Music, nature and structural form

Transcription

1 Music, nature and structural form P. S. Bulson Lymington, Hampshire, UK Abstract The simple harmonic relationships of western music are known to have links with classical architecture, and much has been written over the centuries about the concept of musical proportion in geometrical composition. However, it is possible to extend the link between music and structural form by noting that the well-tempered chromatic scale can be represented in terms of the frequencies of successive notes by a doubling logarithmic spiral. Musical harmonies, sequences and compositions can be given a geometric form and represented by a type of abstract art. These geometric figures resemble man-made structures, and the spirals on which they are based are similar to the shell structures of nature. 1 The structure of music It is well known that attractive, easy on the ear harmonies in conventional western music are linked to simple ratios between the frequency of vibration of notes played or sung simultaneously. Pythagoras showed with vibrating strings that an octave was formed by frequencies in the ratio 2:1; that the dominant or fifth note sounded with the base note of a scale produced a simple harmony with a frequency ratio 3:2; that a fourth and base combination was in the frequency ratio 4:3; and so on. Architects soon realised that the harmonic relationships in music were not unlike the accepted good proportions in the elevations of public buildings, cathedrals, palaces and churches. Much has been written about this over the years and there are many interesting books on the subject in the architectural sections of libraries. The author proposes to move forward from this established position by looking more closely at the chromatic scale of western music and its links with natural and man-made structures. There are twelve intervals within one octave, but these are not formed by subtracting the frequency of the base note from its

2 176 Design and Nature II octave and dividing by twelve. In the most academically exact method of tuning a music instrument the intervals are a true geometric progression, in which the frequency of each successive note on the chromatic scale is 2 1/12 times the frequency of the preceding note, where 2 1/12 is about Thus, if the frequency of the base note of a scale were 256 vibrations per second, the successive notes would have frequencies of 256 x 2 1/12, 256 x 2 2/12, 256 x 2 3/12 and so on. After twelve intervals the frequency of the octave would be 256 x 2 12/12 = 512 vibrations per second. Bach was very keen on the purity of this scale, which is known as the well-tempered scale, and he wrote a number of compositions for his well-tempered clavier, which he deliberately tuned in this geometric progression. Modern methods of piano tuning produce a succession of frequencies that are similar, but not quite the same as the well-tempered scale. The fact that they do not match exactly produces bright and melancholy keys, but that is another subject. The author can find no evidence of the representation of the well-tempered scale by a logarithmic spiral, but this is what is produced when the twelve intervals of the chromatic scale are represented as radial lines with the angular pitch of 30, so that the octave note coincides angularly with the base note. The length of each radial line represents the frequency of the note, the lengths increasing by a constant factor of 2 1/12 to form the logarithmic spiral shown in Fig. 1. The spiral can be extended outwards and inwards to correspond to higher and lower octaves. 30 o Figure 1(a): One octave of the chromatic scale plotted with frequency ratios forming a doubling logarithmic spiral.

3 Design and Nature II 177 Figure 1(b): The doubling spirals of four octaves. Eighty-eight separate lines, all increasing in the ratio 2 1/12, form the 88 notes on a normal piano keyboard, and since the ratio of successive frequencies at any radial line is 2:1, music is a doubling logarithmic spiral, doubling in radial amplitude with every revolution. This explains the attractiveness of music to the human senses. It is the interaction of a geometric and arithmetic progression. Before we explore this in more detail we should review what is known about the properties of spirals. 2 Spirals There was a great interest in spirals in the nineteenth and early twentieth centuries, when mathematicians, philosophers and scientists became fascinated by the properties of these intriguing figures. There was a feeling that spirals were a fundamental aspect of life and the universe, and of the hereafter. Links were formed between the mathematics of spirals and the method of growth of shell, tusk, horn and claw. Each successive increment of growth is similar, and similarly magnified, and similarly situated to its predecessor. Spirals have links with the formation of leaves, petals and seed-heads in plants. The mathematical properties were thought to be so fundamental to life and death that James Bernoulli, following Archimedes, had the logarithmic spiral inscribed on his tomb. Readers wishing to explore spirals in detail are recommended to study three books. The first two are by Sir Theodore Cook: Spirals in nature and art (1903) and the Curves of Life (1914). The third comprises the famous two volumes by d Arcy Thompson entitled On Growth and Form (1917). We will limit ourselves for the moment to the shells of molluscs, in which growth is composed of parts successively and permanently laid down. The material is added to the extremity

4 178 Design and Nature II of the shell, as shown in Fig. 2, where successive periods of growth of a Nautilus shell are shown to be separated by the similar contours of the septa. The shape of shells is governed in side elevation by the ratio of the breadth of successive whorls 360 apart, and in the Nautilus pompilius this ratio is almost exactly three, as indicated in the Figure. However, the prehistoric ammonite shells, particularly Ammonites tornatus, illustrated in Fig. 3, doubled their radial dimensions in one revolution. Figure 2: The trebling spiral of the Nautilus shell. Figure 3: The doubling spirals of Ammonite fossils. The septa or suture lines, where the partitions of the soft body of the animal joined the shell, are clearly seen. Thus, geometrically, the natural spiral of music is exactly reproduced in nature by the ammonites. However, the natural intervals are more frequent in the shells, about 24 periods of growth forming one revolution, as opposed to 12 in music. The shells, therefore, correspond to a chromatic scale consisting of 24 quarter-tones rather than 12 half-tones.

5 Design and Nature II 179 Figure 4: Arpeggios on the Figure 5: The whole tone scale of diminished chords. Debussy. Figure 6: Arpeggio based on the Figure 7: Beethoven s Moonlight Dominant Seventh. Sonata. 3 The geometry of music Returning to the musical spiral, it is possible to represent chords and arpeggios by joining the ends of the radial lines representing the frequencies of the notes that form these musical figures. The patterns formed by this process make geometric figures based on the properties of spirals. They make a form of abstract art, examples of which are reproduced from the works of the author in Figs 4, 5 and 6. These examples include the geometrical representation of augmented, diminished, major and minor chords. The construction lines are all drawn on a spiral base, where the radial dimensions of the spiral are denoted by the sinusoidal form of vibrating strings.

6 180 Design and Nature II It is also possible to represent musical composition in the same way, by connecting the extremities of the radial lines in the order of the notes of the melody. This produces abstract patterns of lines of varying length and position and creates a geometric signature of the melodic theme. Examples are shown in Figs 7 and 8 where the geometric interpretation of Beethoven s Moonlight Sonata and Tschaikowsky s Andante Cantabile are recorded. These abstracts begin to remind us of modern framed structures where struts and ties are freed from the rigid geometric boundaries of circle, rectangle and triangle. If it is possible to convert musical composition into structural forms, is it also possible to convert the geometry of structures into musical composition? Will beautiful, economically designed structures convert to beautiful arpeggios or lyrical melodies? Will poorly designed structures produce discords or ugly, illconditioned melody lines? Figure 8: Tschaikowsky s Andante Cantabile. Figure 9: Simple symmetric framework.

7 Design and Nature II The music of structures One of the simplest framed structures has the symmetric triangular geometry shown in Fig. 9. It is relatively cheap to construct and was widely used in domestic buildings in earlier times. Suppose the span has a dimension of 12 units, so that the truss consists of 3 lengths of 6 units, radiating from the centre of the lower chord, and two lengths of 6 x 2 1/2 units. If the geometry is compared with music we find it rather dull. The repeating of the same base note three times followed by the tritone has a rather flat sound. However, if a small change is made to the geometry an interesting musical world emerges, as shown in Fig. 10. Here the lower chord is split at 0 into units of 5 and 7, and the vertical from 0 is left at 6 units. These proportions are very similar to the frequency ratios of the 3 rd and 6 th notes of the chromatic scale, which we saw earlier were 2 1/4 and 2 1/2. Using 5 as the base note, the music divisions would be 5.2 1/4 and 5.2 1/2 respectively, i.e and So the 5, 6, 7 framework can be represented musically by the diminished chord, which in the key of C would be C, E b, F #. This attractive harmony would be reinforced by the addition of the sixth, in this case the note A. Its length, in terms of our simple frame, would be 8.4 units, which is very close to the diagonal measurement of the first truss (8.46 units), and also happens to be the average of the two diagonals in Fig. 10. So, by averaging the diagonals we can play the framework as the diminished chord C, E b, F #, A. What an interesting prospect if the sight of a roof truss brought pleasant music to our ears. Figure 10: Framework based on diminished chord. Modern suspension bridges, such as the Severn Bridge, are thought of as simple, elegant structures, and at first sight they seem to lend themselves to a musical interpretation, particularly as the regularly spaced hangers give the arithmetic progression of equal intervals, whereas the cables are clearly increasing in their height above the deck at an increasing rate. Under the load of the stiffening girder and decking, and with regularly spaced hangers, the cables change from their unloaded exponential form to something approaching a parabolic curve. In the former the rate of increase of hanger length is

8 182 Design and Nature II proportional to the length of the hanger; in the latter the rate of increase in hanger length is proportional to the horizontal distance of the hanger from the origin. So neither of these curves satisfies the constant ratio, or the compound interest form of music. However, if the height of the top of the towers above the centre of the stiffening girder is taken to include a range of several octaves, we can play the cable as an extended arpeggio. This is illustrated in Fig. 11, which shows how a single parabolic curve lies very close to a sequence of three octaves, when every third note is played. Thus the parabolic curve of a suspension bridge cable lies very close to the geometric series formed by an arpeggio of 12 notes based on the diminished chord. This is also illustrated in Fig. 11, where the diminished chords in the key of C are shown. Figure 11: Playing the cable as an extended arpeggio. It is of interest to note that the diminished arpeggio of the Severn Bridge combines melodically with the successive notes of the Sydney Opera House composition, so we have a single musical link between two structures of very different form and shape. How far did designers suspect that their architectural inspiration was being conditioned by the spiral of music in this way when they thought they were free to do what they liked!

9 Design and Nature II Discussion When spirals were investigated by scientists like Cook and d Arcy Thompson in the early years of the twentieth century there was much talk about the logarithmic spiral as the fundamental conception of the mathematical expression of Nature. The curves of life were thought to be linked to the Fibonacci series and the approach of the ratio of successive pairs of numbers in the series to the Golden Number or Ratio of Pheidas of The composition and spacing of the great classic paintings were thought to combine art and science via this proportion ½(5 1/2-1), which was designated ǿ. We noted earlier that famous and brilliant men have asked for the logarithmic spiral to be engraved on their headstones. We have been concerned in this study with doubling spirals in which the ratio of successive radii at equal angular pitch remains constant, and in one complete revolution the radii double in length. The spiral shape does not bear an immediate relationship to the curves of life based on the ratio ǿ, but does seem to forge a fundamental link between music and some structures of nature and man. There is no indication, however, that this is more than a fortuitous coincidence, and it would be unwise to read more into this work than that. The comparisons are surprising and interesting but are not likely to be part of some hidden natural law that has yet to be discovered.