Arithmetic, that sublime – and only true science, mother to all others – gives rise to the ordered sequence of nature, argued by the Greeks to be the source and origin of all manifest forms. Indeed it is a proven fact that it is through numbers that science is advanced. Genetics would be nowhere without the pioneering work of Gregor Mendel who discovered that crossing gave rise to various expressions in numerical sequences and relationships. There was nothing haphazard or contentious in his discovery. It was, and is, attested by all.
Yet in the material world the use of numbers is in two directions, besides the counting of things – which may be given to Arithmetic itself – it is used in application in measuring things – Geometry (which we may consider elsewhere) – and in the comparison of things – Ratio. It is this last we shall look at here.
In comparing one number with another we determine the relationship between them.
For example 1:2 has a particular relationship which is different from 1:3 or 2:3. some relationships seem more natural to us. 1:2 is somehow more tenable than 1:3, rather in the way that 5 is a simple number in the decimal system, but 6 or 8 are not so easy to identify within the landscape of numbers.
It is obvious that a number is divisible by 5 since it will have either 0 or 5 at the end of it. But as for 6 or indeed any other numbers, save those of a decimal nature themselves, 10, 20, 30 and so on, it is not so simple to identify whether the number is divisible by, or divides into, a second number with which it may be compared. For example does 3 go into 83? But with 2 a number has only to be even for us to know it is divisible by 2, and to divide that number in half is a simple exercise. Much simpler than dividing it into 3 parts, or 6 parts.
But all of this wanders away from Ratio.
When 2 numbers are compared there is a simple relationship between them. Yet when a third number is added it begins a series, or progression. We can often predict what the next number in a series will be.
1, 2, 3, – suggests the progression of the order will run 4, 5, 6. However when the first numbers are 1, 2, 4, we see a different series evolving and can suggest the subsequent numbers will be 8, 16, 32.
The types of Ratio were divided into various characteristics which I shall not go into here but point to the work by Thomas Taylor – The Theoretic Arithmetic of the Pythagoreans – for interested students to follow this esoteric topic.
I have tried to find useful definitions on the internet but find they are all lacking in clarity and prefer to use other definitions, such as fractions or algebra, which only further to obscure the consideration, despite having the same relative meaning. So often writers refer to material objects and are clearly more familiar with the application of Arithmetic to the world of objects rather than any philosophic consideration of numbers in themselves. As Pythagoras said ‘A figure and three steps, never a figure and three objects’.
In the philosophic realm of number such things as negative numbers, and even 0, do not exist. There is never Nothing when One encompasses all things. There cannot be a subtraction from One. It IS. There is no ‘Not IS’ in this realm as there is in banking. ‘Not is’ arises only after the generation of number into the first decimal series. Computer science may choose to differ from this but does not necessarily throw more light on Creation, even if it does on division and dissection.
In comparing any two numbers we can often find that they are similar to another ratio so that all ratios in a Duple relationship will be as 1:2. Such comparisons as 2:4, 3:6, 4:8 and so on.
Or in a triple relationship will hold the same proportion to one another as 1:3. such numbers as 2:6, 3:9 and so forth.
When there is a constant interval between numbers in a series, the progression is said to be Arithmetic. As in 1, 2, 3, 4, where the interval is the unit, 1; or 2, 4, 6, 8 where the interval is equally 2 between each of the numbers, and so on. When the interval between two numbers increases in a symmetrical fashion as 1, 2, 4, 8, where each interval is double the last; or 1, 3, 9, 27 where the interval increases 3fold, each time, then the progression is defined as Geometrical. When the interval between numbers has a relation of – for example the interval between the first two terms is 1 and the interval between the middle term and the last is 2 – as we find in 3 : 4 : 6 the series is said to be in Harmonic proportion.
This combination of numbers is most particular in that it contains the first 3 major ratios. 3 : 6 is as 1 : 2; while 4 : 6 is as 2 : 3 and of course 3 : 4 is the third major ratio. Musically these are significant in that 1 : 2 is an octave, where 2 : 3 is a musical fifth, and 3 : 4 a fourth. These are the basis of much western music from the classic Church musical endings in Amen to Rock and Roll. We shall meet these again shortly.
From here, however, the subject of Ratios and species of Ratio becomes very complex and again Taylor gives the fullest explanation of the various types of progression.
For the Greeks the significant feature was that in the unfoldment of the natural series of numbers the value of 1, the intervening unit, was constantly changing. The relation of the unit in 3 is significantly different from the relation of the unit in 4, or 6. In 3 it is one third of the whole, where in 4 it is one fourth and in 6 one sixth. Clearly the scale of the unit is changing. What then, the Greek thinkers desired to know, was the ideal size of the unit and, through Ratio, they came to the conclusion that it was in the interval from 8 to 9.
8 is the first cube in extension. 1 is the first cube in potential as it is the first of all the forms discovered through the orderly unfoldment of the natural series of numbers; but the cube of 2, 8, is the first in extension. By contrast 9 is the second square in extension, being composed of sides of 3. The solid, cube, represents the material world, the plane figure always represents the mental world. This is the point of division between the material, ‘actual’, and the theoretical, mental worlds. (This is the point at which ‘Not Is’ comes into being.) But why this is supported in Ratio is through the addition of two other numbers, 6 and 12.
The equation 6:8 :: 9:12 contains the most significant ratios.
The first principle is in the form of the duple, the Octave,
6:12 = 1:2
the sesquialter (2:3) is found twice in the forms of
6:9 :: 8: 12
and the sesquitertian ratio (3:4) is found twice also as
6:8 :: 9:12
Here is a delightful thing, placing the relationship of 8:9 at the centre of the composition.
This is further complemented by the relationship between the 8th and 9th harmonics in Music, which sees the tone in a scale appear for the first time in the sequence of harmonics. The 8th harmonic yields the tone three octaves higher, while the 9th harmonic yields the tone immediately above that. It is in this way that the inner harmony of things unfolds, or is generated from the point of origin, the Tone, through reducing frequencies to give the plethora of expressions we see in the world about us.