Discussion of complexity and beauty in music are complicated by three factors. The first is the relationship between musical and physical reality: the second is related to the role of musical notation: the third to the gap between a composer's intentions and what the listener perceives.
The particular set of pitches selected in Western Art Music are very special. The octave (a frequency ratio of 1:2) is divided into 12 exactly equal parts (a ratio of the twelfth-root-of-2 to 1 between successive steps of the scale). More typically (looking across time and across cultures) scales have been made up from pitches in simple numerical ratios to the starting pitch (e.g. 3:2 for the interval known as the `5th' 4:3 for the interval known as the `3rd'). The western (or `tempered') scale contains none of these other `pure' intervals, but does contain pitches very close to those pure intervals. The rationalisation of intervals was part of the evolution of musical language in western Europe, but it is not `natural' or inevitable in any sense. Within the context of the western system it is possible to define a sense of `key' or `tonic' pitch (essentially, which pitch, at any time in a piece, is the reference pitch) and to classify simultaneous-note-groupings (chords) according to their `function' within this key. So long as we understand that this analysis is context-dependent (the context being the western Art Music system) we can argue about the relative sophistication or complexity of particular musical works.
Arguing for the `naturalness' or `inevitability' of musical structures is a different matter. Looking across musical cultures we observe certain recurring patterns. The interval of an octave (2:1) is almost universally regarded as concordant. In western Art Music, pitches separated by an octave are regarded as the same pitch. But the next simplest ratio (3:2) producing the musical `5th' is not present precisely in western Art Music. Being bathed in the sound-world of this music, western musicians tend to accept the approximation in the tempered scale as a true 5th, and it functions within the tonal system as if it were (it is used as if it were the next closest pitch to the `tonic' , in the sense of intervallic tension, or in terms of key relationships). But to someone familiar with `just intonation' or with Indian classical music (which depends upon much finer pitch discriminations), the Western 5th will appear out of tune. The `naturalness' of most musical experience is thus not simply rooted in the physical properties of sounds, but in a set of culturally constructed relationships growing out of elementary physical properties, but bending the natural `rules' wherever the internal development of the language demands it. Most people live within such a rule system, and regard their experience as `natural' in the same sense that they regard their native tongue as `natural'.
Unfortunately music theory (along with much other aesthetic theory) is full of metaphysical speculation attempting to justify the `naturalness' of any particular cultural practice. This is particularly true in relation to time-relationships in music, and towards the end of this paper I will touch again on this issue.
In western Art Music a sophisticated system of analytic notation has been developed. This (in a certain sense) enables music to be written down by composers in a musical score and reproduced by performers who can read the notation. Around this has grown up an extensive institution of academic analysis of musical scores. It seems possible to demonstrate the structure, and hence the complexity of a piece of written music by analysing relationships between objects in the score (pitches, time-signatures, time- patterning, chromaticism etc.).
However, the act of notation is not neutral. Musical notation has grown up to capture those aspects of musical practice which appeared most important within the culture. Thus pitch and duration are captured in an articulated way, dynamics (loudness) in an approximate and relative fashion, `tone colour' and `expression' in a way that depends entirely on the existing technology (i.e. we merely name the instrument and one or two modes of sound production e.g. pizzicato, fluttertonguing) or the player's understanding of performance conventions.
Furthermore, musical notation is idealised in the sense that it portrays abstracted properties of sound-events (`pitch', `duration', `loudness') rather than the true physical characteristics of the sounds. This serves to reinforce a fundamental cultural bias into our understanding of what is, and what is not, musically important. In particular, all those multifarious aspects of sound which were not so easily classifiable or notatable became lumped under the catchall term `tone colour' or `timbre' and were, until the late twentieth century, almost entirely dependent upon current musical technology (instrument design) and performance practice. And timbre was intellectually demoted amongst the hierarchy of musical parameters. Musical analysts (focusing on the score, and hence the parameters it handled) teased out and defined musical structure, substance or complexity in terms of pitch and durational relationships visible in the score. Other sound properties were viewed as a kind of `colouring in' of this real substance. Composers with an ear for colour (Tchaikowsky, Rimsky-Korsakoff, Debussy, Stravinsky) were either demoted in the musical intellectual canon, or only treated seriously when they could be shown to be adopting sophisticated techniques for the organisation of pitch/time.
This is particularly important in relation to the development of music in the twentieth century when sound-recording, and the subsequent ability to analyse fully the physical characteristics of sounds using computers and such applied mathematical procedures of Linear Predictive Coding or the Fast Fourier Transform, means that we can capture all aspects of sound in the greatest detail, and begin to make rational compositional choices in all of these domains. Composers adopting this approach however find themselves facing a wall of prejudice that regards the sophisticated articulation of non-pitch properties of sounds as somehow intellectually inferior, or even unmusical.
Furthermore musical notation has intrinsic limitations. For example it is possible to vary the pitch of a sound over the entire continuum of values, and in actual musical practice (particularly in vocal music), all kinds of subtle articulations and distunings of pitch occur - various (and varying) depths of vibrato, the flat-tuning of a note followed by an upward slide onto the `exact' pitch (a device heard in much popular ballad singing), the random micro-fluctuations of pitch which make the qualitative difference between recognition of a real voice and recognition of a (poor) voice- synthesizer. A written notation system however demands a finite approach to the notation of pitch. We must confine ourselves to a staff-notation of discrete pitch positions. For similar reasons the continuum of time becomes divided into units which are simple multiples (or divisors) of one another. I refer to this restraining grid as the Pitch-Duration Lattice.
Subtler articulations of pitch may be shown in the notation as `ornaments', usually represented by some kind of visual icon indicating that a pitch is to be ornamentally articulated, but this icon contains no information about how this is to be done. This information is conveyed through the performance practice, the non-notated stream of musical knowledge reproduction.
However, the use of musical notation feeds back upon itself. An ornament for the voice (or a bowed stringed instrument), which can articulate pitch over the entire continuum, is not intrinsically restricted to the pitch lattice. However, once instruments are built to conform to the scales of pitch-notation (keyboard instruments, keyed woodwind instruments even ornaments have to be articulated on the lattice. Once music is conceived, by composers, through the score, the articulation of the notatable becomes sophisticated relative to the unnotatable. More importantly, once an academic discipline of analysis of music through the score evolves, evaluating musical complexity or sophistication through what can appear in a score, the musical world is split into a world of objective structure, with intellectual kudos, and performance practice of lesser intellectual clout. Eventually everything that is considered important in music appears to exist on the lattice. The spaces between the lattice appear mysteriously unknowable, beyond the realm of serious musical discourse.
This leads me to the notions of explicit and implicit complexity in musical experience. Explicit complexity is that complexity of musical design which can be elicited from a score, and is founded, in some sense, in the relationships between musical entities we can see and isolate in the score. However, in actual musical performance, the particular quality of a human voice - based partly on its unique spectral makeup, dependent on the physiology of the person, and partly on an individuals way of articulating the off-lattice aspects of musical experience (vibrato and its variation, spectral variation e.g. head to chest voice, variation of noise coloration or breathiness, articulation of pitch through gliding, vocal roughness, subtle loosening of timing etc. etc.) - may be of supreme importance in our appreciation of the musical experience. Sound recording allows us to preserve these aspects of the musical experience, its implicit complexities, in a way that musical notation fails to do. But the cultural inertia built up through centuries of developing a notion of musical structure through focusing on what is notatable, means that musics which depend upon such implicit complexity have a hard time competing for intellectual respectability with more traditionally oriented musics.Transcending Traditional Notations:
The new world of musical possibilities opened up by the computer analysis of sounds and the understanding it brings with it can be tackled in many ways. It is possible to develop entirely new notation systems such as those in my own works Anticredos, Vox-1 and Vox-2 where I attempt to notate in detail the actual sounds produced by the vocalists. This notation grows out of both traditional musical pitch/duration notation and international phonetic notation, combining this with both graphical symbols and transformation notation (showing how one sound may evolve into another) to give a subtle and complex way to specify vocal sounds and their evolution in time.
The voice is a particularly special instrument to work with. Not only can it produce pitches over the entire frequency continuum (within its range) - it is not lattice-bound. It can also vary spectral quality (for example, the different formant characteristics of vowels) and produce a host of unpitched sounds - sibilant consonants, vocal grit, complex multiphonics through ingressive (in-breath) production - and, most importantly, move from one to the other type of sound in a seamless fashion. In effect a whole timbral continuum can be explored, in sharp contrast to the discrete timbral world of the orchestra, where individual instruments are confined to one or a few spectral/articulation types, another dimension of the lattice of traditional musical thought.
With the computer, and recorded sound, we have an even wider pool of possibilities. Abandoning notation altogether, and developing new kinds of musical instruments that exist only as computer software, we can begin to swim in the continuum of sonic space, rather than assemble structures out of technologically given objects (the sounds of existing musical instruments).
We may imagine a new personality combing the beach of sonic possibilities, not someone who selects, rejects, classifies and measures the acceptable, but a chemist who can take any pebble and, by numerical sorcery, separate its constituents, merge the constituents from two quite different pebbles and, in fact, transform black pebbles into gold pebbles, and vice versa.
Musical composition becomes a discipline that handles any sound whatever, recorded from our sonic environment, and not simply those sounds that have been accepted into the canon of the beautiful.
Whichever approach to composition we adopt, however, the organisation of time will be paramount. The way in which time is divided up, regardless of the sonic materials used, is a fundamental aspect of musical organisation. It is also very particular to music. Although we are aware of things like comic or dramatic timing in the theatre, there is still a huge leeway in the way this can be interpreted. With a novel, we move even further away from `real time' as we can read at any pace, put the book down and come back to it later. With music, however, this is unthinkable. The structuring of time itself is intrinsic to the musical experience.
I want to discuss the musical organisation of time from the point of view of the listener, rather than in terms of the aims, or notations, of the composer or performer. In what follows I will use the term `duration' to mean the separation in time of the onsets of musical events.
First of all, it is clear that our perception of musical time depends on the time-scale involved. The most familiar type of temporal perception in music is the experience of rhythm, derived from the (semi-)regular ordering of musical pulses separated by times in the (approximate) range 1/20th of a second to a few seconds, which I will call the rhythmic time-frame. Rhythmic perception would seem to have some physiological or preconscious neurological basis. It is not based on conscious counting (counting is, anyway, a regularly sequenced, not a regularly timed, phenomenon), and seems to lie within the range of some of our repeated bodily functions, like walking, the beating of the heart, down to the scanning rate of the eye.
Once musical events are separated by less than (c.) 1/20th second we may still be aware of whether the events are placed regularly or irregularly in time. But we experience this as a regular or irregular granularity of the sound, rather than as rhythm as such. This is the grain time-frame and this type of perception persists down to a few milliseconds. In time-frames shorter than this however, the articulation of time contributes to our holistic or qualitative experience of the musical event. For example, the quality of a piano tone (in contrast to a flute tone) is (partly) dependent on the brevity of the loud attack with which the sound begins, and the attack time lies below the threshold of direct temporal perception but is accessible to us through out qualitative perception of the sound.
The timing of events in the rhythmic domain can be represented in a score. Setting a tempo (the rate of repetition of the basic unit) for, say a 1/4 note, we can represent whole number multiples or integral fractions of this basic unit using standard notational conventions. This allows us to represent the division of time in the rhythmic time-frame in a highly articulate manner.
Moving upwards, rhythmical units (which we see in musical notation as 1/64 notes up to whole notes) are typically grouped in larger time-frames through musical accentuation, often by the device of barring (grouping say three 1/4 notes into a 3:4 bar), and then of phrase-structure (bars typically being grouped in sets of 4) and sectional structure (through which phrases answer one another and are used to build longer musical `arguments') and the regularity of these larger time-frames can be apparent from the regularity of their constituents. This nesting of time-frames is typical of musical organisation from the rhythmical time-frame upwards. However, beyond a certain upper limit, and particularly if the music does not follow an absolutely regular phrasing (or even barring) pattern, it becomes difficult to judge the relative duration of longer stretches of musical time. Here we move over into the realm of formal time-frames.
I would now like to divide temporal perception into three types, the measured, the comparative and the textural. Given that time-frames may be nested within a musical composition we can ask whether our perception is measured, comparative or textural in any particular time-frame.
Measured perception requires an established (or only slowly changing) reference- frame against which we can `measure' where we are. In the domain of pitch, this might be a tuning system, or a mode or scale or, in Western tonal music, a subset of the (chromatic) scale defining a particular key (e.g. C# minor). Such a reference-frame might be established by cultural norms (e.g. tempered tuning) or established within the context of a piece (e.g. the initial statement of the rag in Indian classical music, or the establishment of tempo and metrical grouping e.g. 4:4 at 120 1/4-notes per minute in a particular piece). If we do not have a reference set we are still able to make comparative judgements. In the pitch domain for example "we are now higher than before" or "we are moving downwards" or "we are hovering around a fixed value".
In the sphere of durations, if we establish a time reference-frame of, say, repeated 1/8 notes at a fixed Tempo (e.g. 120 1/4 notes per minute), we can recognise with a fair degree of precision the various multiplies (1/2 note, dotted 1/2 note = 3/4 note etc.) and integer divisions (1/16 note, 1/32 note, triplet 1/16 notes = 1/24 notes) and combinations (dotted 1/4 notes = 3/8 note, dotted 1/16 notes = 3/32 note) of this unit (and its subunits). This recognition is not a matter of conscious counting, but some kind of direct physiological or neurological response to regularity of pulse. And this is what I call measured perception (see diagram 1).
In a musical phrase made up of 1/8 notes but which is undergoing a ritardando (a deceleration of tempo) (see diagram 2) we are aware of the comparative quality of the succeeding durations. We perceive them as getting relatively longer and we also have an overall percept of `slowing down' but we do not normally have a clear percept of the exact proportions appertaining between the successive `quaver' events. Furthermore, no two live performances of this phrase will preserve the same set of measured proportions between the constituents. This is, then, an example of comparative perception.
In the case of dotted rhythms over a 1/4 note pulse (see diagram 3), if there is a clear underlying 1/16- note reference-frame, and the dotted-1/8-note-to-1/16-note pattern is played "precisely" as written (3/16 to 1/16), we will perceive the 3:1:3:1:3:1_ sequence of duration proportions clearly, as our perception is measured against the 1/16-note frame. However, in many cases, this time-pattern is encountered where the main reference frame is the 1/4 note, and the pattern may be more loosely rendered by the performer, veering towards 2:1:2:1_ at the extreme. Here we are perceiving a regular alternation of long and short durations which, however, are not necessarily perceived in some measurable proportion. The musical score may give an illusory rigour to the 3:1 proportion, but I am concerned here with the percept.
The way in which such 1/4 note beats are divided into long-short patterns is one aspect of a sense of `swing' in certain musical styles. A particular drummer, for example, may have an almost completely regular long-short division of the 1/4 note in the proportion 37:17. We may be aware of the regularity of the patterning and appreciate the particular sense of swing it imparts to the music while remaining completely unaware of the exact numerical proportions involved. Here then we have a comparative perception with fundamental qualitative consequences.
This example also illustrates the importance of time-frames in our perception of musical time. For, in this example, perception at the level of the 1/4 note remains measured - the music is `in time'. Yet , simultaneously, in a smaller time-frame, our perception has become comparative. There are, however, limitations to this time- frame switch phenomenon. If the pulse cycle becomes too long (e.g. 30 minutes) we will no longer perceive it as a measurable time reference frame (some musicians will dispute this). More significantly, if the pulse-frame becomes too small we also lose a sense of measurability and hence of measured perception.
This problem of the disappearing pulse-frame becomes particularly important where different divisions of a pulse are superimposed (as in my work VOX 3). If we compose two streams of duration in the proportion 2:3 (see diagram 4, or the music of Brahms!) we have a clear mutual time-frame in a longer time-frame (equivalent to the rate of repetition of the 1/4 notes in the slower of the two original tempi). If we now regroup the elements in each stream in a way which contradicts this larger-time-frame-mutual- pulse (see diagram 5) we may still be able to perceptually integrate the streams in a measured way as we also have a mutual shorter time-frame. The 1/24 note (or triplet 1/16 notes) pulse in the slower tempo is equivalent to the 1/16 note pulse in the faster tempo. (That faster tempo is 378 events per minute, or around 6 per second, already a little difficult to grasp for the less agile musical mind!).
The more irrational (in the mathematical sense) the tempo relationship between the two streams becomes, the shorter this mutual time-frame pulse becomes. And the smaller this unit, the more demanding we must be on performer accuracy if we are to hear this fast time-frame regularity. (With computer-precise performance we may still retain a sense of synchronised regularity way beyond the realities of a live performance). For example, with a tempo ratio of 7 to 11, using 1/8 notes, the faster mutual pulse rate is approximately 20 events per second, already on the boundary of granular perception. If we overlay tempi in the ratio 7 to 11 to 13, the common faster pulse runs at over 1000 events per second, a millisecond pulse, clearly completely impossible to experience in a measured sense, even with computer precision. For perceptual purposes there is in fact no common faster pulse at all. We can relate the different tempi streams only through their division of a common slower pulse. If we now take pains to remove effective clues as to where the common slower pulse may occur (by using irregular bar-lengths in each stream, avoiding the accentuation of time-events common to 2 or more streams etc.) we can in fact completely destroy the sense of measured perception. Perception then becomes textural i.e. we are aware only of a certain density of events, and perhaps of a flux of density (even a gradual increase or decrease, a comparative perception of `texturality'!) .
This raises an important issue about the nature of temporal complexity in music. Such irrationally-related tempi are found in many scores of contemporary music, and it is argued by some analysts that the musical events represented are highly complex (in fact the genre of music using such devices routinely is known as new complexity). However, I would argue that from a perceptual point of view, much of this music is in fact quite simple. The complexity lies only in the notational devices. Not only are the notated structures often impossible to perform with any great accuracy, but they are perceived in a purely textural fashion. It is as if, on reaching a certain point of complexity, our perception collapses down to a more general level. For example, given a colour field of very many small areas of many different colours, once we make the areas sufficiently small what we observe is a textured brown field - the initial complexity becomes unified under a single simple percept.
At the other extreme, in the organisation of (large scale) formal time-frames, it has been argued that certain time-proportions are both more intrinsically satisfactory than others, and that these time-proportions can be teased out of the works of the great composers by careful analysis. In particular, great claims are made for the efficacy of the Golden Ratio as a `natural' and aesthetically superior device for time organisation on the large scale. This claim is complicated by the fact that some composers consciously use such proportions in their written works.
There are some simple aesthetic counter-arguments to this idea in principle. In particular, the claim that any particular set of proportions is somehow to be favoured at the expense of all others is based on the assumption that all Artistic activity has a similar goal, the achievement of a sense of "cosmic harmony", or some other such transcendent aim. It would seem to me that the proportions favoured within works of Art of any kind are more likely to (a) vary according to the culture in which they originate, (b) vary according to the self-conscious goals of the Artist, in western Art in particular. However, there is a strong metaphysical prejudice towards a notion of `Unity' in musical theory , which is strongly linked to a longing for Unity of Purpose or Design in the Cosmos by many artists. My own view is that creativity is intrinsic to the Cosmos (it creates itself as it goes along, and we are part of that process), and creativity depends on arbitrary variation (as we observe as part of the process of natural selection) because we cannot prejudge what design types might prove to be viable or efficacious. This creativity has no knowable goal, nor any unique source or plan, but is of the nature of things. It is therefore counterproductive to the creative process to base one's artistic plans on some presumed Universal master-plan
A more down to earth objection to the Golden Ratio hypothesis is that we simply cannot judge the time-proportions of long stretches of time. On the one hand, unless the music is regular to the point of banality, there is no pulsed measure basis (see above) for us to use to compare long stretches of time. Furthermore our subjective experience of the passage of long stretches of time is highly dependent on how that time is filled. Everyone has had the experience of hearing a piece of music which seems very long but was, in clock time, quite brief (often an ineffective piece of music) and of (clock) time flying by whilst listening to an absorbing piece of music (or reading an absorbing book).
The illusion of precise proportions may be given by a bar or beat count in a score, but this bears only a loose relationship to our actual experience of the passage of time. Unlike the proportions of a building, or a building plan, where we can observe both arms of the ratio at the same moment in a two-dimensional or three-dimensional space, our appreciation of the time-proportions of successive musical events depends heavily on our memory of the earlier event. This throws the whole issue of defining what is and is not a successful time-proportioning in a piece of music into doubt, especially where we are using the (2-dimensional, spatial, atemporal) score as a basis for our judgements.
My view is that in general the 2nd of two musical sections (where it reutilises or reworks material from the 1st) should, as the material is already familiar, be somewhat shorter than the 1st, because the listener will tire more quickly of that material. However, if it is not long enough the material will not have sufficient time to display itself adequately. Therefore, other things being equal, this type of piece will tend to have a large-small structure falling somewhere between 3:2 and 2:1 in its large-scale proportions. This is the reason why the retrograding second half of the first movement of Bartok's Music for Strings, Percussion and Celeste, balances well with the first half, rather than Bartok's conscious choice to divide the piece at something approaching the Golden Section, in bars/beats (notice in particular that the longer section comes first _ there is no reason why this should be so on the basis of any Golden Section argument). And it certainly balances no better than the parts of other works by equally renowned composers who no-one but a fanatic would claim were using the Golden Section proportions.
The next problem with the Golden Section argument is, what exactly counts as a Golden section. The Golden section is a very precise mathematical ratio and, strictly speaking (because it involves irrational quantities) can never actually be realised using any finite system of time-counting (not 1/64 notes, 1/512 notes or even digital samples at 48000 to the second on a computer). No piece of music can then consciously be made with Golden Section time proportions, and the probability of these proportions occurring by chance is effectively zero. Hence all arguments about Golden Section ratios are in fact about approximations to the Golden Section. But how close does an approximation need to be to fulfil the criteria. If our criteria are sufficiently loose, my own rule of thumb (between 1:2 and 2:3), based on purely pragmatic arguments could be declared a Golden Section proportion, but this declaration would seem to add nothing to the rule of thumb except a little Classical kudos. In practice proponents of the Golden Section theory are apt to `equate' finite proportions between successive members of the Fibonacci series as Golden-Section-ish. It is true that, these proportions in the limit (as we go further and further up the series) approach the Golden Section ratio, but on the way they cover a vast number of possible finite ratios, between 2:1 and 3:2. If all of these are to be given the magic status of Goldenness, then everything is Golden and there seems little point in arguing about it.
Although there are some sound geometrical/biological arguments for the appearance of Fibonacci numbers in the contrary spiral patterns of sunflower seeds, and perhaps in the proportions of growth of certain shells, there are all sorts of other proportions occurring in natural processes Why not for example use the ratio of the mass of the electron to the proton, or the ratio between the numbers of electrons in different atomic orbitals _ all equally, if not more, fundamental to the physical world _ as the basis for organising time proportions. The whole mystification of ratios seem to be a historic throwback to the original Pythagorean brotherhood's mystical view of number.
My hope is that, in the long term, we will come to judge the complexity or effectiveness of musical structures on perceptual criteria teased out by psycho- acoustic research into human perception, rather than on mystical a priori conceptions feeding back into the cultural milieu and into musical practice to become self- fulfilling prophecies.
Trevor Wishart: 1996
The Japanese Shakuhachi is a flute with finger-holes. But there are few of these, and an essential part of Shakuhachi technique relates to the articulation of the space between those pitches obtained simply by closing the various finger-holes.
pp 11-12: Audible Design: Trevor Wishart: published by Orpheus the Pantomime: 1994
There are forms of notated music in which time relations are not specified. However, in listening to the music we experience whatever time-relationships the performer/director chooses to impose on the interpretation. For me, music is the experience of music, not the concept of the music as described in a score. The score is merely a set of instructions for generating a musical experience.
Clearly musical events have there own durations. One event may begin and persist after the initiation of a second event. However, in discussions of rhythm we are concerned with the time between the onsets of individual events.
I'm assuming here that the events are not regularly grouped (e.g. by musical accents on every 4th attack) into larger time-frame units - in which case our perception of rhythmicity may persist down to even shorter time-frames.
I have demonstrated elsewhere (Audible Design) that an error of a few milliseconds - an almost inevitable error in real performance - can even alter the order of occurrence of events in such notational constructs.
Even where the music is thus entirely regularly pulsed, barred and phrased, beyond a certain point we lose any sense of intrinsic duration and can rely only on counting the beats.
This could of course be a particular aesthetic aim!
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