At the heart of the complex issues surrounding contemporary art music and performance, lies the multi-faceted topic of notation. Traditionally, notation has served to direct and inform a performer in the realization of a piece of music. But, "the score and the performer have actually exchanged roles; whereas the score used to be the map designed to guide the performer toward the composer's artistic vision, it now is often completely explicit. . . performances are now often mere stabs in the direction of the composer's envisioned perfection of execution."^[1] The problem is in the increasingly precise notation for pitch, rhythm (meter and note durations), and expressive qualities (e.g. timbre, dynamics, intensity, tempo, attack, intonation, articulation) and the exploration of extended instrumental techniques. Today we are at a crossroads; we need to walk in the direction of clarity, but not necessarily explicity, of music notation. Changes are needed and although standards for notational changes have been proposed, there is as yet little consensus among composers in their usage. Due to our emancipation of pitch and rhythm, our scores have gotten progressively more dense and difficult to accurately interpret. The result of all these situations has been a lack of performer engagement with new music and the inability for the composer to get his or her works fully realized.

Pitch notation has become problematic with the writing of highly chromatic music which requires an accidental on every note. Rhythm notation has been the most difficult and "unyielding problem in recent music."^[2] Whereas with pitch notation it is only a question of density of information, with rhythm the difficulty is with our bipartite system notating complexities of a greater division of the beat than two (or multiples of two). Modern music has been marked by 'irrational' subdivisions of the beat, a situation impossible to clearly show and read with our notational system.

The exhaustive notation of expressive qualities is equally problematic since this is "encroaching on the freedom of the performer, extending control into areas where the performer had previously been free to take decisions, and specifying ever more exactly what is required."^[3] "Composers like Stravinsky and Schoenberg leave the interpreter no freedom whatever, every nuance of dynamic, tempo, phrasing, rhythm and expression is rigidly prescribed, and the performer is reduced to the status of a gramophone record,"^[4]

Another issue is the proliferation of composer generated symbology and pictographs. These are score markings that are freely invented, varying from composition to composition, and need a codex to translate the intentions of the composer. These symbols, replacing or with verbal directions, indicate performance actions, instrumental choices, or the use of extended techniques.

All of these issues are currently problematic for the composer in the realization of his or her works. The issues of education and standardization versus freedom and creativity have come into the arena to be addressed. Where do we draw the line so we don't stifle creativity, and are competently preparing our performers to be able to engage these works actively.

The purpose of this paper is to survey the recent problems with our conventional notation system while focusing on the issues in the areas of rhythm and pitch notation. Along with this survey will be a detailing of proposed new standards, solutions to the problems, and the effects of all this on performance practices.

In this paper, we will observe the problems surrounding issues in contemporary notational reforms. While focusing on how this applies to the notation of pitch and rhythm, we will not look at specific scores, but rather center on the issues, current problems and possible solutions. To accomplish this, the writings of music theorists, composers and performers from the last fifty years will be studied. We will also limit ourselves to the examination of systems that "seek to prolong and complete the existing system."^[5] We will refrain from looking at graphic musical scores, or scores which are "stimuli leading the performers to express themselves through their own sound world . . . without falling back on clichés."^[6] These pieces "are not conceived so as to provide a univocal response to the questions posed by the notation,"^[7] and we will defer these issues for another or more extensive study.

A review of literature was undertaken by this author to investigate the issues surrounding the notation of art music by twentieth century composers. Writings by music professional (music theorists, composers, and performers) were studied to get insight into the issues of notational reform. Articles, books and interviews were selected that have been published in the past fifty years and were studied to find a consensus as to the existing problems with our conventional notation system, proposed changes and the effects on the performers.

There are many specialized terms used to describe contemporary music notation and the issues surrounding it. These definitions were compiled from Gerald Warfield's, Writings on Contemporary Music Notation, Erhard Karkoschka's Notation in New Music, John MacIvor Perkins's "Note Values." in Perspectives on Notation and Performance and this author's own experience.

Conventional (traditional, standard) music notation: a reference to the notation of tonal music. "Traditional notation is notation in which the requirements do not significantly extend beyond those of music of the 19th and early 20th centuries."^[8]

Proportional notation (spatial notation): "when discreet values are not indicated but values are decided upon by the performer, usually by comparing relative distances or sizes."^[9]

Frame notation (Karkoschka): A term originated by Boguslaw Schaffer for notations where the performer has freedom of choice within fixed limits.^[10]

Equitone: Proposed by the Englishman Rodney Fawcett in 1958 and is a system of music notation that only uses lines to indicate octaves. All other notes are placed within the lines, with every other note either black or white. This eliminates accidentals and allows for even spacing to indicate even interval relationships. The horizontal axis is time and vertical is pitch,^[11] as in traditional Western notation.

Klavarscribo: Was created by the Dutchman, Cornelius Pot in 1931 and is a system of music notation that is best suited for the piano because it is actually a kind of tabulature for that instrument. The time axis is vertical and the pitch axis horizontal. The lines (vertical) on which the notes are hung, refer to the black and white keys on the piano. the notes are colored black and white respectively to represent the same.^[12]

Irrational divisions of the beat: Although the name "irrational" is used to denote bracketed note values, this is misleading because mathematically "all conventional durations are rational fractions of the unit duration."^[13]

Since the composer's intents are no longer able to be easily and fully realized with traditional notation, some modifications have been proposed. These proposed changes in notation can be grouped in three general ways: 'changes in traditional notation' (simplifications, additions and elaborations), 'partly new principles' (notation of approximate values, and action notation) and 'completely new principles' (proportional notation, verbal scores, and musical graphics).^[14]

The situations problematizing the performance of contemporary art music using our current music notation system can be summarized by the following five points: 1) With extreme chromaticism, the performer must read an accidental with every note. This doubles the amount of material the performer is required to read. 2). The rhythms used by contemporary composers are frequently too complex to be notated unambiguously in our traditional notation system. 3). Composers are rigidly controlling all expressive aspects of performance, leaving the performer with no expressive freedom. 4). Extended techniques are being explored for all instruments. This includes new techniques for attack, intonation, and special effects, with which individual performers might not be familiar. 5). Composers are inventing new notation symbols with every composition, requiring performers and conductors to learn new systems with each new work.

All of the above, leads to densely over-notated scores that are difficult to penetrate. To the traditionally trained musician, it is sometimes an impossibility, leaving many performers unable to realize the composer's intentions. Performers feel alienated, inexpressive, and mechanistic, abandoning contemporary art music to be performed by an elite few with "relaxed and intelligent performances a rarity."^[15] In this paper, we will look at some of these issues while focusing on the possible solutions, and the effects on performance practice.

Over the last millennium, the evolution of Western art music has led us to develop extreme explicity in pitch, rhythm, dynamics, and timbre. With pitch being the first notational domain to emerge, we have evolved great complexity in this area. But this system evolved from a tonal-based music and some feel it has limited functionality with highly chromatic, atonal, pantonal or twelve-tone music. "As soon as the twelve tones were treated as equal, independent pitch elements . . . that we have accidentals at all - became irrelevant."^[16] This argument states that the use of basic notes with accidental markings presupposes the existence of an 'accidental-free' music, which is music using diatonic scales. If the diatonic scale is not used, as in twelve-tone music, than our notational system should not keep pointing to its importance. Another drawback of our present system is that "we should like vertical displacement to be in direct ratio to the musical interval, which it is not."^[17] Equitone proposed by the Englishman Rodney Fawcett in 1958, and Klavarscribo by the Dutchman, Cornelius Pot in 1931 are two experimental systems proposed to make the reading of our notation simpler by eliminating the use of any accidentals.^[18] As with the many other proposed notational innovations, they have not found a wide acceptance.

While there are many new notational systems reputed to make the reading of complex chromaticism easier, adopting any of these systems would entail re-educating a population of highly trained professionals, an event not likely to occur. "The advantages of universal comprehension (extending back to the music of the past), of ease in writing and reading derived from constant use, to say nothing of the capital invested in printing and engraving equipment and unsold stock, decisively out weigh the remote gains offered by plans for reform."^[19]

Even so, the consensus is that most pitch issues, even microtonalty, can be adequately conveyed with small modifications to the existing notational system.^[20] These modifications could be easily standardized with the general adoption of the comprehensive standards put forth by the International Conference on New Musical Notation in their 1974 report. This slim volume is the "outcome of collaboration between the Index of New Musical Notation which initiated the project and the more than 70 specialists who . . . gathered from all over Europe and North America at the Ghent Conference."^[21] In this informative report is contained their recommendations for notation reforms according to the following criteria: 1. Given a choice, the preferable notation is the one that is an extension of traditional notation. 2). The notation should lend itself to immediate recognition, This means it should be: a) graphically distinct; b) as self-explanatory as possible. 3. Proposals should be made only in cases where a sufficient need is anticipated.. 4. Analogous procedures in different instrumental families should be notated similarly. 5. Given a choice, the preferable notation is the one that has received relatively wide acceptance. 6. The notation should be sufficiently distinct graphically to permit a reasonable amount of distortion due to variations in handwriting and different writing implements. 7. The notation should be the most efficient for the organizational principles that underlie the respective composition.

8. Given a choice, the preferable notation is the one that is spatially economical.^[22]

The International Conference on New Musical Notation grouped their results into General Categories (I. Pitch, II. Duration and Rhythm, III. Dynamics and Articulation, and IV. Score Layout, Conduction and Synchronization) and Instrumental Categories (Woodwinds, Brass Installments, Percussion, Bowed String Instruments, Piano. Voice, Electronic Music). With the general adoption of these standards, some of the current difficulties in realizing new music scores would be alleviated. Of course, this would have to be endorsed and supported by our education system to be effective. For the next generation of composers to understand and employ these compositional tools, they must be taught to use them along with traditional notation.

As the adoption of these standards would be beneficial in facilitating the realization of contemporary music, similarly, the same rigorous conventions could also be adopted in the invention of new notational symbols.^[23] These standards should concomitantly be taught as well.

Lucas Foss comments that the complexities of pitch notation could be eased by using "moments of incomplete notation."^[24] He suggests that there are unessentials in a composition that could be filled in by the performer. "Take a very fast run, for example, low to high and back to low: lowest and highest notes may be essential. Intermediate notes may, under certain circumstances be unessential."^[25] This idea of moments of incomplete notation would enable the performer to be given back some choices in a performance. Foss also suggests using notes to indicate the rhythm, but only approximating the pitch by erasing the staff lines, or alternately, the noteheads.^[26] Another solution that involves the performer in a co-creative role with the composer, is where the performer is required to improvise within varying frameworks. This frame notation has been used by many composers including Stockhausen, Berio, Kagel, Pousseur, Boulez and Ward-Steinman. Ward-Steinman in several of his works allows the performers to improvise freely over a group of pitches in certain sections. He feels this allows the performer freedom within a fairly explicit structure; giving the performer imput while maintaining the integrity of the composition.^[27] All of these solutions require the composers to be very cognizant of performance practice and be willing to give up some control over performance parameters and collaborate with the performers in the creation of their compositions.

Rhythm issues are as yet unresolved. One difficulty is with complex irrational divisions of the beat that are cryptically notated in our Western notational system.^[28] That we are unable to notate these complexities is not surprising, as Western art music has focused, for the past 400 years, on intricacies of harmony not rhythm. Our notation system reflects this situation. Nevertheless, today we are faced with a situation which needs reform.

The rhythmic flexibility of the Middle ages was lost with the advent of the barline. While this rhythmic regularity did provide an "immensely useful scaffolding for musical composition and performance, at the same time it has imposed the tyranny of the steady beat from which composers have been delivering themselves ever since."^[29] "In much contemporary writing, regardless of the theoretical underpinnings, there is virtually a phobia against any reminder of our pulse-driven past, a desperate avoidance of anything suggestive of the old bondage."^[30] Today we use "intricate or irrational, duration relationships through multiple, simultaneous, artificial, divisions, irregular meters, rhythmic modulations, incommensurable tempo changes and/or analog notation. Emphasis is placed on the expansion of resources and on flexibility, often at the expense of traditional cohesive, unifying and organizing forces, and the musical results are thus roughly analogous to the musical results of those earlier developments in the area of pitch relations called by Schoenberg "the emancipation of dissonance"."^[31] "With the decline of the periodic pulse as a structural matrix for music came unprecedented difficulties in performance. These arose from the fact that our notational system, and hence the training of performers, is largely based on progressive halving of a basic unit of time. Other fractional subdivisions on the one hand, and additive rhythms on the other, have had to be fitted into the procrustean notational system."^[32]

One solution was to "notate rhythm precisely, [while] expecting the performer to play it approximately."^[33] Another answer was the use of proportionate notation where the length of the note is indicated by the length of the stem, notehead or spacing of the notes. While this gives an approximate space for the note to sound, it is believed that our space perception is not equal to our pulse perception. Kurt Stone states, "If they are not given something they can count, they will not be able to play in time."^[34] Another problem with proportionate notation is that spatial notation can only be played in time with other performers if each player can see the full score or continuous cue lines. This obviously would become prohibitive with ensembles of any large size.^[35]

Stockhausen has written about how "increasing notational complexity may lead to a state where the performer tends to commit an increasing number of errors."^[36] He solves this problem by taking these "time fields" and notating them approximately rather than exactly. He does this by using what resembles a series of grace notes which he instructs the player to play "as fast as possible, but at all times as clear and important as the other notes."^[37] These notes are played outside of the notated rhythmic structure. And although his aim in doing this is to "break the time continuum of metronomic measures by different events which are unmeasured - or better, measured by action (as fast as possible, different kinds of attacks),"^[38] what he accomplishes is to give the performer some active control over the performance.

Charles Wuorinen states that new music is not as hard as has been said, and if musicians had the proper training the obstacles would disappear. "The problems experienced by performers in dealing with it [contemporary music] are the result of their having been trained in a tradition of no relevance to its performance requirements."^[39] Wuorinen believes that anything that can be heard, can be played. In considering the rhythmic difficulties of contemporary music, he feels that it is no more difficult than the demands of music from the ars subtilior period in France (ca. 1400).^[40] He asserts that our present day musical difficulties are psychological and that contemporary musicians should train themselves in the execution of the complex rhythms.^[41]

Another solution for playing rhythmic complexities comes from the composer, Emmanuel Ghent. He looked for a way to "maintain complete independence as to tempo, meter and positioning of the beat, and yet be precisely coordinated in time."^[42] His solution takes the score and prepares it with signal structures (lines) that show the placement of an audio signal that will be transmitted to the performer via headphones. Each part has different signal structures, and audio signals, in a multi tempo work.^[43] This would replace the periodic pulse with a "predictable but aperiodic pulse that is both seen and heard by the musicians."^[44] This system would also enable exact coordination of widely spatially separated players.

The uses of music notation can be divided into three categories: "1). mnemonic notation . . . in which the notation provides an analogue to the sound of music already learned by ear; 2). sight-reading notation, in which the object is to allow the performers to make at least a reasonably accurate attempt at the music at sight; and 3). notation which cannot be used except by a process of investigation and rehearsal."^[45] Contemporary music, if approached, not from a sight reading perspective, but from the third perspective, "an assumption of careful and detailed preparation."^[46] becomes a less daunting endeavor. "Indeed, in the present century composers have tended to demand from the performer- and have received- a commitment to rehearsal-time that is largely unparalleled in other ages."^[47] If musicians have assumed that they should be able to read contemporary music with the same ease that they can read a classical sonata, then they would be resistant to the extra effort required. If we can understand that new music must be approached on its own terms, then we might not have the resentment and alienation that sometimes accompanies the performance of new music. The time involved might even be construed as a good thing. As Steve Schick points out that while learning a complex piece by Brian Ferneyhough, the "extreme complexity and performative difficulty in his scores enforce a slower pace of learning and allow the natural growth of an interpretive context."^[48]

"Since the 1950s, the ideal of a totally determinate notation has become a somewhat tarnished. It has grown increasingly clear that absolute control, mechanistic response, can never be attained while the human relationship is involved."^[49] Since the development of electronic music, Hugo Cole states, "we have become aware of the incompleteness of conventional notation. Timbre, attack and decay, dynamic and tempo changes, are all vaguely specified. For instance, the timbre of a single oboe note may be varied in 98 ways by the use of different fingerings and methods of blowing."^[50] Cole advocates returning control of these parameters to the performer. Others have simply noticed the situation as "requiring an enormous expansion of notational needs and means."^[51] These can be pictographs, verbal instructions, or variations on standard notations and are used to direct performers toward specific techniques or choices of instruments. In his book, Pictographic Score Notation, Gardner Read has compiled an impressive variety of instrumental pictographs.^[52] For the marimba alone he shows fifty different pictorial variations by individual composers. While most are similar and easily identified, some are cryptic and indistinguishable. In looking in the International Conference on New Music Notation Report, it can be observed that this proliferation of inventions can be cut down to one.

Some feel the imaginative innovations in music notation serve to magnify the inspiration of the composer and that to standardize the notation would mean "the acceptance of a number of established means, and as a result, a falling off of the creative powers."^[53] "For the advanced composer, there is, it seems, what amounts to a moral obligation to reconsider; to add new symbols to those already in existence, or to set up new playing and listening situations by devising new notional methods of appeal."^[54] There is a steady move towards codifying these reforms and changes so that universal literacy will once again reign. Some feel that we are in a perpetual state of notational reform and that the needs of today's music necessitates this constant shifting in the notation.

"In a literate age, music has a double character. Generalizing, we can say that we often find it easier to appreciate structural and intellectual qualities of a piece from a reading of the score, while the emotional impact is only made apparent in live performance."^[55] The problem then is there is little emotional impact if performers are caught up in intellectualizing about the notation. If we are able to compose our music using a more standardized repertoire of expanded notation, then we will be more able to engender expressive and sensitive performances from a broader spectrum of committed, sympathetic and engaged performers. This will in turn create a performance atmosphere that will engage the audiences and eventually better support us in our compositional processes. The other important problem is the implications for pedagogy. If we are to effect changes, it must start with how we educate the next generation. If, as Wuorinen says, the music isn't really that difficult, but that we haven't been properly training our musicians, then it is time we started improving the situation.

There are many issues in new music notation that have had solutions proposed (See Table 1). Many of the solutions to the problems involve giving some freedom back to the performer. Even though these efforts to return something to the

performers is only what Ligeti calls "a superficial aspect of freedom,"^[56] it still is a start in the consideration and involvement of the performer in the creative process of composition.

It has been questioned whether there can exist today a generally excepted system that can serve the music community at large. "The vocabulary of sound is now limitless, to our great benefit today; can it then accommodate a system of writing to be shared by all?"^[57] Historically we have seen that "systems of notation have been invented as they were found necessary, and modified or abandoned as they were found inadequate; so the story of musical notation in Western Europe is one of innovations, changes and disappearances."^[58] We can only expect the same in the present era.

It seems that our pitch notation system will probably not encounter major upheavals. Instead what would be efficacious would be the adoption of standards involving small modifications to the existing system (For example, the standardization of the notating of microtones).
 

Table 1. Notation Issues and Reforms

Notation: Problem	Proposed Notation Reforms/Solutions
Pitch: With highly chromatic music, is very dense and hard to read quickly	New notation systems, for example, Equitone and Klariscribo (is not likely this area will radically change). Use moments of incomplete notation
Microtones: no standard notation	Standardizing microtonal notation symbols
Rhythm: too complex	Improve training in contemporary rhythms, proportionate notation, frame notation, use Stockhausen time fields with groups of notes that resemble grace notes that are outside of rhythm structure.
Multi tempo works, pulseless music, physically separated players: is very difficult to coordinate players	Performers have specially prepared scores with aperiodic electronic sound (in headphones for individual performers to follow).
Expressive elements (timbre, attack, decay, tempo, dynamics) are over-notated: score is cluttered and players have no expressive freedom	Give control of some parameters back to the performers, standardize pictograph symbols and eliminate most verbal instruction.
Extended techniques are used: players are unfamiliar with the techniques and the symbols indicating to use the techniques.	Improve training in contemporary extended techniques, standardize notation and pictograph symbols.
Performers alienated: performances are inexpressive	Change attitudes about how new music should be approached (needing careful and detailed study), improve training in contemporary notation, give back some control to the performers over the performance, use graphic notation and aleatoric techniques, solicit performer input to collaborate on compositions

 

We must prepare ourselves for more changes to come by embedding in the education system appropriate and rigorous contemporary notation tools for the use of future generations. The implications for pedagogy are immense. We must routinely begin to teach standards for innovation. New symbols will always be invented, but we can at least teach the accepted standard deviations.

Much of the impetus for standardizing our notational innovations happened in the 1970s. In the past two decades, there has not been that much done. It seems that while the same issues remain, the focus of scholarship in this area is with an altered perspective. Barry Truax has detailed a paradigm shift away from linear models of acoustics towards multi-dimensional concepts. "In composition, the shift is away from  its literate and deterministic aspects, as well as the notion of art as abstract and context free."^[59] Truax lists the focus on timbre as a principle concern that involves the composer in new types of complexities (gender, environment, and culture) as an example of this paradigm shift.^[60] He continues that this marks the end of the literate composer; That "the shift away from composition as a literate activity will not necessarily mean the abandonment of notation, but rather a change in the traditional thinking that resulted in notation as its sole representation. The post-literate composer may bypass notation entirely by dealing directly with sound or will view notation as a convenient representation of the result of an algorithmic process. Scores won't probably disappear, but they may become just one of many forms of representation of music."^[61] Truax feels that the complexity of the music is a reflection of the complex real world concerns (physical, social, and psychological) that are informing the composer's works.^[62]

Complexity in music was the focus of an entire festival at Darmstadt in 1992. And while not all have the post-modernist perspective of Truax, all are aware of the concerns surrounding the issue. The issues today are not radically different than twenty-five years ago, but the shift is away from the actual notation to conceptions of time, contextual elements and cultural applications. As James Boros states in his article entitles, "Why Complexity?" "As our world views have shifted, as reductionist sciences have come to be challenged by those of complexity, so many of us have been led to the recognition of the futility perhaps even the impossibility, of constructing "determinate and testable statement about musical compositions.""^[63] This means a loss of control but a gain of diversity, participation and the realization of intent (including that of peoples formerly marginalized by established Western traditions).

Earle Brown in the 1964 Darmstadt festival on notation and performance stated the following that is still applicable today: "We do have a crisis of consciousness, and it has changed the nature of the artist's relationship to his work and the relationship of the work to a performer reader viewer or listener. The loosening of notational controls and the conscious introduction of ambiguity and spontaneity in performance were a way to deal with this new situation."^[64]

This author gained some valuable insights from this very preliminary study. The first was a greater understanding of the wealth of creative notational techniques being used today. The second was an understanding of not only how our current system is being constantly modified and continually renewed, but also how the perspectives surrounding essential issues central to that system are also shifting. Today we are perceiving the complex roles that experience and culture have in informing our creations and can observe the resultant complexity as a representative consequence.

Notation functions on many levels. One purpose of notation is to assist the composer in composing and the performer in performing. As we redefine these interactive roles and to better achieve these ends, our notation system will continue to evolve. It will have to integrate our expanded interest in new sounds and timbres, complexities in rhythm, density of pitch information, and cultural or contextual application, while allowing the performer the status of co-creative equal. This will enable the performer to feel involved, committed and supported and the music the be dynamically and expressively realized. These challenges are on going, as the evolution in music and notation are in constant flux. These issues surrounding contemporary music, its notation and performance will not be easily resolved, but, with the informed cooperation of composers, instructors, theorists and performers, we will be able to mitigate the problems and continue to grow with our musical processes.

Notes

[Note 2] Emmanuel Ghent, "Programmed Signals to Performers: A New Compositional Resource." in Perspectives on Notation and Performance. Edited by Benjamin Boretz and Edward T. Cone. (New York: W. W. Norton and Company, Inc., 1976), 134.

[Note 3] Hugo Cole, Sounds and Signs (London: Oxford University Press, 1974), 127.

[Note 4] Thurston Dart, quoted in Cole, Sounds and Signs , 127.

[Note 5] Jean-Yves Bosseur, Sound and the Visual Arts,translated from the French by Brian Holmes and Peter Carrier (Paris: Dis Voir, 1993), 14.

[Note 6] Bosseur, 15.

[Note 7] Bosseur, 15.

[Note 8] Gerald Warfield, Writings on Contemporary Music Notation (Ann Arbor: Music Library Association, 1976), ii.

[Note 9] Warfield, ii.

[Note 10] Erhard Karkoschka, Notation in New Music, translated by Ruth Koenig (New York, Praeger Publishers, 1972), 55.

[Note 11] Karkoschka, 13.

[Note 12] Karkoschka, 11.

[Note 13] John MacIvor Perkins, "Note Values." in Perspectives on Notation and Performance. Edited by Benjamin Boretz and Edward T. Cone. (New York: W. W. Norton and Company, Inc., 1976), 69.

[Note 14] Karkoschka, 5.

[Note 15] Stone, "Problems and Methods ," 9.

[Note 16] Stone, "Problems and Methods ," 10.

[Note 17] Richard Rastall, The Notation of Western Music (London: J. M. Dent & Sons LTD., 1983), 248.

[Note 18] Karkoschka, 11.

[Note 19] Cole, 131.

[Note 20] Stone, "Problems and Methods," 12.

[Note 21] Herman Sabbe, Kurt Stone, and Gerald Warfield, editors, International Conference on New Musical Notation Report, (Amsterdam: Swets & Zeitlinger B. V., 1974), 12.

[Note 22] International Conference on New Music Notation, 33.

[Note 23] The following are suggestions by Karkoschka for conventions in the invention and adoption of new notational symbols: "Unambiguousness 1) the same symbol must not appear with a different meaning. 2) the outward appearance of a symbol must not resemble too closely that of another." 3) "A symbol with a traditionally familiar meaning can only acquire a new one in an entirely new context. 4). A sensible balance of symbols and verbal instructions is to be preferred. 5). As far as possible, a symbol should be able to indicate its meaning directly and without explanation. 6). Abstract symbols and illustrations should be selected according to function, and should never be mixed. "Karkoschka, 5.

[Note 24] Lucas Foss, "The Changing Composer-Performer Relationship." Perspectives on Notation and Performance. Edited by Benjamin Boretz and Edward T. Cone. (New York: W. W. Norton and Company, Inc., 1976), 38.

[Note 25] Foss, 38.

[Note 26] Foss, 39.

[Note 27] David Ward-Steinman, conversation with the composer, August 28,1999.

[Note 28] Perkins, 67.

[Note 29] Ghent, 134.

[Note 30] Ghent, 135.

[Note 31] Perkins, 63.

[Note 32] Ghent, 135.

[Note 33] Ghent, 135.

[Note 34] Stone, "Problems and Methods," 22.

[Note 35] Stone, "Problems and Methods," 22.

[Note 36] Leonard Stein, "The Performer's Point of View," in Perspectives on Notation and Performance. Edited by Benjamin Boretz and Edward T. Cone. (New York: W. W. Norton and Company, Inc., 1976), 47.

[Note 37] Stein, 48.

[Note 38] Stein, 48.

[Note 39] Charles Wuorinen, "Notes on the Performance of Contemporary Music." Perspectives on Notation and Performance. Edited by Benjamin Boretz and Edward T. Cone. (New York: W. W. Norton and Company, Inc., 1976), 51.

[Note 40] The ars subtilior period in France was marked by great rhythmic experimentation, the like of which was not seen again until the twentieth century.

[Note 41] Wuorinen, 54.

[Note 42] Ghent, 135.

[Note 43] Ghent, 136.

[Note 44] Ghent, 142.

[Note 45] Rastall, 257.

[Note 46] Rastall, 257.

[Note 47] Rastall, 257.

[Note 48] Steve Schick, "Developing an Interpretive context: Learning Brian Ferneyhough's Bone Alphabet." Perspectives in New Music, (New York: W. W. Norton and Company, 1994), Volume 32, Number 1, Winter, 132.

[Note 49] Cole, 128.

[Note 50] Cole, 128.

[Note 51] Rastall, 258.

[Note 52] Gardner Read, Pictographic Score Notation (Westport, Connecticut: Greenwood Press, 1998), 8.

[Note 53] Roman Haubenstock-Ramati, "Notation-Material and Form" in Perspectives on Notation and Performance. Edited by Benjamin Boretz and Edward T. Cone. (New York: W. W. Norton and Company, Inc., 1976), 100.

[Note 54] Cole, 132.

[Note 55] Hugo Cole, Sounds and Signs (London: Oxford University Press, 1974), 123.

[Note 56] Stein, 50.

[Note 57] Bosseur, 24.

[Note 58] Rastall, 5.

[Note 59] Barry Truax, "Musical Creativity and Complexity at the Threshold of the 21st Century." Interface (Lisse: Swets and Zeitlinger B. V. 1992), Vol. 21, 29.

[Note 60] Truax, 32.

[Note 61] Truax, 36.

[Note 62] Truax, 39.

[Note 63] James Boros, quoting Brian Ferneyhough, "Why Complexity?" Perspectives in New Music, (New York: W. W. Norton and Company, 1994), Volume 32, Number 1, Winter, 95.

[Note 64] Earle Brown, "The Notation and Performance of New Music." The Musical Quarterly (New York: MacMillan, Inc., 1986), vol. 82, No. 1, 197.

Black, Robert. "Contemporary Notation and Performance Practice: Three Difficulties." Perspectives in New Music. (New York: W. W. Norton and Company, 1984).

Boehm, Laszlo. Modern Music Notation. (New York: G. Schirmer, Inc., 1961).

Boros, James. "Why Complexity?" Perspectives in New Music. (New York: W. W. Norton and Company, 1994), Volume 32, Number 1, Winter.

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