Information Entropy in the Billie Jean and Beat It Vocal Notations

Abstract: This paper examines and calculates the information entropy, as defined by Claude Shannon, in selected songs authored by Michael Jackson, more specifically, their vocal parts. The calculation only takes into account the music, more specifically, its linear notation. The entropies of individual songs are then compared in order to find whether they display similarities.


Article by IVANA RECMANOVA. Ivana is a graduated linguist and communication theoretician from Palacký University in Olomouc, Czech Republic. She works as an information, support specialist at the National Library of Technology in Prague, Czech Republic, and is also an amateur rapper. Her research interests include the use of mathematics, physics, biology, and computer science in linguistics, identity studies, and textual analysis.


REFERENCE AS:

Recmanova, Ivana, “Information Entropy in the Billie Jean and Beat It Vocal Notations”, The Journal of Michael Jackson Academic Studies, 5, no. 2 (2017). Published electronically 07/12/2017 http://michaeljacksonstudies.org/information-entropy-in-the-billie-jean-and-beat-it-vocal-notations/.


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Information Entropy in the Billie Jean and Beat It Vocal Notations By Ivana Recmanova

Information entropy is a key concept within the field of communication theory (also called cybernetics). It was coined by Claude Shannon in the 1940s and has since been used to measure the randomness of a given system, for example, a text (spoken as well as written). The formula for information entropy is:

H = – Σi pi log2 pi,

Where H is the entropy and pi Stands for the probability of occurrence of a construct (in the case of text, it is most likely to be a word). For example, if we have an alphabet of 32 characters and our aim is to measure the entropy of a text consisting of an eight-character word and two four-character words, we count it as:

H = – (1/32)^8 * log2 (1/32)^8 + (- 2(1/32)^4 * log2 (1/32)^4) (assuming all the characters have the same probability of occurrence).

The result is 5 * (2^(-37) + 2^(-17)) bits.

Other important values within the field of information theory are maximum entropy (which measures the maximum value of the entropy of a text) and redundancy (which measures how often signs of a text are repeated). Its calculations are:

 

Hmax = log2 n and

R = 1 – (H/Hmax),

 

respectively, where n is the number of the letter of the used alphabet (or the number of signs of a given notation text). For the purpose of this study, the number of signs was calculated as (21*13*4)+7+26, which covers all the musical pauses and all the notes within Michael Jackson’s vocal range. The notations were obtained from the Musicnotes.com server.

Even though musical notation is considered a text (Eco, 1979, p. 48), the measure of its entropy possesses specific challenges. First of all, we need to define constructs and possible constituents, its smaller units, within the musical text.

For the sake of simplicity, this study will consider musical notes and pauses as constructs. Therefore, each construct will be made of one sign, that is, a specific note of a given pitch and length or a specific pause of a given length. The study will not consider non-linear aspects of songs, such as force or specific expressions of notes (staccato, ligature).

Here are the detailed results:

 

 

 

Billie Jean

 

 

f

 

 

pi

 

 

H [bits]

1/4 pause 19 0,073077 0,27582
1/8 cis2 21 0,080769 0,2932
1/8 h1 23 0,088462 0,30951
1/8 a1 14 0,053846 0,22696
1/16 h1 2 0,007692 0,05402
1/16 a1 2 0,007692 0,05402
1/4 cis2 5 0,019231 0,10962
1/8 pause 16 0,061538 0,24753
1/8 gis1 3 0,011538 0,07428
1/8 fis1 21 0,080769 0,2932
1/4 fis1 3 0,011538 0,07428
1/16 fis1 9 0,034615 0,16797
1/16 gis1 9 0,034615 0,16797
1/2 pause 5 0,019231 0,10962
1 pause 3 0,011538 0,07428
1/8 fis2 34 0,130769 0,3838
1/16 e2 9 0,034615 0,16797
1/16 fis2 15 0,057692 0,23743
1/8 a2 7 0,026923 0,1404
1/16 cis2 2 0,007692 0,05402
1/4 fis2 6 0,023077 0,12548
1/4 gis2 1 0,003846 0,03086
1/8 e2 5 0,019231 0,10962
1/16 pause 2 0,007692 0,05402
1/8 d2 1 0,003846 0,03086
1/16 d2 3 0,011538 0,07428
1/8 eis2 1 0,003846 0,03086
1/16 eis2 3 0,011538 0,07428
1/8 gis2 4 0,015385 0,09265
1/8 h2 6 0,023077 0,12548
1/8 cis3 2 0,007692 0,05402
1/4 h2 2 0,007692 0,05402
1/4 d2 2 0,007692 0,05402
total 260 1 4,42632

 

Beat It f pi H [bits]
1/8 pause 17 0,099415 0,33109
1/8 h1 40 0,233918 0,49027
1/8 a1 25 0,146199 0,40556
1/8 d1 1 0,005848 0,04338
1/4 a1 5 0,02924 0,149
1/4 h1 5 0,02924 0,149
1/8 g1 18 0,105263 0,34189
1/4 pause 17 0,099415 0,33109
1/8 e1 11 0,064327 0,25464
1/8 d1 5 0,02924 0,149
1/4 e1 8 0,046784 0,20668
1/8 fis1 5 0,02924 0,149
1/8 counterh 1 0,005848 0,04338
1/4 d1 3 0,017544 0,10233
1/2 pause 3 0,017544 0,10233
1/8 d2 4 0,023392 0,12673
1/4+ h1 2 0,011696 0,07506
1/4 d2 1 0,005848 0,04338
total 171 1 3,493829

The maximum entropy was 10,13571 bits. The redundancies were 0,563294 and 0,655295, respectively.

It is also worth looking at the ranks of each sign used in the notations because some signs‘ ranks were close to each other across both songs:

 

Billie Jean          f
1   1/8 fis2 34
2   1/8 h1 23
3-4   1/8 cis2 21
3-4   1/8 fis1 21
5   1/4 pause 19
6   1/8 pause 16
7   1/16 fis2 15
8   1/8 a1 14
9-11   1/16 fis1 9
9-11   1/16 gis1 9
9-11   1/16 e2 9
12   1/8 a2 7
13-14   1/4 fis2 6
13-14   1/8 h2 6
15-17   1/4 cis2 5
15-17   1/2 pause 5
15-17   1/8 e2 5
18   1/8 gis2 4
19-23   1/8 gis1 3
19-23   1/4 fis1 3
19-23   1 pause 3
19-23   1/16 d2 3
19-23   1/16 eis2 3
24-30 1/16 h1 2
24-30 1/16 a1 2
24-30 1/16 cis2 2
24-30 1/16 pause 2
24-30   1/8 cis3 2
24-30   1/4 h2 2
24-30   1/4 d2 2
31-33   1/4 gis2 1
31-33   1/8 d2 1
31-33   1/8 eis2 1

 

Beat It          f
1   1/8 h1 40
2   1/8 a1 25
3   1/8 g1 18
4-5   1/8 pause 17
4-5   1/4 pause 17
6   1/8 e1 11
7   1/4 e1 8
8-11   1/4 a1 5
8-11   1/4 h1 5
8-11   1/8 d1 5
8-11   1/8 fis1 5
12   1/8 d2 4
13-14   1/4 d1 3
13-14 1/2 pause 3
15   1/4+ h1 2
16-18 1/8 d1 1
16-18 1/8 counterh 1
16-18 1/4 d2 1

 

Similarities were found among these signs: 1/8 h1 (ranks 2 and 1), ¼ pause (rank 5), 1/8 pause (ranks 6 and 4), and ½ pause (ranks 16 and 14).

In conclusion, the calculations have shown that the information entropy of Billie Jean is higher that that of Beat It. Conversely, redundancy is higher for Beat It. What these songs share is similar rankings for certain signs, especially musical pauses.

Sources:

Eco, Umberto: The Role of the Reader (Bloomington: Indiana University Press, 1979).

Michael Jackson “Billie Jean” Sheet Music (Leadsheet) in F# Minor (transposable) – Download & Print – SKU: MN0069569. URL: http://www.musicnotes.com/sheetmusic/mtd.asp?ppn=MN0069569. Retrieved on 7 October 2017.

Michael Jackson “Beat It” Sheet Music (Leadsheet) in E Minor (transposable) – Download & Print – SKU: MN0075748. URL: http://www.musicnotes.com/sheetmusic/mtd.asp?ppn=MN0075748. Retrieved on 7 October 2017.

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