# A Study on Digital Analysis of Bach’s “Two-Part Inventions”

Authors: Xiao-Yi Song; Dong-Run Huang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

In the field of music composition, creating polyphony is relatively one of the most difficult parts. Among them, the basis of multivoice polyphonic composition is two-part counterpoint. The main purpose of this paper is, through the computer technology, conducting a series of studies on “Two-Part Inventions” of Bach, a Baroque polyphony master. Based on digitalization, visualization and mathematical methods, data mining algorithm has been applied to identify bipartite characteristics and rules of counterpoint polyphony. We hope that the conclusions drawn from the article could be applied to the digital creation of polyphony.


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## Body

## 1. Introduction

### 1.1. Patterns

In the process of composition, the composer will always follow inspirations and then proceed according to a certain mode. Various types of patterns and rules are available in music works, the audience’s understandings on music can be expressed in a formalized way through a series of rules [1, 2]. These rules enables the audience to generate hearing expectation, which exists in different dimensions such as melody, rhythm, and harmony, and produces different patterns and pieces on the basis of constant changes in basic elements [1, 2]. Fractal geometry originated in the nineteenth century. Fractal sets are the geometry of chaos. It is an important branch in modern mathematics. Some famous mathematicians discovered the existence of a special structure and morphology with study on continuous nondifferentiable curves.


### 1.2. Our Works

This paper applied the MATLAB tool to visualize MIDI music data and observed and unveiled pattern features of polyphonic music with intuitive techniques. We emphasized our analysis on the No. 1 to No. 5, No. 6, No. 8, No. 13, and No. 14 of Two-Part Inventions (Johann Sebastian Bach). Experimental analyses were made in terms of the pitch and the tone, and the application of these patterns and rules in computerized digital music composition was discussed in the end.


## 1.1. Patterns

In the process of composition, the composer will always follow inspirations and then proceed according to a certain mode. Various types of patterns and rules are available in music works, the audience’s understandings on music can be expressed in a formalized way through a series of rules [1, 2]. These rules enables the audience to generate hearing expectation, which exists in different dimensions such as melody, rhythm, and harmony, and produces different patterns and pieces on the basis of constant changes in basic elements [1, 2]. Fractal geometry originated in the nineteenth century. Fractal sets are the geometry of chaos. It is an important branch in modern mathematics. Some famous mathematicians discovered the existence of a special structure and morphology with study on continuous nondifferentiable curves.


## 1.2. Our Works

This paper applied the MATLAB tool to visualize MIDI music data and observed and unveiled pattern features of polyphonic music with intuitive techniques. We emphasized our analysis on the No. 1 to No. 5, No. 6, No. 8, No. 13, and No. 14 of Two-Part Inventions (Johann Sebastian Bach). Experimental analyses were made in terms of the pitch and the tone, and the application of these patterns and rules in computerized digital music composition was discussed in the end.


## 2. Experiments

### 2.1. Preparations

The pitch is a very important element in the music; the audiences are very sensitive to changes in the pitch, and they are capable of feeling only 0.5% of change [1, 2]. Constant change in the pitch is reflected in the process of music, and the interval of change between pitches is very important in Western music system. We retrieved information from a MIDI file [3] and only utilized partial information in order to simplify the process. We defined a matrix M as(1)M=DiSiPiTi,where Si is the start tempo, Di is continuing tempo, Pi is the pitch, and Ti is the pitch interval; that is,(2)Ti=Pi+1-Pi.


### 2.2. Pitch Intervals and Statistics

This paper conducted statistics and analysis on semitone spaces of adjacent pitches in Bach’s works, and results were shown in Table1. It is clearly seen that 1, 2, and 3 semitones represent the largest proportion of semitone spaces in the nine works of Bach under study. Effects of melodic interval and harmonic interval are similar, and they arouse different psychological feelings, like harmonic interval does, and let people have expectations, thus developing continuously from music thoughts. We can define a simple rule for composition in accordance with the statistics:when a music event sequenceN is given, the proportion of semitone space between Ni+1 and Ni at 1, 2, and 3 should be greater than 70% (Rule I) in the development of N (Rule I).Table 1
Statistics of semitone spaces of adjacent pitches in Bach’s inventions (intervals bigger than 12 ignored); unit: %.
Works
Intervals
1
2
3
4
5
6
7
8
9
10
11
12
No. 14
0.003
0.177
0.358
0.119
0.058
0.07
0.034
0.068
0.034
0.034
0.014
0.003
No. 13
0.007
0.09
0.08
0.368
0.14
0.067
0.054
0.107
0.033
0.043
0.003
0.003
No. 8
0.003
0.2
0.278
0.119
0.078
0.034
0.027
0.06
0.054
0.047
0.04
0.003
No. 6
0.013
0.294
0.343
0.125
0.086
0.043
0.013
0.013
0.013
0.017
0.01
0.003
No. 5
0.005
0.26
0.455
0.13
0.059
0.026
0.005
0.003
0.01
0.02
0.008
0.015
No. 4
0.025
0.249
0.5
0.036
0.018
0.025
0.007
0.029
0.014
0.04
0.043
0.011
No. 3
0.036
0.266
0.409
0.084
0.033
0.055
0.026
0.007
0.018
0.026
0.036
0.004
No. 2
0.039
0.296
0.4
0.056
0.028
0.09
0.01
0.014
0.017
0.023
0.014
0.011
No. 1
0.016
0.234
0.413
0.163
0.07
0.028
0.004
0.02
0.016
0.01
0.016
0.008


### 2.3. Melody Intervals

In addition to overall statistics on this type of interval spaces, we also want to understand specific laws of changes in pitch interval during the marching process of the melody. We defined a setS, each element of S is a data pair EiEj, representing that the pitch interval Ti between MIDI note event i and i+1 on a voice part P1 is equal to pitch interval Ti′ between P2 MIDI note event j and j+1 on the other voice part; namely, Ti=Ti′. We mapped the pitch interval with MATLAB drawing tool, and the results are shown in Figure 1.Figure 1
Equal pitch interval distribution diagram on two voice parts of Bach’s inventions No. 1. Horizontal and vertical ordinates represent the two voice parts’ MIDI note event sequence numbers; the small circles in the coordinates mean that pitch intervals of two voice parts are the same.The following characteristics can be concluded by analyzing the above diagram: continuous circle lines occur in the linear system (slope: 1), a small starting piece repeatedly occurs at different positions in the second voice part. We can see, according to transitive relation, that lots of repetitive pieces occur on single parts, and by mapping the distribution diagram we can also find out that such pieces are occurring repeatedly in one voice part.With further analysis of the music data, we know that the piece in the slope of Figure1 maps to the note event as shown in Table 2. What is interesting is that the repeated piece is imitated differently. Some are imitated partly, while some fully. As shown in the original music staff in Figure 2, we can see that a piece of different event notes follows the repeated piece. They have different ostinato which gives the listener a sense of change.Table 2
Repeated piece sequence. Each line is, respectively, the start note and end note event number of first voice part and the start note and end note event number of the second.
First part start note
First part end note
Second part start note
Second part end note
1
6
10
15
1
6
33
38
1
7
1
7
1
8
64
71
1
9
52
60Figure 2
The original music staff piece. The blue slope refers to the music sentence, and the red refers to repeated music piece.It can be found from Table3 that repeated interval pieces represent a very large proportion in Bach’s two-part inventions, and most of them (except No. 4) are at least 50%. During further analyses, we were aware that such repeated pieces occurred at different starting points of the pitch, and such repeat occurred at different tonalities. Therefore, we can define a new rule:when given a particular pitch interval sequenceM to form a complete repertoire of music event sequence N, sequence P1,…,Pk can be used to make the pitch interval sequence M repeat k times in N, with P1 being the starting pitch every time (Rule II).Table 3
Total proportion of note events in repeated piece in Bach’s Two-Part Inventions.First represents the first voice part, and Second stands for the second voice part; unit: %.
Part
Works
No. 1
No. 2
No. 3
No. 4
No. 5
No. 6
No. 8
No. 13
No. 14
First
56.39
83.38
46.59
28.10
90.24
49.06
59.73
41.92
52.36
Second
52.37
78.77
56.16
28.22
86.06
46.12
61.10
31.95
51.36


### 2.4. Tonality

As regards researches on tonality, a great many researchers have put forward plenty of models to describe changes in the tonality. Krumhansl proposed an algorithm to measure the music data and to determine perceivable tonality [1, 2, 4] on the basis of relevance with the attribute data of major and minor tonality measured by experience. Krumhansl’s algorithm is called K-Finding algorithm which is used to find out the main tonality of a piece of music. The method has shown great accuracy in measuring classical music such as Bach’s works. In our experiment, we apply Krumhansl’s K-Finding algorithm to analyze the change law of tonal characteristics of creative music in Bach’s inventions which we are going to study.M signifies the matrix of simplified music event; minimum k was calculated from the first note event S1D1P1T1 in sequence, making the total duration from the first note event to the kth note event D=d1+d2+⋯+dk, D>b tempos (b is user defined value). Then we applied the K-Finding algorithm to analyze the tonality key of M1⋯Mk in this piece and repeated the above-mentioned process with the second note event in M as the first music note to get the second piece, until all note events were measured; in this way, we could obtain a set of music piece tonal change data. We used a visual method to map out the data, as shown in Figures 3, 4, 5, and 6:  data distribution of Bach’s Two-Part Inventions No. 1.Figure 3
Tonality distribution diagram of Bach’s Two-Part Inventions No. 1 based on 1 beat per piece (whenb=1). Vertical axis represents the tonality; 1 to 12 correspond to majors extending from Major B to Major C, and 13 to 24 correspond to minors extending from Minor C to Minor B.Figure 4
Summary of tonality distribution diagram for Bach’s Two-Part Inventions No. 1 (whenb=1–8).Figure 5
Summary of tonality distribution diagram for Bach’s Two-Part Inventions No. 1 (whenb=9–25).Figure 6
Summary of tonality distribution diagram for Bach’s Two-Part Inventions No. 1 (whenb=30–65).It can be concluded that melodies converge on several different tonalities when rhythm lengths of benchmark pieces are different. According to the data collected, although Invention No. 1 is a Major C piece, its piece tonality is constantly changing in relation to the tonality of the whole work.In order to further analyze the change rule of the tonality, we use Tables4 and 5 to illustrate the relationship between changing pieces and the tonality.Table 4
Rising and falling keys of various tonalities (Major).
Major
Rising or Falling
Major C
Major F
Falling B
Falling Major B
Falling B, Falling E
Falling Major E
Falling B, Falling E, and Falling A
Falling Major A
Falling B, Falling E, Falling A, and Falling D
Falling Major D
Falling B, Falling E, L Falling A, Falling D, and Falling G
Falling Major G
Falling B, Falling E, Falling A, Falling D, Falling G, and Falling C
Major B
Rising F, Rising C, Rising G, Rising D, and Rising A
Major E
Rising F, Rising C, Rising G, and Rising D
Major A
Rising F, Rising C, and Rising G
Major D
Rising F, Rising C
Major G
Rising FTable 5
Rising and falling keys of various tonalities (Minor).
Relative minor
Rising or Falling
Minor A
Rising G
Minor D
Falling B, Rising C
Minor G
Falling B, Falling E, and Rising F
Minor C
Falling E, Falling A
Minor F
Falling B, Falling A, and Falling D
Falling Minor B
Falling B, Falling E, Falling D, and Falling G
Falling Minor E
Falling B, Falling E, Falling A, Falling G, and Falling C
Rising Minor G
Rising C, Rising G, Rising D, Rising A, and Heavy Rising F
Rising Minor C
Rising F, Rising C, Rising G, Rising D, and Rising B
Rising Minor F
Rising F, Rising C, Rising G, and Rising E
Minor B
Rising F, Rising C, and Rising A
Minor E
Rising F, Rising DWe used the piece tonality distribution diagram to analyze the characteristics of a music piece’s changes in tonality under the Krumhansl model, and the results are shown in Table6.Table 6
Tonality changes in pieces of Bach’s Two-Part Inventions (based on Krumhansl’s model). The spaces listed in the third column are minimum interval spaces of the piece’s tonalities after the major and relative minor formed loops in which the relative minor and major connect together and the first column connects to last column in Tables4 and 5.
Works
Main tonalities
Spaces
Number 1
Major C, Major G, Minor D, and Minor E
1
Number 2
Minor G, Rising Major A, Minor c, and Major C
2
Number 3
Major A, Minor B, and Major D
1
Number 4
Minor D, Minor A, and Major C
2
Number 5
Rising Major D, Minor F, and Rising Major G
1
Number 6
Major B, Major E
1
Number 8
Major F, Major C, Minor D, and Major A
1
Number 13
Minor E, Major C, and Minor A
1
Number 14
Rising Major F, Rising Major A, and Major D
2It can be concluded from Table6 that whenever there are tonality changes, normally a tonality with minimum rising or falling values adjacent to the given main tonality will be selected for change purpose. In line with the above analyses, we can develop a new rule:when composing a complete music event sequenceN, N may consist of k music sequences, and when the piece tonality under the Krumhansl model is no more than 12 (b⩾12), the space between tonalities of k music sequences should be no more than 2 (Rule III).This rule is of great significance, and in the case of connecting repeated pieces, this method of tonality change may be used to analyze possibly connected pieces.


## 2.1. Preparations

The pitch is a very important element in the music; the audiences are very sensitive to changes in the pitch, and they are capable of feeling only 0.5% of change [1, 2]. Constant change in the pitch is reflected in the process of music, and the interval of change between pitches is very important in Western music system. We retrieved information from a MIDI file [3] and only utilized partial information in order to simplify the process. We defined a matrix M as(1)M=DiSiPiTi,where Si is the start tempo, Di is continuing tempo, Pi is the pitch, and Ti is the pitch interval; that is,(2)Ti=Pi+1-Pi.


## 2.2. Pitch Intervals and Statistics

This paper conducted statistics and analysis on semitone spaces of adjacent pitches in Bach’s works, and results were shown in Table1. It is clearly seen that 1, 2, and 3 semitones represent the largest proportion of semitone spaces in the nine works of Bach under study. Effects of melodic interval and harmonic interval are similar, and they arouse different psychological feelings, like harmonic interval does, and let people have expectations, thus developing continuously from music thoughts. We can define a simple rule for composition in accordance with the statistics:when a music event sequenceN is given, the proportion of semitone space between Ni+1 and Ni at 1, 2, and 3 should be greater than 70% (Rule I) in the development of N (Rule I).Table 1
Statistics of semitone spaces of adjacent pitches in Bach’s inventions (intervals bigger than 12 ignored); unit: %.
Works
Intervals
1
2
3
4
5
6
7
8
9
10
11
12
No. 14
0.003
0.177
0.358
0.119
0.058
0.07
0.034
0.068
0.034
0.034
0.014
0.003
No. 13
0.007
0.09
0.08
0.368
0.14
0.067
0.054
0.107
0.033
0.043
0.003
0.003
No. 8
0.003
0.2
0.278
0.119
0.078
0.034
0.027
0.06
0.054
0.047
0.04
0.003
No. 6
0.013
0.294
0.343
0.125
0.086
0.043
0.013
0.013
0.013
0.017
0.01
0.003
No. 5
0.005
0.26
0.455
0.13
0.059
0.026
0.005
0.003
0.01
0.02
0.008
0.015
No. 4
0.025
0.249
0.5
0.036
0.018
0.025
0.007
0.029
0.014
0.04
0.043
0.011
No. 3
0.036
0.266
0.409
0.084
0.033
0.055
0.026
0.007
0.018
0.026
0.036
0.004
No. 2
0.039
0.296
0.4
0.056
0.028
0.09
0.01
0.014
0.017
0.023
0.014
0.011
No. 1
0.016
0.234
0.413
0.163
0.07
0.028
0.004
0.02
0.016
0.01
0.016
0.008


## 2.3. Melody Intervals

In addition to overall statistics on this type of interval spaces, we also want to understand specific laws of changes in pitch interval during the marching process of the melody. We defined a setS, each element of S is a data pair EiEj, representing that the pitch interval Ti between MIDI note event i and i+1 on a voice part P1 is equal to pitch interval Ti′ between P2 MIDI note event j and j+1 on the other voice part; namely, Ti=Ti′. We mapped the pitch interval with MATLAB drawing tool, and the results are shown in Figure 1.Figure 1
Equal pitch interval distribution diagram on two voice parts of Bach’s inventions No. 1. Horizontal and vertical ordinates represent the two voice parts’ MIDI note event sequence numbers; the small circles in the coordinates mean that pitch intervals of two voice parts are the same.The following characteristics can be concluded by analyzing the above diagram: continuous circle lines occur in the linear system (slope: 1), a small starting piece repeatedly occurs at different positions in the second voice part. We can see, according to transitive relation, that lots of repetitive pieces occur on single parts, and by mapping the distribution diagram we can also find out that such pieces are occurring repeatedly in one voice part.With further analysis of the music data, we know that the piece in the slope of Figure1 maps to the note event as shown in Table 2. What is interesting is that the repeated piece is imitated differently. Some are imitated partly, while some fully. As shown in the original music staff in Figure 2, we can see that a piece of different event notes follows the repeated piece. They have different ostinato which gives the listener a sense of change.Table 2
Repeated piece sequence. Each line is, respectively, the start note and end note event number of first voice part and the start note and end note event number of the second.
First part start note
First part end note
Second part start note
Second part end note
1
6
10
15
1
6
33
38
1
7
1
7
1
8
64
71
1
9
52
60Figure 2
The original music staff piece. The blue slope refers to the music sentence, and the red refers to repeated music piece.It can be found from Table3 that repeated interval pieces represent a very large proportion in Bach’s two-part inventions, and most of them (except No. 4) are at least 50%. During further analyses, we were aware that such repeated pieces occurred at different starting points of the pitch, and such repeat occurred at different tonalities. Therefore, we can define a new rule:when given a particular pitch interval sequenceM to form a complete repertoire of music event sequence N, sequence P1,…,Pk can be used to make the pitch interval sequence M repeat k times in N, with P1 being the starting pitch every time (Rule II).Table 3
Total proportion of note events in repeated piece in Bach’s Two-Part Inventions.First represents the first voice part, and Second stands for the second voice part; unit: %.
Part
Works
No. 1
No. 2
No. 3
No. 4
No. 5
No. 6
No. 8
No. 13
No. 14
First
56.39
83.38
46.59
28.10
90.24
49.06
59.73
41.92
52.36
Second
52.37
78.77
56.16
28.22
86.06
46.12
61.10
31.95
51.36


## 2.4. Tonality

As regards researches on tonality, a great many researchers have put forward plenty of models to describe changes in the tonality. Krumhansl proposed an algorithm to measure the music data and to determine perceivable tonality [1, 2, 4] on the basis of relevance with the attribute data of major and minor tonality measured by experience. Krumhansl’s algorithm is called K-Finding algorithm which is used to find out the main tonality of a piece of music. The method has shown great accuracy in measuring classical music such as Bach’s works. In our experiment, we apply Krumhansl’s K-Finding algorithm to analyze the change law of tonal characteristics of creative music in Bach’s inventions which we are going to study.M signifies the matrix of simplified music event; minimum k was calculated from the first note event S1D1P1T1 in sequence, making the total duration from the first note event to the kth note event D=d1+d2+⋯+dk, D>b tempos (b is user defined value). Then we applied the K-Finding algorithm to analyze the tonality key of M1⋯Mk in this piece and repeated the above-mentioned process with the second note event in M as the first music note to get the second piece, until all note events were measured; in this way, we could obtain a set of music piece tonal change data. We used a visual method to map out the data, as shown in Figures 3, 4, 5, and 6:  data distribution of Bach’s Two-Part Inventions No. 1.Figure 3
Tonality distribution diagram of Bach’s Two-Part Inventions No. 1 based on 1 beat per piece (whenb=1). Vertical axis represents the tonality; 1 to 12 correspond to majors extending from Major B to Major C, and 13 to 24 correspond to minors extending from Minor C to Minor B.Figure 4
Summary of tonality distribution diagram for Bach’s Two-Part Inventions No. 1 (whenb=1–8).Figure 5
Summary of tonality distribution diagram for Bach’s Two-Part Inventions No. 1 (whenb=9–25).Figure 6
Summary of tonality distribution diagram for Bach’s Two-Part Inventions No. 1 (whenb=30–65).It can be concluded that melodies converge on several different tonalities when rhythm lengths of benchmark pieces are different. According to the data collected, although Invention No. 1 is a Major C piece, its piece tonality is constantly changing in relation to the tonality of the whole work.In order to further analyze the change rule of the tonality, we use Tables4 and 5 to illustrate the relationship between changing pieces and the tonality.Table 4
Rising and falling keys of various tonalities (Major).
Major
Rising or Falling
Major C
Major F
Falling B
Falling Major B
Falling B, Falling E
Falling Major E
Falling B, Falling E, and Falling A
Falling Major A
Falling B, Falling E, Falling A, and Falling D
Falling Major D
Falling B, Falling E, L Falling A, Falling D, and Falling G
Falling Major G
Falling B, Falling E, Falling A, Falling D, Falling G, and Falling C
Major B
Rising F, Rising C, Rising G, Rising D, and Rising A
Major E
Rising F, Rising C, Rising G, and Rising D
Major A
Rising F, Rising C, and Rising G
Major D
Rising F, Rising C
Major G
Rising FTable 5
Rising and falling keys of various tonalities (Minor).
Relative minor
Rising or Falling
Minor A
Rising G
Minor D
Falling B, Rising C
Minor G
Falling B, Falling E, and Rising F
Minor C
Falling E, Falling A
Minor F
Falling B, Falling A, and Falling D
Falling Minor B
Falling B, Falling E, Falling D, and Falling G
Falling Minor E
Falling B, Falling E, Falling A, Falling G, and Falling C
Rising Minor G
Rising C, Rising G, Rising D, Rising A, and Heavy Rising F
Rising Minor C
Rising F, Rising C, Rising G, Rising D, and Rising B
Rising Minor F
Rising F, Rising C, Rising G, and Rising E
Minor B
Rising F, Rising C, and Rising A
Minor E
Rising F, Rising DWe used the piece tonality distribution diagram to analyze the characteristics of a music piece’s changes in tonality under the Krumhansl model, and the results are shown in Table6.Table 6
Tonality changes in pieces of Bach’s Two-Part Inventions (based on Krumhansl’s model). The spaces listed in the third column are minimum interval spaces of the piece’s tonalities after the major and relative minor formed loops in which the relative minor and major connect together and the first column connects to last column in Tables4 and 5.
Works
Main tonalities
Spaces
Number 1
Major C, Major G, Minor D, and Minor E
1
Number 2
Minor G, Rising Major A, Minor c, and Major C
2
Number 3
Major A, Minor B, and Major D
1
Number 4
Minor D, Minor A, and Major C
2
Number 5
Rising Major D, Minor F, and Rising Major G
1
Number 6
Major B, Major E
1
Number 8
Major F, Major C, Minor D, and Major A
1
Number 13
Minor E, Major C, and Minor A
1
Number 14
Rising Major F, Rising Major A, and Major D
2It can be concluded from Table6 that whenever there are tonality changes, normally a tonality with minimum rising or falling values adjacent to the given main tonality will be selected for change purpose. In line with the above analyses, we can develop a new rule:when composing a complete music event sequenceN, N may consist of k music sequences, and when the piece tonality under the Krumhansl model is no more than 12 (b⩾12), the space between tonalities of k music sequences should be no more than 2 (Rule III).This rule is of great significance, and in the case of connecting repeated pieces, this method of tonality change may be used to analyze possibly connected pieces.


## 3. Discussion

In our experiment, we concluded the patterns and rules in Bach’s Two-Part Inventions, which is typical of polyphony works, and we discovered three characteristic rules (Rules I–III) in Bach’s Two-Part Inventions. Nevertheless, it needs pointing out that these three rules only cover the pitch and the tonality, with no consideration for the rhythm, melody, and harmony. Studies show that global context has an effect on music perception [5]. William did a lot a lot of experiments to study the effects on music perception of the integration of pitch and rhythm. Results show that the integration of the individual music parameter cannot be combined easily. They have effects on each other after integration [6]. So, the modeling of music is difficult; we need to study it further rather than applying the three rules everywhere.


## 4. Conclusion

Computerized musical composition includes auxiliary composition, algorithm composition, and works’ compilation. The three rules we put forward are applicable for basic rules of polyphony works with styles similar to Bach’s; in computerized algorithm composition these three rules can be used for assessing and selecting works with better polyphony styles, and they can be used in auxiliary computer composition to inspire the composer with musical pieces generated from these rules so as to speed up the efficiency of composition. In addition, these three rules can be used for identifying the characteristics of existent works and for categorizing various types of works.



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*Source: 560926-2015-08-27.xml*