nips nips2001 nips2001-91 knowledge-graph by maker-knowledge-mining

91 nips-2001-Improvisation and Learning

Source: pdf

Author: Judy A. Franklin

Abstract: This article presents a 2-phase computational learning model and application. As a demonstration, a system has been built, called CHIME for Computer Human Interacting Musical Entity. In phase 1 of training, recurrent back-propagation trains the machine to reproduce 3 jazz melodies. The recurrent network is expanded and is further trained in phase 2 with a reinforcement learning algorithm and a critique produced by a set of basic rules for jazz improvisation. After each phase CHIME can interactively improvise with a human in real time. 1 Foundations Jazz improvisation is the creation of a jazz melody in real time. Charlie Parker, Dizzy Gillespie, Miles Davis, John Coltrane, Charles Mingus, Thelonious Monk, and Sonny Rollins et al. were the founders of bebop and post bop jazz [9] where drummers, bassists, and pianists keep the beat and maintain harmonic structure. Other players improvise over this structure and even take turns improvising for 4 bars at a time. This is called trading fours. Meanwhile, artiﬁcial neural networks have been used in computer music [4, 12]. In particular, the work of (Todd [11]) is the basis for phase 1 of CHIME, a novice machine improvisor that learns to trade fours. Firstly, a recurrent network is trained with back-propagation to play three jazz melodies by Sonny Rollins [1], as described in Section 2. Phase 2 uses actor-critic reinforcement learning and is described in Section 3. This section is on jazz basics. 1.1 Basics: Chords, the ii-V-I Chord Progression and Scales The harmonic structure mentioned above is a series of chords that may be reprated and that are often grouped into standard subsequences. A chord is a group of notes played simultaneously. In the chromatic scale, C-Db-D-Eb-E-F-Gb-G-Ab-A-Bb-B-C, notes are separated by a half step. A ﬂat (b) note is a half step below the original note; a sharp (#) is a half above. Two half steps are a whole step. Two whole steps are a major third. Three half steps are a minor third. A major triad (chord) is the ﬁrst or tonic note, then the note a major third up, then the note a minor third up. When F is the tonic, F major triad is F-A-C. A minor triad (chord) is the tonic ¡ www.cs.smith.edu/˜jfrankli then a minor third, then a major third. F minor triad is F-Ab-C. The diminished triad is the tonic, then a minor third, then a minor third. F diminished triad is F-Ab-Cb. An augmented triad is the tonic, then a major third, then a major third. The F augmented triad is F-A-Db. A third added to the top of a triad forms a seventh chord. A major triad plus a major third is the major seventh chord. F-A-C-E is the F major seventh chord (Fmaj7). A minor triad plus a minor third is a minor seventh chord. For F it is F-Ab-C-Eb (Fm7). A major triad plus a minor third is a dominant seventh chord. For F it is F-A-C-Eb (F7). These three types of chords are used heavily in jazz harmony. Notice that each note in the chromatic scales can be the tonic note for any of these types of chords. A scale, a subset of the chromatic scale, is characterized by note intervals. Let W be a whole step and H be a half. The chromatic scale is HHHHHHHHHHHH. The major scale or ionian mode is WWHWWWH. F major scale is F-G-A-Bb-C-D-E-F. The notes in a scale are degrees; E is the seventh degree of F major. The ﬁrst, third, ﬁfth, and seventh notes of a major scale are the major seventh chord. The ﬁrst, third, ﬁfth, and seventh notes of other modes produce the minor seventh and dominant seventh chords. Roman numerals represent scale degrees and their seventh chords. Upper case implies major or dominant seventh and lower case implies minor seventh [9]. The major seventh chord starting at the scale tonic is the I (one) chord. G is the second degree of F major, and G-Bb-D-F is Gm7, the ii chord, with respect to F. The ii-V-I progression is prevalent in jazz [9], and for F it is Gm7-C7-Fmaj7. The minor ii-V-i progression is obtained using diminished and augmented triads, their seventh chords, and the aeolian mode. Seventh chords can be extended by adding major or minor thirds, e.g. Fmaj9, Fmaj11, Fmaj13, Gm9, Gm11, and Gm13. Any extension can be raised or lowered by 1 step [9] to obtain, e.g. Fmaj7#11, C7#9, C7b9, C7#11. Most jazz compositions are either the 12 bar blues or sectional forms (e.g. ABAB, ABAC, or AABA) [8]. The 3 Rollins songs are 12 bar blues. “Blue 7” has a simple blues form. In “Solid” and “Tenor Madness”, Rollins adds bebop variations to the blues form [1]. ii-V-I and VI-II-V-I progressions are added and G7+9 substitutes for the VI and F7+9 for the V (see section 1.2 below); the II-V in the last bar provides the turnaround to the I of the ﬁrst bar to foster smooth repetition of the form. The result is at left and in Roman numeral notation Bb7 Bb7 Bb7 Bb7 I I I I Eb7 Eb7 Bb7 G7+9 IV IV I VI at right: Cm7 F7 Bb7 G7+9 C7 F7+9 ii V I VI II V 1.2 Scale Substitutions and Rules for Reinforcement Learning First note that the theory and rules derived in this subsection are used in Phase 2, to be described in Section 3. They are presented here since they derive from the jazz basics immediately preceding. One way a novice improvisor can play is to associate one scale with each chord and choose notes from that scale when the chord is presented in the musical score. Therefore, Rule 1 is that an improvisor may choose notes from a “standard” scale associated with a chord. Next, the 4th degree of the scale is often avoided on a major or dominant seventh chord (Rule 3), unless the player can resolve its dissonance. The major 7th is an avoid note on a dominant seventh chord (Rule 4) since a dominant seventh chord and its scale contain the ﬂat 7th, not the major 7th. Rule 2 contains many notes that can be added. A brief rationale is given next. The C7 in Gm7-C7-Fmaj7 may be replaced by a C7#11, a C7+ chord, or a C7b9b5 or C7alt chord [9]. The scales for C7+ and C7#11 make available the raised fourth (ﬂat 5), and ﬂat 6 (ﬂat 13) for improvising. The C7b9b5 and C7alt (C7+9) chords and their scales make available the ﬂat9, raised 9, ﬂat5 and raised 5 [1]. These substitutions provide the notes of Rule 2. These rules (used in phase 2) are stated below, using for reinforcement values very bad (-1.0), bad (-0.5), a little bad (-0.25), ok (0.25), good (0.5), and very good (1.0). The rules are discussed further in Section 4. The Rule Set: 1) Any note in the scale associated with the chord is ok (except as noted in rule 3). 2) On a dominant seventh, hip notes 9, ﬂat9, #9, #11, 13 or ﬂat13 are very good. One hip note 2 times in a row is a little bad. 2 hip notes more than 2 times in a row is a little bad. 3) If the chord is a dominant seventh chord, a natural 4th note is bad. 4) If the chord is a dominant seventh chord, a natural 7th is very bad. 5) A rest is good unless it is held for more than 2 16th notes and then it is very bad. 6) Any note played longer than 1 beat (4 16th notes) is very bad. 7) If two consecutive notes match the human’s, that is good. 2 CHIME Phase 1 In Phase 1, supervised learning is used to train a recurrent network to reproduce the three Sonny Rollins melodies. 2.1 Network Details and Training The recurrent network’s output units are linear. The hidden units are nonlinear (logistic function). Todd [11] used a Jordan recurrent network [6] for classical melody learning and generation. In CHIME, a Jordan net is also used, with the addition of the chord as input (Figure 1. 24 of the 26 outputs are notes (2 chromatic octaves), the 25th is a rest, and the 26th indicates a new note. The output with the highest value above a threshold is the next note, including the rest output. The new note output indicates if this is a new note, or if it is the same note being held for another time step ( note resolution). ¥£ ¡ ¦¤¢ The 12 chord inputs (12 notes in a chromatic scale), are 1 or 0. A chord is represented as its ﬁrst, third, ﬁfth, and seventh notes and it “wraps around” within the 12 inputs. E.g., the Fm7 chord F-Ab-C-Eb is represented as C, Eb, F, Ab or 100101001000. One plan input per song enables distinguishing between songs. The 26 context inputs use eligibility traces, giving the hidden units a decaying history of notes played. CHIME (as did Todd) uses teacher forcing [13], wherein the target outputs for the previous step are used as inputs (so erroneous outputs are not used as inputs). Todd used from 8 to 15 hidden units; CHIME uses 50. The learning rate is 0.075 (Todd used 0.05). The eligibility rate is 0.9 (Todd used 0.8). Differences in values perhaps reﬂect contrasting styles of the songs and available computing power. Todd used 15 output units and assumed a rest when all note units are “turned off.” CHIME uses 24 output note units (2 octaves). Long rests in the Rollins tunes require a dedicated output unit for a rest. Without it, the note outputs learned to turn off all the time. Below are results of four representative experiments. In all experiments, 15,000 presentations of the songs were made. Each song has 192 16th note events. All songs are played at a ﬁxed tempo. Weights are initialized to small random values. The squared error is the average squared error over one complete presentation of the song. “Finessing” the network may improve these values. The songs are easily recognized however, and an exact match could impair the network’s ability to improvise. Figure 2 shows the results for “Solid.” Experiment 1. Song: Blue Seven. Squared error starts at 185, decreases to 2.67. Experiment 2. Song: Tenor Madness. Squared error starts at 218, decreases to 1.05. Experiment 3. Song: Solid. Squared error starts at 184, decreases to 3.77. Experiment 4. Song: All three songs: Squared error starts at 185, decreases to 36. Figure 1: Jordan recurrent net with addition of chord input 2.2 Phase 1 Human Computer Interaction in Real Time In trading fours with the trained network, human note events are brought in via the MIDI interface [7]. Four bars of human notes are recorded then given, one note event at a time to the context inputs (replacing the recurrent inputs). The plan inputs are all 1. The chord inputs follow the “Solid” form. The machine generates its four bars and they are played in real time. Then the human plays again, etc. An accompaniment (drums, bass, and piano), produced by Band-in-a-Box software (PG Music), keeps the beat and provides chords for the human. Figure 3 shows an interaction. The machine’s improvisations are in the second and fourth lines. In bar 5 the ﬂat 9 of the Eb7 appears; the E. This note is used on the Eb7 and Bb7 chords by Rollins in “Blue 7”, as a “passing tone.” D is played in bar 5 on the Eb7. D is the natural 7 over Eb7 (with its ﬂat 7) but is a note that Rollins uses heavily in all three songs, and once over the Eb7. It may be a response to the rest and the Bb played by the human in bar 1. D follows both a rest and a Bb in many places in “Tenor Madness” and “Solid.” In bar 6, the long G and the Ab (the third then fourth of Eb7) ﬁgure prominently in “Solid.” At the beginning of bar 7 is the 2-note sequence Ab-E that appears in exactly the same place in the song “Blue 7.” The focus of bars 7 and 8 is jumping between the 3rd and 4th of Bb7. At the end of bar 8 the machine plays the ﬂat 9 (Ab) then the ﬂat 3 (Bb), of G7+9. In bars 13-16 the tones are longer, as are the human’s in bars 9-12. The tones are the 5th, the root, the 3rd, the root, the ﬂat 7, the 3rd, the 7th, and the raised fourth. Except for the last 2, these are chord tones. 3 CHIME Phase 2 In Phase 2, the network is expanded and trained by reinforcement learning to improvise according to the rules of Section 1.2 and using its knowledge of the Sonny Rollins songs. 3.1 The Expanded Network Figure 4 shows the phase 2 network. The same inputs plus 26 human inputs brings the total to 68. The weights obtained in phase 1 initialize this network. The plan and chord weights Figure 2: At left “Solid” played by a human; at right the song reproduced by the ANN. are the same. The weights connecting context units to the hidden layer are halved. The same weights, halved, connect the 26 human inputs to the hidden layer. Each output unit gets the 100 hidden units’ outputs as input. The original 50 weights are halved and used as initial values of the two sets of 50 hidden unit weights to the output unit. 3.2 SSR and Critic Algorithms Using actor-critic reinforcement learning ([2, 10, 13]), the actor chooses the next note to play. The critic receives a “raw” reinforcement signal from the critique made by the . A rules of Section 1.2. For output j, the SSR (actor) computes mean Gaussian distribution with mean and standard deviation chooses the output . is generated, the critic modiﬁes and produces . is further modiﬁed by a self-scaling algorithm that tracks, via moving average, the maximum and minimum reinforcement and uses them to scale the signal to produce .

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 In phase 1 of training, recurrent back-propagation trains the machine to reproduce 3 jazz melodies. [sent-6, score-0.488]

2 The recurrent network is expanded and is further trained in phase 2 with a reinforcement learning algorithm and a critique produced by a set of basic rules for jazz improvisation. [sent-7, score-0.779]

3 After each phase CHIME can interactively improvise with a human in real time. [sent-8, score-0.303]

4 1 Foundations Jazz improvisation is the creation of a jazz melody in real time. [sent-9, score-0.374]

5 were the founders of bebop and post bop jazz [9] where drummers, bassists, and pianists keep the beat and maintain harmonic structure. [sent-11, score-0.356]

6 Other players improvise over this structure and even take turns improvising for 4 bars at a time. [sent-12, score-0.161]

7 In particular, the work of (Todd [11]) is the basis for phase 1 of CHIME, a novice machine improvisor that learns to trade fours. [sent-15, score-0.221]

8 Firstly, a recurrent network is trained with back-propagation to play three jazz melodies by Sonny Rollins [1], as described in Section 2. [sent-16, score-0.404]

9 Phase 2 uses actor-critic reinforcement learning and is described in Section 3. [sent-17, score-0.164]

10 1 Basics: Chords, the ii-V-I Chord Progression and Scales The harmonic structure mentioned above is a series of chords that may be reprated and that are often grouped into standard subsequences. [sent-20, score-0.162]

11 A chord is a group of notes played simultaneously. [sent-21, score-0.882]

12 In the chromatic scale, C-Db-D-Eb-E-F-Gb-G-Ab-A-Bb-B-C, notes are separated by a half step. [sent-22, score-0.349]

13 A major triad (chord) is the ﬁrst or tonic note, then the note a major third up, then the note a minor third up. [sent-27, score-1.014]

14 edu/˜jfrankli then a minor third, then a major third. [sent-32, score-0.324]

15 The diminished triad is the tonic, then a minor third, then a minor third. [sent-34, score-0.544]

16 An augmented triad is the tonic, then a major third, then a major third. [sent-36, score-0.591]

17 A third added to the top of a triad forms a seventh chord. [sent-38, score-0.515]

18 A major triad plus a major third is the major seventh chord. [sent-39, score-1.072]

19 A minor triad plus a minor third is a minor seventh chord. [sent-41, score-0.982]

20 A major triad plus a minor third is a dominant seventh chord. [sent-43, score-0.963]

21 These three types of chords are used heavily in jazz harmony. [sent-45, score-0.42]

22 Notice that each note in the chromatic scales can be the tonic note for any of these types of chords. [sent-46, score-0.327]

23 The notes in a scale are degrees; E is the seventh degree of F major. [sent-52, score-0.541]

24 The ﬁrst, third, ﬁfth, and seventh notes of a major scale are the major seventh chord. [sent-53, score-1.148]

25 The ﬁrst, third, ﬁfth, and seventh notes of other modes produce the minor seventh and dominant seventh chords. [sent-54, score-1.211]

26 Roman numerals represent scale degrees and their seventh chords. [sent-55, score-0.334]

27 Upper case implies major or dominant seventh and lower case implies minor seventh [9]. [sent-56, score-0.928]

28 The major seventh chord starting at the scale tonic is the I (one) chord. [sent-57, score-1.147]

29 The ii-V-I progression is prevalent in jazz [9], and for F it is Gm7-C7-Fmaj7. [sent-59, score-0.298]

30 The minor ii-V-i progression is obtained using diminished and augmented triads, their seventh chords, and the aeolian mode. [sent-60, score-0.51]

31 Seventh chords can be extended by adding major or minor thirds, e. [sent-61, score-0.463]

32 Most jazz compositions are either the 12 bar blues or sectional forms (e. [sent-67, score-0.422]

33 2 below); the II-V in the last bar provides the turnaround to the I of the ﬁrst bar to foster smooth repetition of the form. [sent-74, score-0.226]

34 They are presented here since they derive from the jazz basics immediately preceding. [sent-77, score-0.285]

35 One way a novice improvisor can play is to associate one scale with each chord and choose notes from that scale when the chord is presented in the musical score. [sent-78, score-1.542]

36 Therefore, Rule 1 is that an improvisor may choose notes from a “standard” scale associated with a chord. [sent-79, score-0.34]

37 Next, the 4th degree of the scale is often avoided on a major or dominant seventh chord (Rule 3), unless the player can resolve its dissonance. [sent-80, score-1.123]

38 The major 7th is an avoid note on a dominant seventh chord (Rule 4) since a dominant seventh chord and its scale contain the ﬂat 7th, not the major 7th. [sent-81, score-2.203]

39 The C7 in Gm7-C7-Fmaj7 may be replaced by a C7#11, a C7+ chord, or a C7b9b5 or C7alt chord [9]. [sent-84, score-0.514]

40 The C7b9b5 and C7alt (C7+9) chords and their scales make available the ﬂat9, raised 9, ﬂat5 and raised 5 [1]. [sent-86, score-0.284]

41 These rules (used in phase 2) are stated below, using for reinforcement values very bad (-1. [sent-88, score-0.367]

42 The Rule Set: 1) Any note in the scale associated with the chord is ok (except as noted in rule 3). [sent-96, score-0.701]

43 2) On a dominant seventh, hip notes 9, ﬂat9, #9, #11, 13 or ﬂat13 are very good. [sent-97, score-0.441]

44 2 hip notes more than 2 times in a row is a little bad. [sent-99, score-0.365]

45 3) If the chord is a dominant seventh chord, a natural 4th note is bad. [sent-100, score-0.903]

46 4) If the chord is a dominant seventh chord, a natural 7th is very bad. [sent-101, score-0.865]

47 5) A rest is good unless it is held for more than 2 16th notes and then it is very bad. [sent-102, score-0.246]

48 6) Any note played longer than 1 beat (4 16th notes) is very bad. [sent-103, score-0.24]

49 7) If two consecutive notes match the human’s, that is good. [sent-104, score-0.207]

50 1 Network Details and Training The recurrent network’s output units are linear. [sent-107, score-0.16]

51 In CHIME, a Jordan net is also used, with the addition of the chord as input (Figure 1. [sent-110, score-0.514]

52 24 of the 26 outputs are notes (2 chromatic octaves), the 25th is a rest, and the 26th indicates a new note. [sent-111, score-0.345]

53 ¥£ ¡ ¦¤¢ The 12 chord inputs (12 notes in a chromatic scale), are 1 or 0. [sent-114, score-0.89]

54 A chord is represented as its ﬁrst, third, ﬁfth, and seventh notes and it “wraps around” within the 12 inputs. [sent-115, score-0.974]

55 , the Fm7 chord F-Ab-C-Eb is represented as C, Eb, F, Ab or 100101001000. [sent-118, score-0.514]

56 The 26 context inputs use eligibility traces, giving the hidden units a decaying history of notes played. [sent-120, score-0.416]

57 Todd used 15 output units and assumed a rest when all note units are “turned off. [sent-130, score-0.226]

58 Figure 1: Jordan recurrent net with addition of chord input 2. [sent-157, score-0.582]

59 2 Phase 1 Human Computer Interaction in Real Time In trading fours with the trained network, human note events are brought in via the MIDI interface [7]. [sent-158, score-0.202]

60 Four bars of human notes are recorded then given, one note event at a time to the context inputs (replacing the recurrent inputs). [sent-159, score-0.606]

61 The machine generates its four bars and they are played in real time. [sent-162, score-0.321]

62 An accompaniment (drums, bass, and piano), produced by Band-in-a-Box software (PG Music), keeps the beat and provides chords for the human. [sent-164, score-0.18]

63 This note is used on the Eb7 and Bb7 chords by Rollins in “Blue 7”, as a “passing tone. [sent-168, score-0.177]

64 It may be a response to the rest and the Bb played by the human in bar 1. [sent-171, score-0.432]

65 ” In bar 6, the long G and the Ab (the third then fourth of Eb7) ﬁgure prominently in “Solid. [sent-173, score-0.191]

66 ” At the beginning of bar 7 is the 2-note sequence Ab-E that appears in exactly the same place in the song “Blue 7. [sent-174, score-0.201]

67 At the end of bar 8 the machine plays the ﬂat 9 (Ab) then the ﬂat 3 (Bb), of G7+9. [sent-176, score-0.181]

68 In bars 13-16 the tones are longer, as are the human’s in bars 9-12. [sent-177, score-0.278]

69 3 CHIME Phase 2 In Phase 2, the network is expanded and trained by reinforcement learning to improvise according to the rules of Section 1. [sent-180, score-0.361]

70 The same inputs plus 26 human inputs brings the total to 68. [sent-184, score-0.275]

71 The plan and chord weights Figure 2: At left “Solid” played by a human; at right the song reproduced by the ANN. [sent-186, score-0.817]

72 The same weights, halved, connect the 26 human inputs to the hidden layer. [sent-189, score-0.23]

73 The original 50 weights are halved and used as initial values of the two sets of 50 hidden unit weights to the output unit. [sent-191, score-0.161]

74 2 SSR and Critic Algorithms Using actor-critic reinforcement learning ([2, 10, 13]), the actor chooses the next note to play. [sent-193, score-0.254]

75 The critic receives a “raw” reinforcement signal from the critique made by the . [sent-194, score-0.314]

76 is further modiﬁed by a self-scaling algorithm that tracks, via moving average, the maximum and minimum reinforcement and uses them to scale the signal to produce . [sent-199, score-0.245]

77 The critic inputs are the outputs of the hidden layer of the actor network; it “piggy-backs” on the actor and uses its learned features (see Figure 5). [sent-210, score-0.46]

78 in that a note played too many times in a row may result in punishment, and if 2 notes in a row coincide with the human’s it is rewarded. [sent-214, score-0.45]

79 The critic also uses eligibility traces of the inputs, so is actually where . [sent-217, score-0.204]

80 Rules 1-4 are based on discussions with John Payne, a professional jazz musician and instructor of 25 years 1 . [sent-222, score-0.257]

81 The rules by no means encompass all of jazz theory or practice but are a starting point. [sent-223, score-0.345]

82 The notes in rule 2 were cast as good in a “hip” situation. [sent-224, score-0.249]

83 The notion of hip requires human sophistication so for now these notes are reinforced if played sporadically on the dominant seventh. [sent-225, score-0.721]

84 is limited, , providing stability, and deliberate action uncertainty so different notes are played, for the same network state. [sent-229, score-0.259]

85 In a typical example using the phase 1 network prior to phase 2 improvement, the average reinforcement value according to the rule set is -. [sent-232, score-0.454]

86 28 after 30-100 off-line presentations of the human solo of 1800 note events. [sent-235, score-0.211]

87 ¥ ¦y v ¢ v ©§ ¥ ge 6¨y ¦y ¢ g Figure 6 shows 12 bars of a human solo and 12 bars of a machine solo. [sent-236, score-0.396]

88 The note durations 1 The rules are not meant to represent John Payne Figure 4: Recurrent reinforcement learning network with human input used in phase 2. [sent-237, score-0.547]

89 Figure 5: Phase 2 network with critic “piggy-backing” on hidden layer. [sent-238, score-0.237]

90 The machine plays chord tones, such as Bb and D in bars 1 and 2. [sent-240, score-0.691]

91 In bars 3 and 4 it plays C sharp, a hip note (the #9 of Bb7) and high G. [sent-242, score-0.322]

92 In bars 5 and 6 the 9 and 13 (F and C) are played on Eb7. [sent-244, score-0.27]

93 Hip notes show up in bar 9 on Cm7: the 13 (G) and the 9 (D). [sent-246, score-0.32]

94 In bar 11 G is played on G7+9 as is the hip ﬂat 9 (the Ab). [sent-247, score-0.41]

95 In bar 12, the Eb (ﬂat 7 chord tone) is played on the F7+9. [sent-248, score-0.788]

96 In bars 2, 4, 7, 9, and 10 the machine starts at the G at the top of the staff and descends through several chord tones, producing a recurring motif, an artifact of a “good” jazz solo. [sent-249, score-0.942]

97 The phase 2 network has been used to interact with a human in real time while still learning. [sent-250, score-0.303]

98 A limitation to be addressed for CHIME is to move beyond one chord at a time. [sent-252, score-0.514]

99 There are plenty of improvisation rules for chord progressions [8]. [sent-254, score-0.697]

100 Because CHIME employs reinforcement learning, it has a stochastic element that allows it to play “outside the chord changes. [sent-255, score-0.681]

similar papers computed by tfidf model

tfidf for this paper:

wordName wordTfidf (topN-words)

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