Compositional planning of “Nomos” according to Gestalt principles

Helder Alves de Oliveira – [email protected]
  Universidade Federal do Rio de Janeiro

Liduino Pitombeira – [email protected]
  Universidade Federal do Rio de Janeiro

This article aims to describe the compositional planning of Nomos, based predominantly on Gestalt principles of closure and good continuation. The musical suggestions of these principles, related to the analytical studies by Meyer (1956) and Tenney (1988), were used in a prescriptive fashion in the elaboration of musical gestures and structures of the work, added to the compositional propositions by Bordini (2004), as well as original suggestions by the authors.

Introduction

In this article, we will detail the compositional planning of Nomos1, for chamber orchestra, by Helder Oliveira, with respect to structure and pitch parameter (melody and harmony). Similar planning was used in the elaboration of the first movement of Segmentos, for symphony orchestra (Oliveira 2014). Greater attention will be given to the structural aspects that will have a direct connection with two principles of Gestalt perceptual organization (closure and good continuation), since they will be used to guide the various compositional decisions.

Initially, a succinct basis will be provided on the principles of Gestalt Theory and its application in the musical field. Then, the compositional planning of Nomos will be shown in a detailed fashion, focusing primarily on aspects related to gestalt principles.

Gestalt Principles and Music

The Gestalt Theory is associated with the psychologists of the Berlin school, to which belonged Wertheimer and his associates Kurt Koffka and Wolfgang Köhler. According to this theory, among the various subjects it addresses, an orderly arrangement of objects exists “[…] when every object is in a place which is determined by its relation to all others” (Koffka 1936, 15). Principles or laws that define groupings and divisions of the units of a whole govern this ordering. These principles were first presented by Wertheimer (1997, 71-88), and briefly are: 1) Proximity: The most natural forms of grouping are those involving smaller intervals; 2) Similarity: similar parts tend to unite themselves; 3) Uniform destiny (or common destiny): stimuli that move in the same direction tend to group together; 4) Pregnancy: “[…] there are certain Präganzstufen [or regions of Figurel stability] with their appropriate realms or regions, and intermediate stages typically appear ‘in the sense of’ one of these characteristic regions” (Wertheimer 1997, 79); 5) Objective set: When a way of grouping is established (repeated), this way tends to hold; 6) Direction (good continuation): is the continuation of the movement of successive parts; 7) Closure: perception of unity closed in on itself, rather than units that do not present themselves naturally as closed, 8) Good curve: the combination of independent figures form a new, totally different figure; 9) Past experience or habit: Arrangement of a series or constellation is determined, in principle, only by extrinsic circumstances (e.g., training). 10) Segregation: In a homogeneous field, an object will only be perceived if there is differentiation of stimuli.

Several writers have attempted to apply these principles in the musical field, amongst them Wertheimer himself, as well as Polansky (1978), Lerdahl and Jackendoff (1983), Lipscomb (1996). However, the compositional planning of Nomos will be guided by the observations of Bordini (2004), Meyer (1956), Tenney (1988)—specifically their musical suggestions for the laws of closure and good continuity—, and by some of our additions (Table 1)2. The choice of these theoretical-analytical studies was made considering the fact that these laws, in addition to being followed for the creation of coherence and organization, are “broken” to bring musical interest to the listener.

In this article, we will use the terms reiteration and recurrence as distinct forms of repetition. “Recurrence is repetition which takes place after there has been a departure from whatever has been established as given in the particular piece. There can be a return to a pattern only after there has been something different which was understood as a departure from the pattern. Because there is departure and return, recurrence always involves a delay of expectation and subsequent fulfillment. Reiteration, whether exact or varied, is the successive repetition of a given sound term, even if it is very extensive, is nevertheless perceived as a unit. Reiteration does not necessarily give rise to expectations of further repetition. On the contrary, if repetition is fairly exact and persistent, change rather than further repetition is expected, i.e., saturation sets in.” (Meyer 1956, 151-52).

Table 1. Musical suggestions for Closure and Good Continuation
ClosureGood Continuation
BordiniMelody with increasingly modified reiterations in the rhythm parameterPhrase contour planning to promote good continuation
Use for virtual axis by approximation and departure from symmetrical pitches Abrupt or gradual transition between parts
Use of melody with incomplete reiterationsGood continuation through reiteration with minimal variation (e.g. timbre)
Good continuity through variations of the sequence of pitches of a melody
MeyerIncompleteness through the use of the beginning of a theme as a repeating fragmentContinuation through the use of conjunction by common pitch-class
In the construction of a melodic line, some notes of a scale are used, and then the remaining onesTransposed reiteration of prior material (Tenney's objective group) to promote continuation
Incompleteness from minimum saturation (preparation for thematic return). We added in this Meyer's suggestion the anticipation of a theme not yet presentedContinuation of movement of the highest pitches in conjunct motion or fixed intervals (or their discontinuity)
Interruption of melodic pattern to generate incompletenessSuccessive repetition of rhythmic grouping type (Tenney's objective group) or its deviation
Delay in the continuation of melodic pattern (higher pitches) to generate incompletenessContinuity by the dominance of metric or discontinuity by the use of hemiolas, syncopation, polyrhythm, and change of time signature (we added this last resort to the theoretical body proposed by Meyer)
Long note to generate incompleteness (conclude with ascending or descending conjunct motion)
Long pause to generate incompleteness (our addition)
Skip compensation by contrary and conjunct motion to generate completeness
Metric incompleteness from the absence of the measure first beat
Thematic recurrence to generate completeness (which is also associated with Tenney's objective group)
Cumulative reiteration fragment (our addition)

Compositional Planning of Nomos

The basis for the compositional planning of Nomos is predominantly based on the Gestalt laws of closure and good continuation, which guided the structural aspects of the work, involving the creation and connection of musical gestures. The title of the work is of Greek origin (νόμος), and means “law”, making reference to the Gestalt laws of perceptual organization applied to the music.

During the planning of the piece, a prospective work was carried out on the musical suggestions indicated in the previous section in order to identify those that potentially need thematic material for its effectiveness, and those that do not require this type of resource. Table 2 lists these two cases. This division was fundamental for the planning of subsection i1 of the introductory section of Nomos, as we shall see later.

Table 2. Grouping Gestalt suggestions
Thematic Material RequiredThematic material not required
Melody with increasingly modified reiterations in the rhythm parameterUse for virtual axis
Melody with incomplete reiterationsInterruption of melodic pattern
Use of the beginning of a theme as a repeating fragmentDelay in the continuation of melodic pattern
In the construction of a melodic line, some notes of a scale are used, and then the remaining onesLong note to generate incompleteness
Minimum saturationLong pause to generate incompleteness
Thematic recurrenceSkip compensation by contrary and conjunct motion
Phrase contourAbsence of the measure first beat
Variations of the sequence of pitches of a melodyAbrupt or gradual transition between parts
Cumulative reiteration fragmentReiteration with minimal variation
Conjunction by common pitch-class
Transposed reiteration of prior material
Continuation of movement of the highest pitches in conjunct motion or fixed intervals
Successive repetition of rhythmic grouping type or its deviation
Continuity by the dominance of metric or discontinuity by the use of hemiolas, syncopation, polyrhythm, and change of time signature

Table 3 presents the macro-structural planning of Nomos. The introductory section will be marked by small musical gestures and by the appearance of Theme 1, section A will consist of Theme 2, section B will contain overlapping themes 1 and 2, section C will present Theme 3, and the Coda will be marked by the small musical gestures similar to the beginning.

Table 3. Macro-structure of Nomos
SectionSubsectionCharacteristics
Introi1No theme
i2Theme 1
Aa1Theme 2
a2Theme 2 varied
Bb1Theme 2 varied
Theme 1 var. in ostinato
b2Fragments of Theme 2
Theme 1 var. in ostinato
b3Theme 2 varied
Theme 1 var. in ostinato
CTheme 3
CodaNo Theme

Between the introductory section and section A, we will apply the Gestalt law of good continuation in the form of a smooth transition between these parts. Unlike Bordini (2004), who exemplifies the smooth transition by overlapping materials from both sections, this type of transition will be effected by the anticipated use of the initial fragment of the material of section A, and will be gradual and regular, adding a note for each repetition of the initial fragment of the anticipated theme 23. The neglect of the law of good continuity, that is, the use of abrupt transition between parts of music, without overlapping of materials, will be present between sections A and B, and between sections C and Coda. There will be in this transition, therefore, a drastic change of character and musical materials, that is, there will be only juxtaposition of materials. Finally, between sections B and C, there will be a smooth transition through the overlap of materials.

With respect to the pitch parameter, especially with regard to its vertical dimension (harmony), the Gestalt principle of closure will come into force through an approach that consists of employing complementary pitch-classes. Two basic criteria will be used, considering that all the material related to this parameter will come from a single structural set class, which is Messiaen’s (1993, 87-95) mode 4 that corresponds to set class [01236789]4 shown in Figure 1 in the transposition T0. The melodic material also comes from this very source.

Figure 1

Figure 1. Messiaen’s mode 4

The procedure works as follows: after composing a melodic material with the pitches freely chosen from Messian’s mode 4, we will apply two criteria to harmonize this material. The first harmonic criterion, which deals exclusively with homorhythmic structures, consists of pilling-up pitch-classes from Messiaen’s mode 4, considering only the eight pitches of a specific transposition.5 This pilling-up of pitches can be made above or below the melodic material. The rule for choosing the pitches follows a strict vertical voicing that consists of selecting the scale degrees according to an arithmetic progression with a predetermined ratio before it reaches a transposition of the initial pitch. In FIG 2, from an initial pitch C4, if the ratio is 2, the pitches to be pilled-up are D4, F#4, and G#4. We should notice that the next pitch after G#4 would be C5 (reached through the dotted line), and, therefore, we stop before it. Conversely, for the same ration (2) if the initial pitch is G#4, the pitches to be pilled-up are F#4, D4, and C4. If, on the other hand, the ratio is 3, the initial pitch C2 will be pilled-up with D#2, G#2, C#3, F#3, A3, D4, and G4. The next pitch would be C5 (22+3), which will be disregarded because it is already present, in another transposition.

Figure 2

Figure 2. Selection of the pitches of Messiaen’s mode 4 according to arithmetic progressions with ratios 2 (black) and 3 (red).

FIG 3 shows the pilling-up of pitches for different ratios from 1 to 7. Ratios 1 and 7 will not be used. The former has too dense of a harmony; the latter presents a range that surpasses the instrumental force we are employing. Ratio 8 will be disregarded because it would simply produce the same pitch-class represented in various octaves. Also, the ratios that are multiple of 2, 3, 4, 5, 6, and 7 will be disregarded since they produce similar harmonic content compared with the originals, but with increasing rarefaction, which also creates a range problem.

Figure 3

Figure 3. First harmonic criterion for Nomos

In the second harmonic criterion, exemplified in Figure 4, a melodic fragment constructed wholly or partially with the pitches of Messiaen’s mode 4, will be divided freely into parts. The harmonic support for each of these parts will be provided by the complementary pitch-classes, that is, by the pitch-classes that, added to the pitch-classes of the melodic line, produce an aggregate.

Figure 4

Figure 4. Second harmonic criterion Nomos

In places where the first harmonic criterion comes into force, there may be the chromatic complement of the mode as well as the free use of the pitches of the melody themselves in other gestures. There will be in all criteria, therefore, completeness at the aggregate level, because at all times the aggregate is used in full.

Introductory section

The introductory section, of which a detailed planning is described in Table 4 and 5, will be divided into two parts (i1, i2) that contain different events. The first part (i1) will not contain thematic material, but will concentrate only on small gestures based predominantly on directives derived from the Gestalt law of closure that do not need a theme.

For the second part of the Introduction (i2), a theme will be elaborated, using as grid scale Messiaen’s mode 4, which will be a mode used throughout the piece. Themes 1, 2, and 3 will be constructed in the pre-compositional phase therefrom.

Table 4. Structure of subsection i1 of the introductory section of Nomos
MeasureMeyer’s Suggestions for Law of ClosureTransposition
1–3Long notes followed by resolution in step motion; rest for completeness after sound returnT0 + T4
4–6Varied recurrence of tutti; skip compensation by contrary and conjunct motion; shorter pause recurrence

The work will begin with an orchestral tutti and will continue with long notes followed by ascending and descending resolution in conjunct motion. After this, there will be a long pause to generate incompleteness and expectation for the listener. It is worth mentioning that this procedure is an addition to the suggestions proposed by Meyer in Table1. The sensation of completeness will only occur with the return of the sonority, which in this case will consist of the tutti restated. The recurrence of the chord in orchestral tutti will refer to Meyer’s suggestion for the Gestalt law of closure through the recurrence of material. Then notes in opposite direction and conjunct motion will compensate large skips. These compensated skips refer to Meyer’s suggestion of completeness. That same principle of using the long rest to generate incompleteness will be applied again, this time with a rest less long than the first one.

For the subsection i2, whose structure is in the Table 5, Theme 1 was prepared in the pre-compositional stage and will mark the beginning of this subsection. This theme will be predominantly arranged in a mirror counterpoint, which contains two simultaneous voices in contrary motion. In this way, the upper melody was initially elaborated (Figure 5) using only pitches from Messiaen’s mode 4, and then a lower melody in a mirror contour was constructed. The pitches of this melody were adjusted correspondingly to this same mode according to Figure 6. Thus, for example, the pitch-class D will have as counterpoint the pitch-class Bb; the pitch-class Eb, will have the pitch-class A, as counterpoint, and so on.

Table 5. Structure of subsection i2 of the introductory section of Nomos
MeasureClosureGood continuationTransposition
7–10Theme 1 with virtual axis (C#) (Bordini); Incompleteness due to lack of previous material reiterationTransposed reiteration of prior materialT1
11–13Completeness from the pitch C# (Bordini); beginning of Theme 2 as a fragment in cumulative reiteration (our suggestion)

Figure 5

Figure 5. Theme 1 of Nomos

Figure 6

Figure 6. Messiaen’s mode 4 in mirror counterpoint

The use of the mirror counterpoint aims to effect the perception of the virtual axis, which corresponds to the pitch-class C#. This pitch-class only appears at the end of Theme 1, producing completeness. The set class [01236789], for Theme 1, was used only in the transposition T1, which corresponds to the set (1234789A). The suggestion of transposed repetition of previous material associated with Gestalt law of good continuation is used in measure 10 (motive formed by two eighth notes and a quarter note), and soon deviated due to the absence of repetition of that material. At the end of this introductory section, there is regular gradual use of the initial notes of Theme 2 in repetition, causing an effect of incompleteness. The event will only be completed with the full appearance of Theme 2 already in section A.

Section A

Table 6 presents in detail the structure of section A of the work Nomos. This section will consist of two subsections (a1 and a2). In order to characterize the subsection a1, Theme 2 (Figure 7) was elaborated in the pre-compositional phase based on the same set class of Theme 1 (i.e. [01236789]) in transposition T2, which corresponds to set (234589AB). The appearance of the pitch-classes of this set occurs gradually. Thus, the suggestion for the Gestalt law of closure comes into effect, which consists in the use of some notes from a pitch repository, and then in the use of remaining notes to complete this repository. A good continuation in the movement of the highest notes in ascending conjunct motion to the arrival of the climax of the theme is presented in Figure 6, which also shows the delay of this continuation by the recurrence of Bb. The attention with the melody contour (musical suggestion for the law of good continuation) was present in the construction of Theme 2, as in Theme 3 (section C). At the end of Theme 2, there is discontinuity of movement of the highest notes through the change of direction.

Table 6. Structure of section A of Nomos
SubsectionMeasureClosureGood continuationTransposition
a114–22Theme 2 with gradual completeness of pitches; delay in continuation of melodic pattern of the highest notesContinuation of the highest notes and subsequent discontinuity; melodic contour planning (Bordini)T2
a222–30Reiteration with minimum variation (other timbre), transposition and inversion (Bordini)T3 + T5

Figure 7

Figure 7. Theme 2 of Nomos

In subsection a2, a reiteration of Theme 2 will occur in inversion canon, with change of timbre and transposition (T3 = (34569AB0)), so that the Gestalt law of good continuation comes into force.

Section B

Table 7 presents in detail the structure of section B, which will consist of three subsections (b1, b2 and b3). In b1, in order to apply the Gestalt law of good continuation, there will be recurrence of Theme 1 in another timbre (pizzicato) and in the transposition T3 of the set class [01236789], which corresponds to the set (34569AB0). Now, Theme 1 in pizzicato and ostinato will support the reiteration of Theme 2, which will be in transposition T4, in another timbre (suggestions for the law of good continuation), and with subtle rhythmic variations (suggestion for the law of closure). This ostinato will last practically throughout the section. Figure 8 presents the initial excerpts of Themes 1 and 2 modified in this subsection.

Table 7. Structure of section B of Nomos
SubsectionMeasureClosureGood continuationTransposition
b131–52Reiteration of Theme 2 with slight rhythmic modifications (Bordini)Theme 1 (ostinato) in transposed recurrence and in another timbre; Theme 2 reiterated with change of timbre and transpositionT4; T3
b252–61Fragments of the beginning of Theme 2 in transposed repetition; saturation; delayed movement of highest notesFragments of the beginning of Theme 2 in transposed repetition; saturation; delayed movement of highest notesT3; T5; T0; T1; T2
b361–75Recurrence of Theme 2 with rhythmic variationRecurrence of Theme 2 with rhythmic variationT4; T3

Figure 8

Figure 8. Themes 1 and 2 in subsection b1 of Nomos

Subsection b2, shown in Figure 9, will have episodic feature in order to develop the initial fragment of Theme 2 in a short period of time. The main melody of this subsection was elaborated in the pre-compositional phase. After the initial fragment of Theme 2, two transposed repetitions occur. Two more repetitions of the initial fragment of Theme 2 in inversion, interval contraction, and rhythmic decrease occur next. The use of repetition of an initial fragment of a melody to generate incompleteness is one of the musical suggestions of the Gestalt law of closure. What follows is a repetition of the previous material without transposition, which is the minimum saturation to generate incompleteness and expectation. In the final part of the subsection, there will be change of timbre from woodwinds to strings. In addition, a rhythmic change by repetition of each note dictates another type of variation for saturation. In subsection b2 and beginning of b3, there is continuation of the upward movement in conjunct motion of the highest notes, which is delayed by the connecting segment formed by eighth and sixteenth notes (measures 57-61). This delay is also one of the musical suggestions to promote incompleteness.

Figure 9

Figure 9. Excerpt of subsection b2 and the beginning of b3 in Nomos

The Gestalt law of closure, in the form of recurrence of Theme 2, will mark the beginning of subsection b3. There is good continuation from a new presentation of this theme in another timbre and in the transposed form of the set class [01236789], in T4, that corresponds to the set (4567AB01), and there is the sensation of completeness from the subtle rhythmic variation of this theme presented again.

Section C and Coda

Theme 3 (Figure 10) of section C, like the previous ones, was elaborated in the pre-compositional stage of the work, and it begins with the type of rhythmic grouping called trochee (rhythm of two beats, of which the first one is accented and the second one not accented), which is repeated until measure 87, where the grouping type called iamb (two-beat rhythm with accentuation at the end) begins in the last beat.6 The symbology for each of these groups is: trochee (— ⌣) and iamb (⌣ —). Despite the syncopation in measure 86, the grouping type remains trochee. This reiteration of rhythmic grouping type and its change correspond to a musical suggestion for the Gestalt law of good continuation and its disturbance. The return to the initial metric pattern (trochee) will occur only from measure 88, with the reiteration of the theme in another voice, since this section will have the structure of a fugue.

Figure 10

Figure 10. Theme 3 of Nomos

The Coda will be marked by the small initial musical gestures similar to the beginning of the musical work, but with different harmony, with the addition of more overlapping pitches. This return to the initial idea to generate completeness is a musical suggestion proposed by Meyer and fits very well for the sensation of completion of the work.

Conclusion

Nomos has characteristics of music that works with motivic materials, procedure that has been used for a long time in the history of music. This is a consequence of the musical suggestions used in the planning of this piece, deriving mainly from the analytical studies of tonal works according to Meyer (1956). What should be emphasized in Nomos is that, starting from something with which the listener is already familiarized and frustrating his/her expectations at various moments, there is generation of interest and greater expressiveness. Although this musical work contains typically traditional motivic treatment, other musical aspects have been presented according to contemporary characteristics. This is the case of harmony, which starts, for example, from the principle of completeness through the seeking for aggregates. Thus, a dense arrangement of pitches, close to that sound mass, was obtained throughout the piece.

References

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Koffka, Kurt. Principles of Gestal Psychology. London: Kegan Paul, Trench, Trubner & Co., LTD, 1936.

Lerdahl, Fred, Ray Jackendoff. A generative theory of tonal music. Cambridge, Mass.: MIT Press, 1983.

Lipscomb, Scott D. “The cognitive organization of musical sound.” In Handbook of music psychology, edited by, D. Rodges, 133–75, 2nd ed. San Antonio: IMR Press, 1996.

Messiaen, Olivier. Tecnica de mi lenguaje musical. Translated by Daniel Bravo López. Paris: Alphonse Leduc, 1993.

Meyer, Leonard B. Emotion and meaning in music. Chicago and London: The University of Chicago Press, 1956.

Oliveira, Helder A., Liduino Pitombeira. “Aplicação de princípios gestálticos no planejamento de estruturas composicionais utilizadas em obras para piano expandido.” In XX Congresso da associação nacional de pesquisa e pós-graduação em música. João Pessoa, 518–25, 2012.

Oliveira, Helder A. “Aplicação de princípios gestálticos no planejamento composicional de Segmentos.” In II Simpósio brasileiro de pós-graduandos em música, Rio de Janeiro, 211–20, 2012.

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Oliveira, Helder A. “Aplicação de princípios gestálticos no planejamento de estruturas composicionais utilizadas na peça Segmentos.” Master thesis, Federal University of Paraíba, 2014.

Polansky, Larry. “A hierarchical analysis of Ruggles’ Portals.” Proceedings of the ICMC. Evanston (1978): 790–852.

Tenney, James. Meta+hodos: A Phenomenology of 20th-century Musical Materials and an Approach to Study of Form; META meta+hodos. Edited by Larry Polansky. 2nd ed. Oakland: Frog Peak Music, 1988.

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Notes

1 Composed in 2015, it was awarded 5th prize in the OPUS Composition Competition, promoted by the Student Philharmonic Orchestra. It was premiered on November 7, 2015 on Nova Acrópole, Curitiba, Brazil, at 5:00 PM under the direction of William Lentz.

2 The application of musical suggestions from Gestalt principles in the planning of musical works has already been tried in other articles we have written, such as the one related to the creation of a piano four-hands piece (Oliveira and Pitombeira 2012), other discussing the elaboration of Segmentos (Oliveira 2012; Oliveira et al. 2013), and in a master thesis (Oliveira 2014).

3  A somewhat similar example lies at the beginning of the fourth movement of Beethoven’s Symphony No. 1. This process, excluding the numerous repetitions of each element, also resembles the minimalist compositional technique called linear additive process, which “articulates repetition processes based on addition of figures from a base pattern.” “This process can be regular [which is the case of Nomos] with the addition of a regular number of units during the repetition process (e.g. 1, 1-2, 1-2-3), or irregular with addition of an irregular number of units (e.g. 1, 1-2-3, 1-2-3-4, 1-2-3-4-5-6) during the repetition process.”(Cervo 2005, 52).

4 In this paper, set classes are indicated by their prime forms inside brackets. Normal forms are indicated inside parenthesis.

5  The exception is found in the first two vertical blocks in tutti of Nomos, which contain two overlapping transpositions (T0 and T4) in order to form vertical aggregates.

6 There are three other types of rhythmic groupings deriving from prosody—all with three beats—mentioned by Meyer (1956, 103): 1) dactyl (with accentuation at the beginning), 2) anapest (with accentuation at the end), and 3) amphibrach (with emphasis in the center).