Musical Analysis using statistical methods 권상일

Contents 1. Overview 2. MIDI 3. Theories 4. Samples 5. Results 6. Limits 7. Conclusion 8. Reference

1. Overview What I want to do is…  Analyze music with statistical approach.  Search or define quantity that shows characteristics of music.  Find the factors that determine the BEAUTY of famous songs.

2. MIDI (1) Musical Instrument Digital Interface Digitalized Score  Time, channel, note, volume, instruments, and various effects… Table of few channel voice messages Channel Voice Messages Status D7----D0 Data Byte(s) D7----D0 Description 1000nnnn 0kkkkkkk 0vvvvvvv Note Off event. (kkkkkkk) is the key (note) number. (vvvvvvv) is the velocity. 1001nnnn 0kkkkkkk 0vvvvvvv Note On event. (kkkkkkk) is the key (note) number. (vvvvvvv) is the velocity.

2. MIDI (2) Table of MIDI Note Numbers Octave Number Note Numbers CC#DD#EFF#GG#AA#B

3. Theories (1) 1/f law (musical Zipf’s law)  Almost every music have 1/f dependence.  Frequency spectrum  Pitch interval distribution Scatter diagram  It shows how strongly or weakly related one piece of data is to the previous one.  The x-axis is labeled n and the y-axis is n-1

3. Theories (2) Fractal dimension  Scatter Diagram’s fractal dimension is given by

3. Theories (3) Entropy  Treat each pitches as accessible states and the number of appearance as probabilities. Then  High entropy : there are many chromatic notes… Fractal dimension and entropy tells us  Degree of correlation and ratio of chromatic scale

4. Samples (1) Why many Beatles?  Lennon and McCartney’s songs have SIMPLE and VARIOUS style.  They are so FAMOUS! Why Debussy?  His melody line was very UNUSUAL form for that time. Why Bach?  Many people says, “Bach’s music has esthetical BEAUTY!” ComposerTitleTonic J. S. Bach Cello Suite No. 1 in G major - BWV 1007, Prelude43 (G major) Cello Suite No. 3 in C major - BWV 1009, Courante48 (C major) Cello Suite No. 6 in D major - BWV 1012, Courante50 (D major) The Art of Fugue - BWV 1080, Contrapunctus I62 (D minor) C. Debussy Clair de lune73 (C# major) Prelude a l'Apres-Midi d'un Faune71 (B major) J. Lennon (The Beatles) Across The Universe74 (D major) Girl72 (C minor) Julia60 (C major) Norwegian Wood64 (E major) Nowhere Man64 (E major) Strawberry Fields Forever70 (A# major) P. McCartney (The Beatles) And I Love Her68 (G# minor) Here, There And Everywhere67 (G major) In My Life69 (A major) Let It Be72 (C major) Michelle62 (D minor) Penny Lane72 (Cmajor) Yesterday65 (F minor)

4. Samples (2)

4. Samples (3) Programs  Note counts  Deviation  Interval counts  Interval distribution (scatter diagram)  Pitch counts  Fractal dimension  Entropy

5. Results – Zipf’s Law (1) Well-known factors satisfy Zifp’s law  Frequency spectrum  Pitch interval distribution  Etc…

Bach – CS No. 1 in G major - BWV 1007, Prelude Debussy - Prelude a l'Apres-Midi d'un Faune Lennon - Nowhere ManMcCartney - Yesterday

Bach – CS No. 1 in G major - BWV 1007, Prelude Debussy - Prelude a l'Apres-Midi d'un Faune Lennon - Nowhere ManMcCartney - Yesterday

5. Results – Scatter Diagrams (1) SD shows how close the notes are. How can we know? Look at 1/f β !  0 < β < 0.5 : white noise, nearly random  0.5 < β < 1 : pink noise, most songs are in here!  1.5 < β < 2 : brown noise, too correlated Compare with y=x graph.  Near : repetitious  Far : varied

Debussy – Clair de lune (-1.3)Bach – AF BWV 1080 Contrapunctus I (2) (-1.6) Lennon – Strawberry Field Forever (-1.2)McCartney – Yesterday (-1.3)

5. Results – Relative Pitch (1) Relative Pitch shows…  How chromatic a passage is? Why we observe relative pitch?  To calculate entropy  Most of people recognize tonic, major third, perfect fourth, and perfect fifth better than other pitches  To give the answer : What makes comfortable music be COMFORTABLE?

Bach - Suite No. 1 in G major - BWV 1007, Prelude Debussy - Prelude a l'Apres-Midi d'un Faune Lennon – Norwegian WoodMcCartney – Let it be

5. Results – Dimension & Entropy (1) ComposerTitleDimensionEntropy J. S. Bach Cello Suite No. 1 in G major - BWV 1007, Prelude Cello Suite No. 3 in C major - BWV 1009, Courante Cello Suite No. 6 in D major - BWV 1012, Courante The Art of Fugue - BWV 1080, Contrapunctus I (2) C. Debussy Clair de lune Prelude a l'Apres-Midi d'un Faune J. Lennon (The Beatles) Across The Universe Girl Julia Norwegian Wood Nowhere Man Strawberry Fields Forever P. McCartney (The Beatles) And I Love Her Here, There And Everywhere In My Life Let It Be Michelle Penny Lane Yesterday

5. Results – Dimension & Entropy (2) NumberTitleDimension 1Bach - Cello Suite No. 1 in G major - BWV 1007, Prelude Bach - Cello Suite No. 3 in C major - BWV 1009, Courante Bach - The Art of Fugue - BWV 1080, Contrapunctus I (2) Debussy - Prelude a l'Apres-Midi d'un Faune McCartney – Michelle McCartney - Penny Lane Lennon – Julia McCartney - Here, There And Everywhere Lennon – Girl Lennon - Strawberry Fields Forever Lennon - Nowhere Man Debussy - Clair de lune McCartney - Let It Be Lennon - Across The Universe McCartney – Yesterday Lennon - Norwegian Wood McCartney - In My Life Bach - Cello Suite No. 6 in D major - BWV 1012, Courante McCartney - And I Love Her

5. Results – Dimension & Entropy (3) NumberTitleEntropy 1Bach - Cello Suite No. 3 in C major - BWV 1009, Courante Debussy - Prelude a l'Apres-Midi d'un Faune Bach - The Art of Fugue - BWV 1080, Contrapunctus I (2) Bach - Cello Suite No. 1 in G major - BWV 1007, Prelude Debussy - Clair de lune2.16 6Bach - Cello Suite No. 6 in D major - BWV 1012, Courante Lennon - Norwegian Wood McCartney - Here, There And Everywhere Lennon - Strawberry Fields Forever McCartney - Yesterday McCartney - Penny Lane McCartney - And I Love Her Lennon - Nowhere Man Lennon - Girl McCartney - Michelle Lennon - Across The Universe McCartney - In My Life Lennon - Julia McCartney - Let It Be1.601

5. Results – Dimension & Entropy (4) ComposerBachDebussyLennonMcCartney Dimension Entropy The entropy of impressionist Debussy is higher than that of baroque composer Bach. Easy-listening pop song has very low entropy  It is a SONG.  Bach and Debussy’s sample music is orchestra pieces.

6. Limits (1) Statistical approach  Notes are NOT INDEPENDENT particles. Complexity  Changing key makes entropy higher.  Polyphony music is pretty hard… Dimension  It’s not easy that consider other factors (such as volume, rhythm, etc.) Various composition goal  There are so many genre! (such as rap)

6. Limits (2) Catching the exact key is not so easy… Example (McCartney – Yesterday)

7. Conclusion Statistical approach can give us MOST OBJECTIVE data. So it can be a good music analysis in spite of many limits. Beauty of music is dependent on  1/f (of course!)  Tonic, major third, perfect fourth, perfect fifth  But they are just NECESSARY condition. So, what can we do with that methods?  Give a quantitative value of certain music  Artificial compose

8. Reference 이석원, 음악심리학, 심설당, Madden, C. "Fractals in Music: Introductory Mathematics for Musical Analysis", High Art Press, Manaris B., McCormick, C. and Purewal, T. "Can Beautiful Music be Recognized by Computers? Nature, Music, and the Zipf-Mandelbrot Law," Technical Report CoC/CS TR# , March