Mirio Cosottini Five Improvisations: non linear aspects of an improvisation Five Improvisations is the title of a performance that took place in Udine (Italy) on October 10th, 2013 as part of a convention on Philosophy of music: atto, oggetto, opera. For this event I submitted five improvisations1 and wrote some indications that were subsequently performed by Marcello Giannandrea on bassoon, Andrea Tinacci on bass clarinet, Francesco Cigana on drums and myself on trumpet. These improvisations were conceived on the basis of a number of musical invariances, musical elements that remain constant in time, that don’t change under a series of transformations. My aim here is to conduct an analysis of the linear and non-linear aspects of the improvisation Shift Harmony by using the graphic score and referring back to the preparatory phase of the performance. This analysis will clarify how the invariances represent the constant point of reference of the improvisation and, along with the linear elements, are instrumental in determining the end result2. 1. The conceptual tools My musical esthetic has evolved and come into focus on the basis of the linear/non-linear dichotomy. This dichotomy is not new in the philosophic landscape, it has been explored by Jonathan Kramer in his book The Time of Music (Kramer [1988]). The concept of Invariance as well has been discussed in the realm of science and philosophy. For the scope of my research I’ve referred to Nozick’s ideas on the subject and particularly his book Invariances (Nozick [2003]). Besides, my personal esthetic is a separate issue from the research on the esthetic of music and on improvisation; the two are not interchangeable. For this reason I will shine some light on the theoretical frame of reference just enough so that we understand the type of analysis conducted on a concrete musical improvisation and I will save a more detailed philosophical analysis for the future. According to Kramer (1988): 20, “Virtually all music utilizes a mixture of linearity and nonlinearity. Linearity and nonlinearity are the two fundamental means by which music structures time and by which time structures music”. Linearity and non-linearity are present in every structural phenomenon (structural force). They don’t operate on the same level, linearity deals with the organization of musical events within the piece while non-linearity has to do with principles that regulate the whole. Here are the definitions of the two terms: linearity is “the determination of some characteristic(s) of music in accordance with implications that arise from earlier events of the piece”3, while non-                                                                                                                1  Shift   Harmony,   Pulse,   Listen   to   What?,   Core   and   The   Core.   The   improvisation   The   Core   was  conceived  and  proposed  by  Marcello  Giannandrea  and  Francesco  Cigana  2  The  content  of  this  article  is  the  result  of  a  twenty  yearlong  activity  as  a  musician  and  improvisor.  The   theoretical   and  philosophical   arguments   supporting   these   ideas,   still   in   a   embryonic   state,   are  the  first  outcome  of  a  research  I’m  conducting  as  part  of  a  PhD  program  at  the  University  of  Trieste  and  Udine,  which  scope  is  the  study  of  non-­‐linearity  in  relationship  to  musical  improvisation  and  the  concept  of  invariance.  3  Where  not  specified  the  translation  is  mine.    

linearity is “the determination of some characteristic(s) of music in accordance with the implications that arise from principles or tendencies governing an entire piece or section” (Cfr. Kramer [1988]: 20). The notion of linearity is useful in which it brings to focus all those aspects of a musical experience that present implications, consequentiality, and in its broadest sense, narrative and drama4. We have linearity when a certain degree of consequentiality characterizes the succession of musical events; a musical event follows another that implies it more or less relevantly5. The use and the theoretical development of these relationships have been consolidated in time by the tonal system which, due to its array of rules for composition, other than harmonic, melodic and formal construction, represented the paradigm of linear music in the twentieth century. On the other hand, the non-linear features of a piece have nothing to do with the succession of events, nor with implicative type relationships that connect different events. Here the musical events are rather considered in their own autonomy. What these events determine, individually or in their totality, is a principle or a rule that remains constant in time for the duration of the entire piece (or part of it). In general I propose to call this principle an invariance. By the term “musical invariance” I mean the feature of one or more musical events that remains constant under a certain number of allowed transformations. The relationship between non-linearity and invariance deserves a more in depth analysis that can’t be conducted in the context of this article6. For our scope we can assume that the notion of invariance plays a fundamental role in the discussion on non-linearity and that it forms the basis to the understanding of those musical elements that remains constant with respect to a series of transformations. Is it possible to identify a musical invariance while listening to a piece? Let’s consider the first Prelude from the Well Tempered Clavier by Bach.

A western ear has a tendency to listen to the prelude in a linear way and, for example, it captures the implicit melodic texture (the same tendency generated the famous Ave                                                                                                                4  For  a  discussion  on   the  relationship  between  music  and  narrative  and  between  music  and  drama  see  Bertinetto  (2012a):  72  ss.  5  In   his   The   Time   of   Music,   Jonathan   Kramer   utilizes   the   theory   of   Markov’s   chain   to   explain   the  consequential  mechanism  linking  musical  events  in  the  listening  experience  (Kramer  [1988]:  22).  6  For  the  notion  of  invariance  see  Nozick  (2001)  (Nozick  [2003])  and  of  non-­‐linearity  Kramer  (1988).  

Maria7 in the mind of Charles Gounod). Such tendency is surely implicit in Bach’s composition: the rules of counterpoint, and thus, the movement of the voices, permeate all of Bach’s music. Now let’s try to identify that element that stays constant, that doesn’t change in time. This element readily manifests itself: it is the arpeggio, generating a perceptive invariance that remains always constant, despite many other musical aspects - the pitch of the notes for example - change and transform themselves. We could imagine alternative developments to the prelude in which the pitches change in a way so radical as to compromise the identity of the composition8. However such a perceptive invariance remains always present as equal to itself9. Therefore, to listen to the prelude from a non-linear point of view it means to be open to the invariances contained in it. Improvised music exhibits non-linear characteristics as well. Let’s consider a solo piano improvisation by Tonino Miano10. If we listen by paying attention to the elements that remain constant we’ll immediately identify an invariance that permeates the whole improvisation: the timbre invariance of the clusters11. This invariance is the basis of their homogeneity and it allows for identifying these clusters like analogous sound lumps, very similar to each other, cut from the same “cloth”12. The pianist develops his improvisation maintaining such invariance for the entire duration of the piece. 2. Introduction to Shift Harmony Given these premises we can now return to the Five Improvisations and understand, thanks to the analysis of Shift Harmony, in what way the invariances determined the performance of the musicians and the relative auditory result13.

                                                                                                               7  The  Ave  Maria  was  composed  by  Charles  Gounod  in  1859  and  it  consists  of  a  superimposed  melody  on  the  Prelude  N.1  from  the  Well  Tempered  Clavier  by  Bach  (BWV  846).  The  original  version  of  the  piece  was  composed  by  Gounod   in  1853,   it  went  by   the   title  Meditation   sur   le  Premier  Prélude  de  Piano  de  S.  Bach  and  it  called  for  a  violin  instead  of  a  voice.  8  This  is  an  interesting  phenomenon  for  the  study  of  the  relationship  between  identity  of  a  piece  and  invariances.  Some  transformations  are  admissible  with  respect  to  an  invariance  but  not  with  respect  to  identity.  9  As   an   example   I   improvised  a  piece   starting  with  Bach’s  Prelude  which   can  be   listened   to   at   this  link:  https://www.dropbox.com/s/p5xlvznlw4ov3ht/bach%20preludio%20reloaded.mp3.  10  Tonino  Miano,  Metaphrase,   solo  piano   improvisation,   recorded   at   the  Abrons  Art  Center   in  New  York,  May  2010.  Listen  at:  https://www.dropbox.com/s/y0met412nv5u773/Tonino%20Miano%20-­‐%20piano%20solo%20improvisation_%20Metaphrase%20I.mp3.  11  According   to   the   New   Grove   Dictionary,   the   cluster   is   «A   group   of   adjacent   notes   sounding  simultaneously»,  (Chew  [1995],  Vol  4,  p.  504).  12  Note   how   the   wise   use   of   dynamics   becomes   the   glue   that   holds   together   these   frequencies.  Sometimes  you  have  the  impression  that  a  cluster  is  not  a  group  of  notes  but  one  sound.  13  Listen  at:  https://www.dropbox.com/s/m8a3tf39gkvaq8t/shift%20harmony%20web.mp3.    

Shift Harmony is an improvisation suggested by a graphic score that indicates useful ways to achieve a collective idea of sound14. There are four measures with a line and one note. In the lower part is suggested an idea of rhythmic pulse that acts subjectively on the musicians; and a graphic item in the middle. What is immediately clear is the lack of explicit indication about the sequence of musical events. At the same time we notice the absence of formal indications regarding the sequence of events as a whole or according to meaningful units, like musical phrases, periods, sections or larger formal units. Finally, the score gives no instructions on how to linearly connect the musical events, nor does it establish relevant formal units. What these indications suggest is nothing else than an idea of sound, or better, a set of invariances in the perception of sound that remain the same in time. Such invariances are dependent upon the polyphonic and dynamic structure. The metronome mark refers to the organization of each musician subjective time. In theory the improvisation should contemplate the phase displacement of pulses of the same duration. In reality, what happens in normal listening conditions among the musicians is the shift from a subjective perception of the pulse to an intersubjective one, and the subsequent overlapping of all pulses in a single rhythm. “In a “free” improvisation, when nothing has been previously decided [...] it is the ensemble that has the responsibility of timing. Individuals, with their eyes closed, will attempt to reproduce simultaneous attacks without warning. Strangely these attacks will get closer to one another, to then end up together. Timing has become collective” (Globokar [1986]: file 26c, Vol. 5). In essence, the pulse becomes common to all musicians in a short time. The ondulated graph item that stretches horizontally across the score was discussed by the ensemble                                                                                                                14  A  collective  idea  of  sound  is  a  set  of  non-­‐linear  characteristics  of  the  sound  adopted  by  an  ensemble  of  musicians  as  a  condition   for   the  performance  of  a  piece.  They  can  be  suggested,  prescribed  by  a  score  or  verbally  discussed.  

before the performance, and it was decided that it would be interpreted as a moment of emphasis, of auditory relevance in the otherwise rather static path of the musicians (with the condition that it would be executed only once during the entire improvisation). Therefore, this graphic item as well, contributes nothing to explaining how to connect the musical events, which logic to follow, what type of implications; it is not interpreted as an incentive or a spark for a linear articulation, given that it is impossible to discern in it discrete elements and a defined directionality in time, nor elements that can be gathered under the umbrella of some type of musical syntax15. We’re then in the presence of a symbol expressing a non-linear characteristic of music. It requires a non-linear reading. How do you carry out the non-linear reading of a graphic symbol? As I have written elsewhere (cfr. Cosottini [2012]: 24), « the non-linear reading of a graphic symbol is intended to be an interpretation where to each element of graphic invariance corresponds an element of musical permanence»: as an example, a symbol that shows a very dense color can be interpreted musically as a rather uniform timbre16. The musical improvisation Shift Harmony acquires consistency by following a set of indications of non-linear nature and it is articulated linearly by keeping the invariances into account. Every choice made by the musician is born primarily from satisfying a non-linear characteristic of the sound and secondarily from the development of musical events. The score can be interpreted in various ways (individually, collectively, before or during the performance, etc.). In the particular case of the performance given at the Udine convention we opted to interpret the symbols in advance and identify a set of auditory invariances upon which to organize our improvisation. We agreed on two basic invariances, a dynamic based type and a timbre based type. The dynamic mark p (piano) was set for the entire duration of the improvisation. The dynamics of a piece are contextual parameters that depend on acoustic needs (dimension of space, reverb), orchestration needs (the dynamics of a section of the orchestra is regulated by the dynamics of individual musicians) and lastly, music needs (in some cases it is advisable that the third of a triad split among three musicians be played “more quietly”). They are in strict correlation with all the other parameters, and can influence the sound response (a lower note played more loudly seems higher in pitch), it could influence the intonation (raising the intensity of the sound changes the intonation of reeds instruments differently than the flutes) and lastly on the timbre (a piano sound is normally less full than a forte one). In an improvisation all this needs to be taken into account. This happens with various degrees of awareness, but given that the invariance is based on the control of dynamics, before we began the performance we went through the just mentioned checklist of potential issues. Maintaining this invariance was not an easy task, due to the rather peculiar and diverse character of the ensemble (trumpet, bassoon, bass clarinet and percussion).

                                                                                                               15  The   musical   syntax   concerns   the   study   of   musical   structures   and   harmonic   rules,   along   with  rhythm.  These   rules  have  mainly   a   linear  value  because   they  examine   the  possibility  of   combining  musical  events  as  implied  by  previous  ones.  16  The   way   every  musician   choses   to   interpret   is   different,   what’s   important   is   to   couple   the   two  types  of  invariances,  the  graphic  one  and  the  musical  one.    

The trumpet is a brass instrument that requires good skills to be played softly and in some respects the same can be said of double reeds like the bassoon. It gets a bit easier on the bass clarinet, but the risk is often that of compromising the intonation. Similar consideration can be made for the second invariance, the timbre type. Clearly this invariance, like the dynamic type, concerns the overall sound of the ensemble and it contributes into the making of such sound as well. In this case too, the heterogeneous body of instruments doesn’t ease the task: a brass, double reed, single reed and percussion are not easy to blend together. It’s much easier to obtain a compact and uniform collective sound if we blend together instruments of the same family (for example four brass: trumpet, french horn, trombone and tuba). In our case the instruments all belonged to different families. Therefore the timbre had to find its place within the strict correlation of sound parameters. As a general rule, a louder dynamic tends to lighten the sound and this influences the register in which the musicians choose to play. It is possible to find balance in the mid-high frequency range provided the dynamics are contained, but louder dynamics will upset the balance, the sound will distort and create strong beats. This net of acoustic and psychoacoustic conditions is the outcome of just some of the possible transformations the sound goes through as a consequence of the choices made by the musician. The act of changing one auditory parameter triggers transformations that influence other parameters as well. There is no doubt that the knowledge of these transformations enhances the ability to control the sound and its use. This is possible because there is a relationship between the transformation triggered by the musician and the invariance retained in time. Moreover, not all transformations are admissible, some can end up compromising the invariance to the point of ceasing it. And this is particularly relevant in Five Improvisations: if one of the musician decides to play much louder than the others the dynamic invariance is compromised. If the transformation really takes the sound to excessive loudness the invariance disappears. The result is the substitution of one invariance with another that emerges and redefines the new set of admissible transformations. In such cases the listener perceives that something ends and something else begins. Karlheinz Stockhausen wrote something similar with regards to Moments17: “When certain characteristics remain constant for a while – in musical terms, when sound occupy a particular region, a certain register, or stay within a particular dynamic, or mantain a certain average speed – then a moment is going on: these constant characteristics determine the moment […] and when these characteristics all of a sudden change, a new moment begins. If they change very slowly, the new moment comes into existence while the present moment is still continuing”18. Even Five Improvisations can be considered a Moment, and more precisely an Improvised Moment. In fact, the invariances determine that which remains constant in the                                                                                                                17  A   moment   can   be   defined   as   a   «self-­‐contained   (quasi-­‐)   independent   section,   set   off   from   other  sections   by   discontinuities».   Cfr.   Kramer   (1988):   453.The   concept   of   moment   is   born   with   the  composition   Kontakte   (1958-­‐60)   di   Karlheinz   Stockhausen.   He   defined   it   as   a   «formal   unit   of   a  particular  composition  recognizable  by  its  own  specific  and  unique  character  »  (Stockhausen  [1963]:  200).  18  Finally   Stockhausen   clarifies   his   idea   of   composition   of  Moment-­‐Forming:   «I   form   something   in  music  which  is  as  unique,  as  strong,  as  immediate  and  present  as  possible.  Or  I  experience  something.  And  then  I  can  decide,  as  a  composer  or  as  a  person  who  has  this  experience,  how  quickly  and  with  how  great  a  degree  of  change  the  next  moment  is  going  to  occur».  Cfr.  Maconie  (1989):  63  s.  

music, as is the case with moment; however the degree of change (the transformations) inherent the moment is not established through composition procedures, but it is decided collectively by the improvisors as the performance unfolds. The way improvisors make musical decisions has been studied by Canonne (2013): 40-55. When considering the Collective Free Improvisation, he describes it as «a musical phenomenon produced by at least two persons improvising simultaneously and freely, i.e. trying to leave un-decided every compositional aspects until the very moment of the performance», (Canonne, Garnier [2011]: 29-41) and shows as the concept of focal point (Schelling [1960]) and team-reasoning (Sugden [2003]) are the basis of improvisation and of the decision making processes of the musician. Despite the fact that Shift Harmony is not a free improvisation («referent-free improvisation»), because it’s based on a set of previously given non-linear indications, it displays both those elements analyzed by Canonne which I will highlight later. Let’s look at my transcription of the improvisation (I have transcribed only the wind instruments) and see what choices the musicians have made19.

                                                                                                               19  The   transcription   deliberately   overlooks   sound   duration   and   adopts   a   proportional   value   in   the  writing.  

3. Linear analysis of the improvisation The piece begins with the trumpet and the bassoon playing an octave apart20. The trumpet “widens” the octave by playing a Db. The melodic interval of a minor second will characterize the trend of the trumpet until the end of the piece. Its largest melodic interval is shown within the ellipse “E” which precedes the “solo” indicated by the graphic item and included in the ellipse “F”. The rectangle “A” contains the notes of the wind instruments that form a diminished triad (G, Bb, Db), that could most likely lead to F min or Ab maj. The F of the trumpet in circle “B”, tracing back to the Db, seems to give way to a semidiminished chord (G, Bb, Db, F) without the Db (albeit retained by the ear from                                                                                                                20  It’s  not  improbable  to  begin  in  unison  or  at  an  octave  distance  apart,  especially  when  the  musicians  know  each  other  well  and  have  been  playing  together  for  a  long  time.    

the previous melodic line of the trumpet). Therefore, the harmonic tension is weakened by the transition from a diminished chord to a semidiminished one. The overall harmony undergoes a significant tonal shift immediately following the D natural played by the bass clarinet (circle “C”). The bassoon, playing a Db, triggers a decisive dissonance hardly ascribable to the previous tonal landscape. The trumpet descends chromatically to E natural marking a harmonic tightening of the improvisation. The ellipse “A” contains the most harmonically unstable phase of the entire improvisation. The possible tonal options previously postulated all fail as a consequence of the bass clarinet’s natural D. Every unstable phase generally tends to a resolution (especially in a tonal context): in this case the turning point is given by the crescendo of the trumpet along with the larger melodic motif, two novelties that are relevant enough to achieve significance within the counterpunctal texture and introduce a different scenario. The trumpet player’s choice shows a clear example of Focal Point because the strategies of the individual musicians converge towards that solution. The other instruments seems to stop21. The trumpet finds its own space and suggests that «melodic flower» which is notated as the central graphic item. After the short trumpet solo the bass clarinet plays a C (circle “G”) that restores the harmonic setting of the beginning. What follows this is the «melodic flower» of the bassoon in the high register, an interesting solution that “drags” the sound space upward. And indeed the outcome is surprising: the mezzoforte of the bass clarinet, at the end of the second line, pushes the harmony towards C minor, and the result is that of closing a multidirectional harmonic itinerary that since the beginning has shown a “restless” character. The musicians “find” themselves again on the C minor chord, and this marks a point of arrival as well as one of departure: it’s the beginning of the end of the improvisation. The «melodic flower»22 of the bass clarinet, in the ellipse “I”, moves along by minor seconds (a pattern reminiscent of the behavior of the trumpet and that uses the “characteristic” interval of the piece) and like a snake wraps itself in a coil, it begins in Eb and ends in Eb. The harmony’s static behavior pushes the musicians to embrace melodic choices. The melodic line of the bass clarinet (rectangle “M”) is clearly the coda that precedes the end of the piece, and the improvisation ends on a dyad that timidly points to Eb major. 4. Non-Linear analysis of the improvisation In general, the makeup of the beginning of an improvisation is determined by several factors; these are a set of known elements and a set of possibilities. As put by Alessandro Bertinetto [2012b]: 132), “there is always a background upon which improvisation will take place. In an improvisation, pre-existing forms and shaped materials are worked out and re-shaped in new or different ways (are ‘interpreted’, ‘combined’, ‘transformed’).

                                                                                                               21  Even  the  reaction  of  cautiousness  by  the  musicians  after  the  focal  point  is  described  by  Canonne:  «A  typical  reaction  is  the  use  of  a  waiting  strategy  (play  suddenly  piano,  repeat  a  note,  sustain  a  note,  have  a  more  sparse  playing...)  during   the  apparition  of   the  accident.  But   the   initial   idea   is  pursued,  with  no  or  just  a  few  changes.  The  accident  produces  a  simple  adjustment».  Cfr.  Canonne  (2013):  5.    22  With   the   term  Melodic   flower   I  mean  a  sequence  of   sounds   linearly  relevant   that  emerge   from  a  non-­‐linear   setting.     The  melodic   flower   is   nourished   and   grows   in   its   non-­‐linear   environment,   but  progressively  blossoms  and  drifts   away.  This   is  possible   the  moment   its   level  of   significance   shifts  from  non-­‐linear  to  linear,  depending  on  the  context.    

Explicit or implicit, conscious or unconscious rules, conventions, precepts, instructions, abilities, habits, styles, patterns guide the improvisational performing process, which anyway occurs not only in virtue of these contextual constraints, but also against and in spite of them”. In the specific case, within the known elements there are some invariances. First the dynamics of the piece. As prescribed by the graphic score, the attack has to be piano. Let us take into account that the attack of a sound is never separated from its development, but it is in strict correlation with its sustain and its decay. From its inception a sound carries within its own future23, it envisions its own development and therefore its own possible end24. In the case of Shift Harmony, the life of the sound is organized according to an additional invariance, that of the tenuto sound. Every musician in the ensemble knows that the attack of a note is the beginning of a long sound (that entails that the invariance “tenuto sound” is part of the known elements). Lastly, to these we add a timbre invariance that regulates the development of the improvisation, describable as timbre homogeneity. Therefore, the invariances “tenuto sound”, “dynamics of the piece” and “timbre homogeneity” are the three fundamental invariances that make up the known elements in possession of each musician. Obviously these known elements have to come to grips with actual moment the sound is produced and the invariances organize themselves according to the collective sound of the ensemble25. Alongside the set of known elements there are also a set of possibilities. The musicians can attack a sound in the high, medium or low register. All of them, or just some, can attack the sound. The duration and the articulation of the sound can vary considerably from one musician to the other, and the attack itself (for a given dynamics) can be accented differently (tenuto, accentato, sforzato etc.). These possibilities as well enter the realm of useful considerations the musicians have to ponder when beginning the improvisation.

                                                                                                               23  This  idea  was  developed  by  Luigi  Pareyson,  according  to  whom  «the  spark  is  the  seed  of  the  work,  it’s  the  work  itself  in  an  embryonic  stage,  therefore  it  has  an  intent  of  its  own,  a  tendency  to  achieve  its  own  form,  an  organizational  destiny»  (Pareyson  [2010]:  82).  24  In   2012   I   invited   the   German   pianist   and   educator   Reinhard   Gagel   to   teach   during   the  Improvisation  Lab  held  in  Arezzo.  Reinhard  suggested  to  the  students  a  very  interesting  exercise.  A  musician   attacks   a   sound   and   the   others,   part   of   the   ensemble,   have   to   immediately   identify   the  significant  invariance  in  that  sound  and  consequently  attack  another  sound,  as  soon  as  possible,  that  satisfies  the  invariance.  It  is  astonishing  to  see  how  quickly  the  ensemble  acquires  a  coherent  sound  and   a   great   homogeneity   as   a   consequence   of   identifying   the   invariance.   I   think   this   is   possible  because  the  sound  prefigures  its  own  development  in  the  attack  and  therefore  it  anticipates  possible  invariances.  The  following  phases  of  the  development  of  the  sound  allow  the  musician  to  retain  some  of  those  anticipations  and  verify  the  actual  structure  of  the  invariance.  25  With  respect  to  the  invariance    “dynamics  of  the  piece”,  the  musicians  are  not  required  to  simply    “play  piano”,  but  compare  each  dynamics  in  a  way  that  satisfies  the  established  invariance,  which  has  not  an  individual  value,  but  a  collective  one.    

The trumpet and the bassoon begin on the same note an octave apart. Incidentally the octave is a good departure point to investigate the blending of the sound (it is the closest interval to the unison in terms of consonance) and to establish how intonation and dynamics can determine an homogeneous sound 26 . The two instruments, as a consequence of that attack, immediately establish a “gravitational center” around which the improvisation is developed and unravels its own conclusion. That sound at the end of the second line (performed again by the bassoon) is crucial because it determines the end point of the harmonic development of the piece and the beginning of its end. Immediately following the initial octave the trumpet plays an interval of minor second (C-Db) and the sound “blossoms” slowly.

The trumpet moves up as the bassoon descends, while the bass clarinet widens the sound space with a low note (G). In this beginning section take shape other non-linear elements that will characterize the whole piece. Let’s see what they are and what do they hinge upon. The minor second will become a recurrent interval. Overall the minor second intervals performed by all instruments are 24 on a total of 43 (55%). The trumpet executes 30% of the total and 72% individually (13 out of 18 intervals played are minor seconds), while the bass clarinet plays 23% of the total and individually 62% (10 out of 16). The bassoon plays only one minor second, therefore 2% of the total, and 11% individually (1 out of 9). Other than that the melodic pattern of the bassoon is characterized by conjunct motion and therefore following a behavior very similar to that of the other instruments. The minor second will also characterize the «melodic flowers» as indicated by the graphic item in the score (particularly trumpet and bass clarinet). Trumpet:

                                                                                                               26  Playing   the   same   note   (in   unison   or   an   octave   apart)   is   an   excellent   exercise   to   investigate   the  sound   and   particularly   the   relationship   between   frequency   and   timbre,   between   intonation   and  sound  qualities,  between  the  body  of  sound  and  dynamics.  

Bass clarinet:

Bassoon

The bassoon uses predominantly the major second interval, not the minor second. Otherwise its melodic behavior follows the type of development inaugurated by the trumpet (and by imitation the melodic intervals follow the same direction). In all, the three winds favor the interval of a second in going from one note to the next; therefore the presence of the second (and particularly of the minor second) generates an invariance in the articulation of the notes that permeates the whole piece. This invariance is evident both in the flowing and the gathering of the sounds that begin and end one inside the other, as in the moments of melodic blossoming of the various instruments. The result is a remarkable uniformity in the articulation of sounds, which then constitutes a new invariance. It’s interesting to notice how al the «melodic flowers» of the winds are “inscribed” at the beginning of the piece. It follows that both the movement by minor and major seconds are an index of great coherence. The melodic articulation of the piece pertains the individual voices of each instrument, but not exclusively; it is represented as well by he melodic line generated the joining of the voices and therefore distinguishes itself as a melody of heterogeneous timbre (the attack of the winds is never concurrent, which leads to form an oblique melody). It is the case of a melody “hidden” within the counterpoint outlined by the subsequent entering of the voices. On the one hand it comes to mind Schumann’s Innere Stimme (internal melody) inaugurated in Humoreske op. 20 composed in 1839, and on the other the «ghost melody» investigated by György Ligeti in the etude for piano Desordre (1985 ca.), in which the continuous displacement of accents creates in the listener the perception of emerging melodies that are not written in the score. I then transcribed the attacks of the various instruments in the order in which they appeared therefore reconstructing the oblique melody within the harmony (taking the license to gather the voices in a uniform register).

As we can see from box “A” the first part of the oblique melody contains all the ingredients that will make up the «melodic flowers» of the wind instruments: the first minor second, which will reappear many times, and the major second. Notice how the last four notes within “A” are the same ones we see towards the end of the bassoon’s melodic flower. All the «melodic flowers» therefore testify to a great degree of coherence with the initial part of the improvisation, from which all seem to blossom; in other words both the trumpet and the bass clarinet’s melodic patterns pick up on the horizontal articulation of the voices, while the bassoon’s melodic pattern will recall the oblique melody. The “vertical” relapse of this melodic articulation is evident. The harmonic movement hinges on the behavior and the articulation of the parts, and the harmonic changes flow into one another with ease. The conjunct movement of the voices creates horizontal fluidity while vertically it generates gradual transitions in the harmonic texture; it’s like witnessing the transition between shades of the same color rather than one between different colors. The binding of these invariances along with the linear choices made by the musicians leads to what the title of the improvisation predicted, a progressively sliding harmony, shifting slowly. 5. Conclusions Shift Harmony is a clear example of how it’s possible to improvise starting from a set of non-linear indications, like the timbre and dynamics invariance. Soon, other invariances weave themselves alongside these, and they stay constant until the end: the melodic articulation and the tenuto sound. The linear elements of the improvisation “lay out” these invariances in chronological time on the basis of musical choices mindful of their permanence. Every transformation in the development of the piece is admissible provided the invariances are maintained. For example, the musicians imitate linearly each other’s melodies (the intervallic succession of the musical «flowers» is very similar in all three cases), but at the same time they’re sensitive to the oblique trend of the melody of timbres, which imbues the collective sound with a sense of cohesion. From a harmonic standpoint the piece follows a linear logic built around a tonal center of C minor, but not enough harmonic tension is generated to upset the dynamics invariance. A detailed linear analysis will uncover the counterpunctal and harmonic richness of the piece, while the overall perception of the music is subordinate to a pervasive static sense. This is additional proof of the fact that

the perceived stillness of a sound hides a wealth of significance to be found within the tight network of transformations generated in the musical development, which constantly points to one or more invariances. Every musical parameter is therefore connected to the others: the frequencies to the dynamics, to the timbre, the tempo, to the attacks and the silence. Every linear choice happens within the context of a non-linear frame of reference; the net that holds together the linear development of sounds. This is not to say that every improvisation is built upon invariances. Quite the contrary, our musical heritage is deeply permeated by a linear approach in all fields of research: from Composition to Improvisation, from Pedagogy to Philosophy and Aesthetics of music. However, I believe that emphasizing non-linearity and the importance of the musical invariances would give us an opportunity to rethink our musical horizon, contributing in the enrichment of our experience and knowledge of music and, primarily, musical improvisation. Bibliography Bertinetto, A., 2012a: Il pensiero dei suoni, Bruno Mondadori, Milano. Bertinetto, A., 2012b: Performing the unexpected Improvisation and Artistic Creativity, “Daimon”, Revista Internacional de Filosofía, 57, pp. 117-135. Chew, G., “Cluster”, 1995: The New Grove Dictionary of Music and Musicians, a cura di S. Sadie, Macmillan, London, Vol 4, p. 504, Canonne, C., 2013: Focal Points in Collective Free Improvisation, “Perspectives of New Music”, 51/1, pp. 40-55. Canonne, C., Garnier, N., 2011: A Model for Collective Free Improvisation, in Proceedings of the Third international conference on Mathematics and computation in music, Springer Verlag, Berlin, pp. 29-41. Cosottini, M., Pisani, A., 2012: Nonlinearità e Segno Grafico, “Musica Domani”, 164-165, Edizione EDT, pp. 22-26. Globokar V., 1986: Individuum – Collectivum, Quaderni di Bequadro, Edizioni Unicopli, Milano. Kramer, J., 1988: The Time of Music, Schirmer Books, New York. Maconie, R., 1989: Stockhausen on music, Lectures and Interviews, a cura di Robin Maconie, Marion Boyars Publishers, New York. Nozick, R., 2001: Invariances: the Structure of the Objective World; Harvard University Press Cambridge (MA ), 2001. Trad. it. Invarianze. La struttura del mondo oggettivo, Fazi Editore, Roma, 2003. Pareyson, L., 2010: Estetica. Teoria della formatività, Bompiani, Milano (prima ed. 1954). Schelling, T., 1960: The Strategy of Conflict, Cambridge: Harward University Press. Stockhausen, K., 1963: “Momentform: Neue Beziehungen zwischen Aufführungsdauer, Werkdauer und Moment”, Texte zur Musik, vol. 1, pp. 189-210, DuMont Schauberg, Cologne. Sugden, R., 2003: The Logic of Team Reasoning, “Philosophical Exploration”, 6, pp. 165-181.