A Quantitative Model of Tension and Resolution in Turkish Makam Music Using an Entropy‑Based Approach

2.1 Consonance and dissonance in music

Research on consonance began 2500 years ago with Pythagoras, who asked, ‘Why is consonance connected to small numbers?’ and continues to this day. The famous mathematician Euler, in order to explain the feeling of consonance, states that the more the numbers forming the frequency ratios (FRs) decrease, the more easily the order in nature is perceived, and that humans enjoy this easy perception (Can, 2001: 145).

Tenney (1988), one of the pioneers who addressed the subject of consonance and dissonance in music in detail, proposed five Consonance and Dissonance Concepts (CDCs) and addressed them chronologically. Melodic Consonance (CDC‑1) is the oldest concept, focusing on the consonance and dissonance of sequential melodic intervals. Polyphonic Consonance (CDC‑2), which arose with the birth of polyphonic music, focuses more on vertical structures than on melodic contour. Contrapuntal Consonance (CDC‑3) defines consonance by its role in counterpoint. Functional Consonance (CDC‑4) explains consonance based on the simplicity of each sound’s relationship with the ‘tonic’ or ‘root’ sound. The most recent approach, Psychoacoustic Consonance (CDC‑5), focuses on the perceptual mechanism of the auditory system, discussing two fundamental components: ‘sensory dissonance’ and ‘tonalness.’ The concept of ‘sensory dissonance,’ commonly associated with Helmholtz and later developed by Plomp and Levelt, is explained by the ‘roughness’ (opposite of smoothness) caused by beats (Helmholtz, 1954). The concept of ‘tonalness,’ based on the work of Rameau and Terhardt and later developed by Parncutt and Erlich, is expressed in terms of proximity to harmonic series (Sethares, 2004: 75–78).

The theorists mentioned in the context of the psychoacoustic consonance concept created harmonic dissonance curves while explaining the psychoacoustic foundations on which they base consonance and dissonance (Figure 1). Although the theorists used different methods and approached the issue of consonance and dissonance in different ways, the peaks in all curves correspond to simple integer ratios.

Figure 1

Looking at more recent studies focusing on music perception, it is seen that the issue of consonance and dissonance in music is frequently discussed. Some of these studies approach music perception from a biological perspective, focusing on the auditory system. McDermott and Oxenham (2008) acknowledge that music perception is dependent on cultural factors but also state that the characteristics of the auditory system constrain it. They emphasize that the psychoacoustic foundations of consonance and dissonance can be found in Ancient Greece, where consonance was associated with simple integer ratios (McDermott and Oxenham, 2008). Another study addressing consonance and dissonance in music from a neurobiological perspective (Ushakov et al., 2010) conducts a theoretical and probabilistic analysis that distinguishes between consonance and dissonant chords. Analyses conducted within the auditory perceptual system and at the neuronal level reveal that simple integer ratios are associated with consonant chords. Bowling and Purves (2015), who investigate the biological reasons for musical consonance, address the concept of consonance on three bases: mathematical simplicity obtained with simple integer ratios (Pythagorean approach), physics (Helmholtz’s roughness theory), and biology.

There are a limited number of studies addressing consonance and dissonance in Turkish makam music, with one recent study conducted by Günalçin and Oter (2023). This study, which examines the understanding of consonance from antiquity to the present day, establishes connections between the understanding of Ancient Greek philosophers, beginning with Pythagoras, and that of Islamic philosophers, beginning with Al‑Kindi (9^th c.). Philosophers from this period, such as Farabi (9–10^th c.) and Ibn Sina (10–11^th c.), frequently use superparticular intervals (e.g., 3/2, 4/3, 9/8, 10/9) while explaining musical consonance.¹ The Edvar tradition, which began with Urmevi, was influenced by Islamic philosophers and placed consonance at the center of makam music. Following Urmevi (13^th c.), Kutbuddin Mahmut Şîrâzî (13^th c.), Alişah bin Hacı Büke (15^th c.), Fethullah Şirvânî (15^th c.), Benâî (15^th c.), and Abdülkadir Merâgî (14–15^th c.) are recognized as Systematist School Theorists of the 15^th century. These theorists followed the Edvar tradition, focusing on the concept of melodic movement to explain makam structures (Levendoğlu, 2021). By the late 19^th century, the concept of consonance faded with the scale‑based understanding of makam that emerged with Rauf Yekta (1871–1935) (Günalçin and Oter, 2023).

The melodic movement‑based understanding of makam, which has recently come back into focus, aims to re‑center consonance and dissonance by tracing the Edvar tradition (Bayraktarkatal and Güray, 2023; Güray, 2017). The melodic nucleus approach also aligns with this understanding and has been incorporated into the method proposed in this study.

2.2 Erlich’s harmonic entropy model

The concept of entropy has been used in music research for many years. Pinkerton, one of the pioneering researchers in this field, considered entropy as a measure of the predictability of a melody. Pinkerton expressed the relationship between the process of composing and entropy as follows (Pinkerton, 1956):

Very few studies have used the concept of entropy in analyzing Turkish makam music works. One such study, conducted by Gündüz and Gündüz (2005), used entropy to quantify the information dynamics of melodies in selected Turkish classical and folk music works. In the analyzed melodies, the current randomness and maximum diversity values were calculated by considering how many different notes a given note controlled and the frequencies, durations, and sudden changes in ‘order–disorder’ transitions were observed. In another study conducted in subsequent years (Aydemir and Gündüz, 2014), entropy was used to quantitatively examine the ‘order–disorder’ structure of Meragi’s compositions. The study examined 21 Meragi pieces and determined how predictable or complex the melodies were as time series.

Özmenteş (2018) proposes that the concept of entropy lies at the nucleus of the tension‑and‑resolution behavior observed across all musical practices. According to this approach, the tendency toward disorder, and thus entropy, is minimized by achieving resolution through the central pitch. It is emphasized that, unlike natural systems, music theory systems are designed to reduce entropy continuously. In addition to explanations based on tonal and central relations, examples are given of how the regular rhythmic structures in makam music reduce entropy and how certain structural factors (such as thematic contrast) are effective. A recent study published by Gündüz (2023) states that melodies are suitable for studying using the concept of entropy due to their patterns of order and disorder. It is evident that research applying the concept of entropy to music began shortly after the formulation of Shannon entropy and continues to this day. Paul Erlich is one such researcher who has used the concept of entropy to quantify consonance and dissonance.

The ‘Harmonic Entropy’ model, developed by Erlich and evaluated within the concept of Psychoacoustic Consonance, quantitatively expresses the uncertainty in pitch perception by relating it to entropy. While entropy, as a law of thermodynamics, expresses disorder and uncertainty, Erlich’s harmonic entropy model quantifies the degree of uncertainty in the perception of pitch (Sethares, 2004: 88).

Following studies on consonance and dissonance based on simple integer ratios in the literature, Erlich uses the Farey series to develop his model.² He employs the mathematical fact that the distance between simple integer ratios in this series is greater than the distances between others. Accordingly, the probability of an interval ‘i’ being perceived as the ‘j’^th element of the Farey series is expressed in Equation (1) (Sethares, 2004: 355–357).³

By introducing Equation (2) as the definition of entropy used in information theory, the calculated probability values are used as the relevant function. The reason for using the ‘minus’ sign at the beginning of the equation is the inverse relationship between consonance and entropy value (Sethares, 2004: 355–357).

All psychoacoustic models focusing on sensory dissonance are options that can be considered in this study. The main motivation for using Erlich’s model is that it is associated with the concept of entropy, which is also the focus of this study. This model can also generate an entropy value for each 1‑cent interval, making it applicable to Turkish makam music. Furthermore, Sethares (2004: 355–357) clearly presents the mathematical formulation of Erlich’s model, making its application practical. However, it should be noted that this study could also be conducted using roughness or critical‑bandwidth approaches adopted in other models.

2.3 Melodic nuclei approach for makam structures

In the long‑standing research on makam music, various methods and approaches have been proposed to analyze it. Until the 15^th century, a scale‑based approach was predominant, and makams were explained through the scales and melodic intervals thought to define them. After the 15^th century, the melody‑based approach came to the fore, and makams were defined by the characteristic melodic structures that constitute them (Bayraktarkatal and Güray, 2023; Bayraktarkatal and Öztürk, 2012; Öztürk, 2014).

The makam‑based melodic nucleus model, proposed by Bayraktarkatal and Güray (2023), was developed to understand, analyze, and classify makam structures, adopting a melody‑based approach. According to this model, the melodic nucleus is the smallest structural unit that carries the characteristic structure and melodic identity of a makam. It is usually a group of three or four pitches, organized around a central pitch (axis). The melodic nucleus has a tension–resolution structure arising from the relationship between balance and imbalance; therefore, it can reveal the basic structure of the makam on its own. In other words, the most effective way to understand the melodic character of a makam is to analyze these small but functional melodic nuclei that define the related makam (Bayraktarkatal and Güray, 2023).

The melodic nucleus (framed on the right side) reflecting the character of the Hüseyni makam, some possible Hüseyni melodies produced from this nucleus, and the notation of the Hüseyni makam pitches used in this study are shown in Figure 2.

Figure 2

Melodic nuclei are melodic patterns that contain information specific to the relevant makam, arising from the hierarchical movement of the pitches they contain.⁵ Bayraktarkatal (1997) is one of the pioneers of the idea that makam melodies can be understood through hierarchical relationships. Other researchers who worked on the makam melodic nucleus in later years also adopted an approach centered on this hierarchical relationship. Among these researchers, Öztürk (2018) states that the concept of makam can be understood as a melody‑oriented model rather than a structure‑oriented one. According to this approach, temporary centers may form within the melody, thereby reestablishing the pitch hierarchy.

The method proposed in this study brings together three fundamental theoretical frameworks. In this context, the applicability of psychoacoustic consonance and dissonance to Turkish makam music is discussed first. Then, Erlich’s entropy model is investigated in more detail. Lastly, the role of the melodic nuclei approach in determining hierarchical relationships and temporary centralizations within a melody is discussed. In light of this information, the method was designed to calculate entropy based on hierarchical relationships within a melody and to explain the melody’s tension–resolution behavior. Each theoretical background mentioned in this section has contributed to the development of the proposed method.

3.1 Melodic segmentation

Öztürk (2014: 140–144) examines makam melodies by creating analytical sections. In this context, he has examined the positional and orientational behaviors in makam melodies using some basic figures he has defined. These basic figures, which can be considered the smallest units that make up melodies, are extremely useful in explaining both the positionality at a center and the orientationality between different centers. In this way, the relative hierarchical relationships formed throughout the melodic movement can be revealed through these figures.

In the first stage of the method proposed in this study, melodic analysis is performed using the melodic figures defined by Öztürk (2014: 144–155) (Figure 4). For this purpose, the figures in question are sought in the melody to be analyzed, and the combination of figures with which the melody is formed is revealed. Melodies can consist solely of positional figures, solely of orientational figures, or be based on various combinations of the two.

Figure 4

The melodic figures will vary depending on how the melody is segmented. Therefore, before identifying the figures that build the melody, we need to know how our mind segments the melody. At this point, the concept of perceptual grouping becomes particularly important.

In this study, the authors identified the segments by listening to the melody under examination and applied the segmentation based on preexisting knowledge on the makam in question. This approach, which could be referred to as perceptual grouping, is subjective and open to interpretation. During the application of this method, the segmentation of the melodies in discussion may differ from individual to individual to result in some alternative identification for the melodies. This subjective method that is open‑to‑interpretation should be enhanced in the future through studies to further examine the perceptual principles underlying the perceptual grouping to increase the objectivity of the approach.

3.2 Relative hierarchy tree

One of the most recent studies on the melodic nuclei is by Bayraktarkatal and Güray (2023), which explains the structure of the makam not through scales and intervals but through the combination of melodic nuclei. These melodic nuclei are formed by the hierarchical relationship between the central pitches and the specific group of pitches, and they create melodies.

Öztürk (2014, 2018) uses the positional and orientational figures he has defined when examining a melodic nucleus. He states that the makam hierarchy is established within these figures and that the temporarily centered pitch determines a hierarchical relationship. According to this approach, the hierarchical relationships in question are established relative to temporary central pitches and reveal the definition of ‘relative hierarchy,’ which is the focus of this study as well.

To demonstrate the relative hierarchical relationships, a relative hierarchy tree is constructed in the second stage of the method proposed in this study. Various examples of the use of relative hierarchy tree representations in Western and Turkish makam music studies can be found in the literature⁷ (Baysal, 2011; Lerdahl, 1996).

In the positional figures identified as a result of the analysis of the melody sections, the pitch (A) that provides positionality (through the cyclic motion) is hierarchically superior and has a temporary central function. The other pitch (B) or pitches in the figure are hierarchically linked to this temporary central pitch. This relationship also explains the tension arising in B from the desire to resolve to A. The increasing tension in B decreases when the resolution occurs in A.

Since the identified hierarchical relationship is ‘relative,’ it is only valid for the relevant positional figure, and these relationships are reestablished for the next figure. Let us consider that, in the next positional figure, the hierarchically superior pitch is B, and the pitch hierarchically linked to it is pitch C. In this case, the pitch C will be connected to pitch B rather than to pitch A. In this case, the increasing tension in C will be partially resolved in B; when it reaches A, it will be completely resolved. The tree representation of the relative hierarchical relations for a sample notation is given in Figure 5.

Figure 5

In positional figures, the temporary central pitch is clear, so it is relatively easier to resolve the hierarchical relationship. In this case, if the analyzed pitch is the center of the positional figure, it is connected to the pitch at the top of the hierarchy. If it is different than the center, it is connected to the center of the positional figure. Whereas, defining the hierarchical relationships is more challenging for orientational figures, orientational figures are used to indicate directions either from the very beginning of the melody to a center, or from one center to another. If the pitches are at the beginning of the melody, they are hierarchically connected to the center of the first positional figure which they are directed to. In the other case, the pitches that form this figure establish a tension–resolution relationship with both the center from which the direction begins and the center to which it ends. In this case, it is necessary to construct the hierarchical relationship for this type of orientational figures based on the centers of the preceding and/or the following positional figures. Especially for consecutive orientational figures, the question of which center the pitches are connected to becomes more ambiguous and open to interpretation. The process for determining the central pitches required to construct the relative hierarchy tree is shown in Figure 6.

Figure 6

The central pitch, which is hierarchically at the top, to which all pitches forming the melody are ultimately connected, and to which all other pitches are directly or indirectly connected, forms the main trunk of the relative hierarchy tree. In summary, the main trunk and its side branches, as described above, form a tree representation that clearly reveals the hierarchical relationships within a makam melody.

3.3 Entropy calculation

The relative hierarchy tree is used to model the tension and resolution relationships in a melody. Accordingly, the branches in the tree represent tension, while the trunks represent resolution. The concept of ‘entropy’ is used to quantitatively express the tension in the pitches forming the branches of the tree or the resolution occurring in the pitches forming the trunks. Entropy, used as a measure of tension, is designed to be directly proportional to tension and inversely proportional to resolution.

With the analysis of the figures in the melody and the creation of the relative hierarchy tree, the pitch pairs for which entropy calculations will be performed are also obtained. Each pitch in the melody and the temporary central pitch to which it is hierarchically linked form the pitch pair in question. When calculating the entropy, the frequencies of this pitch pair are first divided to obtain the FR, as shown in Equation (3).

This value is then converted to a cent value using Equation (4). This expresses the distance between the two pitches in cents.

Since Erlich’s entropy model can generate an entropy value for each cent interval, it is sufficient to use the outputs of this model directly to obtain the entropy values of the pitches that constitute the melody. The application of Erlich’s harmonic entropy model, described theoretically in the previous section, was implemented in this study. The pseudo‑code of the entropy calculation algorithm is given in the ‘Reproducibility’ section (Section 6).

As a result of applying this algorithm, an entropy value has been generated for each cent value within the 0–1200‑cent range (octave). The entropy values corresponding to the cent values representing all possible intervals can be found in the graph provided in Figure 7.

Figure 7

These entropy values can be used directly for the pitch pairs formed by each pitch within the melodic figure and the central pitch. Entropy values for ‘dügah–hüseyni’ and ‘gerdaniye–hüseyni’ pairs are shown in the graph as a sample illustration in Figure 7.

In orientational figures, since there are two different centers, the entropy values for each pitch are separately calculated for the two centers, and the arithmetic mean of these values is calculated.⁸

‘Cumulative entropy’ represents the total energy accumulated from the first pitch of the melody, reflecting the overall tension perceived from the moment the melody begins. Cumulative entropy is zero where event‑based entropy is zero and is calculated as given in Equation (5) for other cases.⁹

3.4 Tension–resolution graphs

The graphs created by the obtained entropy and cumulative entropy values can also be called tension–resolution graphs. With the creation of these graphs, which is the final stage of the method, the tension and resolution felt in the makam melody are mathematically modeled.

Since the method described so far produces an entropy value for each pitch, the resulting tension–release graphs will be event‑based. When these graphs are to be created on a time basis, the method to be followed is as follows:

1. First, a unit time value is determined (such as a 16^th or a 32^nd note).

2. Then, the amount of this unit time contained in each pitch is calculated.

3. Next, the entropy value calculated for each pitch is multiplied by this number.

4. Finally, a graph of the resulting repeated entropy values is plotted. The x‑axis of this graph represents time in the unit determined in step 2.

To summarize the method proposed in this study, the melody to be analyzed is divided into segments according to perceptual grouping performed by the authors, and each segment is named according to its positional or orientational nature. Positional figures in the melody reveal temporary centers, while orientational figures express the transitions between these centers. With this approach, the hierarchical relationship between each pitch in the melody and its corresponding center is determined. The interval between each pitch and the hierarchically connected pitch(es) is calculated, and the corresponding entropy values are used to generate an entropy graph that explains the tension–resolution behavior of the melody.

Numerous studies on Turkish makam music highlight the variability of pitch and the discrepancies between theory and practice (Bozkurt et al., 2009, 2014; Kanık, 2024; Özek, 2014). In the present study, the proposed entropy model operates on discretized pitch representations. As the current limitation of the study, the model is not compatible with the context‑dependent pitch behavior observed in practice. However, the proposed model is still applicable in the studies which use symbolic music data and for those which focus on more theoretical concepts rather than practice. The model has the potential to be enhanced for capturing pitch variability by more sophisticated algorithms and by using audio data which represents the variable pitch practice.

4.1 Melodic segmentation

In the first stage, the melodic figures of the Hüseyni melody to be analyzed were examined. This analysis allowed the melody to be separated into its smaller subcomponents, enabling the identification of positional and orientational melodies.

Figure 8 represents orientational figures, while the red frames represent positional figures.¹⁰ The names of the numbered figures are as follows:

1: Ascending step (orientational)

2: Repetition (positional)

3: Ascending leap (orientational)

4: Descending step (orientational)

5: Ascending direction (orientational)

6: Repetition (positional)

7: Ascending–descending compound directional (positional)

Figure 8

In the fifth melodic figure, the pitches are closely spaced and moving in the same direction, which makes these three pitches perceived as a group. In the seventh melodic figure, the first and the last pitches are the same, with a symmetric movement, which makes these seven pitches perceivable as a group.

4.2 Relative hierarchy tree

By identifying the figures that form the nucleus of the melody, it is possible to examine the constantly changing, relative hierarchical relationships within the melodic movement. In the Hüseyni example under analysis, all pitches are connected to the hüseyni pitch, which stands at the top of the hierarchical structure. When we examine the first figure, we see that the rast pitch is not directly under the hüseyni pitch hierarchy but instead under the dügah pitch hierarchy, and that the dügah pitch is under the hüseyni pitch hierarchy. In the third figure, the neva pitch, which comes with the dügah–neva leap, is hierarchically connected to both the previous center dügah and the main center of the melody, hüseyni. The ascending direction in the fifth figure indicates that the çargah and neva pitches in the figure are directly under the hüseyni hierarchy. All of the pitches in the seventh figure at the end of the melody (eviç, gerdaniye, muhayyer, and acem) are directly under the hierarchy of the hüseyni pitch due to the positional feature of this figure. The tree representation of the relative hierarchy in the Hüseyni example is shown in Figure 9.^11,12

Figure 9

4.3 Entropy calculation

With the creation of the relative hierarchy tree, we can see the pitch pairs to use in entropy calculation. In the Hüseyni example, all pitches forming the melody and their associated central pitches are tabulated horizontally in Table 1.¹³ The ratios of the frequency values of these pitches (Freq. ratio) and the distance between the pitches (cent) were calculated as described in the Method section. Then, the entropy values (Entropy) corresponding to this interval value were added to the table. Finally, the cumulative entropy values (Cumulative entropy) were calculated to show the total energy loaded from the beginning of the melody. For example, the calculation performed for event 4 is given in Table 2.

Figure 10

4.4 Tension–resolution graphs

The entropy and cumulative entropy values in Table 1 are displayed as a line graph in Figure 11. In this graph, the x‑axis represents the event number, while the y‑axis represents the entropy values. Both event‑based and cumulative entropy values are displayed on the same graph to enable comparison.

Figure 11

According to the event‑based entropy graph, the melody starts with an entropy value, and the entropy values decrease in the second and third events, indicating a decrease in tension. Then, in the fourth, fifth, and sixth events, the tension increases again. In the seventh and eighth events, a complete resolution is achieved in the central pitch hüseyni to which the entire melody is hierarchically linked, and the entropy value is reset to zero. In the subsequent events, the tension increases again; a temporary decrease is observed in the 11^th event; and, in the final pitch of the melody, the entropy value is reset to zero, with complete resolution.

According to the cumulative entropy graph, the energy (tension) accumulated from the beginning of the melody is reset to zero with the complete resolution in the seventh and eighth events. Then, the energy loading that starts again and continues until the end of the melody ends with the final pitch hüseyni, and all the accumulated energy is discharged, resetting the entropy value to zero.

The event‑based entropy graph can be converted into a time‑based entropy graph using pitch‑duration values. The time‑based cumulative entropy graph of the Hüseyni example is given in Figure 12. The unit time value used in this graph is a 32^nd note. Furthermore, it is assumed that the tension remains the same during the duration of a pitch. The tension–resolution behavior in this graph is identical to that of the event‑based entropy graph, only adapted to the time axis.

Figure 12

At this point, comparing the cumulative entropy graphs of two different melodies may help demonstrate the model’s consistency. In the opening section of Hüseyni Sofyan Şarkı, the melody immediately resolves to the central pitch, then creates tension for a short time before returning to it and releasing all the accumulated tension. This behavior is evident in the entropy graphs in Figure 11.

In the opening section of Hüseyni Peşrev (Figure 13), which is the second analyzed melody to compare with the first one, tension increases throughout the section under examination, and the melody resolves by reaching the central pitch at the very end. In addition to the difference at the moment of resolution, the accumulated tension level is also higher in the second melody than in the first. In summary, the tension–resolution behavior between the two melodies differs, as observed in the cumulative entropy graphs in Figure 13.

Figure 13