Benny Sluchin
email: ircambenny.sluchin@ircam.frircam-stms

Mikhail Malt
email: iremusmikhail.malt@ircam.fr

GEOMETRIC-BASED EXTENSIONS TO THE TIME-BRACKETS NOTATION, JOHN CAGE AND BEYOND…

In the last period of his life, John Cage used time brackets in order to define the musical event placement on the timeline. In previous work we modeled these time brackets by parallelograms to build computer interfaces for interpretation assistance in the context of Cage's Two 5 and Five 3 . Over time, we understood that the shape, used to represent time brackets, brought important information for the interpretation and musical analysis. The main purpose of this paper is to generalize this time bracket parallelogram representation, as a tool for musicological analysis, to enhance performance and, possibly, as a composition notation.

INTRODUCTION

In his works, John Cage used time brackets in order to define musical events placement on the timeline. The large variety of works he has composed in the last decade of his life is surprising. This technique permitted him to continue his usual undetermined way of composition, while giving the performer ways to exhibit his (her) skill in real time situations. John Cage used only few types of time brackets, and almost exclusively symmetrical ones. He also filled them with very little musical material (generally only one note). In Cage's Number Pieces each individual part contains musical events with time brackets. Generally, an event consists of a score endowed with two pairs of numbers: time brackets (Figure 1). This gives the interpreter lower and upper-time bounds to begin and end each event. The composition has a defined total duration and the events are placed on a time line, with a degree of freedom, depending on the performer. Although there are only individual parts, a score for the group is implicitly present and leads to a form.

EARLIER RESEARCH

In previous work [START_REF] Sluchin | A computer aided interpretation interface for John Cage's number piece Two 5[END_REF] we modeled these time brackets by parallelograms to build computer interfaces for interpretation assistance in the context of Cage's Two 5 and Five 3 . Over time ( [START_REF] Cage | On John Cage Two 5[END_REF], [START_REF] Cage | On John Cage, Music for Two[END_REF], [START_REF] Cage | On John Cage, Ryoanji[END_REF], [START_REF] Sluchin | John Cage's Number Pieces, a geometric interpretation of "time brackets" notation[END_REF]), we realized that the shape used to represent time brackets, brought important information for the performance and musical analysis. As for example, managing time in an efficient way to perform the given musical material in the time constraints defined by the time-brackets. Having the choice to shift or position some materials in order to create musical relations with other performers, even if we are aware that this was not Cage's intent.

The unusually long duration of these pieces, 40 minutes, and the use of time brackets show that the temporal question, and its representation, is essential in the Number Pieces.

The first step in the process was to model a graphic representation of each part as a succession of musical events in time. For this purpose, the temporal structure of the piece has been represented as quadruples on a timeline. (𝑠 # (𝑘), 𝑠 ( (𝑘), 𝑒 # (𝑘), 𝑒 ( (𝑘) ) . The graphic event obtained by connecting the four points has a quadrilateral shape. The height has no particular meaning. The starting duration 𝛿 -(𝑘) is defined as the difference * 𝑠 ( (𝑘) -𝑠 # (𝑘)+, which is the time span the performer has to start the event. In the same way the ending duration 𝛿 / (𝑘) will be the time span given to end the event * 𝑒 ( (𝑘) -𝑒 # (𝑘)+. In the general case, these values are not the same, and the form we get is asymmetrical. When dealing with Cage's Number Pieces, one generally has: 𝛿 -(𝑘) = 𝛿 / (𝑘), both durations are the same, and the figure to represent an event is a trapezoid (Figure 3). This is the case in the majority of the corpus we are treating. Special cases will be mentioned later on.

Figure 3. Graphic representation for a time event in Cage's Number Pieces

There is mostly an overlapping of the two time zones, *𝑠 # (𝑘), 𝑠 ( (𝑘)+ and *𝑒 # (𝑘), 𝑒 ( (𝑘)+ but it can happen that those are disjoined. We have defined a variable 𝛾(𝑘) where: 𝑠 # (𝑘) + 𝛾(𝑘) = 𝑒 # (𝑘) [START_REF] Sluchin | John Cage's Number Pieces, a geometric interpretation of "time brackets" notation[END_REF]. In Cage's Number Pieces, 𝛾(𝑘) depends generally on the event duration. Thus, we don't have a big variety of forms in this case.

An alternative way to present a quadruple will be: (𝑠 # (𝑘), 𝛿 -(𝑘), 𝛿 / (𝑘), 𝛾(𝑘)) where 𝛾(𝑘) is the value previously discussed. This representation can easily display the regularity in the time brackets construction (Figure 4). We make a distinction between a generic musical event and a real (or determined) musical event. A real musical event is the one whose starting points (s) and end points (e) are defined, that is, where there is a concretization of choice. One could represent this by a straight line from 𝑠(𝑘) to 𝑒(𝑘) (Figure 5).

MUSICAL SCORE, AN APPLICATION

A musical score is a timeline representation of a musical composition. Each source of sound gets a "track," on which the particular elements are represented. In its classic form, a stave (a five-line musical staff) is used for the "track," but any graphical representation can serve for that. The tempo indications and the measures help in the horizontal organization, but also time indications can use to specify the relationship between the instrumental tracks. A vertical scale on each track represents the precise pitch.

The music score is necessary to understand, analyze and perform a musical work. While each performer uses generally only an individual part for the actual performance, the score gives all the relationships between the elements of the composing tracks. (Figure 6) Figure 6. Beethoven septet op. 20, opening measures.

John Cage, deliberately omitted a score on many occasions. Not only the parts were composed separately and independently to include parameters fixed by the performer in real time, but also, he aimed at a performance to be a onetime event, and a discovery for both the public and the performers. For the sake of studying the composition, a score representation is needed, even if the precision of the tracks' relationship has to consider the free parameters the composer had incorporated in his work. For this reason, we have developed a score-like presentation to cover John Cage's Number Pieces and related works. (Figure 7) This "score" presentation permits us to determine pitch relationships and event particularities (as the first two events in contrabass part).

Starting Time Zone sl(k) el(k) eu(k) su(k)

EXTENDING THE NOTION OF TIME BRACKETS

The idea to generalize comes naturally when one is first presented with the geometrical figures presented earlier. The composer will have in his (her) disposition a larger variety of forms, which could enhance the graphical approach to the composition. A large variety of tasks for the performer will thus be available. The filling of the figures could also be approached with different ways. See example in the end of this text (Fig. 17).

Positioning the starting and ending time zones

In Cage's works, the two time zones are symmetrical, the values are equal, and the overlap is positive, as for example Figure 3.

From here on we consider the asymmetrical case. The first generalization consists of shifting these zones on the timeline. By the definition of 𝛾(𝑘), its value is smaller than 𝛿 -(𝑘) in Cage's works. This fact creates the abovementioned overlapping (which is in fact 𝛿 -(𝑘) -𝛾(𝑘)) as in Figure 8.

Figure 8. A generic non-symmetric music event

Another case is "no overlapping", when (𝛿 -(𝑘) -𝛾(𝑘)) ≤ 0 (Figure 9). An event of this kind must occur, as there is no way the ending point will be earlier than the starting point. 𝛾(𝑘)>𝛿 -(𝑘). All these considerations are geometric oriented, studying abstractly the figures. The practical implications will follow later. For example, in the last case (Figure 10), the trapeze's far-left, right-angled triangle is superfluous unless the event won't occur.

Combining fixed and flexible time brackets

The Music for___ corpus presents two types of time brackets: the usual flexible ones (called pieces), and the fixed one (called interludes), Figure 11. In what follows we consider generalizations consisting of a combination of the two-time brackets' types, leading to a triangle rather than a parallelogram.

The musical consequences are multiple. If we start from a graphical conception of a musical work, we have a lot of geometrical figures to use. If we start from the musical material we want to align on the timeline of the composition, we can be more precise and obtain a planned organization, which will depend on the performers' skill. This will be illustrated later (Figure 17).

Fixed start

In this case, we have: 𝑠 # (𝑘) = 𝑠 ( (𝑘). Thus *𝑠 # (𝑘), 𝑠 ( (𝑘)+, the Starting Time Zone, is one fixed point on the timeline. The Ending Time Zone, is a positive time interval, and the geometric figure to represent could be one of the following Figure 12. Naturally, the actual arrival time can't be smaller than the starting time (at least if we want the musical event to exist). In these cases, the minimum event duration will be equal to 𝛾(𝑘), for 𝛾(𝑘) > 0.

Fixed ending

In this case we have: 𝑒 # (𝑘) = 𝑒 ( (𝑘). Thus *𝑒 # (𝑘), 𝑒 ( (𝑘)+, the Ending Time Zone, is one fixed point on the timeline. The Starting Time Zone is a real-time interval, and the geometric figure to represent could be one of the following Figure 14. In this case, the starting point can't be chosen later than the ending point. The event simply will not occur. From a practical point of view, when 𝑠 ( (𝑘) > 𝑒 ( (𝑘), either the event won't occur or the effective figure will be a right-angled rectangle, with 𝑠 ( (𝑘) = 𝑒 ( (𝑘) (Figure 14).

This case is tricky to perform. When musical event is a held tone, the fixed ending could be easily executed. But when an event is composed of a varied musical material, the planning of a fixed arrival is not a simple task.

The following Figure 16 illustrates this point. In this case, the duration of an event is 5, 10, or, 15 seconds. It is easier for the eye to grasp a graphical indication of the duration than to read time indications, and be aware what the real duration is. This is done in real time, while the playing of the figure has been started. In event 3, the musician has 5 seconds for 8 notes, which indicates a high speed. On the other hand, event 7 (see Fig. 11) has three tones for 15 seconds, which also require time to phrase.

CONCLUSIONS

We use here a graphical representation of Time Brackets notation, in order to generate a score of John Cage's Number Pieces, works conceived individually, having only separate parts. This is done aiming at performing, studying, and analyzing the works in consideration.

John Cage's use of just a few types of Time Brackets, leads to the idea of generalizing the figures, creating a large variety of geometrical figures, thus creating musical tasks for the performers that extends those time management asked by Cage. At the same time, the compositional techniques involved are not limited. One could use various graphical figures and different filling methods (Figure 17).

This last example is a compositional exercise using geometric figures to extend those of John Cage. However, the material to fill the figures is taken from different John Cage's pieces. The two violins represent musical material passing from one to another in a precise way dependent on the performer. The low strings form a background, whose time definition is not strictly defined.

This example was conceived in the traditional way: starting from a score (for string quartet) and extracted individual parts using time brackets.

Figure 1 .

 1 Figure 1. John Cage's Two, flute, first event.

Figure 2 .

 2 Figure 2. Graphic representation for a generic time event To obtain a graphic representation of each event in time we consider the quadruple: (𝑠 # (𝑘), 𝑠 ( (𝑘), 𝑒 # (𝑘), 𝑒 ( (𝑘)) where *𝑠 # (𝑘), 𝑠 ( (𝑘)+ is the Starting Time Zone and *𝑒 # (𝑘), 𝑒 ( (𝑘)+ the Ending Time Zone. As the two intervals have, in our case, a designed superposition, we

Figure 4 .

 4 Figure 4. An event represented as (𝒔 𝒍 (𝒌), 𝜹 𝒔 (𝒌), 𝜹 𝒆 (𝒌), 𝜸(𝒌))

Figure 5 .

 5 Figure 5. A real music event represented by a straight line, joining the starting to ending time zones

Figure 7 .

 7 Figure 7. Cage, Seven 2 , first 4 minutes.

Figure 9 .

 9 Figure 9. A music event with disjoint starting and ending time zones. No overlapping.A last case implies 𝛾(𝑘) < 0.

Figure 10 .

 10 Figure 10. A music event represented as a trapeze form. 𝜸(𝒌)<0.

Figure 11 .

 11 Figure 11. Interlude from Music for__, trombone part.

Figure 12 .

 12 Figure 12. A music event represented as a triangle form. A fixed start.

Figure 13 .

 13 Figure 13. Particular cases. A fixed start.

Figure 14 .

 14 Figure 14. A music event represented as a triangle form. A fixed ending.

Figure 15 .

 15 Figure 15. Particular cases. A fixed ending.

Figure 16 .

 16 Figure 16. Interludes from Music for__ trombone part.

Figure 17 .

 17 Figure 17. Example of a theoretical string quartet


Benny Sluchin
email: sluchin@ircam.fr

Mikhail Malt
email: mikhail.malt@ircam.fr

John Cage's Number Pieces, a geometric interpretation of "time brackets" notation

Keywords: Computer Aided Performance, Notation, Musical Graphic Representation

Conceptual musical works that lead to a multitude of realizations are of special interest. One can't talk about a performance without considering the rules that lead to the existence of that version. After dealing with similar works of open form by Iannis Xenakis, Pierre Boulez and Karlheinz Stockhausen, the interest in John Cage's music is evident. His works are "so free" that one can play any part of the material; even a void set is welcomed. The freedom is maximal and still there are decisions to consider in order to make the piece playable. Our research was initially intended to develop a set of conceptual and software tools that generates a representation of the work as an assistance to performance. We deal here with the Number Pieces Cage composed in the last years of his life. Over time, we realized that the shape used to represent time brackets, brought important information for the interpretation and musical analysis. In the present text, we propose a general geometric study of these time brackets representations, while trying to make the link with their musical properties to improve the performance.

Introduction

The interpreter who approaches the music of John Cage composed after the middle of the 20th century is often disconcerted by a great freedom of execution, associated with a set of precise instructions. The result is that, each time, the musician is led to determine "a version," and to decide on a choice among the free elements proposed by the piece. A fixed score is thus created, which can be used several times. The musician interprets "his version" while thinking that it conforms to the composer's intentions. But in fact, most works of Cage composed after the 1950s should not be preconceived, prepared, "pre-generated" for several executions. Each interpretation should be unique and "undetermined." It is in this sense that the use of the computer can help the performer: a program will allow the latter to discover without being able to anticipate what and when he plays. The performance of the work thus escapes the intention of the musician to organize the musical text.

John Cage's Number Pieces

The corpus of John Cage's late compositions (composed between 1987 and 1992) is known today as Number Pieces. Each work is named after the number of musicians involved; and the exponent indicates the order of the piece among the other compositions containing the same number of musicians [START_REF] Haskins | The Number Pieces of John Cage[END_REF].

Silence and Indeterminacy

In the course of his creative research as a composer, Cage has laid down essential structural elements. Thus, silence has been posited as an element of structure to be thought of in a new and positive way; not as an absence of sound, but as a diachronic element, a presence, an acoustic space. This innovative work concerning silence has itself evolved: at first it was conceived as giving the work its cohesion by alternating with sound, then Cage extended the reflection to a spatial conception: the silence is composed of all the ambient sounds which, together, form a musical structure. Finally, silence was understood as "unintentional," sound and silence being two modes of nature's being unintentional [START_REF] Chilton | Non-intentional performance practice in John Cage's solo for sliding trombone[END_REF].

Moreover, in this desire to give existence to music by itself, Cage has resorted to various techniques of chance in the act of composition and principles of performance.

The principles of indetermination and unintentionality go in that direction. The principle of indetermination leads the musician to work independently from the others, thus introducing something unexpected in what the musical ensemble achieves. The performer, unaware of the production of his fellow musicians, concentrates on his own part and on the set of instructions. This requires great attention, even if the degree of freedom of the playing is high [START_REF] Pritchett | The Music of John Cage[END_REF].

Time Brackets

In Cage's Number Pieces each individual part contains musical events with time brackets. Generally, an event consists of a score endowed with two pairs of numbers: time brackets (Fig. 1). The composition has a defined total duration and the events are placed inside a pair of the time brackets. Although there are only individual parts, a score for the group is implicitly present and leads to a form.

Earlier research

In previous work [START_REF] Sluchin | A computer aided interpretation interface for John Cage's number piece Two 5[END_REF] we modeled these time brackets by parallelograms (see Figures 2 and3) to build computer interfaces for interpretation assistance in the context of Cage's Two 5 (Fig. 2). Over time ( [START_REF] Cage | Two 5 . On John Cage Two 5[END_REF], [START_REF] Cage | Music for Two. On John Cage, Music for Two[END_REF], [START_REF] Cage | Ryoanji. On John Cage[END_REF]), we realized that the shape used to represent time brackets, brought important information for the interpretation and musical analysis. The unusually long duration of this piece, 40 minutes, and the use of time brackets show that the temporal question, and its representation, is essential in the Number Pieces, in general, and in Two 5 in particular. The computer interface whose use has become obvious, has created for us a climate of confidence in our relationship to the piece. Random encounters of synchronicity as well as intervals bring unexpected situations… [START_REF] Rappaport | On Panorama [CD[END_REF] In the present text, we propose a general geometric study of these time brackets representations, while trying to make the link with their musical properties to improve the performance.

The Geometry of Time Bracket

The first step in the process was to model a graphic representation of each part as a succession of musical events in time. For this purpose, the temporal structure of the piece has been represented as quadruples on a timeline. (𝑠 # (𝑘), 𝑠 ( (𝑘), 𝑒 # (𝑘), 𝑒 ( (𝑘) ).

In order to place an event k on the timeline, time brackets are defined as quadruples to indicate the time span allocated to it. Each quadruple consists of two pairs. More precisely, each pair gives the interpreter lower and upper time bounds to start (𝑠 # (𝑘), 𝑠 ( (𝑘) ) and to end (𝑒 # (𝑘), 𝑒 ( (𝑘) ). Theses closed time intervals give to the performer, a choice of the pair (𝑠(𝑘), 𝑒(𝑘) ), where (𝑠 # (𝑘) ≤ 𝑠(𝑘) ≤ 𝑠 ( (𝑘) ) and (𝑒 # (𝑘) ≤ 𝑒(𝑘) ≤ 𝑒 ( (𝑘) ). One could choose the starting time (𝑠(𝑘)), while performing and, then accordingly, the end time (𝑒(𝑘)). This is the way one would employ when actually performing the work.

To obtain a graphic representation of each event in time we consider the quadruple:

(𝑠 # (𝑘), 𝑠 ( (𝑘), 𝑒 # (𝑘), 𝑒 ( (𝑘) ) where (𝑠 # (𝑘), 𝑠 ( (𝑘) ) is the Starting Time Zone and (𝑒 # (𝑘), 𝑒 ( (𝑘) ) the Ending Time Zone. As the two intervals have, in our case, a designed superposition, we prefer to distinguish starting and ending zones by using two parallel lines (Fig. 3).

Fig. 3. Graphic representation for a generic time event

The graphic event obtained by connecting the four points has a quadrilateral shape. The height has no particular meaning. The starting duration 𝛿 , (𝑘) is defined as the difference: ( 𝑠 ( (𝑘) -𝑠 # (𝑘) ), which is the time span the performer has to start the event. In the same way the ending duration 𝛿 . (𝑘) will be the time span given to end the event ( 𝑒 ( (𝑘) -𝑒 # (𝑘) ). In the general case, these values are not the same, and the form we get is asymmetrical. When dealing with Cage's Number Pieces, one generally has: 𝛿 , (𝑘) = 𝛿 . (𝑘), both durations are the same, and the figure to represent an event is a trapezoid (Fig. 4). This is the case in the majority of the corpus we are treating. Special cases will be mentioned later on.

Starting Time Zone s l(k) el(k) e u(k) su(k)

Ending Time Zone Time John Cage's Number Pieces, a geometric interpretation of "time brackets" notation 5

Fig. 4. Graphic representation for a time event in Cage's Number Pieces

There is mostly an overlapping of the two time zones, (𝑠 # (𝑘), 𝑠 ( (𝑘) ) and (𝑒 # (𝑘), 𝑒 ( (𝑘) ) but it can happen that those are disjoined. We can define a variable 𝛾(𝑘) where: 𝑠 # (𝑘) + 𝛾(𝑘) = 𝑒 # (𝑘). In Cage's Number Pieces, 𝛾(𝑘) depends generally on the event duration. Thus, we don't have a big variety of forms. For example, in Five 3 , we have only 4 different time brackets sorts, for a total number of 131 events for the five instruments and 𝛾(𝑘) = 2 3 𝛿 (𝑘) for all quadruples. We make a distinction between a generic musical event and a real (or determined) musical event. A real musical event is the one whose starting points (s) and end points (e) are defined, that is, where there is a concretization of choice. One could represent this by a straight line from 𝑠(𝑘) to 𝑒(𝑘) (Fig. 5). There are certain properties of a generic event that can easily be deduced from the trapezoidal graphic representation:

1. The starting or ending durations: 𝛿 , (𝑘) or 𝛿 . (𝑘) are a kind of a nominal duration that Cage gives to an event.

2. The maximum duration, 𝑒 ( (𝑘) -𝑠 # (𝑘) = 𝛿 456 (𝑘), is the maximum length (duration) an event can have.

3. The fact that, 𝑠 ( (𝑘) > 𝑒 # (𝑘) means that we can choose a starting point s(k) placed after the end, which leads to an empty musical event ∅ (an important idea of Cage: he often indicates that the artist can choose, all of, a part of, or nothing of the material placed at its disposal). In this case, s(k) > e(k). 5. An implicit parameter that is important is the straight line's slope of the concrete event (Fig. 5). This value is inversely proportional to the concrete event duration. The slope is strongly related to performance: it shows how much time the performer has for a particular event k. In regard to a wind instrument part, often only composed by held notes, knowledge of this parameter allows the artist to better manage his air capacity, in order to respect the composer's indications. As far as the pianist is concerned, the slope gives some information that allows him to manage his interpretation with reference to the time indications. When the straight line of a concrete event is close to the vertical, the event will be short and concentrated.

Starting Time Zone

The relationships of the generic events

Concerning the placement of two contiguous events k and k+1 we can define a variable 𝜀(𝑘), the gap between the elements k and k+1 where: 𝜀(𝑘) = 𝑠 # (𝑘 + 1) -𝑒 ( (𝑘) (Fig. 7). We will observe five typical placements of two contiguous events.

1. e > 0.

The two events are separated on the timeline. There is a minimum length of silence between the two events, which will probably be longer according to the choice of 𝑒(𝑘) and 𝑠(𝑘 + 1). In Five 3 for example, we have events 1 and 2 of violin 2 separated by more than 8 minutes, or 3 minutes between events 6 and 7 of violin 1. Here the piece could also be considered from the point of view of the relative density of the musical elements. One should mention the global statistical approach done elsewhere [START_REF] Popoff | Indeterminate Music and Probability Spaces: the Case of John Cage's Number Pieces[END_REF] [START_REF] Popoff | John Cage's Number Pieces: The Meta-Structure of Time-Brackets and the Notion of Time[END_REF].

2. e = 0.

The two events are adjacent (Fig. 8). Again, a gap may occur between the two events as the actual ending of event k: 𝑒(𝑘), and/or the actual starting of event k+1, 𝑠(𝑘 + 1) will differ from 𝑒 ( (𝑘), and 𝑠 # (𝑘 + 1) correspondingly. For example, Two 5 , trombone, events 21 and 22 (Fig. 9), events 27 and 28. In this case, the performer's opinion and attitude can determine the performance. There are many remarkable cases of interest in this situation; we could mention some cases that presently occur in Cage's Number Pieces (Fig. 10). For example, Two 5 , trombone events 28 and 29, and piano events 6 and 7. While performing event k, the player could start the event k+1 when not yet ending event k. We can encounter a superposition as shown in Fig. 11. For example, Two 5 , trombone events 37 and 38; piano events 9 and 10, events 12 and 13. And even the same starting time for the two events: 𝑠 # (𝑘 + 1) = 𝑠 # (𝑘) (Fig. 12). For example, Two 5 , piano, events 14 and 15 (Fig. 13). As the events have an order given by Cage, one may assume that the sequence of events is to be respected. But the performer may consider mixing the two events and choosing the respective ending times, 𝑒(𝑘) and 𝑒(𝑘 + 1).

In some case one has the configuration shown in Fig. 14. For example, Two 5 , trombone events 31 and 32, events 39 and 40. This may be a mistake, in calculation or in printing. Again, without change the order of events, one could start with the event k, and continue with the event k+1, mixing or separating. Starting with the event k+1 would mean that mixing has to happen, or the event k, should be skipped, that an idea dear to Cage: the event k wouldn't be performed.

The presentation of the time brackets as geometric figures and the variables we have defined lead to calculate some constants related to each of the instruments involved. The average filling rate (𝐹𝑟 III ) gives an indication of how much a particular instrument is present during the piece. This value will be the ratio of the sum of all the events' duration by the overall length of the work (Δ), where the event duration, 𝛿(𝑘), is the arithmetic mean between 𝛿 , (𝑘) and 𝛿 . (𝑘) [START_REF] Haskins | The Number Pieces of John Cage[END_REF].

𝐹𝑟 III = ∑ 𝛿(𝑘) L M Δ (1)
In the analog way, if we set: 𝜀(0) be the gap before the first event, and 𝜀(𝑛) the gap after the last event n, the average silence rate (𝑆𝑟 III ) will be the ratio of the sum of all the gaps between the events by the overall length of the work [START_REF] Chilton | Non-intentional performance practice in John Cage's solo for sliding trombone[END_REF].

These interesting values are based on the lengths of events, the gaps between them and their number, independent of the contents of the events. If instead of using 𝛿(𝑘), the event duration, we consider 𝛿 456 (𝑘), then:

R 𝛿 456 (𝑘) + R 𝜀(𝑘) L Q = ∆ L TUM (3)

Musical Analysis Application

Table 1 shows the values for the 21 events of violin 1 in Five 3 , and the constants we just defined. The time values, onsets and durations, are defined in seconds. The following Table 2, compares these constants for the five instruments. We can observe how these two constants (𝐹𝑟 III 𝑎𝑛𝑑 𝑆𝑟 III ) are strongly related to the presence of the instruments. For example, trombone will be more present, more active than the string instruments. One can see that 𝑆𝑟 III may be negative. This occurs when many of the events are superposed (All cases with 𝜀 < 0). These values are clearly reflected in the form of the piece seen in the upper part of Fig. 15. We had implemented several models, some offline in "OpenMusic"1 computer aided composition software, and in a real-time "Max" software [START_REF] Sluchin | A computer aided interpretation interface for John Cage's number piece Two 5[END_REF]. Fig. 15 presents a generic computer interface we are exploring, to perform most part of Cage's Number Pieces. The medium part of this figure, displays one of the instruments chosen (here violin 1) and bottom part displays the musical score corresponding to the time (here 30 seconds after beginning). The global view displays a presentation of the entire duration of Five 3 , using the trapezoidal event representation. It allows the performer to have a global view of the piece at a glance. As Cage mention about the context-specific character of his time-bracket notation: Then, we can foresee the nature of what will happen in the performance, but we can't have the details of the experience until we do have it. [START_REF] Retallack | Musicage: Cage muses on words, art, music[END_REF] This global representation enables another perspective of the piece. The printed score orients a natural local view. More than being a graphic representation for each time bracket, it allows us to identify similarities between generic musical events. Fig. 16, a detail from Fig. 15, presents the first ten minutes of the global representation of Five 3 . In an analog way Table 3 presents 𝐹𝑟 III and 𝑆𝑟 III constants for Two 5 , and Fig. 17 shows the global structure of the piece. One can clearly distinguish the difference in the presence of the two instruments.

Conclusions

At the present time we work to offer the musicians a way to approach other pieces from the same family, constructing a generic interface. The task may be somewhat complicated. The works called Number Pieces, share the same principal described earlier, but often contain particularities and exceptions in the instructions for performance. The interface then has to be adapted to cover these.

The interface is a substitute to the printed score. It reveals the structure of the work and provides the performer with the tool to achieve the "meditative concentration" needed. The few instructions given by Cage are integrated in the interface.

Considering the graphic representation, we presented above, our main goal was to find geometric properties and strategies to enhance the performance of these pieces through computer interfaces. John Cage's works have been the target of our work for several years now. We have developed computer tools for the interface, and used it in practice. Both concerts and recordings have been the tests for the usefulness of the approach towards performance. The modeling process is transformed in a pragmatic analysis of the musical phenomena that leads us, step by step, to model some of Cage's concepts. Mentioning first the Concert for Piano and Orchestra (1957), an earlier work that has become important step of his output [START_REF] Sluchin | Interpretation and computer assistance in John Cage's Concert for piano and Orchestra[END_REF]. Followed by two of his number pieces for a small number of performers [START_REF] Sluchin | A computer aided interpretation interface for John Cage's number piece Two 5[END_REF]. These works were also the object of a recording and performance sessions ( [START_REF] Cage | Two 5 . On John Cage Two 5[END_REF], [START_REF] Cage | Music for Two. On John Cage, Music for Two[END_REF], [START_REF] Cage | Ryoanji. On John Cage[END_REF]).

SLUCHIN,

  Benny et MALT, Mikhail. John Cage's Number Pieces, a geometric interpretation of "time brackets" notation. In : International Symposium on Computer Music Multidisciplinary Research. Springer, Cham, 2019. p. 129-141.

Fig. 1 .

 1 Fig. 1. John Cage's Two 5 , piano, 9th event

Fig. 2 .

 2 Fig. 2. Cage's Two 5 main computer interface

Fig. 5 .

 5 Fig. 5. A real music event represented by a straight line, joining the starting to ending time zones

4 .

 4 An alternative way to present a quadruple will be: (𝑠 # (𝑘), 𝛿 , (𝑘), 𝛿 . (𝑘), 𝛾(𝑘) ) where 𝛾(𝑘) is the value previously discussed. This representation can easily display the regularity in the time brackets construction (Fig.6). It is easy to see that 𝛿 456 (𝑘) = (𝛿 , (𝑘) + 𝛿 . (𝑘)) 2 + 𝛾(𝑘).

Fig. 6 .

 6 Fig. 6. An event represented as (𝒔 𝒍 (𝒌), 𝜹 𝒔 (𝒌), 𝜹 𝒆 (𝒌), 𝜸(𝒌) )

Fig. 7 .

 7 Fig. 7. 𝜺(𝒌), The gap between the elements k and k+1 sl(k) sl(k+1) e l(k) el(k+1) e u(k) eu(k+1) s u(k) su(k+1)

Fig. 8 .

 8 Fig. 8. e = 0

Fig. 9 .

 9 Fig. 9. Two 5 , trombone, events 21 and 22

Fig. 10 .

 10 Fig. 10. e < 0, 𝒔 𝒍 (𝒌 + 𝟏) = 𝒔 𝒖 (𝒌)

Fig. 11 .

 11 Fig. 11. e < 0, 𝒔 𝒍 (𝒌 + 𝟏) < 𝒔 𝒖 (𝒌)

Fig. 12 .Fig. 13 .

 1213 Fig. 12. e < 0, 𝒔 𝒍 (𝒌 + 𝟏) = 𝒔 𝒍 (𝒌)

Fig. 14 .

 14 Fig. 14. e < 0, 𝒔 𝒍 (𝒌 + 𝟏) < 𝒔 𝒍 (𝒌)

Fig. 15 .

 15 Fig. 15. Computer interface used for performing Five 3

Fig. 16 .

 16 Fig. 16. The first ten minutes of the global representation in Cage's Five 3

Table 1 .

 1 Data for Five 3 , first violin

Table 2 .

 2 Comparison values in Five3

		#Events	𝐹𝑟 III	𝑆𝑟 III
	Violin 1	21	0.34	0.43
	Violin 2	12	0.16	0.74
	Viola	26	0.34	0.44
	Violoncello	25	0.23	0.5
	Trombone	47	0.74	-0.24

Table 3 .

 3 Comparison values in Two5

		#Events	𝐹𝑟 III	𝑆𝑟 III
	Piano	29	0.33	0.15
	Trombone	40	0.46	-0.14

Fig. 17. Two 5 global structure

"OpenMusic" is a software developed by Ircam by Gerard Assayag, Carlos Augusto Agon and Jean Bresson. See: http://recherche.ircam.fr/equipes/repmus/OpenMusic/.


Sampling

Operating Conditions

Isothermal

-Objectives of the present work

Understanding of the macrokinetics of the hydroxidation of aluminum. Measurements of the H 2 gas release rate as a function of time and temperature. Investigating the effect of salt water compared to pure water.

-Results

-Introduction

Aluminum nanoparticles react with water and produces hydrogen. As hydrogen is envisaged as an alternative energy vector, is it possible to consider Ultrafine Aluminum Powder (UFP) as a in situ hydrogen source ?

-Experimental setup

X-ray analyses of the solid reaction residues reveal aluminum oxyhydroxide AlOOH (boehmite) Gas-phase chromatography confirms the production of gas dihydrogen (H 2 )

Al(s) + 2 H 2 O(l) AlO(OH)(s) +3/2 H 2 (g)
Induction time and maximal flow rate follow an Arrehnius law. t ind for saline solutions are higher than for pure water, with lower W max .

After this period, instantaneous H 2 flow rate increases rapidly, exhibits a maximum then decreases.

Reaction starts after an induction period (t ind ) S-shaped gas evolution for the total volume of hydrogen observed.

Temperature dependence of the induction period.

The theoretical value of maximum gas release (1350 cm 3 H 2 /g Al) is not reached, but conversion is quasi-complete.

Rat'ko et al [2] suggest that this induction period corresponds to the hydration of the natural oxide film (Al 2 O 3 ).

-TEM Analysis

Initial Alex powder Quenching of the reaction by freezing mixture samples at different times. Samples are then analyzed by TEM.

-Conclusions

The temperature dependence and the influence of salinity have been investigated on the low temperature reactivity of aluminum nanoparticles with water .

The production of hydrogen is preceded by an induction period corresponding to the hydration-dissolution-precipitation of the oxide film initially covering the particles.

This induction period is thermally activated (Ea= -80 kJ/mol).

The hydrogen released by the reaction (hydroxidation) is characterized by W max , which is also thermally activated with similar values of Ea found in the literature.

The hydroxidation of the nanopowder is slowed down in the presence of chloride ions.

  conditions during experiment Temperature parameter range [45 -90 °C] Alex powder, air passivated (100 nm mean diameter) Mixture composition fixed as: • 60 g distilled water • 0.1 g of Alex powder Saline solution 3% w. (KCl or NaCl) Experimental Techniques Temperature ➩ Thermocouple graduated test-tube H 2 gas release ➩ Mass flowmeter Transmission electron microscopy X-ray diffraction Institut de Combustion, Aérothermique, Réactivité et Environnement, Centre National de la Recherche Scientifique 1C, avenue de la Recherche Scientifique, 45071 Orléans Cedex 2, France Low Temperature Reactivity of Aluminum Nanopowders with Liquid Water V. Sarou-Kanian*, S. Ouazar, P. Escot Bocanegra, C. Chauveau, I. Gökalp sarou@cnrs-orleans.fr

3 NaCl

 3 Ea (kJ/mol) H 2 O 6.41 10 -11 -79.9 ± 4.1 NaCl, KCl solutions 1.12 10 -10 -80.4 ± 4.3 Solution Ivanov et al.[1] 7.93 10 15 89 ±1.5 H 2 O, Lur'e et al. [3] 3.02 10 14 85 ± 4.Diffraction diagram of the Al+H 2 O reaction residues. •: AlOOH (JCPDS 00-049-0133); □: Al (JCPDS 01-085-1327) of an experimental result at T=72°C