Towars Complexity Stuies of Inonesian Songs Hokky Situngkir [hs@compsoc.banungfe.net] Dept. Computational Sociology Banung Fe Institute Research Fellow Surya Research International August 8 th 2007 Abstract We see some complex properties from Inonesian music iscography by means of music as perceive by Inonesian people. This covers the folk songs, national anthems, popular songs by Inonesian moern artists an performers an also from western popular an classical music as reference. The self similarity is rawn by using the moel of gyration an the internal ynamics of the pitches an urations use in songs is observe by using the logarithmic spiral moel. The employe entropy moel is also iscusse as well as introuction to the calculate ynamic complexity of meloic structure. Some generalization on the flow of music respect to the ynamic complexity is also shown. We iscover that at least there are two phases in the playe song: the shorter introuctory phase that ens in the peak of complexity of the song an the attenuating phase of complexity in which the multiple equilibria of the song is measure. The paper raws some interesting aspects regaring to those parameters an variables on Inonesian meloic corpora. Keywors: music, self similarity, spiraling effect in music, meloic entropy, complexity. 1
wrote a song for everyone, wrote a song for truth, wrote a song for everyone, an I couln t even talk to you John Fogerty Creeence Clearwater Revival 1. Prolog Inonesia is place with various ethnicities with also various aspects of culture. An interesting artifact in Inonesian moern culture is music. To a moern Inonesian, there are at least four types of songs popular among the people: ethnic songs, national anthems, inonesian popular songs, an some popular international songs which is ominate by the western inustrial music. In the previous work [10], we have iscusse the possibility of observing meloy as language in the statistical analysis in the properties of Zipf Manelbrot plot. Music is a complex artifact mae by many components incluing tone, pitch, rhythm, tempo, contour, timbre, louness, spatial location, an reverberation [12]. However, in orer to simplify our approach, we only put meloy into account as the most important elements of music comprise by pitches an urations in a song [7]. The paper reports the statistical observation regaring to the complexity of Inonesian musical corpora, of course, along with few classical western iscography as reference. What we meant here regaring to the statistical observation is the iscussions about some features revealing the complexities of some Inonesian meloies. The complexity can be seen for instance by reaing the unique musical contours of the meloies, the self similarities of some songs with some possibilities to see the respective fractal geometry, the ynamic entropy of the meloies an the measurement of its inherent complexity. The motive of the paper is to see that unique types/genres of meloies show specific features. There shoul be a lot of interesting research irections can be mae upon this report in orer to grasp the meloic structures of Inonesian musicality. There have been some research reports relate to the work that also invigorates the observation presente here. The work of cross cultural cognitive musical aspects as iscusse by Toivianen an Eerola [13], the mathematical analysis to some Turkish songs [4] are among those. There have been a lot of concern in musicology an even wier, in anthropology, researchers yearn for some possible classification an categories of music, ethnicity, an even iniviuals or groups involve in the creation of music [1], [3]. This paper moestly expects that some specific quantitative features expresse in the observation might open a goo entry an irection for such purposes. In sociology an communication stuies, it is wiely accepte that inter culture meetings (e.g.: yieling 2
social aspects we unerstan as acculturation) might affect the newly artifact prouctions, an meloies as the most important element in music, is nonetheless an aspect of it. For example, the presentation an prouction of traitional music nowaays are not merely using the genuine cultural artifacts. The interesting campur sari in Javanic folk songs, etc. might be one from a lot more musical interactions among cultures. A conjecture of this was shown by showing in the report the very istinguishable aspects between iscoveries in some music in other part of the worl in some pioneere observations with similar statistical approaches we are bringing. SONGS (sometimes polyphonic) SONGS (sometimes polyphonic) SONGS (sometimes polyphonic) monophonic meloy pitches urations (in secons) binary igit representation binary igit representation DECIMAL REPRESENTATION ANALYSIS Figure 1. The ata preparation for analysis The structure of the paper can be escribe as follows. We introuce some basic concepts relate to the ata structure use in the research an remining some aspects of the ata preparation as it has been iscusse in etail in the previous works [11]. This section is followe by iscussions about some meloic aspects relate to self similarity an raial representation of meloy in some corpora. The latter section iscusses some concepts of entropy we might incorporate in our analysis an information extraction from the iscography we observe. Some 3
general iscussions about complexity in music an especially Inonesian musicology as the result of our observation iscusse in the latter sections while some future possible irections are also rawn. 2. Meloy in Inonesian Songs When we talk about Inonesian songs, actually we are ealing with more than anyone can imagine relate to music. There are a lot of musical types an genres are enjoye by Inonesian, from ethnic music, national anthem, popular songs, an of course, the internationally recognize songs from the multinational music inustries. It is obvious that the hunger for some statistical aspects an features of music woul never be satisfie for the rapi music prouction is always be there with innovative an artistic ieas from musicians or artists as well as the rapily growing computational methos an moeling that capable to be exploite for further analysis. Figure 2. The representation of the 16 th 35 notes of Bubuy Bulan (Sunanese folk song) in its representation of stanar meloic contour (top) an the transformation as use in the paper (below). 4
In this report, we use the monophonic note sequence for the song analysis. We must clean the songs before we are able to analyze the songs an the cleaning methos are omitting some polyphonic aspects of the available songs an then converting the songs. The proceures we o before the analysis is epicte in figure 1. The more etaile iscussion about preparing the ata can be referre to [10]. The heart of this ata representation is to having single representation for a pitch an the uration of it, a contrast to major works in this fiel istinguishing the pitches an the uration elements in musical corpus. Thus, we woul have a set of ecimals θ θb which is comprise by the respective pitch an uration in the form of binary. An argument to have such moel for songs is base on realization that it is more comfortable to talk about a note irectly with its respective urations for analysis of a song. [10]. Figure 2 shows that by using our representation, the yiele contour of sequences of notes in a song are not very ifferent with those shown by the stanar meloic contour representation (plotting the notes as represente by ecimals in its orinate an the time beats in the horizontal axes). We use the representation of θ (which is normalize with its maximum value in each sequence so that θ = [0,1] ) to analyze more than a hunre of Inonesian iscography in the efinition as songs that are wiely recognize by Inonesian in classifications as follows, Inonesian Folk Songs, the songs that are mostly accompanie by lyrics an meloic pattern sensitive to certain ethnic languages in Inonesia. However, some of the ethnic or traitional songs are with the national theme while some other are about local life ynamics as perceive by traitional an ethnic artists or people. The songs of this category are calle with their respective ethnicities, for example, Javanese songs (e.g.: Suwe Ora Jamu), Minangnese songs (e.g.: Ayam en Lapeh), Sunanese songs (e.g.: Bubuy Bulan), Malukunese songs (e.g.: Hela Rotan) an so on. National Anthems, the songs with patriotic themes expressing the joy of life as Inonesian people, mostly with Inonesian lyrics sung at many places an occasions in Inonesia. Some songs in this category are Gugur Bunga, Inonesia Pusaka, Satu Nusa Satu Bangsa. Some songs in this category has an interesting an well known pattern of Keroncong Musik, the music popular in Java, e.g.: Sepasang Mata Bola. Inonesian Popular Songs, the moern popular songs broacaste in mass meia an become a part of popular culture mostly among youths. However, as well as some international popular music an songs, there is sensitivity of time in this category of songs. Some songs in our iscography for this category, e.g.: Kugaaikan Cintaku (sung by Gombloh), Buna (sung by Potret), Juwita Malam (classic popular song with several times to be re release by ifferent artists), Dibalas engan Dusta (sung by Aui). 5
International Songs, this category is use as a matter of fact as referential to Inonesian popular songs. Some songs in this category, e.g.: I want to Break Free (Queen), Morning Has Broken (Cat Stevens), Now an Forever (Richar Marx), It Must Have Been Love (Roxette). Meloy from Classical Corpus, while the original form of this category are in the form of symphonies, operas, etc., we use some limite songs as reference the previous categories. 3. Meloic Self similarity Figure 2 shows the ups an owns of the notes a long with their urations, while we can also represent the meloic contour in the phase iagram epicting the transitional pattern of a note to another note, θ ( τ ) vs θ ( τ + 1) [10]. Günüz & Günüz [4] shows that we can see the internal structure of the song by starting our observation from here. A previous work on time series ata has also incorporate this kin of metho [9]. Figure 3. The tren line of the normalize in the phase space of θ ( τ ) vs θ ( τ + 1) from Inonesian popular song, Kaulah Segalanya popularize by Ruth Sahanaya. Any song shows the tren line as illustrate in figure 3. The tren line can actually showing the structure of pitches an urations emerging the meloy as we enjoy. We can see the tren line, as some kin of axis of rotation in the song ynamics. The line is in the form of stanar linear equation of y = mx + c, we coul have the axis of rotation of R g in the form of 6
R g = n i= 1 ( n y ) i+ 1 i+ 1 ( n 1) cosα 2 (1) 1 where α = tan m. Thus, simply speaking we can say that the concept of gyration raius, R g, is mae up by vertical istance between the position of θ relative to the tren line. This variable shows how compact the meloic structure of a song is. The smaller value of R g shows the scattering along the iagonal axis of the tren line: the smaller it is, the more compact the meloic structure is. Table 1 shows some results representing the categories. Figure 4. Visualization of raial movement of two popular songs, Kaulah Segalanya sung by Ruth Sahanaya an It Must Have Been Love by Roxette. The circle in the mile is the value of R an the re otte line is the graient of the tren line. g The iscussion about the abstract gyration ynamics in meloic movement has brought us to visualization of the meloy in a song into raial visualization. The visualization is quite simple by transforming the plot of θ ( τ) θ ( φ) where ϕ = [0,2 π ]. An example of the result is shown in figure 4 where we compare two popular songs: Inonesian Kaulah Segalanya an international hit It Must Have Been Love. Figure 5 shows the visualization for some of Inonesian folk songs an Inonesian National Anthems both groups are plotte in the scale raial iagram so that we concern the form of the θ ( φ ) visualization only. Interestingly, we can see the pattern of selfsimilarity within songs an in other cases the similarity between songs in the same group. 7
Figure 5. Scale Raial Movement of Songs in Folk Song Category (above) an National Anthems (below). 8
If 0 Figure 6. Illustration of logarithmic spiral in terms on α. α = π it turns out to be simple spiral. α = it turns out to be a circle an if /4 From what picture in figure 4 an 5, we can see the self similar patterns in Inonesian songs, an in orer to see more properties of the songs, we can visualize the raial istribution of notes an pitches in each song. We o this by sorting θ an plotte the ranke notes an pitches in a song. Interestingly, the pattern of logarithmic spirals woul be emerge as seen in the examples in figure 7. The logarithmic spirals can be written mathematically as [6, 14] ρ = aexp( bφ ) (2) where a an b are constants an as shown in figure 6, we can write r cotα = (3) rφ thus, 9
r ab exp( bφ ) br φ = = (4) an we have b = cotα (5) so that we have another parameter inicating the spiraling effect of a song [4], α 1 = cot b (6) The greater the value of α, the more notes in the song tens to move outwar in spiraling effect. In the visualization in figure 7, it is obvious that the song Kaulah Segalanya moves outwar relatively more than the song It Must Have Been Love by recognizing the singer of the former song is well known with her ability to cover high amplitue notes. Figure 7. The comparation of two popular songs regaring to the raial istribution of. θ Of course the usage of the notion about the spiral effect must be straightene here for in the common practice of art, this may gives polysemy. Nonetheless, the spiral effect is relate to the strength of repelling effect of the gyration when notes are in a song. 10
Classical Inonesian Anthem Songs International Moern Pop Songs Inonesian Popular Songs Inonesian Folk (ethnic) Songs Table 1 1 Gyration Raius an spiral coefficient in some songs in our iscography SONGS Oe To Joy (from Beethoven s 9th Symphony) The First Violin of Mozart s Symphony 40 Syukur (H. Mutahar) Gugur Bunga (Ismail Marzuki) Halo halo Banung (Ismail Marzuki) Dari Sabang sampai Merauke (R Suharjo) Hari Mereka (H. Mutahar) Satu Nusa Satu Bangsa (Liberty Manik) Inonesia Pusaka (Ismail Marzuki) Now an Forever (Richar Marx) I want to Break Free (Queen) Gyration Coefficient R g Spiral Coefficient α Song Negentropy Song Complexity Coefficient 0.1974 2.7751 3.1989 0.4996 0.1895 1.9222 6.0026 0.4404 0.1546 1.6187 3.8193 0.4470 0.1707 1.6258 4.0859 0.4563 0.0901 1.4742 2.6424 0.5544 0.1273 1.4719 3.0481 0.4957 0.1316 1.7011 3.5608 0.4739 0.1151 1.567 4.1278 0.4235 0.1112 1.5873 3.8517 0.4726 0.164 1.6164 4.2437 0.4371 0.116 1.4251 4.8442 0.3901 Kugaaikan Cintaku (Gombloh) 0.2298 3.3066 3.8640 0.5087 Kaulah Segalanya (Ruth Sahanaya) 0.1891 2.1041 3.9634 0.5076 Kemesraan (Iwan Fals) 0.1862 2.0966 3.1411 0.5428 DIbalas engan Dusta (Aui) 0.0916 1.0616 4.0104 0.4163 Buna (Potret) 0.1601 1.2636 5.2837 0.3427 Suwe Ora Jamu (Javanese) Bubuy Bulan (Sunanese) Serma Dengan engan (Simalungunese) Ayam en Lapeh (Minangnese) Dago Inang Sarge (Bataknese) Tanuk Majeng (Mauranese) Hela Rotan (Malukunese) O Ina Ni Keke (Minahasanese) 0.2366 2.8833 1.9253 0.6639 0.195 2.0399 3.3494 0.4970 0.2086 1.7246 4.5985 0.3604 0.1318 1.441 5.0253 0.3810 0.086 1.0431 3.5493 0.4448 0.1742 1.5598 3.9498 0.4251 0.2237 3.4344 3.7477 0.5083 0.1389 1.7504 3.2032 0.5124 1 Only few songs from our iscography are shown here for the limite space. 11
The stronger the spiral coefficient (α ) the more tenency the usage of more notes of which higher values of θ, practical speaking the higher frequency of pitches. Most national anthems o not have relatively very high spiral effect contraste to some of folk song an popular song for instance. This makes sense for national anthems are not suppose to be too ynamic in its variations of pitches an urations. However, some popular songs might also become not too spiraling by its theme an genre. In table 1, we can see apparently that the popular song like Buna sung by Inonesian music ban, Potret, has the lowest spiraling effect while other popular song, even though within really ifferent genre, I Want to Break Free sung by Queen, also have small coefficient. 5. The Entropy an Complexity of Meloic Structure The exploitation of the concept entropy to musical analysis an song analysis are always interesting. Some previous works has iscusse about the complexity of music in the terms of entropy of pitch class istribution an entropy of interval istribution [12] an in return to be iscusse in cross culture musical analysis in [12]. The entropy of a meloic structure is usually measure respect to the number of notes [11] as the number of the microstates in a song [7]. Here, we woul put into account not only the number of notes solely. Our measurement woul be more like the one propose in [4] but yet we woul use variable θ as the basis of the calculation instea of the notes only. The iea is to capture the level of possible micro states in the terms of numbers of notes an pitches controlle by other notes an pitches. Figure 8. The ynamic calculation of entropy (First parts of Tanuk Majeng, Mauranese folk song) In a sequence of meloy, if we enote k i as the number of types of notes after a specific note θ then, the entropy of the single pitch an uration is i k p = (7) θ N 12
where N = k, thus the entropy of all the meloic structure can be written as, i i S( p ) p log 2 p θ = θ θ (8) i i i i Obviously, the entropy of a meloic structure eals with what note is controlle by a former one [4]. In avance, the maximum entropy of all meloic structure can be calculate as, S = log 2 N (9) Here, the complexity of a song can be calculate as [cf. 12], S( p ) p log 2 p θ song θ θ song i H = = S log N max 2 song (10) It is also worth to iscuss about negentropy [4], as the parameter showing the egree of organization, or simply speaking, the orer of the song. Neg = S S( p max ) (11) θ song The interesting part of the offer from equation 8, 9, 10, an 11 is that the possibility to see the real time orer an isorer state of the song as per time, per aition of figure 8. θ, like the one exemplifie in Glancing over the entire entropy an complexity plots, we coul see some patterns in terms of entropy, negentropy, as well as complexity of a song. From our observation over all the songs, the songs seems to have the entropy rapily higher in first notes, until later the entropy fluctuates in certain values showing the multiple equilibria. The orer an isorer phases are shown here in the long run of a playe song. The maximum entropy is always growing bigger as unerstoo from the nature of the aitions of possible microstates as long as there is a tone playe. However, the ifferences across songs are the way the song ens whether in the state of organization ( Neg > S ) or vice versa. In figure 9, we can see some result obtaine from observation to the work of Mozart an Inonesian classic composers to the national anthems. The pattern of the ynamics showe by the 13
entropies is similar but the ifference arouse in the way the complexity grows within a song. Mozarts seems not ecreasing very rastically over the long part of the symphony, a contrast to the other three epicte in the figure. Figure 9. Meloic Contour, Entropy, an Complexity of some songs Mozart s an Inonesian Anthems. An interesting pattern arouse as we see the folk song corpora (figure 10) in which we foun that most of them are ene with higher negentropy relative to its ynamic entropy. This means that the song seems to buil an organization throughout the song. The folk songs seems to be laste in the state of orer, a little bit ifferent with ones in other categories of song. However, there are also some other songs with ene in the state where entropy is bigger than the negentropy (Suwe Ora Jamu an Ayam en Lapeh). This is interesting facts that we can see apparently in the ynamics of the folk songs. The more various patterns occur in the playing of popular songs. This is visualize in figure 11, the entropies an complexities of Inonesian popular songs. Like in figure 8, the songs in the popular songs are ominate by the ening of the lower organization inex (negentropy) relative to its respective entropy. 14
Figure 10. Meloic Contour, Entropy, an Complexity of some Inonesian Folk Songs. Figure 11. Meloic Contour, Entropy, an Complexity of some Inonesian popular songs. 15
In table 1, we can see the value of some songs negentropy an the overall complexity as calculate in the en of the song. The more a song varies in the usage of pitches an urations, the more complex the song is, while the more sequence of pitches an urations use more frequently the more the organizational structure of the song is reveale (as epicte by the negentropy). Of course if in a song the negentropy is very high the song becomes boring an the lesser it is, the more ifficult the songs to be enjoye. However, it is not always the bigger negentropy woul be linearly relate to the calculate complexity. An interesting example is the long song from Mozart s #40 th Symphony comprise by more than 1500 notes has relatively low complexity but very high in the inex of organization. To have an interesting an easy to be graspe song probably lai upon these two constraints, while the inner ynamics of the songs portraye by the two previous parameters, the gyration an the spiral inex. 6. Discussions The self similarity of a song can be visualize in the raial iagram of a song as shown in figure 5. The latter observation has brought us to the spiral ynamics that can be moele by using the logarithmic spirals. The boring meloic structure woul be yiele by the structure of very low values of spiral coefficient while in return, too big spiraling effects within a song mae the song becomes harer to be enjoye. However, there is something interesting as we try to make a sort of generalization as we observe a lot of corpora of songs. It is in the ynamic complexity of the song from the first playe note till the last tone it plays. A simple generalization is epicte in figure 12. Most of songs seem to follow the way the ups an owns of the complexity inex as shown in the figure. First notes flow to grow the complexity ( τ 1 ) or can we say the phase to introuce the pattern in the whole songs until they accumulate to the maximum complexity of the songs. The next steps are the attenuating phase of complexity as the complexity (sometimes rippling) ecreases till the en of song is reache ( τ 2 ). We can formally write that the secon phase is always taking longer time than the former one. τ < τ (12) 1 2 There might be several peaks in the ecreasing phase of complexity but yet, the secon peak seems to be smaller than the previous one, or vice versa, there might be a smaller peak is coming right before the general peak of complexity throughout the song. In the sense of ynamic entropy of a song, we coul see that the ecreasing phase is the multiple equilibria of the song as enjoye by the 16
listener. Furthermore, the inequality of formula 12 might become the thing that shoul be relate to human cognitive system since the longer phase of τ 1 might make the listener having more effort to enjoy the beauty of the song. This very interesting general pattern is foun in our observation over our iscography. Nonetheless, more empirical results is encourage to have the generality that might obtaine in the ynamical analysis of the song. Complexity (H) τ 1 τ 2 peak of complexity time steps song begins song ens Figure 12. The flow of complexity within a song. 6. Concluing Remarks We have shown some complex property of the music as perceive by Inonesian people: folk songs, national anthems, an popular songs, as well as some reference to the arts of classical an popular western music. We show that the self similarity is shown in most of the songs, while there is also some self similar pattern in the same categories of music. The way the use of the variations of pitches an urations is represente by the gyration coefficient epicting the ynamicity of the song an the (logarithmic) spiral coefficient reflecting the tenency of each songs to show the spiraling effect by means of the numbers of variations an tonal movements within. From our observation we foun that meloic structures of Inonesian songs are varies through both gyration an spiral coefficient. However, some conjectures are iscovere that Inonesian anthems are more likely to have smaller spiraling effects relative to folk songs an popular songs, even the classical western meloies we use as reference. However, from the observation by using the concept of entropy, we foun that there is a great similarity between the 17
Inonesian anthems with the classical western meloic structures giving us some insight of the process of the creative process of the making of the songs in the category. Another interesting finings are that the Inonesian folk songs tens to en up in the form of organize structure with the negentropy tens to be bigger with the entropy as the inex of regularity in each song. This is contrast with those we coul apparently observe in the popular an national anthem songs. We also iscover some conjecture for further generalization for a like further research on songs. The flow of the most songs seems to be rapily hiking the complexity as the meloic introuctory phase until the note flows foun the maximum complexity. This phase is then followe by the attenuating phase of complexity as reflecte by rippling ecreasing complexity. Further research irections can be conucte by incorporating some aspects regaring to the enlargement of the corpora, not only covere Inonesian music but a lot more from various cultures an musical sources. Works Cite [1] Cope, D. (1998). Signatures an earmarks: Computer recognition of patterns in music. In W. B. Hewlett & E. Selfrige Fiel (Es.). Meloic similarity: Concepts, proceures, an applications (pp. 129 138). MIT Press. [2] Fiske, J. (1990). Cultural an Communication Stuies 2 n eition. Routlege. [3] Freeman, L. C. an Merriam, A. P. (1956). Statistical classification in anthropology: An application to ethnomusicology. American Anthropologist 58: 464 472. [4] Günüuz, G. & Günüz, U. (2005). The Mathematical Analysis of the Structure of Some Songs. Physica A 357: 565 92. [5] Peitgen, H O., Jürgens, H., & Saupe, D. (2004). Chaos an Fractals: New Frontiers of Science 2 n eition. Springer. [6] Planes, A. & VIves, E. (2002). Entropic Formulation of Statistical Mechanics. Journal of Statistical Physics 106: 827 50. [7] Selfrige Fiel, E. (1998). "Conceptual an representational issues in meloic comparison". In Hewlett,W.B. & Selfrige Fiel, E. (es.), Meloic Similarity: Concepts, Proceures, an Applications. MIT Press. [8] Lambiotte, R. & Ausloos, M. (2006). On the Genre Fication of Music: a Percolation Approach. The European Physical Journal B 50: 183 8. [9] Situngkir, H. & Surya, Y. (2003). Neural Network Revisite: Perception on Moifie Poincare Map of Financial Time Series Data. Physica A 344: 100 3. [10] Situngkir, H. (2007). "An Alternative Postulate to See Meloy as 'Language'". BFI Working Paper Series WPK2007. 18
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