The Subtle Mathematics of Music
The Subtle Mathematics of Music
Abstract and Keywords
What distinguishes music from other sounds is symmetry. Only in music are sounds related to one another by precise, quasi-mathematical ratios, and even there only in melodic pitch and rhythmic duration. The mind groups these ratios into subjective patterns (as opposed to objective shapes). These are hierarchically grouped, often according to the principle of self-similarity, i.e., the similarity of the whole to the part, or of the larger part to the smaller part. Over time, these patterns grow more and more complex in an almost organic way, but there is always a tendency to revert to their simple original forms.
Life is not an illogicality; yet it is a trap for logicians. It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait
G. K. Chesterton1
At an open‐air service on the island of Skye, Ralph Vaughan Williams once heard a sermon in Gaelic, a language he did not understand. As he later explained, this ignorance enabled him to devote all his attention to the tones of the preacher's voice: ‘The fact that he was out of doors forced him to speak loud, and that, coupled with the emotional excitement which inspired his words, caused him gradually to leave off speaking and actually, unconsciously of course, to sing.’2 At first the preacher was content with a monotone, but as his excitement grew he gradually developed certain clearly defined melodic formulas, which Vaughan Williams quotes in musical notation as e‐a–b‐a, a–b–c+-b‐a, and a–b‐a–g‐a.3
He was probably mistaken in assuming that this gradual heightening of speech into song was unconscious, but perfectly justified in drawing a connection between musical pattern and emotional emphasis. Not only preachers and orators, but all human beings incline towards song when excited. Similar forms of oratory might have been observed in Africa, in the southern United States, and in many other parts of the world. What is much more remarkable, these speech tunes bear a strong family likeness to one another, regardless of language or culture. Again and again we find the same limited set of rhythms and intervals, expressible in the same simple mathematical ratios.
This proves the closeness of speech to song and the impossibility of drawing a clear distinction between the two. It proves, too, the fundamental importance (p.8) of certain melodic and rhythmic patterns, and the mathematical nature of these patterns. Finally—and surprisingly—it proves that music is emotional not in spite of but because of, its mathematical nature. It is a language for communicating emotion, largely, though far from entirely, by means of mathematics. As Leibniz observed, it is a form of subconscious computation.4 And if, as most linguists now believe, verbal language has a ‘universal grammar’ instinctively understood by every infant, the same must surely be true of music. There is, however, one important respect in which the language of music differs from that of words: its ‘vocabulary’ is self‐explanatory. No one ever had to look up the meaning of a cadence in a dictionary.
THE EVOLUTION OF SYMMETRY
How did human beings develop the capacity to communicate in this astonishing way? Unfortunately, our remote ancestors are beyond direct observation. But by putting together what we know of rudimentary music or near‐music, as performed by human infants, primitive peoples, monkeys, and apes, we can get a fair idea of what happened. Here an important study is The Dancing Chimpanzee by Leonard Williams, who, as both a musician and an expert on animal behaviour, is in a good position to evaluate the musical abilities of our ape‐like ancestors.
His verdict is unequivocal. Chimpanzees do not dance. Nor do they, or other apes or monkeys, sing, or drum in any musical sense. On a purely emotional level, such behaviour may be the ancestor of true music or dance, but it lacks an essential intellectual component. What is absent is any feeling for what scientists call ‘symmetry’: that property, ‘by virtue of which something is effectively unchanged by a particular operation’.5 In the everyday sense of the word, that operation takes place around a central axis. Thus, a building is ‘symmetrical’ if we can mentally draw a line down the middle, take one of the halves, reverse it, and fit it precisely over the other half. But in the broader scientific sense, this reversal is optional. An ornamental pattern of roses or flying ducks is equally symmetrical, in that any given rose or duck will fit perfectly over any of its fellows.
Moreover, symmetry exists in time as well as space. A rock (or any other object) exhibits temporal symmetry if it remains precisely the same on Tuesday as it had been on Monday. The most symmetrical objects and processes are also the simplest: a straight line or circle; a plane or sphere; an evenly sustained note; a regular (p.9) pulse. In every case, we can mentally detach any part of the whole, put it down anywhere else, and get a perfect fit.6
In this special sense, a thing must be symmetrical to exhibit exact proportions. This is what enables us to say that two sides of a square are the same length, for instance, or that one note goes on for twice as long as another. Where the sense of symmetry is lacking—as, for instance, in comparing the roughness of different fabrics—we can make no such precise comparison. And here we may notice how rare this ability is. Symmetry can be perceived by sight, by hearing, and by the sense of time—and that is all. At least in human beings, it is inaccessible to the more ancient senses of touch, taste, and smell. Even our eyes and ears can detect it only in certain circumstances: in length, breadth, and depth, but not in area or colour; in pitch, but not in volume or timbre. In the entire sensory world, it occurs in only three properties: spatial dimension, melodic pitch, and rhythm. The first of these belongs to the eye, the second to the ear, and the third again mainly to the ear. Other senses may perceive rhythm—in the silent beating of a bird's wings, in a regular stroking of the skin—but never with the same mathematical precision.
Moreover, the symmetry of space is much simpler than the symmetry of sound. Any ‘subconscious computations’ we make while contemplating classical architecture, or abstract decorative design, are vague and rudimentary indeed compared with those of music, the symmetrical art par excellence. We may guess that it was the gradually developing sense of symmetry that transformed the unmelodic cries and arrhythmic movements of our remote ancestors into music and dance. Most likely the perception of symmetry in all the senses arrived as part of a single evolutionary package, together with the ability to reason. At all events, the gradual transition from the non‐musical to the musical did not deprive these instinctive actions of their emotional content, but rather gave to that content a partially mathematical expression. A striking example, cited by Williams, is the development of cries similar to those of the woolly monkey into the ‘tumbling strains’ of certain primitive peoples.7 The descending contour, wide range (up to two octaves, in the case of the monkey), and background of strong emotion are all the same. But, even at its wildest, the human cry has something that its simian equivalent lacks: it is symmetrical, and therefore musical.
And, if the emotion is expressed through mathematics, it is equally true that the mathematics is controlled by emotion. The simplest patterns of all, those of musical rhythm, are interpreted against the background of the beat, that regular, moderate pulse that first began to be impressed on the human brain when our remote ancestors learnt to walk on their hind legs. (At least, that is the most likely explanation. (p.10) The fashionable theory that it derives principally from the heartbeat is much less plausible.) This beat may be speeded up or slowed down; it may be subdivided, multiplied, syncopated. But through all these changes it remains a norm. We carry in our heads both a ‘tempo giusto’ (estimated by Curt Sachs at between 76 and 80 pulses a minute)8 and what might be called a ‘ritmo giusto’ of regularly alternating strong and weak beats. All else is felt to be a transformation of this.
In the same way, the pitch ratios of melody are perceived with a uniquely human bias. Just as our bipedal gait is the source of all rhythm, so the arbiter of all melodic propriety is the human voice. Like the woolly monkey, we find it more natural to fall than rise in singing. We also prefer fairly narrow intervals, ranging from the major second to the perfect fourth. Intervals outside this range, while quite possible, are felt to be just a trifle abnormal, so that the octave and fifth are in some ways less fundamental to human melody than the acoustically more complex major or minor third.
Harmony, too, has its norms, derived from the bottom five notes of the harmonic series, which together make up the major triad. Chords within this range are felt to be euphonious; those outside (notwithstanding a barrage of propaganda to the contrary), increasingly discordant.
SHAPES, PATTERNS, AND FIGURES
The ratios of music, though mathematical, are also subjective. Only intermittently do they exist in nature. But that does not bother us in the least. We are content to apply, to the approximations actually heard a set of ideal ratios carried in our heads. Rhythms in the ratio of 2.017 : 1.103 : 0.972, let us say, are construed as being ‘really’ 𝅗𝅥 𝅘𝅥 𝅘𝅥 vibrations in the ratio of 2 : 3.14 as ‘really’ a perfect fifth.
What are we to call such an ideal pattern? It has something in common with Kant's ‘schema’, intimidatingly defined by the Oxford English Dictionary as ‘any one of certain forms or rules of the “productive imagination” through which the understanding is able to apply its “categories” to the manifold of sense‐perception in the process of realizing knowledge or experience’.9 But there is an important difference. Kant believed that true reality—‘the thing in itself’—is unknowable; all we have to go on is the schema. In music, we are aware of both ‘the sound in itself’ and the schema. In fact, much of our enjoyment derives from comparing the two—in noticing, for instance, how the performer's subtle rubato diverges from metronomic regularity. In Origins, I used the term ‘matrix’ for the musical schema, (p.11) not without misgivings that have since grown steadily worse. For the present volume, I decided on the simple term ‘pattern’.
Such patterns reside in the head of the listener. They should be distinguished both from what I call ‘figures’—what we actually hear—and also from mere ‘shapes’, which lack their intellectual and emotional significance. Two notes a perfect fifth apart, for instance, make up a shape; but once we begin to think of the lower as a tonic and the upper as a dominant (though probably without being aware of those terms), they turn into a pattern. No clear line separates shapes from patterns. In the course of musical evolution, shapes are constantly combining into patterns, and patterns dissolving into shapes.
Nevertheless, the distinction between the two is crucial, and its neglect leads to some dangerous fallacies. The publisher Ernst Roth tells us how, among Schoenberg's early followers, the theorist Erwin Stein held the position of ‘official panegyrist and publicist’. ‘His 1924 article on “New Formal Principles” had among the initiates almost the same standing as Albert Einstein's first few pages about relativity in physics.’10 Here is an extract from that article, on the vast possibilities of atonal harmony:
Whereas the old harmony teaching knew only a few dozen chords which, transposed on to the various degrees of the scale, amount to a mere few hundred, every possible combination of the twelve tones is possible. Hence we dispose of 55 different three‐part chords, 165 four‐part, 330 five‐part, 462 six- and seven‐part, 330 eight‐part, 165 nine‐part, 55 tenpart, 11 eleven‐part and 1 twelve‐part—altogether over 2,000 chords, and 4,000 when transposed on to other degrees of the semitonic scale.11
Alas! The few dozen old chords are patterns, but the thousands of new ones are only shapes.
To observe pattern recognition at its most basic, one must reduce music to the regular, unaccented, toneless pulse of a ticking clock—supposing, that is, one manages to find a clock that still ticks. As Percy Scholes pointed out long ago:
It appears that the human ear demands of music the perceptible presence of a unit of time—the feeling of a metronome audibly or inaudibly ticking in the background… And, the ticks being felt, it is a further necessity that they shall be grouped into twos or threes. Indeed, the mind cannot accept regularly recurring sounds without supplying them with (p.12) some grouping, if they have not already got it: in listening to a clock ticking the mind hears either tick-tack or tick-tack‐tack; this is so definite that it is hard to believe that the ticking is really quite accentless, yet that this e ffect is purely subjective is seen in the fact that a very slight conscious effort turns the effect from the one grouping to the other and then back again.12
This simple observation brings out several important points. The first is that, simple as this bare ticking may seem, its musical potential is far greater than that of a sustained note. Such a note would be purely symmetrical, in the sense explained above. The alternation between tick and intervening silence introduces an element of asymmetry into the scheme that makes it much more interesting. As at every other level of musical complexity, interest and beauty are generated, not by symmetry or asymmetry alone, but by a conflict between the two.
On this simple sound‐shape, as Scholes points out, the ear superimposes a purely subjective asymmetry of its own: ‘Hick’ (silence) ‘tack’ (silence) ‘Hick’ (silence) ‘tack’ (silence) … Notice that the accented tick comes first. One could switch the order round to ‘tick tack tick tack …’, but there would be something forced in this. Once again, the ‘Hick tack’ pattern is rooted in human psychology rather than mathematical abstraction. The accent comes first because effort naturally precedes relaxation. It is for essentially the same reason that the cries of the woolly monkey (and the tunes of primitive peoples) proceed from high to low.
This still leaves the question of why we feel impelled to impose the ‘tick tack’ pattern at all. The answer lies in the instinctive human need for perceptual hierarchy. Like symmetry, this is confined to the senses of sight and hearing, but with a difference. Broadly speaking, visual hierarchies already exist in nature—in trees, clouds, rivers, rocks, and a thousand other natural features—and are merely discerned by the eye. Aural hierarchies are read into sounds by the ear.
So, as we read those accents into the even pulses of the clock, we are arranging them in a two‐level hierarchy, with the accented ‘Hicks’ on top (as it were) and the unaccented ‘tacks’ at the bottom. This separation into closely connected ‘levels’ is what distinguishes the true hierarchy from the simple combination, such as is perceptible to the other senses. On entering a church, we may be conscious of the odours of stone, wood, incense, leather, and paper, but this ecclesiastical medley is not a hierarchy. The scent of wood, for instance, may be fainter than that of stone, but lacks the essential hierarchic property of being part of it. The test is always: what can be removed from what? The part can be removed from the whole, but the whole cannot be removed from the part. You can prune a twig from a branch, but not a branch from a twig. This proves both that they are hierarchically related, and that the branch is on a higher level than the twig.
(p.13) Applying the same test to the ticking of the clock, we find that we can mentally suppress the unaccented ‘tacks’, but not the accented ‘ticks’. The former is of course a fundamental rhythmic procedure, the equivalent of replacing ¦ 𝅘𝅥𝅘𝅥 ¦ with ¦ 𝅗𝅥 ¦. The attempt to do the opposite—to suppress the upper level of the rhythmic hierarchy—has interesting consequences, to which we shall return. For the moment, let us observe that there is no need to limit ourselves to two levels. If we silence or ignore the ‘tacks’, the remaining ‘ticks’ will again group themselves into twos, and this process can then be repeated, if not ad in finitum, at least to the point where successive pulses are too far apart to be grouped at all, while in the opposite direction pulses can be subdivided to the point where they merge into a drum roll. Experiment with a metronome shows that the grouping instinct operates comfortably at the rate of between 40 and 400 pulses per minute, becoming gradually stronger as the tempo increases. At a rate of about 320 pulses per minute, it is quite possible to hear the four hierarchic levels depicted in Fig. 1.1.
This rhythmic hierarchy has an important and interesting property: it is ‘selfsimilar’. Indeed, a simpler case of self‐similarity could hardly be conceived. Selfsimilarity is usually defined as the similarity of the part to whole, but, as we have just seen, there is not always a clear‐cut ‘whole’. Hierarchic systems may be open at the top, at the bottom, or (as here) at both ends. A more satisfactory definition might therefore be ‘similarity of form or function across hierarchic level’. In this case, every level has exactly the same form—a series of evenly spaced pulses—but on a different scale.
As a feature of hierarchic systems, self‐similarity is perceptible only to the senses of hearing and sight. In this it resembles symmetry, though it is important to understand that symmetry is not essential to it. Countless natural examples of self‐similarity—mountains, coastlines, ferns, leafless trees, the branching networks of the vascular or nervous systems—are not symmetrical, or at least not in this strict sense (the term ‘average symmetry’ is sometimes used). The combination of self‐similarity with symmetry is found predominantly, perhaps even exclusively, in creations of the human mind; and even there its scope is severely limited, since it depends on a delicate combination of simplicity and sophistication.
(p.14) For it is precisely the simplest patterns that are most easily repeated across hierarchic levels. Consider again that ticking clock. As Scholes remarks, it is not necessary to group the ticks in twos. Threes will do just as well, but in that case our ability to pile up self‐similar levels is greatly restricted. In practice two is the limit, as in 9/8 time, with three—a hypothetical 27/8 time—just conceivable as a tour de force.13 More irregular groupings, such as the 5/8 or 7/8 metres that occur quite naturally in many musical cultures, are even more restricted, being confined to a single level. There is no such thing as 25/8 or 49/8 time.
In contrast, the self‐similar grouping of duple pulses can multiply to an astonishing degree. They attained probably their greatest heights in the early eighteenth century, notably in the works of J. S. Bach. The Allemande from the ‘English’ Suite No. 1 in A, for instance, consists, without repeats, of thirty‐two commontime bars of evenly flowing semiquavers, sometimes subdivided into demisemiquavers (or thirty‐second notes, for the benefit of American readers). This makes a staggering total of ten levels, or eleven if one observes the repeats.
MUSIC AS ORGANISM
A page or two back we encountered the question of what would happen if one suppressed the strong ‘ticks’ rather than the weak ‘tacks’. This is not something that can easily be done with an actual clock, so let us imagine, rather, that the ticks are produced by two metronomes acting in alternation: the ‘tick’ and ‘tack’ metronomes, as we may call them. The ‘tick’ metronome is fitted with a volume control, so adjusted that its ‘ticks’ at first match the ‘tacks’ in loudness. Now let us gradually turn down the volume. As it sinks, a new pattern challenges our preconceived one. The ‘tacks’ at first strike us as off‐beat sforzandos, but later, as the ‘ticks’ recede into inaudibility, as main beats. In other words, the ‘tacks’ have turned into ‘ticks’. A hierarchic level has successfully rebelled against its immediate superior.
This may seem a fancifully metaphorical way of putting it. After all, these sounds are not alive. No—but we human listeners are, and we are inveterate personifiers. In placing the accent at the start of the group—‘tick tack’ instead of ‘tack tick’—we are, in effect, turning the pulses into tiny organisms, which must relax after effort. And if we can breathe life into these irreducibly primitive patterns, could we not do the same for more complex ones? Can chords, modes, or keys behave as if alive?
On this question, musical theorists have long been sharply divided. One school (p.15) of thought will have none of this cloudy anthropomorphism, preferring to draw its metaphors from the hard, practical world of engineering: ‘pivot chord’, ‘bridge passage’, ‘harmonic foundation’, ‘melodic superstructure’, ‘rhythmic framework’, and the like. But another school, particularly strong in Germany about a century ago, regards the biological view as almost self‐evident. Thus, we find Schoenberg observing that
And in much the same way, Schenker
Every chord … that is set beside the principal tone has at least as much tendency to lead away from it as to return to it. And if life, if a work of art is to emerge, then we must engage in this movement‐generating conflict. The tonality must be placed in danger of losing its sovereignty; the appetites for independence and the tendencies toward mutiny must be given opportunity to activate themselves …14
continually admonishes us to see tones as creatures. ‘We should learn to assume in them biological urges as they characterize living beings.’ Each tone, he argues, has its own ‘egotism’ and, as the bearer of its generations, strives to exert its will as the tonic, as the strongest scale step, by struggling ‘to gain the upper hand’ [Lebenskräfte reichen] in its relationships with others.15
These notions, so fantastic‐seeming to those used to the engineering model of musical analysis, belong to the venerable German tradition of Naturphilosophie. Beginning in the late eighteenth century as a reaction against the mechanistic universe of Newton and Descartes,16 this was from the first the antithesis of the conventionally scientific, concerned as it was with wholes rather than parts, becoming rather than being, qualities rather than quantities. Where the Newtonian tradition had treated living creatures as machines, the Naturphilosophen were inclined to treat the inanimate world as alive.
In such an attitude the dangers of pantheistic mumbo‐jumbo are all too evident. Nature philosophy has had a bad press, deservedly on the whole. Yet its ideas have never ceased to appeal to a long line of dreamers, misfits, cranks, and occasionally geniuses. Among the works in which something of its spirit lives on are D'Arcy Thompson's On Growth and Form (1917), Smuts's Holism and Evolution (1926), Koestler's The Act of Creation (1964), and Mandelbrot's The Fractal Geometry of Nature (1982). Often under heavy disguise, naturphilosophische ideas persisted through the holism of the 1920s; the Gestalt psychology of the 1930s and 1940s; (p.16) the general systems theory of the 1950s and 1960s; and, most vigorously of all, the chaos theory of the 1970s and 1980s and the complexity theory of the 1990s.17 Subjected to scientific discipline, the intuitions of the nature philosophers have made important (if seldom acknowledged) contributions to many branches of learning, including, as we have seen, musicology. The present study owes much to them, in both their recent and earlier guises.
I must acknowledge a special debt to Koestler. Unusually for a central European Jewish intellectual, he seems not to have been particularly musical, so it is ironic that, of all the arts, music is probably the one that most fully bears out his ideas. And, quite apart from this, his now almost forgotten non‐fiction works are still good reading. Time and again, one comes upon ideas that have since become fashionable. Independently of Thomas Kuhn, he developed a theory of ‘paradigm shifts’;18 and, long before Richard Dawkins, a theory of ‘memes’:
New ideas are thrown up spontaneously like mutations; the vast majority of them are useless crank theories, the equivalent of biological freaks without survival‐value. There is a constant struggle for survival between competing theories in every branch of the history of thought. The process of ‘natural selection’, too, has its equivalent in mental evolution: among the multitude of new concepts which emerge only those survive which are well adapted to the period's intellectual milieu.19
Since the present volume is a study of musical ‘memes’, it is only right that I should quote this passage, written almost two decades before Dawkins coined the term.20
Although Koestler did occasionally mention the Naturphilosophen, he seems to have been unaware (or unwilling to acknowledge) how much he owed to them. In particular, he was preoccupied throughout his life by the nature of hierarchies, and more generally by the relation of parts to wholes. One of his insights was that most hierarchies have no natural top or bottom. In such cases, there is no definite whole, and therefore no clear‐cut relation between the whole and its part. To resolve this difficulty, he invented the word ‘holon’ for a hierarchic component.21 (p.17) Though this term has yet to make it into the Oxford English Dictionary, it has the great merit of forcing us to recognize the ‘Janus‐faced’ properties of such a component. Every holon is both a part and a whole, but the extent to which it is one or the other is not fixed. Wholeness and partness, or what Koestler called the ‘selfassertive’ and ‘integrative’ tendencies (compare the above quotations from Schoenberg and Schenker), are in perpetual conflict.
The best‐known Koestlerian coinage is ‘bisociation’, signifying the process whereby new ‘matrices’ (roughly corresponding to the Kantian ‘schemata’) are produced by the association of old ones. A simple example is the contraption, compounded from the typewriter, electronic calculator, and television set, with which I am writing these words. Many inventions, both material and conceptual, do undoubtedly arise in this manner, the only question being whether ‘multisociation’ would not be a more accurate word. In music, particularly, such processes appear to be ubiquitous. It is not only that, in the old adage, nothing comes from nothing; nothing comes from anything single. The antecedents of musical patterns are always multiple—often bafflingly so.
The Naturphilosophen were struck by the paradoxical way in which orderly processes coexist with the irregularities of nature. Recent theorists have given these processes various names: ‘self‐regulating open hierarchic orders’, ‘dynamical systems’, ‘complex adaptive systems’, and so on. Whatever one calls them, they exhibit a form of change quite unlike that of the Newtonian clockwork: a selfregulating irregularity, a ‘deterministic chaos’, an endless conflict between order and disorder.
What keeps such systems from flying off into total chaos is the ‘attractor’, a state of complete order which is periodically approached but never quite attained. In the somewhat different form of the ‘magnet’, this too, goes back to the early musings of the nature philosophers.22
In music, the most obvious attractors are certain primordial rhythmic, melodic, and harmonic patterns. Whenever music departs too much from them, whether on the scale of the individual movement or of the entire genre, we begin to feel uncomfortable. In either event, it is time for a return to simplicity.
Music is ambiguous for at least two reasons: first, because the patterns or schemata that we bring to it are never totally precise, and often extremely vague; and second, because the process of matching actual sounds against these patterns is instinctive rather than logical. As Koestler says, ‘The so‐called law of contradiction in logic⥭hat a thing is either A or not‐A but cannot be both—is a late acquisition in the growth of individuals and cultures … The unconscious mind, the mind of the child and the primitive, are indifferent to it.’23 So, too, is the musical mind, which feels no obligation to choose between contradictory patterns. Rather than being either X or Y, a figure may be—and very often is—both X and Y. In addition, all ‘holons’ are ambiguous—‘Janus‐faced’—by their very nature. A G major chord in a C major movement, for instance, is simultaneously dominant and tonic. Depending on which face is more prominent, we may say that the ‘key’ is C or G. But this distinction is misleadingly precise. In fact, there is an unbroken range of emphasis between the two.
Ambiguity, far from being an occasional subtlety, is part of the fabric of music. It is what makes music beautiful and musical analysis difficult. Any conceivable system of musical terminology or notation runs the constant risk of making things seem more clear‐cut than they actually are. I make no claim to have escaped this danger, but have at least done my best to warn the reader against it. If words like ‘tendency’ and ‘inclination’ crop up with irritating frequency, they are a mark not of woolly thinking but, on the contrary, of a regard for accuracy.
One further warning. Music has been aptly described as a ‘rum go’. It is a baffling mixture of opposites: of the mathematical and instinctive, the emotional and intellectual, the precise and vague. It would be unreasonable to expect something so odd to be reasonable. In evaluating any musicological theory, it is well to bear in mind the words attributed to the nuclear physicist Wolfgang Pauli, when presented by a colleague with a particularly bright idea: ‘It's crazy—but is it crazy enough?’
(1) ‘The Paradoxes of Christianity’ (third sentence), in Orthodoxy.
(3) In another passage, evidently referring to the same incident, Vaughan Williams gives the notes as being “a, b, a, g, a, with an occasional drop down to e”. See National Music and Other Essays, ch. 2, “Some Tentative Ideas on the Origins of Music”, 17.
(4) His precise words, in a letter to Christoph Goldbach (1712), are: ‘musica est exercitium arithmeticae occultum nescientis se numerare animi’ (‘music is a secret exercise of the mind in which it computes without being aware of doing so’). See Rudolph Haase, ‘Leibniz, Gottfried Wilhelm’, New Grove II, xiv. 500.
(5) OED xvii. 456, article on ‘symmetry’, definition 3.b.(a).3 In another passage, evidently referring to the same incident, Vaughan Williams gives the notes as being ‘a, b, a, g, a, with an occasional drop down to e’ See National Music and Other Essays, ch. 2, ‘Some Tentative Ideas on the Origins of Music’, 17.
(6) For a wide‐ranging and readable discussion of symmetry in nature, see Ian Stewart and Martin Golubitsky, Fearful Symmetry.
(7) Quoted in musical notation in The Dancing Chimpanzee, 77.
(8) See Rhythm and Tempo, 32.
(9) OED xiv. 615, article on ‘schema’, definition 1.a.
(10) The Business of Music, 153.
(11) First published as ‘Neue Formprinzipien’ in the avant‐garde magazine Anbruch. Quoted in this translation by Cecil Gray in Predicaments, 171.
(12) The Oxford Companion to Music, ‘Rhythm: 3. Rhythm as “Grouping”’, 878.
(13) For an example of ‘27/8’ time (written as 9/16) see the last movement of Beethoven's Piano Sonata in C minor, Op. 111, variation 4. Notice that he is careful to build up this complex rhythm incrementally, first dividing the pulses into 3s, then 9s, and finally 27s.
(14) Theory of Harmony, 151.
(15) Eugene Narmour, Beyond Schenkerism, 35. This passage is accompanied by footnote references to Schenker's Harmony, 6, 30, 29, 256, and 84.
(16) The seminal works were Goethe's Versuch die Metamorphose der Pflanzen zu erklÄren (Attempt to Explain the Metamorphosis of Plants, 1790) and Schelling's Ideen zu einer Philosophie der Natur (Ideas on a Philosophy of Nature, 1797). It was the latter who introduced the term Naturphilosophie.
(17) Eric Hobsbawm points out the links between ‘chaos’ theory and nature philosophy in Age of Extremes, 541–2 n. For a brief account of nature philosophy itself, see the same author's The Age of Revolution, 355–7.
(18) The Kuhnian ‘paradigm’, for those not yet acquainted with it, is briefly described in the next chapter.
(19) The Sleepwalkers (1959), ‘Epilogue: 1. The Pitfalls of Mental Evolution’, 525.
(20) According to Edward O. Wilson, writing in 1997, ‘The notion of a culture unit… has been around for over thirty years, and has been dubbed by different authors variously as mnemotype, idea, idene, meme, sociogene, concept, culturgen, and culture type. The one label that has caught on the most, and for which I now vote to be winner, is meme, introduced by Richard Dawkins in his influential work The Selfish Gene in 1976’ (Consilience, 149). If Wilson's chronology is correct, Koestler must have been among the first to arrive at this idea (or meme?).
(21) This word, which probably owes something to Smuts's ‘holism’, was introduced in The Ghost in the Machine (1967): ‘It seems preferable to coin a new term to designate these nodes on the hierarchic tree which behave partly as wholes or wholly as parts, according to the way you look at them. The term I would propose is “holon”, from the Greek holos = whole, with the suffix on which, as in proton or neutron, suggests a particle or part’ (p. 48).
(22) ‘One of Schelling's most powerful poetic images was that of the magnet, a paradigm for the idea of the coincidence of opposite forces in Nature, for which parallels were found in such phenomena as the contraries of acid and alkali in chemistry’ (Roger Cardinall, German Romantics in Context, 82).