). 18) Stolcke,A.: SRILM An Extensible Language Modeling Toolkit, in International Conference on Spoken Language Processing (ICSLP), pp (2002). 10 c 2011 Information Processing Society of Japan

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