Table 1.
Fundamental symbols leading to minimal entropy computed for short texts.
On the right, the object text is a tiny segment of the MIDI representation of the 4th movement of Beethoven’s violin concerto. On the left, the object text is an English text written for the purpose of this example. Symbolic models of the original texts were obtained by applying the Fundamental Scale Algorithm to the text’s objects.
Table 2.
Music classification tree MusicNet and the data associated with top levels of the tree.
Fig 1.
Diversity as a function of piece length measured in symbols for different classes of music.
Each bubble represents a piece of music. The vertical axis represents the symbolic diversity D expressed in thousands of symbols. The horizontal axis represents the length L of the piece description expressed in symbols.
Fig 2.
Entropy as a function of specific diversity for different classes of music.
Each bubble represents a piece of music. The vertical axis represents the entropy h. The horizontal axis represents the specific diversity d = D/L. Top and bottom row show the same graphs presented at different scales.
Fig 3.
Variation of frequency profiles for several degraded scales and information profiles calculated for Hindu-Raga.Miyan ki Malhar.
Fig 4.
Symbol-ranked frequency profiles for eight performances grouped by pairs associated with the same music piece.
Pairs of graphs show two performances of Bach’s Toccata & Fugue, Ravel’s Bolero, Rachmaninov’s Piano Concerto #2 and the Venezuelan waltz ‘El Diablo Suelto’. Each circle represents a symbol. As a helpful reference, each graph shows a Zipf profile represented by grey ‘+’ signs. The vertical axis is the probability of encountering a symbol with the text. The horizontal axis shows the rank of the symbols according to their frequency. Both axis scales are logarithmic. Links to sound: JSBach.ToccataFuga.Organ, JSBach.ToccataFuga.Piano, Ravel's Bolero.1, Ravel's Bolero.2, Rach.PC2.PianoSolo.mid, Rach.PC2.PianoAndOrchestra.mid. DiabloSuelto1.mid, Diablo Suelto 2.mid.
Table 3.
Distances between profile pairs taken from profiles shown in Fig 1.
Pairs of interpretations of music pieces are compared against each other and with respect to the Zipf reference profile. The comparison consists of the computation of the Euclidian distance over the logarithmic difference for each dimension, as indicated in Eq (2).
Fig 5.
Symbol-ranked frequency profiles for Impressionistic music.
Graph (a. Left) shows the traditional symbol profile. Graph (b. Right) shows the profile for the 2nd order symbols.
Table 4.
Entropy and distance to Zipf’s reference for academic music grouped by composer and period.
Table 5.
Entropy and distance to the Zipf’s reference for traditional and popular music grouped by composer and style.
Fig 6.
Entropy vs distance to Zipf’s reference for separable groups of music.
Left: academic music. Right: traditional and popular music.
Table 6.
Student t-tests for entropy and distance to Zipf’s reference for academic music sub-groups.
Table 7.
Student t-tests for entropy and distance to Zipf’s reference for traditional and popular music sub-groups.
Table 8.
Univariate ANOVA tests for entropy and distance to Zipf’s reference for academic music sub-groups.
Degrees of freedom = Number of groups– 1, SS = Sum of squares of variance errors, MS = Mean of squares of variance errors, F = MS / Degrees of freedom.
Table 9.
Univariate ANOVA tests for entropy and distance to Zipf’s reference for traditional and popular music sub-groups.
Degrees of freedom = Number of groups– 1, SS = Sum of squares of variance errors, MS = Mean of squares of variance errors, F = MS / Degrees of freedom.
Table 10.
Multivariate ANOVA tests for entropy and distance to Zipf’s reference for traditional and popular music sub-groups.
Λ value = Wilks’ lambda value. F = Mean of Squared Errors / Degrees of freedom.
Fig 7.
Three views of the representation of selected MIDI pieces in the space specific diversity, entropy, and 2nd order entropy (d, h[1], h[2]).
Each bubble represents a MIDI piece.
Fig 8.
Three views of the representation of music period/style groups in the space specific diversity, entropy, 2nd order entropy (d, h[1], h[2]).
Each bubble represents a group of music pieces sharing the same style/period.
Table 11.
Properties of western academic music.
Table 12.
Properties of some traditional and popular music.
Fig 9.
Change of the second order entropy over the last few centuries for genres and styles of music.