Music: Vibration and Key Relationships.
p.38 Vibration
My attention now was turned towards Music, and the relationship between music and number.
Robert Lawlor writes The experience of hearing the octave played contains the mystery of sameness and difference’. He continues ‘ this quality of perceiving sameness and difference is part of the poise of mind that Sacred Geometry means to cultivate, one which is precisely discerning yet harmoniously integrating’.
The observation above seems to be a study in relationships, which is one of the themes running throughout this book as it unfolds. The emphasis of being able to concentrate on the precise detail of something at the same time as being aware of how it resonates with others and with the whole.
The octave arises between just two notes, and it is interesting to observe (at the top of the page) that the wave length of the higher note of the octave is exactly double that of the lower. And the measurement of the string or pipe is reciprocal to the to the frequency of vibration. i.e. half the string length gives double the frequency of the original tone: 1/2 x 2/1=1.
The long drawing on the left-hand side of the page illustrates more clearly what this means, and includes other important notes in the octave. If the long line represents the open string with a frequency value of 6 for sake of convenience, then the relationships of string length to frequency can be seen. This observation is carried out further in the small chart at the top, and illustrated in the 3 circles in the centre.
The Harmonic Series. Simultaneous to the sounding of the fundamental note, a sympathetic sounding of the vessel (string or pipe) creates resonances - the overtones - climbing up the scale in a series of decreasing proportions: 1:1 fundamental note 2:1 octave 3:2 perfect fifth 4:3 perfect fourth 5:4 major third.
The last little chart at the bottom of the page. (I had not built this into the rough draft!) The material was discovered when I was half through the finished page, and I was not sure if I could fit it in. This problem quite often happened. In doing my research in various books, I would come across other information that I would like to include.
Key Relationships: The Harmonic Series
p. 39
Continuing where p.38 finishes off, this page expands the field of harmonic relationships. Overtones or upper partials establishes the tonal spectrum of a particular key. For instance, the note C in isolation generates resonances going up the scale that imply C major. As the harmonic overtones resonate upwards many of the notes repeat themselves and confirm the particular key.
The foundation of harmony in music is derived from sound itself. In the diagram at the top right of the page records the resonances of the fundamental note, in this case C. Though the notes marked with a cross indicate that are not in tune with the tempered scale, but with the mathematical laws.
The Key Wheel.
The diagram of the key wheel is placed in the middle of the page. Starting at the top of the wheel is C major with no sharps or flats notes. Travelling clockwise, G is the next key with one sharp, D with two sharps - and so on to half way round the circle. Starting again from the top going anticlockwise, a similar progression is seen starting with F minor , and progressing from one to two and three flats etc.
Smooth transitions occur between closely related keys, thus a key change between C to G major involves only F sharp. Key changes across the board give a much more dramatic and complex resonance, and composers are very alive to the importance of key in the music they write.
Transition and Modulation.
As with some of the information on the previous page, I came across the sequence later shown on the bottom of the page. It postulates that If C was taken to be 1.000 cycles per second and the octave above was resonating at 2.000 per second, then F sharp (in the middle of the sequence) would be 1.4141. This is so very nearly √2 which is 1.1412….. ( In reality Middle C is 262 Hz - cycles per second).
I remembered the diagram of the principles of growth as shown in the square on pp. 30/31. In this diagram it describes how the diagonal √2 is the mediating principle of transition from one state to another. - from one to two. In music F sharp fulfils the same role in modulating the key of C major to G major: from no sharps to one sharp. Both examples are taken from the beginning of the sequence.
To me this seemed an extraordinary coincidence and a thrilling discovery, and another example of how things seem to be connected across seemingly different disciplines; though to a mathematician it might seem obvious.
Expressions of Reciprocity, Mediation and Alternation
p.40.
By now, having written about 40 pages of the book, I was beginning to sense that many structures of the natural law begin with very simple principles. Some of these principles generate a multiplicity of derivatives (across many different domains) which are then grouped under such headings as Reciprocity, Mediation and Alternation.
Reciprocity. The dictionary reads: a Mutuality, Existence on both sides, principle or practice of give and take. Inverse correspondence. Function or expression so related to each other that their product is Unity.
In the first column on the left hand side of p.40, this theme is continued. It shows the mathematical equation in music of the relationship of length of instrument and frequency of vibration : 1/2 x 2/1 =1 (Unity). The plucked string divided in half produces double the frequency of vibration. This musical equation expresses the essential laws of harmony. Much of modern physics too are based around the interchange of matter and energy: e.g. e=mc squared. It is extraordinary that something of such great importance can be simple enough to be written on a grain of rice.
Continuing the column downwards, the focus is on Kepler’s laws. Taking Kepler’s 2nd law, The radius vector of a planet sweeps out equal areas in equal times about the focus containing the Sun. Kepler’s 3rd law states that for any planet, the square of its period of revolution (its year’s orbit) equals the cube of its mean distance from the sun.
The second column is concerned with music. I read that the inverse of every harmonic progression is an arithmetic progression: i.e 2.3.4.5.( ascending series) and 1/2 1/3 1/4 1.5 ( descending series) respectively, which gives rise to the little chart at the centre showing how the frequency of vibration is related to the length of the string or pipe. So the measurement of the vibrating body is reciprocal to the vibrational frequency.
The shorter third column mostly contains quotations from the Bible. In all of them there is a relationship between the two halves of the teaching. The Buddhist law of Karma echoes this, where we receive back what we give.
p.41
The theological theme is continued into the next page, column 1. It brings to our attention The Pythagorean doctrine of proportion which underlies the following passage of Jesus’s sayings from St. John’s Gospel. Simone Weil has high-lighted the interactive and reciprocal relationships. For instance,‘ As the Father (A) has loved me(B), so have I (B) loved you (C). This is a three term progression (see explanation at the top centre). Christ here is the Mediator between God and Man. The sequence flows from the God the Father.
The column concludes with the drawing of the double circles, where in the central overlap is the Vesica. Traditionally Christ sits as the mediator between the worlds of the unchanging and the changing, inner and outer, spirit and matter: the place of reconciliation between apparent opposites.
However, perhaps there is another possible interpretation from a different perspective - though not realised here in the above text - that God, man and everything is made of the same stuff; so the two term proportion of the Golden Proportion could be used. Thus A is to B, as B is to A+B. God is to Christ as is Christ is to God + man. So man(C) is in essence A+B, not separate from God and Christ.
Mediation continues into the second column of this page, which is described as ‘Connection, not directly, but through some other person or thing’. Involving intermediate agency. Intervention for purposes of reconciliation. Formation of connecting link, or transitional stage in between’.
The irrational numbers were seen as ‘transcendental’ as they were ultimately unknowable and immeasurable, and were the hidden transforming processes that changed things from one state to another:Pi and ⏀Phi, and √2,√3,and√5…etc.
What is seen in the diagram of the two squares ( shown in the central column) is the working together of the whole numbers and the irrational, transcendental numbers - providing both alternation and mediation in the development of the growth of the square. √2 is the mediating agency that transforms Square 1 into Square 2.
Similarly, F sharp fulfils the same role between the keys of C major and G major, in music. √2 can be seen then, as the hidden power of potential: the Generative Principle. The diagonal is implicit in Square 1, thus Square 2 is the flowering of that which is implicit. Also the mid-point of the octave is the ‘root’ from which the key of G comes into being.
Furthermore, many traditions provide us with a canon of proportions of the human body (e.g. Leonardo’s drawing ‘Vitruvian’ man), which suggests that if the body’s height and outstretched arms form a square, the diagonals of this square(√2) cross at the sex organs, where the generative potential lies implicit.
Alternation is the focus of the third column on this page. The principal of alternation is expressed by the ancient Taoist symbols of Yin and Yang, created by the dynamic flow between two circles which expand and diminish within a larger encompassing circle. The sum of the circumferences of the smaller circles - however many there are - equal the circumference of the larger one. The processes of division can be carried on towards infinity, where the alternating line becomes indistinguishable from the line of the diameter, creating a paradox where the diameter and circumference of a circle are equal.
In biology, D’arcy Thompson’s researches into organic forms of growth suggest that in cell division, the more stable arrangement of four cells is not at 90 degrees, as one might expect, but 120 degrees. So creating the polar furrow. By inference, bi-lateralism is established near the very beginning of cell division, and thus the Yin and Yang functioning of our right and left brains. ( However this arrangement surely only applies to the later development of bi-lateralism for instance in the mammals? Earlier evolution of plant and animal life may not bear this out).
Ratios and Proportion. Turning to the top section at the centre. The section begins: It was the goal of many esoteric teachings to lead the mind back to the sense of Oneness through a succession of proportional relationships. A proportion is formed from ratios.
Below these words is written the progressive sequences of the Golden Proportion (Phi), starting at the centre of both pages with One. The right hand page (40) displays Phi as increasing in value - and externally. On the left, the same sequence is inverted and diminishing, as being implicit and less that One. Both sequences show that the relationships are self similar, up and down the scale.
Moving downwards, further examples of different ratios are explored. A four term ration included two ratios / two comparisons thus a: b :: c:d It has proportion but not mediation.
A three term relationship lifts us to a higher level, where a mediator intercedes between the two conditions. a:b :: b:c. There is one, and one only two term relationship, that is the Golden Proportion: its structure always includes itself, thus a:b:: b: a+b. 1/Phi: 1:: 1: Phi - a mirror image, on a descending and an ascending scale.
‘The inner creative dream of God is exactly reflected in its own outer reflection’. And ‘Implicit to Explicit: a Universe of infinite diversity and self likeness in constant creation’. Robert Lawlor