Automatic Melody Composition Using Path Constraints of Nonchord Tone Rules

Transcription

1 Automatic Melody Composition Using Path Constraints of Nonchord Tone Rules Satoru FUKAYAMA, 1 Tauya NISHIMOTO, 1 Nobutaa ONO 1 and Shigei SAGAYAMA 1 In this paper, we discuss a model that can generate a melody which satisfies non-chord tone rules given by the classical methods of composition. Since non-chord tones are important when musicians analyze the relationship between the melody and the chord, rules of non-chord tones are relatively well-organized in music theories. However there were some difficulties in appling those rules to automatic composition because of the variety and complexity of non-chord tone usages. We show that non-chord tone rules can be represented as local constratins on the melody path and the melodies can be automatically generated by finding the optimal melody paths using dynamic programming under constraints of those local constraints. 1 Graduate School of Information Science and Technology, the University of Toyo 1. 1) 2) 3) 4) 5) 8) 12)13) c 2009 Information Processing Society of Japan

2 2 neighbor tone) : 1 passing tone) : : 3 : 1 I i,iii,v) ii ii i 1 ii iii i passing tone) 4 : neighbor tone) 3 appoggiatura) suspension) escape tone) anticipation) 4 9) 1 2 c 2009 Information Processing Society of Japan

3 )13) 5 : 10) X 5): X = argmax PX) 3) X 8 i i δ i, ) : PX) = b x0 a x0 x 1 b x1 a x1 x 2 a xn 1 x n b xn 1) n = b x0 a xi 1 x i b xi 2) i=1 b, a j j PX) δ i, ) = max P X, x i = ) 4) X ={x 0,,x i 1 } X = {x 0, x 1,, x n} PX) = max max P X, x i 1 = j ) a j b 5) j X ={x 0,,x i 2 } = max δ i 1, j) a j b 6) j 1 δ i 1, j) 11) X = {x 0, x 1,, x n} 3 c 2009 Information Processing Society of Japan

4 X Step.1 j = 0,, M δ 0, j) = b j 0) ψ 0, j) = 0 Step.2 i = 1,, n = 0,, M δ i, ) = max δ i 1, j) a j i) b i) j ψ i, ) = argmax δ i 1, j) a j i) j Step.3 ˆP = max δ n, ) x n = argmax δ n, ) Step.4 i = n, n 1,,1 X = {x 0,, x n} 6 : δ i, ) = max P x 0,, x i 3) P x i 2, x i 1, x i = ) 7) X ={x 0,,x i 1 } 1 = max δ i 3, x i 3) a xi s x i ) b xi ) 8) x i 3,x i 2,x i 1) x i 1 = ψ i, x i ) K a j 6) b m 3.3 n n m 8) 2 3 4) s=3 K x i 3, x i 2, x i 1) 6) 1 3 l 4 c 2009 Information Processing Society of Japan

5 Step.2 8 i = 1,, n = 0,, M 7 l ) ) 1 l) 0 = 0, l) 1, l) 2, l) 3 8 l = 1) δ i, ) = max δ i n l), + l) a l = { l) 0, l) 1, l) 2, l) 3 } xi 3 = + l n l) l) + b s + l) l) l) 3, xi 2 = + l) 2, xi 1 = l) 1 s=n l) + 8) δ i, ) = max + l) 3,+ l) 2,+ l) 1 ) δ i 3, + l) 3 ) 1 s=3 a l) + b s + l) l) + 9) Step.3 ) 1 ˆP = max δ i 3, + l) = max δ n, ) 3 a l) l + b s + l) l) + 10) s=3 x n = argmax δ n, ) ) δ i 3, + l) 3 Step.4 l) = { l) 0,, l) n } i = n i = 0 n l n l) X = {x 0,, x n} ) 1 δ i, ) = max δ i n l), + l) ˆl = φi, x 11) i ) l n l) a l) + s + l) s=n l) b + l) δ i, ) l δ i n l), + l) i := i n ˆl) n l) ) X Step.1 j = 0,, M δ 0, j) = b j 0) φ0, j) = 0 φi, ) = argmax δ i, ) l i n l) 0 x i 1 = x i + ˆl) 1, x i 2 = x i + ˆl) 2 x i n ˆl) = x i + ˆl) n ˆl) 5 c 2009 Information Processing Society of Japan

6 CREST 9 : 1) L. Hiller, L. Isaacson: Musical Composition with a High-Speed Digital Computer, in Machines Models of Music, M. Schwanauer, and D. Levitt Eds, MIT Press pp , Reprint of original article in Journal of Audio Engineering Society, ) W. Shottstaedt: Automatic counterpoint, In M. Mathews and J. R. Pierce, Current 4.1 Directions in Computer Music Research, pp , Cambridge, Massachusetts, The MIT Press, ) R. Ramirez, J. Peralta: A constraint-based melody harmonizer, ECAI 98 Worshop on Constraints and Artistic Applications, Brighton, ) M. Allan, C. Williams: Harmonizing Chorales by Probabilistic Inference, Proceedings of Advances in Neural Information Processing Systems, 17, ) D. Levitt: A melody description system for jazz improvisation, M. S. Thesis, Artificial Intelligence Laboratory, Chambridge, Massachusetts, ) F. Pachet, P. Roy: Musical harmonization with constraints: A survey, Constraints 4.3 Journal, 61):7-19, ) M. Baroni, R. Brunetti, L. Callegari, and C. Jacobini: A grammar for melody: relationships between melody and harmony, In M. Baroni and L. Callegari, eds. Musical Grammars and Computer Analysis, Florence: L. Olshi, pp , ),, :,, 2008-MUS-74, pp.77 82, ) :, pp.8 9, pp.18 20, pp.37 39,, ) :, pp.68 89,, ) Bellman, R. E.: Dynamic Programming, Princeton University Press, Princeton, New Jersey, ),,, :,, pp , ),,,,,, : Orpheus,, 2008-MUS-76, pp , c 2009 Information Processing Society of Japan