INTRODUCTION

A listener may experience music as "difficult," "complex," or even "incomprehensible." Many listeners will put such music aside, and look for alternatives, unless they are explicitly interested in the reasons why a particular musical piece is perceived as it is. This paper explores this specific interest in the context of children's folk songs. It specifically examines—and refutes—the assumption of many musicologists that children's folk songs are simple or trivial.

Behind each piece of music is the story of the creation of a musical structure, covering different musical elements and dimensions, and a musical syntax, a more or less formal characterization of rules that define the permissible structure—that is, how the constituent parts in a piece may be formed and combined over time (Rohrmeier & Pearce, 2018).

Thus, Western tonal music is often considered to be fundamentally syntactic. A syntactic feature is, e.g., the hierarchical structure that is achieved by combining perceptually discrete related elements, on a range of timescales. Another syntactic feature is the sequential relation between musical elements, because the "syntactic functional psychological qualities" of some musical elements depends on the relation with others (Bigand et al., 2014, p. 2).

Several studies (e.g., Bigand & Poulin-Charronat, 2006; Koelsch & Jentschke, 2008; Bigand et al., 2014; Mihelač et al., 2018; Mihelač & Povh, 2020a) point to the significance of musical syntax, and its contribution to the notion of structure which is regular—i.e., which conforms to syntactic rules—as opposed to irregular—non-conformant—structure. Broadly, these studies take either a traditional empirical approach (viz., involvement of listeners in experiments and in the evaluation of music), or, recently, a computational approach. The studies explore how musical syntax, and deviation in a musical structure, which occurs when structure does not conform with syntax, are related to the listener's perception of (ir)regularity in musical structure, and their acceptance, pleasure, and understanding of musical pieces and/or musical genres.

Taking the more traditional approach can be costly, and also scientifically challenging, because it is difficult to avoid subjectivity in evaluation tasks, both in musical experts and in listeners without musical training. Furthermore, it is difficult to replicate the results, as the response in a task can vary significantly between hearings, even if the same participants are involved in the replication (Mihelač & Povh, 2020b).

In recent decades, computational approaches to the study of music have offered a more objective analysis of music (Potter, Wiggins, & Pearce, 2007), enabling identification of significant features (such as semiotic structure: Wiggins, 2010) in a musical structure, and comparison with the processing and production of music in humans. These computational methods are now becoming capable of simulating the human perception of music to some degree, especially where they are based on an underlying cognitive theory.

The fact that computational models can be efficiently used in the simulation of the human perception of music was the motivation for a recent study by Mihelač and Povh (2020a). Human experts involved in the detection of (ir)regularity in the musical structure in a previous study (Mihelač & Povh, 2020b), and the evaluation of (ir)regularity of musical excerpts by listeners were replaced by simulating their responses with a computational model, Information Dynamics of Music (IDyOM: Pearce, 2005).

The artificial model of the perception of (ir)regularity obtained by Mihelač and Povh (2020a) is shown to accord with human perception of irregularity in the previous study (Mihelač & Povh, 2020b), suggesting that expert-based detection of (ir)regularity in musical structure can usefully be replaced by a suitable computational model. In the current study, we used a similar approach to the (ir)regularity in musical structure, by simulating and detecting the (ir)regularity in musical examples with the computational model IDyOM, and our own algorithm, Ir_Reg, which classifies melodies according to regularity of the musical structure. The data for this study consisted of monophonic children's folk songs. We used this corpus to show that children's folk songs, often presumed to be simple and regular in structure (Herzog, 1944; Romet, 1980; Pond, 1981; Nettl, 1983; Ling, 1997), can be also complex.

BACKGROUND AND RELATED WORK

Music is an art of "humanly organized sounds" (Godt, 2005, p. 84). Thus, sounds are not merely randomly distributed, but are organized in a specific order: discrete units (sounds) into smaller parts, smaller parts into larger parts, up to the level of a musical piece, creating a hierarchical structure, which is one of the cornerstones of music theory and music cognition (Levitin & Menon, 2000). A set of permissible and rigid rules, defined as musical syntax, defines how this structure has to be organized, covering a wide range of different fundamental musical elements (Berezovsky, 2019).

According to studies (e.g., Cohen, 2003; Marcus, 2003; Cohen & Katz, 2013), musical syntax differs not only between different cultures (in non-Western and Western tonal music traditions), but also in different styles within Western tonal music (e.g., the Middle Ages, the Renaissance, the period of tonal music, from ca. 1600–1918, and the Modern period after 1918), due to the evolution of musical-syntactic rules over time (Meyer, 1989; Klein & Jacobsen, 2014; Vuvan & Hughes, 2019). Changes within these periods are considered as compositional choices (Meyer, 1989), rather than as changes in musical language.

Several empirical studies suggest that musical syntax is represented cognitively (e.g., Bharucha & Stoeckig, 1986; Tillmann, Bharucha, & Bigand, 2000) and that observable neural correlates exist (e.g., Janata et al., 2002; Koelsch & Siebel, 2005). Manipulating any of the fundamental musical elements in a structure impacts the reported feeling of regularity (Rohrmeier, 2011; Pole, 2014; Rohrmeier & Pearce, 2018; Mihelač & Povh, 2020a,b) and its identification (Bruner, Wallach, & Galanter, 1959). Patel (2003) argues that, following manipulation of the order of elements (parts), a piece of music loses its identity, which accords with the processing of sequential information and structure in non-musical domains (Garner, 1974).

A piece of music is considered to be regular (from the perspective of musical syntax and/or the perception of listeners), when a strong structure exists, with periodic dominant musical elements (patterns), and there are strong relationships between these patterns (Wu et al., 2000). Conversely, an irregular piece contains a non-structured or weakly structured texture, with a small number of detectable patterns with weak interrelationships (Kramer, 1988). Klein and Jacobsen (2014) point out that for every compositional rule in a given tonal style a complementary regularity is generated for a complementary tonal style: a given structure, perceived as syntactically regular in context of one piece, can appear to be irregular in another (e.g., frequent endings on dominant triad in folk songs from Serbia, perceived as a syntactically "regular", however as "irregular" in children's folk songs from the same country, as these songs ends on the tonic triad).

Findings from various studies outline that a stronger feeling of regularity is perceived when similar parts are periodically repeated, either in an "absolute repetition condition", where parts are repeated identically, or in a "relative repetition", where parts are considered to be conditionally identical (Bader, Schröger, & Grimm, 2017). 1 Relative repetition is important in the development of motifs that form the structure of longer pieces (Deliège, 1987; Cambouropoulos, Crawford, & Iliopoulos, 2001). Thus, the prevalence of repetition in music, found in all known human cultures, is unsurprising, as is the fact that listeners tend to listen repeatedly to familiar musical pieces (Margulis, 2014). Repetition is a fundamental characteristic of music, a "design feature" of music (Fitch, 2006), contributing significantly to our understanding of music (Schoenberg, Strang, & Stein, 1967).

Bruner, Wallach, and Galanter (1959) point out that perception of recurrent regularities (parts/patterns repeated at pitch or transposed repeated patterns) is impacted either by elements that mask the identification of recurrent regularities (e.g., an input-stimulus, which does not conform the recurrent series in a sequence) or by the regularity itself, in the case where it "exceeds the memory span" of an observer (Bruner, Wallach, & Galanter, 1959, p. 84). The identification of recurrent regularity in a musical structure requires that a listener either construct a model to represent this regularity or deploy a model, which they have previously constructed. In either case, the success of identification depends on separating recurrent regularities from interfering structures. The greater the number of elements interfering the recurrent regularities, the more noise is found in the stimulus, and consequently the more difficult is the identification of recurrent regularity (Bruner, Wallach, & Galanter, 1959). The perception and identification of recurrent regularities and structural particularities in a musical structure depend on how well the musical structure and its syntax are internalized in listener. The internalization of music, which can be implicit, during mere exposure to music without listening, or explicit, when interacting with music (Pearce & Wiggins, 2012), depends on several factors, such as the listener's musical experience and/or training (Lappe, Steinsträter, & Pantev, 2013; Bangert & Altenmüller, 2003). Lu and Vicario (2014) show that human infants and adults identify recurrent sound patterns even when exposure to musical sequences is passive.

No matter how internalization occurs, internal models of the (permissible) musical structures are generated and applied whenever listening to novel music (Deliège, 1987; Agres, 2019). If the internal models somehow confront the (prescribed) rules of the arrangement of the constituent parts of a musical structure, the structure is perceived not only as irregular, but also as more complex and less enjoyable (Sauvé & Pearce 2019; Mihelač & Povh, 2020b). This is because more effort has to be put into the listening process, as the learned musical syntax and its rules have to be adjusted in order to understand the novel structures (Kramer, 1988).

Musical genres vary from the perspective of the recurrent regularities in musical structure, and from the perspective of musical elements that are emphasized in a repeated at pitch or transposed repeated condition, and therefore contributing more or less to the regularity. In some musical genres (e.g., modernist, and expressly avant-garde approaches), repetition is avoided (Margulis, 2014), and any feeling of surface regularity is deliberately lacking. In some other genres, e.g., in minimalist music, regular structure is achieved by (e.g.) emphasizing the rhythm, using repetitive rhythmic patterns (Johnson, 1994). Another example of music with recurrence regularities in its structure, employing both repeated at pitch or transposed repeated conditions, is children's (folk) songs (Jožef-Beg & Mihelač, 2019), to which we return in Section Information Dynamics of Music: IDyOM.

Several approaches have been used to measure (ir)regularity in musical structure, including subjective evaluation by listeners (e.g., Deutsch, 1980; Tillmann & Bigand, 1996; Mihelač et al., 2018; Mihelač and Povh, 2020b), measurement of neural response (mismatch negativity, MMN) and functional magnetic resonance imaging (fMRI) (e.g., Ulanovsky, Las, & Nelken, 2003; Grahn & Rowe, 2012; Yu, Liu, & Gao, 2015), or simulation of human perception of (ir)regularity using a computational system (e.g., Hansen & Pearce, 2014; Mihelač & Povh, 2020a). The current study uses a computational model to enable a novel approach to the examination of (ir)regularity in the musical structure of children's folk songs.

CHILDREN'S FOLK SONGS

Music "happens to" children all the time, often in singing songs during unrelated activities, and as part of education. However, this seems not to have been understood by adults in earlier decades (Campbell, 1998). As a result, many children's folk songs (defined also as traditional songs), contributions by and for children, may not have been collected. Such collection could contribute to a better understanding of these songs within and between cultures, and of their relation to adult music, within culture. A renewed interest of ethnomusicologists, sociologists, educators, and folklorists in children's folk songs can be seen from 1940 onward, centering on studies of musical content, on the social and cultural significance of children's folk songs and their relationships to the music of adults, on how these songs contribute to the preservation of a particular culture, on how the children's folk songs are transmitted from one generation to another, and so on (e.g., Herzog, 1944; Brailoiu, 1954; Newell, 1963; Blacking, 1973; Nettl & Béhague, 1980; Pond, 1981; Nettl, 1983).

Collections of children's folk songs, from all over the world, include songs created by adults for children, folk songs, as well as songs created by children and passed along to each other. It is sometimes extremely difficult to trace the origin of children's folk songs, as children's songs created by children are often changed and adapted by children in different generations, or even published by adults, due to lack of musical skill in children (e.g., how to transcribe the songs). A similar problem occurs with folk songs (to be found among children's songs), as it is not clear whether these songs have been created by children, or by adults for children. The single exception is lullabies, existing in nearly every culture, which are created by adults for children (Jožef-Beg & Mihelač, 2019). No matter who created them, children's folk songs exhibit many features which also exist also in (other) folk songs, and which are passed from generation to generation (as folk songs), via oral tradition.

Children's folk songs, for which there is still no clear definition (Sutton-Smith, 1999), are tonally and structurally simple, with similar properties to be found all over the world in this genre. The content of these songs is close to the child's world, presumably to make them maximally accessible children (Romet, 1980; Pond, 1981; Nograšek & Virant Iršič, 2005; Voglar & Nograšek, 2009). Meyer (1989) argues that the simplicity (in terms of structure and performance) of the genre can be explained by the influence of parameters which are external to music (e.g., ideology, social history, conditions of performance, audience, etc.).

Tonal material in children's folk songs usually covers a pitch range within a perfect fifth, and smaller, descending intervals (seconds and thirds) prevail in the melody (Borota, 2013). Proto-melodies including only two tones, a descending third G–E (so–mi), interacting with rhythmic variation, and proto-melodies using three tones from a pentatonic tetrachord (the third interval with an added fourth) are very frequently used (Gortan Carlin, Pace, & Denac, 2014).

Children's folk songs are syllabic (that is, they lack melisma), free from ornamentation, with many absolutely or relatively repeated motifs, and with minor motivic variations on account of the text (Herzog, 1944; Ling, 1997). The frequent repetition of content in this genre is due to the audience for which it is intended: repeated motifs are more readily comprehended and memorized (Borota, 2013). Duple and quadruple meter prevail over triple, and binary rhythms are preferred over ternary. The prevalence of binary relations suggests that the fundamental perceptual requirements for such relations also exist in this genre, and are intuitively deployed when creating new songs (Fraisse, 1982, explains more about binary and ternary relations).

INFORMATION DYNAMICS OF MUSIC: IDYOM

We used the computational perceptual model IDyOM 2 (Information Dynamics of Music: Pearce 2005, 2018) to measuring the (ir)regularity in children's folk songs. The model has proven to be an accurate predictor of melodic expectancy (Pearce & Wiggins, 2006a; Sauvé & Pearce, 2019), of behavior and neural measures (EEG) of melodic expectedness (Pearce et al., 2010; Agres, Abdallah, & Pearce, 2018), of phrase boundaries (Pearce, Müllensiefen, & Wiggins 2010a,b). It has been shown to provide a good quantitative model of cultural distance (Pearce, 2018).

Entropy, Information Content and (Ir)regularity in Children's Folk Songs

In IDyOM, the internal representations of a musical piece/style are acquired and processed by using statistical learning and probabilistic prediction. While observing and analyzing the content in a corpus consisting of musical pieces, IDyOM learns about the syntactic structure merely by exposure. On the basis of sequential regularities, the likelihood of a forthcoming event is determined from the perspective of the feature which is analyzed with a viewpoint or viewpoints (Pearce, 2005, 2018).

IDyOM simulates a listener's expectations in music (which is based on the knowledge acquired during the entire lifetime) with a long-term model (LTM), which accumulates statistical information about musical structure from a large corpus. As listeners are sensitive to repeated patterns in an on-going listening experience, a second, short-term model (STM) is also used, in which the information about the musical structure of the current piece is learned dynamically and incrementally (Pearce, 2005, 2018). LTM and STM predictions are then combined, and it has been shown that better prediction performance is achieved by combining LTM and STM dynamically (Conklin, 1990a; Pearce & Wiggins, 2004, 2006b; Pearce, 2005).

In IDyOM, two information-theoretic measures, entropy, and information content (MacKay 2003), are used for the examination of the musical content and for the prediction of events. Although these measures are fundamentally related, and have been used synonymously by Shannon and Weaver (1949), they can be considered as usefully different. Shannon's entropy, or simply entropy, related to the idea of entropy in physics, is a measure of uncertainty. It is the average number of bits, required to transmit/represent the statistical uncertainty about the character of an event, randomly selected from a probability distribution.

Less uncertainty is experienced in case when one event dominates in a probability distribution. Larger or maximum entropy is expected when no event dominates another, for example in an equal or approximately equal probability distribution. Entropy, as defined by Shannon (2001), and shown in equation 1, is used as baseline model for quantifying the uncertainty in the prediction of a musical event before it is heard, using a specific viewpoint. Hansen and Pearce (2014) provide evidence that entropy is correlated with perceived uncertainty; mathematically, it is related to the balance of likelihoods in a distribution: a uniform distribution has maximum entropy (because there is no information in the distribution) while zero (i.e., minimum) entropy entails that exactly one value of the distribution has probability 1,

1. $H\left(e_i\right)=-\sum _{x\in x_i}{p}_x{\log }_2\left(p_x\right)$

where X_i is the set of all possible continuations of a given musical sequence after event e_i (in our case, the alphabet of a viewpoint) and p_i is the probability of event e_i.

The idea behind quantifying information content (MacKay, 2003), which is significant from the perspective of compressibility (Bell, Cleary, & Witten, 1990), is to estimate how unexpected an event is in the context in which it happens. Thus, information content is a measure of unexpectedness (sometimes called "surprisal") of an event which actually appears in the sequence. For a discrete event e_i, it can be calculated as shown in equation 2. Rare events in context have a low probability, are more surprising, and have a higher information content. Frequent events in context have a high probability and low information content, and so are less surprising. If the probability of an event is 1 (when there is no surprise), the information content is zero.

1. $IC\left(e_i\right)=-{\log }_2{p}_i$

where p_i is the probability of event e_i. In the rest of this subsection, we present, as an example, the information content (unexpectedness) and entropy (uncertainty) by exploring the viewpoint cpitch in three children's folk songs, selected from the much larger dataset used in this study. Note that for each event, e_i (in our case a tone in a sequence/melody), the information content and entropy is calculated from a probability distribution.

In Figure 1, the Turkish children's folk song, Cumhuriyet Çocuklarıyız, is presented. It consists of 78 events, in which motifs, hereafter defined as patterns (p), are denoted by p₁, p₂, and p₃. From the perspective of pitch span (within a perfect fifth), rhythm, and meter, the song is simple. Furthermore, the song has a clear and understandable musical syntax, and with a structure that is strong, consisting of three dominant and often at pitch repeated patterns with substantial relationships, as p₂ always follows p₁, and p₃ follows p₂. Below we introduce algorithm Ir_Reg to classify songs into regular and irregular based on the mean values of the viewpoints computed by IDyOM. This algorithm classifies this song (Cumhuriyet Çocuklarıyız) as "regular" and therefore complies with our observations.

Figure 1. An example of a regular, structured melody. The Turkish children's folk song Cumhuriyet Çocuklarıyız, consists of three at pitch repeated patterns (the patterns are marked with p), which are repeated throughout the entire song. From the perspective of their frequency of appearance, these patterns can be considered as "recurrent", "dominant", and the melody appears to be "regular".

In Table 1, the information content (IC), and entropy (E), obtained for the viewpoint cpitch, and assigned to each event in pattern p₁ is shown. The pattern p₁ appears at the beginning of the song (from e₁ to e₄), and is repeated four times, in e_25–28, e_35–38, e_59–62, and e_69–72. The order of the events (from e₁ to e₄) is always the same, which means, that each repeated version of pattern p1 starts with the same note presented as event e₁, and afterwards as e₂₅, e₃₅, e₅₉, and e₆₉ in the repeated patterns.

Table 1: Information content (IC) and entropy (E) defined for each event (from the distribution at that point in the sequence), for the pattern p1 in the Turkish song Cumhuriyet Çocuklarıyız.

Event	1st (IC/E)	Event	2nd (IC/E)	Event	3rd (IC/E)	Event	4th (IC/E)	Event	5th (IC/E)
e1	3.62/3.87	e25	3.96/2.56	e35	2.90/2.50	e59	1.24/2.38	e69	1.05/2.20
e2	1.91/3.60	e26	2.44/2.76	e36	2.88/1.80	e60	1.05/2.29	e70	1.04/2.20
e3	2.99/3.37	e27	1.66/2.62	e37	2.05/2.42	e61	1.41/2.29	e71	1.43/2.30
e4	0.56/1.62	e28	0.51/1.66	e38	0.44/1.52	e62	0.37/1.38/	e72	0.33/1.27

Note. The IC values in the first and second appearance of this pattern are more or less similar, but then decrease in the third, fourth, and fifth appearance.

Tables 2 and 3 present the IC and E values for pattern p₂ and p₃, respectively. As in pattern p₁ the events in these two patterns always appear in the same order in the repeated versions. A tendency to decrease is visible in the IC of the events in all three patterns (depicted also in Figure 2), from their first to their last (repeated) appearance.

Table 2: Information content (IC) and entropy (E) assigned to each event in the pattern p₂ in the Turkish song Cumhuriyet Çocuklarıyız.

Event	1st (IC/E)	Event	2nd (IC/E)	Event	3rd (IC/E)	Event	4th (IC/E)	Event	5th (IC/E)
e5	1.42/2.13	e29	1.20/2.09	e39	0.63/1.62	e63	0.62/1.65	e73	0.45/1.37
e6	6.89/1.51	e30	2.03/2.44	e40	1.85/2.05	e64	0.69/1.82	e74	0.57/1.68
e7	2.35/3.02	e31	1.74/2.47	e41	0.97/2.12	e65	0.73/1.87	e75	0.59/1.68
e8	2.16/2.79	e32	1.09/2.35	e42	0.74/1.96	e66	0.59/1.74	e76	0.49/1.57
e9	1.81/2.14	e33	0.98/2.07	e43	0.68/1.78	e67	0.54/1.58	e77	0.45/1.42
e10	6.16/2.45	e34	1.67/2.54	e44	0.89/2.10	e68	0.66/1.83	e78	0.53/1.63

Note. The IC values and entropy values in this pattern decrease after its first appearance in each further repetition.

Table 3: Information content (IC) and entropy (E) assigned to each event in the pattern p₃in the Turkish song Cumhuriyet Çocuklarıyız.

Event	1st (IC/E)	Event	2nd (IC/E)	Event	3rd (IC/E)	Event	4th (IC/E)
e11	4.13/2.94	e18	4.78/2.54/	e45	2.03/2.41	e52	1.56/2.46
e12	1.65/2.67	e19	1.81/2.72	e46	1.03/2.19	e53	0.92/2.17
e13	1.51/2.33	e20	1.28/2.45	e47	0.58/1.69	e54	0.43/1.45
e14	2.16/1.97	e21	1.46/2.34	e48	0.84/1.98	e55	0.60/1.75
e15	1.63/2.04	e22	1.37/2.04	e49	0.80/1.82/	e56	0.63/1.70
e16	3.74/2.06	e23	2.28/2.39	e50	0.89/2.06	e57	0.64/1.79
e17	2.53/2.54	e24	1.53/2.52	e51	0.88/2.08	e58	0.67/1.83

Note. The IC values in this pattern (exception is the second repetition compared to the first appearance) decreases, as in pattern p₂, with each new repetition.

Figure 2. Information content (IC) and entropy (E) for the viewpoint cpitch in the Turkish children's folk song Cumhuriyet Çocuklarıyız. Both values decrease through this song. The peaks in the graph (event e6, e10, e18, and e25), correspond with notes with low-probability.

The second song, Schneeglöggli, which is from Switzerland (see Figure 3) consists of 50 events and 11 patterns, again labeled as p_i.

Figure 3. An example of a structure with weak relationships between the patterns (motifs). This Swiss children's folk song Schneeglöggli has even 11 motifs, and none of them is dominant, repeated at pitch or transposed repeated, therefore giving the impression of an irregular, non-structured melody.

Ir_Reg classifies this song as "irregular". It has a weak structure, without any dominant (repeated at pitch or transposed repeated) patterns, and it has large intervals (e.g., event e35–36) and no special relations between the patterns. The information content (IC) and entropy (E) obtained for the viewpoint cpitch are presented for each event in this song in Figure 3 (in the musical score, in the boxed text IC and E) and depicted also in Figure 4. We can see clearly very high IC values in each of the 11 patterns, giving the impression of highly unexpected and complex musical content.

Figure 4. Information content (IC) and entropy (E) obtained for the viewpoint cpitch in the Swiss children's song Schneeglöggli. The entropy values (E) are slightly decreasing. The values for the information content remain very high, showing that the events in this song are highly unexpected.

The third song, Doidas andam as galinhas, from Portugal (Figure 5), has been classified by Ir Reg as "neutral".

Figure 5. An example with transposed repeated patterns in the Portuguese children's folk song Doidas andam as galinhas.

The song consists of 48 events with three distinct patterns (p₁, p₂, p₃), and three transposed repeated patterns, of which the pattern t₁ is a transposed repetition of pattern p₁, t₂ a transposed repetition of pattern p₂, and pattern t₃, a transposed repetition of pattern p₃. As can be seen in Figures 5 and 6, the information content of the patterns t₁, t₂, and t₃ suggests that the content of these patterns is perceived as unexpected, even though they are a transposed repetition of the three main patterns.

Figure 6. Information content (IC) and entropy (E) obtained for the viewpoint cpitch in the Portugal children's song Doidas andam as galinhas

Table 4 presents the mean values for information content and entropy (the mean value across all the events in a particular song), for each viewpoint (ten variables) and the three example songs separately. The mean values of IC and E are taken as a measure of (ir)regularity: songs with a high IC and E, are classified as more irregular than those with low values.

Table 4
Average values for information content (IC) and entropy (E), for each viewpoint (ten variables in total), computed by IDyOM on the three presented songs.

Viewpoint	Cumhuriyet Çocuklarıyız	Doidas andam as galinhas	Schneeglöggli
	(regular)	(neutral)	(irregular)
IC_cpitch	1.36	2.46	4.40
E_cpitch	1.99	2.49	3.37
IC_cpint	1.70	2.12	3.73
E_cpint	2.12	2.41	3.03
IC_cpintfref	2.05	3.15	3.83
E_cpintfref	2.61	3.18	3.68
IC_cpitch⊗dur	1.46	2.81	4.80
E_cpitch⊗dur	1.78	2.41	3.24
IC_contour	4.15	4.02	4.96
E_contour	3.68	3.72	3.90

THE DETECTION AND PREDICTION OF (IR)REGULARITY IN CHILDREN'S FOLK SONGS BY IR_REG

Algorithm Ir Reg

We now describe our algorithm Ir_Reg. Ir_Reg takes as an input two sets of songs: the entire data set D, a subset D̅ of D, and a set V of viewpoints that we want to analyze. Firstly, Ir_Reg computes by IDyOM information content and entropy for all viewpoints from V, for all songs from D̅, using the entire data set D. Next, Ir_Reg calculates the maximum values of these viewpoints across the dataset D̅. Finally, for each song from the subset D̅ the algorithm counts the number of viewpoints above the 75% of the maximum value of the viewpoint (upper threshold) or below the 25% of the maximum value of the viewpoint (lower threshold). The Ir_Reg algorithm is actually defined for general set of information content and entropy viewpoints V, but in this paper, it was applied with V consisting of {cpitch, cpint, cpintfref, cpitch⊗dur, and contour}.

If a given song has at least 5 values (entropy or information content) above the upper threshold it is classified as irregular. Otherwise, if it has at least 5 values below the lower threshold, it is classified as regular. If neither of these cases happens, it is labeled as unclassified. The number 5 in the algorithm was determined empirically. Ir_Reg is specified below as Algorithm 1.

Results

In order to analyze the (ir)regularity of the musical structure in our data, we firstly analyzed the entire data (736 children's folk songs) by examining the information content and entropy of each of the set of viewpoints V, as explained above. In other words, we ran Ir_Reg with input data sets D (the entire data set) and D̅ = D. For each song from D we computed by IDyOM the mean values of IC_cpitch, E_cpitch, IC_cpint, E_pint, IC_cpintfref, E_cpintfref, IC_cpitch⊗dur, E_cpitch⊗dur, IC_contour, and E_contour.

Then we classified the songs as regular, irregular, or unclassified using the global threshold values (i.e., the values computed across D) and counted the numbers of songs that were classified as regular, irregular, or unclassified, for each country separately. These totals are presented in Table 6 and visualized on Figure 7. Regarding the relative frequencies, the most irregular in the structure are the songs from Norway, since 17 out of 23 (74%) are classified as irregular. The most regular songs are the songs from Serbia, where 11 songs out of 13 (85%) were labelled as regular. Other countries with no regular songs in our dataset are Switzerland and Great Britain, while the countries with no irregular songs are Croatia, Hungary, Portugal, Romania, and Turkey.

Table 6: Number of irregular, regular, and unclassified children's folk songs found in the dataset of 736 songs.

Country	Irregular	Regular	Unclassified	Country	Irregular	Regular	Unclassified
Bulgaria	2	3	13	Croatia	0	12	4
Denmark	4	2	9	France	16	9	46
Germany	16	28	80	Great Britain	23	0	15
Greece	3	10	13	Hungary	0	12	15
Italy	3	6	13	Latvia	2	4	17
Netherlands	6	14	38	Norway	17	0	6
Poland	1	5	16	Portugal	0	8	19
Romania	0	5	13	Russia	4	5	12
Serbia	0	11	2	Slovenia	8	7	29
Spain	16	13	25	Sweden	17	2	10
Switzerland	18	0	5	Turkey	0	16	8

Figure 7. The percentage of irregular, regular, and unclassified children's folk songs in each country. Total number of children's folk songs is 736.

Table 7: Upper (75%) and lower (25%) threshold for each viewpoint, and mean values for all 10 viewpoints, separately for regular, irregular, and unclassified children's folk songs, computed over the entire dataset of 736 songs.

	Irregular	Regular	Unclassified	Treshold	Treshold
Viewpoint	(mean)	(mean)	(mean)	(25%)	(75%)
IC_cpitch	3.06	1.84	2.37	2.00	2.72
E_cpitch	2.79	2.41	2.61	2.45	2.75
IC_cpint	3.07	1.97	2.49	2.13	2.82
E_cpint	2.66	2.41	2.57	2.40	2.71
IC_cpintfref	3.45	2.61	3.04	2.71	3.73
E_cpintfref	3.39	3.15	3.30	3.05	3.57
IC_cpitch⊗dur	3.17	1.77	2.41	1.93	2.87
E_cpitch⊗dur	2.71	2.18	2.43	2.21	2.63
IC_contour	4.23	3.63	3.86	3.51	4.27
E_contour	3.60	3.48	3.50	3.32	3.73

We also performed principal component analysis (PCA) with the same variables. Figure 8 shows irregular, regular and unclassified children's folk songs, projected to the 2-dimensional subspace spanned by the first two principal components, of which the first explains 54,7% variance, and the second explains 18.4% of total variance. We can actually observe that these three groups could be well separated also using only the first principal component.

Table 8 contains data about importance of the principal components. We can see that we need 5 principal components to explain 94 % of variability (conventional goal in PCA is 95%), which reveals that the data has 5 important dimensions.

Figure 8. Visualization of irregular, regular, and unclassified children's folk songs with the first two principal components.

Component	PC1	PC2	PC3	PC4	PC5	PC6	PC7	PC8	PC9	PC10
Standard deviation	2.34	1.36	1.01	0.82	0.63	0.42	0.41	0.33	0.28	0.25
Proportion of Variance	0.55	0.18	0.10	0.07	0.04	0.02	0.02	0.01	0.01	0.01
Cumulative Proportion	0.55	0.73	0.83	0.90	0.94	0.96	0.97	0.99	0.99	1.00

Secondly, we analyzed the (ir)regularity of the musical structure in data from each country separately by examining the information content and entropy with the same viewpoints, and by applying our algorithm to our data sets of songs from each country. Therefore, we ran Ir_Reg for each country separately with the inputs D – the entire dataset, and D̅– the dataset of songs from particular country. More precisely, we took the values of viewpoints, computed across the whole data set D, but the threshold values were computed using only the viewpoints from the songs of particular country D̅. These threshold values were basis for classification of songs from D̅. This gave as a new classification, presented in Table 9. In total, from 736 children's folk songs, we detected with this approach 158 irregular and 173 regular children's folk songs.

Table 9: Number of irregular, regular, and unclassified children's folk songs found in 736 songs, examined in each country separately.

Country	Irregular	Regular	Unclassified	Country	Irregular	Regular	Unclassified
Bulgaria	5	3	10	Croatia	4	4	8
Denmark	3	4	8	France	15	16	40
Germany	29	27	68	Great Britain	7	10	21
Greece	5	8	13	Hungary	6	5	16
Italy	6	7	9	Latvia	3	4	16
Netherlands	12	16	30	Norway	5	4	14
Poland	3	5	14	Portugal	3	7	17
Romania	4	5	9	Russia	3	6	12
Serbia	3	3	7	Slovenia	11	10	23
Spain	11	11	32	Sweden	7	8	14
Switzerland	7	5	11	Turkey	6	5	13

In these 331 children's folk songs, we analyzed, with two additional and independent musical experts, patterns (motifs) by using the musical score. Each pattern (p), either repeated at pitch or transposed repeated (t), was firstly annotated in the musical score. Secondly, based on the frequency distribution of patterns in each song, we computed the entropy of unigrams for each song (Mihelač & Povh, 2020b give details about computing the unigrams). The Welch two-samples t-test was used to compare the difference in mean values of unigrams between the irregular and regular children's folk songs. The difference is statistically significant, (p < 6.651e-11), i.e. the mean value of unigram was significantly smaller on the set of on regular children's folk songs, mainly because the transposed repeated patterns were more recurrent in regular children's folk songs compared to irregular.

A chi-square test of independence was performed to examine the relation between transposed repeated patterns (t) and the (ir)regularity in musical structure. More precisely, we considered only the classified songs (i.e., regular, and irregular) and have introduced new binary variable having value 1 if the song contained transposed repeated patterns (t) and 0 otherwise. The relation between these two variables was significant, X²(1, N=331) = 27.09, p = 1.942e-07, which means that transposed repeated patterns were more frequent in the irregular children's folk songs (in 80 songs out from 158) than in regular one (only in 39 songs out from 173).

Implied harmony was examined with IC_cpintfref and E_cpintfref (Table 7) and analyzed in musical score. In Table 7, we see that no significant differences were found between irregular and regular children folk songs. The majority of the children's folk songs start with the tonic triad as its first chord, and ends with the tonic triad. However, dominant triads were found in some songs (0.68%), even subdominant triads (0.41%) at the beginning of the songs, and dominant (1.49%), and subdominant (0.13%) triads in the endings (Table 10).

Table 10: Dominant and subdominant triads found at the beginning and end of 736 children folk songs.

Beginning of songs		Ending of songs
dominant triad	subdominant triad	dominant triad	subdominant triad
0.68 %	0.41 %	1.49 %	0.13 %

DISCUSSION AND CONCLUSIONS

A plausible explanation for a higher or lower presence of (ir)regular songs in some countries could be that children's folk songs exist in each of the 22 countries, whose origin can be traced to folk songs. Folk songs are a true amalgam of different historical, cultural, and musical processes (Golež Kaučič, 2003), carrying thus all the specificities of a (musical) culture from a particular country. This means that all the diversity found in folk songs, passed to the children population due to its (suitable) content, presumably without simplifying the musical structure, either from the perspective of musical dimensions, or musical elements, can be found also in children's folk songs. This could clarify either a very high or low information content and entropy found in children's folk songs in some countries.

According to our results, frequency of repeated patterns contributes to a more regular musical structure and to a stronger feeling of regularity, if the patterns are repeated at pitch, which is in accordance with the findings from Bader, Schröger, and Grimm (2017), although even transposed repeated forms (in our case transposed repeated patterns) can be perceived as different, depending on the context which precede or follow it (Margulis, 2014, p. 27–54). Musical structure is perceived as more regular, if the recurrence of a pattern is not "masked" (see more about it in the study from Bruner, Wallach, & Galanter, 1959), as it was the case with the transposed repeated patterns found in irregular and regular children's folk songs.

It seems that the perception of these relative repeated patterns is affected somehow by the duration of children's folk songs, which are approximately 8–12 bars long. If the children's folk songs were longer, the transposed repeated patterns could have a more significant role in the identification of these patterns as a transposed repetition of patterns repeated at pitch, and contribute more significantly to the regularity of the musical structure (more about the significance of repetition in longer musical pieces in the studies from Deliège, 1987; Cambouropoulos, Crawford, & Iliopoulos, 2001). As the duration of a children's folk songs is on average short, then the mixture of patterns repeated at pitch and transposed repeated patterns in a very short time period is actually interfering with the recurrent regularities, i.e., it is not contributing to a more regular structure, but rather to more noise, which is in accordance with the findings of Bruner, Wallach, & Galanter (1959).

The principal component analysis (PCA) performed on all the variables (IC_cpitch, E_cpitch, IC_cpint, E_cpint, IC_cpintfref, E_cpintfref, IC_cpitch⊗dur, E_cpitch⊗dur, IC_contour, E_contour), has shown that regular and irregular musical songs were linearly separated into a 2-dimensional subspace spanned by the first two principal components. The salience of pitch in the perception of (ir)regular structure, can be understood, that each pitch, even when being acoustically identical to another one, can be perceived as different, depending on the context, as each pitch can differ in its tonal function, i.e., each pitch can have assigned a unique function in two different tonalities (Krumhansl & Shepard, 1979).

Small pitch intervals are prevailing in children's folk songs, as also in a considerable number of folk songs across the world (Huron, 2001; Von Hippel, 2000), of which some, included in our data, were found to exist as children's folk songs in different countries. Some of these folk/children's folk songs in the data were assigned as extremely irregular and complex (especially the songs from Great Britain, Norway, and Switzerland, according to our results), not only because of highly unexpected distributions of pitch in the melody, but also because of the use of large leaps (intervals) between successive notes. Thus, not only pitch, but also pitch intervals are affecting the regularity of a musical structure, which was found also in the study of Beauvois (2007).

Differences in rhythm (duration) were found between irregular and regular children's folk songs (see Table 7). The regularity in metrical structure, is according to different studies affected by the absence/presence of a regular beat, which groups different rhythms in time intervals (Bouwer et al., 2018). However, with the exception of some children's folk songs found in e.g., Bulgaria, Romania and Turkey, the in-depth analysis of the musical scores showed that rhythm, based on recurring pulse/meter was found more or less in the majority of children's folk songs included in our data. Recurrent pulse/meter has obviously contributed to a more regular musical structure, in which the events are more predictable, which is also in accordance with the findings in the study of Lappe, Steinsträter, & Pantev (2013).

No significant impact on regularity was found in the analysis of (implied) harmony examined with IC_cpintfref and E_cpintfref (Table 7). This result was expected, as basic harmonic progressions are prevalent in the melodies, either in irregular or regular children's folk songs (e.g., progressions I-V-I, I-IV-V-I, I-IV-I, …). The use of dominant/subdominant functions and triads at the beginning and end of songs is to be understood more as a rare exception rather than a rule in children's folk songs. Furthermore, three basic harmonic functions (tonic, dominant, subdominant) are predominantly found in children's folk songs (Berget, 2017), suggesting that harmony, as a secondary parameter, is in this genre established and heavily dependable on the syntactical constraints and rules formed by the primary parameter pitch (see more about primary and secondary parameters in Meyer, 1989, and Bauer, 2001).

Contour has not been shown to affect the regularity of the musical structure in children's folk songs (Table 7). A plausible explanation could be that the information about contour is present in other viewpoints (for example in cpitch and cpint). Furthermore, in the majority of children's folk songs, ascending and descending directions of the melodies (arch-like structure) have been found. This is probably related to the origins of some children's folk songs traced to folk songs with an arch-like melody. The same tendency of ascending-descending (convex) melodies has been found in a computerized analysis from Huron (1996) in 40% of approximately 10,000 phrases (5–11 notes in length), and also in combined phrases (two phrases together, with a low midpoint, producing an overall convex shape) in over 6000 European folk songs. Furthermore, contour has proven to be a powerful identifying factor in melodic recognition, and to contribute to a better memorization of melodies (Dowling, 1978; Bartlett & Dowling, 1980; Trainor & Trehub, 1993; Trehub, Bull, & Thorpe, 1984), which could also explain the prevailing arch-shaped contour in children's folk songs, which is more or less predictable and probably contributing to a higher feeling of regularity.

In this study, we have shown that irregularity exists in children's folk songs, and that this genre can be complex. Future research with additional examples of children's folk songs in each country, and with more musical feature variables in defining (ir)regular songs could extend the explanations of (ir)regularity. An in-depth analysis of the origin of children's folk songs would certainly contribute to the understanding of (ir)regularities found in the musical structure of this genre, and why irregular structure is more frequently found in some European countries than in other European countries.