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Mapping Pitch Classes and Sound Objects: a Bridge Between Klumpenhouwer Net- Works and Schaeffer’S Tartyp

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Mapping Pitch Classes and Sound Objects: a Bridge Between Klumpenhouwer Net- Works and Schaeffer’S Tartyp View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ZENODO MAPPING PITCH CLASSES AND SOUND OBJECTS: A BRIDGE BETWEEN KLUMPENHOUWER NET- WORKS AND SCHAEFFER’S TARTYP Israel Neuman Iowa Wesleyan University [email protected] ABSTRACT compositional process. While we approach this task from a generative compositional perspective, we are neverthe- We present an interactive generative method for bridging less aiming to find a meaningful method consistent with between sound-object composition rooted in Pierre the analytical context from which it evolves, i.e., trans- Schaeffer’s TARTYP taxonomy and transformational formational theory and the TARTYP. pitch-class composition ingrained in Klumpenhouwer Transformational theory, a prominent method for ana- Networks. We create a quantitative representation of lyzing pitch structures in twentieth-century music, is de- sound objects within an ordered sound space. We use this rived from musical set theory, in which the collection of representation to define a probability-based mapping of 12 pitch classes is considered to be the pitch space of pitch classes to sound objects. We demonstrate the im- music. Klumpenhouwer Networks, a major component of plementation of the method in a real-time compositional transformational theory, reside within this ordered space process that also utilizes our previous work on a of 12 pitch classes. The space of electronic music and TARTYP generative grammar tool and an interactive K- concrete music in particular is much less defined. Never- Network tool. theless, one may look at Schaeffer’s TARTYP and identi- fy in this taxonomy the potential for defining several 1. INTRODUCTION sound spaces based on the table’s division into sub- collections of sound objects [1,2]. Klumpenhouwer Net- For hundreds of years, musical analysis focused almost works’ isographic principles are facilitated by the quanti- exclusively on pitch in the description of musical struc- tatively comparable characteristics of the ordered ele- tures and compositional unity. The appearance of elec- ments of the space, i.e., the pitch classes. These charac- tronic music in the twentieth century shook these founda- teristics, however, do not easily translate to the textually tions, revealing a palette of non-pitch sounds or not- defined sound objects of the TARTYP. entirely-pitch sounds that can play a role in music com- In this paper, we first present a method for organizing position. Following the creative work within this expand- sound objects, i.e., concrete sound samples, in ordered ed world of sounds came attempts to formalize its theo- spaces based on quantitative analysis of the sounds’ spec- retical understanding, most notably by Pierre Schaeffer tral characteristics. Following the definition of such or- and his idea of musique concrète (concrete music). Con- dered sound spaces we present a mapping between pitch temporary composers, in particular of compositions for classes and sound objects. This mapping uses additional live performers and electronics, commonly use pitch- aspects of quantitative spectral analysis of sounds to de- based composition alongside sound-based composition. termine probability-based relationships between the fixed However, few attempts were made to connect these two pitch-class space and arbitrary collections of sound ob- approaches in one unifying musical framework. jects. Since the latter may refer to any size collection of This study has evolved from previous studies in which sounds, our mapping is not 1-to-1 but rather 1-to-many or we introduced two generative compositional tools: the many-to-many, depending on the size of the collection. TARTYP generative grammar tool [1,2] and the interac- With this mapping, we can use the pitch-class based in- tive K-Network tool [2]. The former is for use within the teractive K-Network tool to generate musical structures domain of sound objects and is based on Pierre of sound objects as well as the TARTYP generative Schaeffer’s TARTYP taxonomy. The latter is for use grammar tool to generate musical structures of pitch clas- within the domain of pitch classes and is based on David ses. Lewin’s transformation theory and Henry Klumpen- houwer’s networks. Both tools support real-time compo- 2. BACKGROUND sition and computer improvisation. Both tools share simi- lar mechanics and computational principles, as they are In [3], David Lewin formally introduces the Generalized both based on traversing through search trees. This simi- Interval Systems (GIS). A GIS is defined by the tuple (S, larity put forward the idea of finding a method to bridge IVLS, int) in which S is a space of elements, IVLS is a between the two domains of pitch classes and sound ob- group of intervals and int is a function mapping S to IVLS jects, to produce a single musical structure and a unified [3]. In chapters two through four of [3] Lewin provides multiple examples of GISs describing a variety of musi- Copyright: © 2018 Israel Neuman. This is an open-access article distrib- uted under the terms of the Creative Commons Attribution License 3.0 Unported, which permits unrestricted use, distribution, and reproduction SMC2018 - 476 in any medium, provided the original author and source are credited. cal elements and musical material in the pitch, time and the time domain [1,2]. Moreover, Schaeffer divides the timbre domains. In all of these GISs, the elements in the table into sub-collections of sound-objects, some of spaces S are obtained by a quantifiable process or an un- which are notated in the table while others are clearly derlying formula that makes them compatible with a discussed in Schaefferian studies [5]. Sound objects in a mapping to measured intervals. For example, the chro- sub-collection may be similar in some characteristics but matic scale space is obtained by integer addition and its differ in other characteristics. For example, the sound reduction to 12-pitch-class space is obtained by integral objects marked N’ X’ and Y’ in the Balanced sub- multiple of 12 [3]. A similar example is the Just intona- collections all have the same time domain characteristic, tion GIS of which S is a space all pitches generated by i.e., impulse (or a very short duration). However they Just intonation from a given pitch and IVLS is a “group differ in their frequency domain characteristic: N’ is defi- under multiplication of all rational numbers that can be nite pitch; X’ is complex pitch; and Y’ is not very varia- written in the form 2a3b5c, where a, b, and c are integers ble mass. While the meaning of Schaeffer’s terminology [3].” is the subject of some debate by scholars, the TARTYP Lewin’s approach to timbre analysis is similar. His first taxonomy is a prominent theory of electronic music and example of timbral GIS describes classes of harmonic is constantly being explored for analytical and practical steady-state sounds [3]. A class of harmonic steady-state uses [6]. sounds includes all the harmonic sounds of which the The differences between Schaeffer’s approach and first, third and fifth partial have the same power, de- Lewin’s approach are striking. Schaeffer’s terminology is scribed by the tuple s=(s(1),s(3),s(5)). IVLS and interval all descriptive, subjective and incompatible with any un- transformations in this GIS are defined by proportional derlying quantitative formula. Schaeffer provides an al- amplification or attenuation of the power of each partial. phanumeric notation that represents sound object classes More specifically, the interval transformation tuple i [1,2], however, this is not a quantitative representation =(i(1),i(3),i(5)) when applied to s will produce a sound in suited for mathematical definitions. On the other hand, which the power of the first partial is i times s(1), the Schaeffer considers a wide variety of possible timbres, power of the third partial is i times s(3) and the power of while Lewin’s two timbral GISs deal only with harmonic the fifth partial is i times s(5). Lewin regard this GIS as sounds. As such they describe only one category out of commutative, i.e., all sound classes in this space are re- the four categories of timbre defined by Schaeffer in the lated under the int(s,t) = i function. In other words, any spectral domain of the TARTYP, the category of definite sound in class s can be transformed to a sound in class t pitch. In the time domain Schaeffer uses descriptive defi- by multiplying its first, third and fifth partial by i. In nitions of duration with no measuring scale. In particular, practical terms, Lewin’s choice of elements, i.e., the Schaeffer uses the term undefined to describe very long steady state of sounds and three harmonic partials, are sounds. Lewin’s temporal and rhythmic GISs are all limited in their ability to describe differences in timbre. based on measurable time units such as beats and se- Together with the restriction of proportional power rela- conds/milliseconds. tions the sound classes in Lewin’s timbral GIS are all of Brain Kane [7] describes the difference between similar timbre. Schaeffer’s concrete music and pitch-based music (ab- In his second example, Lewin expand the sound class stract music) with the following words: definition to include eight partials of harmonic sound, Abstract music, which Schaeffer contrasted with mu- s=(s(1),…,s(8)) [3]. He then combines the timbral GIS sique concrète, was music that began with the note, with a time-point GIS to create a direct-product GIS that organized its musical thinking in terms of the note, describes the change of power (amplitude) over time in and then draped it in the guise of acoustic or elec- regard to each partial.
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