Bach Is the Father of Harmony: Revealed by a 1/f Fluctuation Analysis across Musical Genres

Abstract

Harmony is a fundamental attribute of music. Close connections exist between music and mathematics since both pursue harmony and unity. In music, the consonance of notes played simultaneously partly determines our perception of harmony; associates with aesthetic responses; and influences the emotion expression. The consonance could be considered as a window to understand and analyze harmony. Here for the first time we used a 1/f fluctuation analysis to investigate whether the consonance fluctuation structure in music with a wide range of composers and genres followed the scale free pattern that has been found for pitch, melody, rhythm, human body movements, brain activity, natural images and geographical features. We then used a network graph approach to investigate which composers were the most influential both within and across genres. Our results showed that patterns of consonance in music did follow scale-free characteristics, suggesting that this feature is a universally evolved one in both music and the living world. Furthermore, our network analysis revealed that Bach’s harmony patterns were having the most influence on those used by other composers, followed closely by Mozart.

Multiple scaling behaviour and nonlinear traits in music scores

“…Power laws or scaling laws are manifestations of self-similarity in the world around us [ 12 – 15 ]. In music, the 1/ f β spectra with β =1 has been interpreted in [ 9 , 16 ] as a trade-off between predictability and surprise; if β tends to lower values (zero is the case of white noise), the temporal sequence of notes is highly uncorrelated and sounds unpleasant. On the other hand, if β becomes too large, the music becomes monotonous.…”

“…Scaling behaviours, in particular 1/ f β with 1< β <2 have been found frequently in music [ 2 , 4 , 5 , 9 – 11 ]. However, few of the studies have focused on music scores [ 4 , 10 , 16 ], and to the best of our knowledge, none presents a detailed analysis of the scaling behavior in the pitch fluctuations of pieces from different composers. It should be noted that unique scaling laws in musical pieces are not always present, indeed, scaling exponents may vary on different timescales [ 2 , 4 ].…”

Novelty and influence of creative works, and quantifying patterns of advances based on probabilistic references networks

“…The availability of large-scale musical databases and advances in scientific, analytical methods continue to enable novel and interesting findings on their properties [10]. Recent examples include researches on the topology and dynamics of the networks of musicians for the discovery of human and stylistic factors in the creation of music [15][16][17][18][19][20] and stylometric analyses of music that lead to corroborations or fresh challenges to established musicological understanding [21][22][23][24][25].…”

Melodies as Maximally Disordered Systems under Macroscopic Constraints with Musical Meaning

“…Many quantitative analyses in music have been carried out using different elements as building blocks, or "units of context", which allow the message of a musical piece to be apprehensible at different time scales [1]. Common choices for these "units of context" are single pitches (ignoring or taking into account the chroma properties [2,3]), single musical notes (i.e., pitch and rhythm values), pairs of pitches or musical intervals (either harmonic or melodic), triplets of pitches between contiguous notes, and chords [1][2][3][4][5][6][7][8][9]. In the case of musical intervals (from now on referred to as intervals), quantitative analyses frequently employ parameters that describe their psychoacoustic properties, such as the sizes of intervals (commonly measured in tones or semitones), the ratio of the fundamental frequencies of both pitches (commonly measured in units of cents), and the difference between the fundamental frequencies [10,11].…”

“…Analyses based on statistical methods can capture information about musical features, such as the style of a musical piece, the composer, and even the emotions conveyed [1,[3][4][5][6][7][8][9][12][13][14][15][16][17][18][19][20]. Several statistical analyses employ successive pitches as units of context.…”