Combinatorial Principle in the Use of Drum ( Kendhang ) Formulae in Gamelan Music of Yogyakarta . From Prescriptive Models to the Interpretation of a Composition : the Case of Kendhang Pinatut

In this paper, I address the question to the use of drums (kendhang) in the traditional Gamelan music of Yogyakarta, by presenting some prescriptive models (or formulas). I illustrate how, the use of different prescriptive models in a composition follow what I labeled as “Combinatorial Principle”. In order to describe the essential elements of this principle, I will analyze the modalities of interaction between a very flexible drum formula (known as pinatut) and three other prescriptive models for drum within some exemplary pieces of traditional Gamelan music. The concept of combinatorial principle illustrated in these pages, on the one hand explains the way of interaction between the drum’s rhythmic formulas and their capacity to influence the choices made by the entire orchestra during a performance; on the other hand, through this principle we are able to trace a path that attempts to understand the “deep structures” that are the basics of making music in Gamelan tout court. Through the perspective of the combinatorial principle it is possible to analyze the prescriptive models and techniques of many other instruments of the Gamelan of central Java.


INTRODUCTION
Listening to different versions of the same piece of traditional music for Gamelan is something that happens frequently.This is not only about the ability or the expressiveness of the musicians, but it also concerns aspects such as the duration, the execution of one or more sections of a song, the use various instrument techniques, etc.Thus, a piece of traditional music for Gamelan is a flexible and interpretable entity.While observing how a piece of traditional music is realized, we have to consider that each musician doesn't necessarily follow a score, but instead, he calls to mind and elaborates prescriptive models and playing techniques for each instrument starting from the same nucleus of organized sounds (balungan).Through the creative application of shared rules of composition based on manipulation of melodic and rhythmic formulas inherited from tradition, musicians generate a collective melody, which is the result of the overlap of multiple layered melodies, rhythms and timbres.Therefore, a piece of traditional music is recreated every time.Although the identity of a composition is maintained and, in general, there are exemplary performances in which musicians are inspired, a simple variation of a sound or a melodic-rhythmic pattern can be distinguished as a regional or personal style.
Each musician can contribute with his own instrument to the overall character of a song.In the case of the drum (kendhang), the choice of a certain model during the performance is decisive for the final result of the composition.Among all the instruments that compose a Gamelan orchestra, the drum plays a role of particular significance: it controls and manages all the dynamic processes that come into play during a performance.Viewed through the perspective of a particular "conductor", the kendhang player influences the way of playing of the other musicians; for example, through his rhythmic formulas, he is able to indicate which section of a song must be performed, what kind of technique can be adopted by the other instruments, when the singers can enter the composition, and giving signals for the speed variation, for the opening and the closing of a piece, etc.Therefore, the study of the rhythmic formulas of the drum leads us to the heart of the performance of traditional Gamelan music.

MATERIALS AND METHODS
In this paper I will start to talk about the concept of "kendhang formula", focusing on the aspects that can qualify a certain succession of sounds as a "formula"; then I will move to the presentation of some prescriptive models used in Gamelan music in Yogyakarta; finally, through the analysis of some Gamelan pieces, I will show how the interaction and manipulation of prescriptive models for the drum follow a combinatorial principle, which is articulated in three interconnected processes: 1) caesura of a rhythmic formula in one of the moments of the composition marked by some instruments with accentual function; 2) re-use and re-contextualization of a formula (or a part of it) both in the same and in different musical forms; 3) time variation.

DISCUSSION 1. Kendhang Formulae.
There are many factors that determine the rhythmic sequence of a prescriptive model for the drum.Although today the prescriptive models are learned and circulate more and more in written form, they still represent the result of consolidated performance practices transmitted orally for a long time.For this reason we can find a lot of variations (regional or personal).In this study I consider only some formal-type variables (through which the identity of a formula is maintained) leaving out those "surface" aspects such as specific sounds or patterns used in a certain rhythmic sequence.The rhythmic formulas that can be used in a composition depends primarily on the type of drum used.In the big Gamelan orchestras of Central Java there are three types of double-head drums of different sizes: the largest one takes the name of kendhang Ageng; the medium size drum is named kendhang Batangan; the smallest one is called kendhang ketipung.The largest of the three drums can be played individually (satunggal, lit."one") or together with the smaller drum: this configuration is called kendhang kalih (lit." two drums ").In this study, I focused on the latter type (or combination) of drum.Thus, the formulas that will be presented and analyzed in the following pages are all performed with the kendhang kalih.

Structure of a composition and rhythmic formulae.
The second aspect that determines a rhythmic formula is the musical form (bentuk) of a piece of traditional music.This is defined by a particular distribution of "accents" represented by an organized succession of a group of instruments of the orchestra, i.e. gong, kempul, kenong, kethuk, kempyang.The first two instruments mentioned form the group of the large gongs suspended vertically; the others form the medium and small size group of gongs arranged horizontally.In general these instruments have a limited register (only one sound in the case of gong, kethuk and kempyang) and their use during the performance is to create engravings in the melody.Because of their specific use, the ethnomusicologist Jaap Kunst labeled these instruments as "colotomy".( Kunst, 1973).In this paper I analyze only two of the many musical forms in which Gamelan repertories are divided in, i.e. the form lancaran and ladrang.These forms are presented below in the cipher notation used in the Gamelan music (note: they are layed out following the methods of transcription commonly adopted by the Javanese): Although there are cases of formulas developed for a specific song, in general the same prescriptive model is valid for any composition which has the same structure made by the succession of the colotomic instruments.They are common prescriptive models (umum) that take their name from the musical form they belong to: we say kendhangan lancaran (lit.drumming lancaran) in the case of the form lancaran, and kendhangan ladrang (lit.drumming ladrang) in the case of the form ladrang.

Level of irama and rhythmic formulae.
A third element that determines the type of rhythmic sequence used to work out composition is represented by the "level of irama".The Theory of Central Java Gamelan defines four levels of irama.In the tradition of Yogyakarta these levels are distinguished by using the roman numerals I, II, III and IV.Irama is a concept that deserves its own detailed discussion, however, in this paper we will focus only on some essential aspects, which are as follows: • Time: a level of irama can be in a certain way associated to the time speed.In gamelan music theory there are three types of time: cepat (or seseg = fast); sedang (or tanggung = medium, moderate); lambat (= slow).These are general indications, not exactly quantifiable and must be contextualized in the song played, while respecting the character or the spirit of a composition.For instance, depending on the case, playing in "irama I" could mean playing on a fast or moderate time; playing in "irama II" coluld mean playing on a medium or slow time.

•
Proportionality: each irama is the result of a different rhythmic layering level within it is possible to conceive a certain melodic and rhythmic formula.Different levels of irama are in a proportional relationship with each other: a level of irama is in relation to the next with a ratio of 2:1.Consequently, even the melodicrhythmic elaborations of an instrument in different levels of irama will have the same proportional relationship in quantitative terms.This can be also applied to the drum, which adopts different rhythmic formulas based on the level of irama in which such models are executed.

•
Instrument technique: depending on the level of irama many instruments of the orchestra realize different melodic-rhythmic techniques.
To summarize, we can say that the level of irama expresses the relationship between time, stratified dimension of the instrumental parts and the playing technique of an instrument.
Type of drum, form of a composition, and level of irama are three elements that determine the identity of a formula and do not depend on stylistic variables.The drummer, whatever the rhythmic formula he chooses, must consider these elements during the performance.

Prescriptive models umum.
I illustrate below the prescriptive models used to work out the form of a composition (see 1.2) I learned during my studies of traditional music in Yogyakarta.Such models admit of course variants in terms of played sounds, but as mentioned, we are not going to consider that, as it does not influence the mechanisms that are going to be analyzed in the following pages.Formulas for the form lancaran (irama I) and ladrang (irama I and II) are presented below in cypher notation, by using one of the many ways of drum sounds transcription adopted in Yogyakarta.In addition to the use of letters to indicate the sounds, the reader also must pay attention to the use of one or more horizontal bars placed over two sounds, which indicates fractional durations.
Drum sounds (kendhang kalih) P = Phung (or thung): a sound produced by striking the big membrane of the ketipung, leaves the membrane vibrating after the stroke.t = Tak: a sound produced by striking the little membrane of the ketipung, without releasing the hand.k = Ket: a sound produced by striking the big membrane of the ketipung, without releasing the hand.b = Dhang: a combined sound produced by striking the big membrane of the ageng releasing the hand, concurrently with the sound tak.Kendhang Pinatut.
Compared to prescriptive models shown previously, which are strictly linked to the form of a composition, the kendhang pinatut (or kendhangan pinatut) is a more flexible formula, even if it is mostly used in pieces in the form ladrang.The term pinatut comes from the javanese patut, which means "appropriate/matched with the situation"; the infix in is used to give the meaning of "to make something"; thus the term pinatut can be interpreted as "to make something in an appropriate way".While talking with one of my teachers, Pak Didik Supriyantara, I learned how this formula is somehow connected to that is known in Yogyakarta as semarangan style.Elaborated by Ki Nartosabdho, this style had wide circulation in the 60s and 70s, mainly due to cassette recordings and radio transmissions.Native of Klaten and trained as dhalang (puppeter) in Surakarta, Ki Nartosabdho was one of those musicians that from the second half of the last century has introduced many innovations both in the shadow puppet theater (wayang kulit) and in traditional Gamelan music.From the stories of Pak Didik, what made Ki Nartosabho so special was his ability to assimilate various musical styles and playing techniques and editing them, thus creating his own unique style.
I have not yet had the opportunity to learn more about aspects of the history of Ki Nartosabdho, so I cannot say if the kendhangan Pinatut is his own exclusive creation, or rather, the re-elaborated version of rhythmic formulas already used in the area of Yogyakarta.The fact is that the kendhangan pinatut is a rather anomalous formula within the whole prescriptive models of Gamelan music in the tradition of Yogyakarta, eventhough is frequently used.One more interesting aspect about this formula is that, even if at the beginning it could be a free interpretation of a single musician, it soon became a specific formula for other musicians.Consequently, today it is performed without significant variations.The pinatut formula can be observed as a recent case of crystallization of a personal style, which can possibly explain the evolution of many other formulas, such as the lancaran and the ladrangan before they became common models used by everyone.Below there is the prescriptive model taught to me by my first Gamelan teacher here in Yogyakarta, Pak Abujana: The use of the pinatut formula and the others illustrated above will be the subject of the analysis in the following pages.I will try to show how the interaction between these formulas is based on principles of combinatorial nature.I will also try to underline how the choices of the drummer can influence the final result of a piece of traditional music.

Interaction between rhythmic formulae: combinatorial principle.
Speaking about improvisation in Gamelan, the ethnomusicologist Mantle Hood already realized that if there had to be rules (implicit or explicit) followed by musicians during a performance, these rules were "1) reasonably simple (when viewed as deep structures), 2) but also capable of generating endless variety and complexity (when Realized as surface structures)".(Hood, 1988 : 150).By observing drum formulas as "deep structures", I tried to identify the generative elements of dynamic processes implemented during the performance of a composition, while also trying to show off the "reasonable simplicity" of the rules that allow the musician to generate infinite variety and complexity of the formulas.I defined "combinatorial" the procedure used in a piece of traditional music for Gamelan.This is based on three elements: a) caesura of a rhythmic formula in those moments of composition marked by the colotomic instruments (gong, kempul, kenong, kethuk, kempyang).b) Re-use and re-contextualization of a formula (or a part of it) in different levels of irama and in different musical forms.c) time variation.
To understand how the prescriptive models for the drum are subject to manipulations based on a combinatorial principle, I have chosen to present some Gamelan pieces, illustrating the ways of interaction between the ladrangan and lancaran formulas and the pinatut one.The three aspects summarized above will be argued separately, but the reader must keep in mind that they are not organized hierarchically, but rather interconnected.
Kendhangan Pinatut (ladrang) First, on the one hand we have rhythmic formulas, on the other hand we have a traditional piece that needs to be interpreted.As previously said, the formulas are not associated directly to a specific song, but to the musical form.This form is defined by the succession of instruments such as gong, kempul, kenong, kethuk, kempyang.In the case of drum formulas, the moments of a composition marked by these instruments constitute the break points: in other words, the most simple process of manipulation of formulas in a composition is the interruption of it in a structural joint in the form of a composition, which follows the grafting of a new formula.Many of the dynamic processes that take place during the performance of a song depend on the application of this caesura mechanism, such as the change of playing technique by the other instruments, the change of a section of a song (if this one is divided into two or more parts), or even the entry of the vocal part.
During my studies I observed two ways in which the break and graft of two formulas from A to B by inserting in A the rhythmic sequence of B; the second one is realized by inserting in A a special rhythmic sequence "c" which facilitates the passage to the formula B; this intermediate formula can also contain the sequence of strokes of the formula B. To give a first look at what happens, I propose the following two generic models (lines marked with the colors represent the parts of the formula played during the transition process): The case of kendhangan pinatut falls into the second model shown above.I will illustrate two cases, represented by the pieces ladrang Asmåråndånå and ladrang Gonjing miring.Here the kendhangan pinatut interacts differently with the prescriptive model kendhangan ladrang -irama I. Apart from the opening section (bukå) -which is common to all the traditional pieces of Gamelan music -both the pieces are structured into two sections, dados and ndhawah.These kind of compositions are known as lirihan, in which the vocals are co-leader with instruments such as rebab and gender.I present them below in cypher notation (we have previously introduced the diacritical marks and the point; the numbers from 1 to 6 indicate the notes, a point above the number indicates the higher octave): I only consider the section dados of both compositions, because the pinatut formula is used in there.I learned the two ways of working out from my first gamelan teacher in Yogyakarta, pak Abujana.
From the example below the reader can observe that the rhythmic formula kendhangan ladrang (A) is cut in correspondence to the third kenong; the succession c is inserted after this point; this one precedes the entire execution of the pinatut formula (B), which takes place after the gong.Caesura of a rhythmic formula: Ladrang Gonjing Miring.
In this case the break point of the formula kendhangan ladrang is in correspondence to the fourth kenong, where the formula has already ended.At this point, immediately after the gong, the sequence c is inserted; the pinatut formula is performed after the first kenong and it is a part that is a cut off of the entire rhythmic sequence (see next page).
As done in the previous case, we can draw the following scheme: At this point an observation can be made: why does the pinatut formula have two ways of interaction in compositions that are formally equivalent?It on the particular use of kendhangan pinatut in compositions that also contain the vocal part (lirihan).Here, this formula is used to accompany the vocals (gerongan), as an alternative to the use of rhythmic formulas of kendang batangan.In ladrang Asmåråndånå the vocal part is collocated after the gong, as we can see in the following example, where the moment of transition between formulas is represented: From the two examples illustrated above, we can add by saying that the drummer decides when the vocals can enter the composition by using the pinatut formula; moreover, the rhythmic sequence c has not only the function of linking two different formulas, but also becomes a signal that indicates to the choir (gerong) when it is time to sing.

Re-use and re-contextualization of a rhythmic formula.
This is probably the most interesting aspect of the combinatorial principle.Re-use and re-contextualization of a rhythmic formula can happen in two situations: 1) the same formula can be used in different levels of irama in the same musical form, taking advantage of the relation of proportionality between different levels of irama; 2) a formula can be used in different musical forms taking advantage of the relation of proportionality between different forms.The difference between two musical forms consists in the level of density of the strokes sequence of gong, kempul, kenong, kethuk.
When the distribution of a certain accentual schema is "condensed", it produces a short musical form; when vice versa the distribution of the same accentual schema is more "rarefied" we will have a longer musical form.
Between two levels of irama of a same composition or between different musical forms there is a substantial ratio of similarity, as there is for two geometric figures.Moreover, in the music of Gamelan this relationship of similarity is strictly proportional and in a strictly arithmetic sense.This similarity represents the second "key" in the process of working out of a piece of traditional Gamelan music.In this process, a rhythmic formula is subjected to a re-framing that offers the possibility to the musician to operate a manipulation on it, in order to make it consistent with the new level of irama or with the new musical form in which it is realized.In the following pages I will present: a) the case of ladrang Jatikumårå, in which a part of the pinatut formula is used in irama II; b) two cases of re-use of some rhythmic patterns of the pinatut formula in a piece belonging to a different musical form, i.e. the lancaran Manyar Sewu.

Re-contextualization in a different level of irama.
The ladrang Jatikumårå illustrated below is a composition which consists of two sections (A, B) that can be played both in irama I and irama II.If played in irama II, the part played by the saron can change.This change is connected to the use of kendhangan pinatut.
Let us observe now the behavior of the formula pinatut in this song.Here the drummer must to consider the relationship of proportionality between the levels of irama I and II.First of all, we call attention to the sequence of transition from the formula ladrangirama II.The reader will notice that the sequence is inserted after the kempul, where the notes "3 5 3 2" are played.The sequence c in the form "ladrang -irama I" works out the space of eight notes (see 2.11., 2.12.); now, due to the process of expansion of the structure in the level of "irama II", it works out the space of four notes.
If then we look at the rhythmic sequence C used in the first section of the song, where the melody changes, we can trace the rhythmic material of the pinatut formula, as follows: The rhythmic sequence C is repeated twice, however the second time, in correspondence to the kethuk, it is cut off and followed by the rhythmic sequence of the ladrang formula for irama II.
Once again, we can draw a scheme as we did in the previous cases, considering the moment of transition between the ladrang formula and the one traced in the kendhangan pinatut: Similarly to what we have seen for the previous pieces, even in the ladrang Jatikumårå the drummer's choices are crucial for the other instruments.If he adopts the formula C, the saron group must play the variation in B. Thus, the formula of transition c is the signal to which the other musicians will have to pay attention.Re-use and re-contextualization in a different musical form.
The lancaran Manyar Sewu is one of those compositions defined as soran (vigorous) where the saron group plays the most relevant role.In my experience I have encountered several variations of this composition.I show below a version taught to me by Pak Abujana; it is made by two sections: the first one in which the the melody played by the saron consists of a regular succession of notes; the second one in which the succession is variable (ngracik); each section consists of four gong cycles.
I illustrate two different cases of re-contextualization of the pinatut formula rhythmic material, which show the relations of proportionality between the form ladrang and the form lancaran.In the first one the reader will observe that the rhythmic material of the formulas used to generate the composition is traced within the pinatut formula, even if there are special rhythmic patterns inside (i).In the second one, the formula pinatut and the relative formula of transition are reproduced entirely (ii).
(i) In the case presented below, the formula kendhangan lancaran (A) is repeated three times, working out three of the four cycles of the first section of the song.In the last cycle, a variant of the formula c viewed in the previous cases is inserted.In the example, this formula is indicated as c1: The formula c1 has the function of a bridge to the second section of the piece (ngracik), in which two rhythmic sequences have been illustrated (D1, D2).These sequences can be traced in the pinatut formula: The formula indicated as D1 is exactly a part of the pinatut formula.In the formula D2, a specific sequence is instead used.This sequence is also performed in the transition formula c1.
(ii) Looking at second case illustrated below, we can see clearly how the recontextualization of the formula pinatut shows the relationship of proportionality between the form ladrang and lancaran:  If the whole pinatut formula works out a gong cycle in the ladrang form (irama I), it works out four gong cycles in the lancaran form: thus, the proportional relationship between the two forms is 1:4.
As seen in the previous cases, in Lancaran Manyar Sewu the use of pinatut formula (or the rhythmic material contained in it) has the to work out the second section of the song.Consequently, the execution of this section depends on the choice of the drummer, who suggests to the other musicians the change of section by playing the formula c (or c1).

Time varation and manipulation of formulae.
The third essential element that comes into play in the combinatorial principle is the "time variation".As said for the concept of irama, even the concept of time should deserve its own discussion.In fact, time (laya) and irama are different and, at the same time, interconnected concepts.These concepts occupy a central role in gamelan music theory and they are subjects of dissonant interpretations.Here, I want to focus the attention only on two aspects, which are relevant for the manipulation of drum formulas, as follows: • Generally, in Gamelan music all the moments of a performance follow one another without interruption; therefore, time variations are necessary to combine two rhythmic formulas.Time variation is -in other words -the lubricant through which the graft of two formulas is realized in the most balanced way possible.For instance, all the transitions between the different  • Time variation may determine contractions and expansions of the form of a composition, which has large repercussions on the performance.Since the techniques of each instrument in the gamelan is bound by its specific collocation in terms of proportionality and melodic-rhythmic layering, contractions and expansions of the accentual schema marked by gong, kempul, kenong, kethuk or kempyang (colotomy) determine changes in the "logical space" of a composition within it is possible to conceive different techniques, as well as new relations of proportionality between the parts.We have previously observed that a formula can be re-contextualized in different levels of irama, as in the case of pinatut one in the ladrang Jatikumårå.

CONCLUSION
In conclusion, when observing the ways of interaction between prescriptive models of the drum, the reader will have noticed a strong arithmetic component in the elaboration of a composition.Although music as art cannot be reduced only to arithmetic calculations, we can say that the work out of a composition through the manipulation of formal models in Gamelan seems to follow a rather strict logic.Perhaps this could be for the "refusal of anarchy" that Hood was talking about in his Paragon of the Roaring Sea, a rejection that leads to the diametrically opposed land of numbers, proportions and formal analogy; or perhaps, because thanks to this particular vision, musicians finds their way to interact with each other and generate a rich complexity of interlayed melodies, rhythms and timbres.The concept of combinatorial principle illustrated in these pages, on the one hand explains the way of interaction between the drum's rhythmic formulas and their capacity to influence the choices made by the entire orchestra during a performance; on the other hand, through this principle we are able to trace a path that attempts to understand the "deep structures" that are the basics of making music in Gamelan tout court.
Through the perspective of the combinatorial principle it is possible to analyze the prescriptive models and techniques of many other instruments of the Gamelan of central Java.Furthermore, if we consider the term "combinatorial" in a wider meaning, we could also apply it to the repertories themselves.For example, let us think about the re-use of the same group of notes in different composition, which is crucial for some authors to define the concept of pathet.Or we could even examine the re-use of the same composition into the many contexts (such as the wayang kulit, or the kethoprak -the Javanese traditional theater -) in which these repertoires are performed until today.
. Self-management Strategies of Music Teacher Husen Hendriyana, et.al.. Meanings and Symbols of Dalima Relief Luca Pietrosanti.Combinatorial Principle in the Use of Drum (Kendhang) Formulae                                      Kendhangan Pinatut:                                                                                          "                                                                                "  " . Self-management Strategies of Music Teacher Husen Hendriyana, et.al.. Meanings and Symbols of Dalima Relief Luca Pietrosanti.Combinatorial Principle in the Use of Drum (Kendhang) Formulae formulas described in the previous pages are always accompanied by a decrease of time.

1.1. Type of Drum and rhythmic formulae.
Ayu Niza Machfauzia.Self-management Strategies of Music Teacher Husen Hendriyana, et.al.. Meanings and Symbols of Dalima Relief Luca Pietrosanti.Combinatorial Principle in the Use of Drum (Kendhang) Formulae Meanings and Symbols of Dalima Relief Luca Pietrosanti.Combinatorial Principle in the Use of Drum (Kendhang) Formulae Ayu Niza Machfauzia.Self-management Strategies of Music Teacher  Husen Hendriyana, et.al..

et. al..
Meanings and Symbols of Dalima Relief Luca Pietrosanti.Combinatorial Principle in the Use of Drum (Kendhang) Formulae Meanings and Symbols of Dalima Relief Luca Pietrosanti.Combinatorial Principle in the Use of Drum (Kendhang) Formulae Ayu Niza Machfauzia.Self-management Strategies of Music Teacher  Husen Hendriyana, et.al..
Self-management Strategies of Music Teacher Husen Hendriyana, et.al.. Meanings and Symbols of Dalima Relief Luca Pietrosanti.Combinatorial Principle in the Use of Drum (Kendhang) Formulae Meanings and Symbols of Dalima Relief Luca Pietrosanti.Combinatorial Principle in the Use of Drum (Kendhang) Formulae                                                                                                                      