The figure above is a set of three objects, with a special ('musical') relationship, from Hawkins' A General History of the Science and Practice of Music, 1776.

Probably the most well known source of esoteric 'harmonic' symbolism is Plato's Timeaus, in which there is a description of the creation of the world 'structure' in harmonic proportions, actually using numbers. This appears to be a 'Pythagorean' influence in Plato.

Number was very important to the Pythagoreans, for reasons that are now interpreted as mystical, numerological or scientific. Today Number is essential to modern science.  Modern science is very much concerned with the mathematical laws that 'govern' the relationships between objects. 

Objects, however, do not have to be physical, as every object oriented language computer programmer knows. An object can be any named concept - even something like a mathematical function. Naming, and thinking with concepts and their relationships is a fundamental action of object oriented thinking, which is not just something computer programmers do, but is the generic process that most 'intellectual' activity engages. This mental action is so reinforced by training, practice and habit, it becomes practically an automatic reaction to the world in many thinkers - a kind of reflex action to whatever is presented, in order to say something about it, evaluate it, critically judge it, or solve it. It doesn't matter whether it's 'vertical' thinking, 'lateral' thinking, 'emotionally intelligent thinking', 'thinking outside the box' or whatever - it's all object oriented, because it is impossible to think of anything in relation to something else without both things already beings concepts - without both things first being instantiated as objects in the thinking mind.

Even thinking about oneself is object oriented, because 'I' and everything associated with 'me' become instantiated as objects in that thought process, even though those objects may have the special status of being a living, self identity. Object oriented thinking is a tool, an existential tool for dealing with the relationships between objects (concepts), but it cannot be used to approach the real meaning of esoteric symbols.

Mathematical symbolism, particularly as it appears in the study of harmonics is not object oriented in its true purpose. Timeaus's description of God's creation of the world by 'tearing off strips' and constructing an enigmatic object with harmonic 'proportions' between the parts, is an esoteric symbol, yet it has been plundered and plundered by object oriented thinkers for its meaning, which of course remains elusive, because that is not its purpose. It is so often left unrecognised that what appears to be an object oriented communication is not at all what it appears to be on its object oriented surface, but in truth its function is simply to increase, without concepts, the perception of intelligent order in existence.

Symbolic Correspondence

Correspondence, an ancient idea, precedes modern scientific method, and is not part of science's entirely object oriented world. It is concerned with the relationship between objects, or what are presented as objects, but in a totally different way. The principle of correspondence is always presented in object oriented terms, through objects and the relationships between them, but it is not meant to be apprehended in the object oriented way. Correspondence is ultimately a symbol for something that is meaningful only in a subject oriented way. That means, the subject can perceive something meaningful in the 'display' of correspondence, without concepts and 'formulae'. The relationships between the objects of natural phenomena can indeed be called 'symbols' for the 'higher' mathematical laws that govern them, when they are perceived in an object oriented way. But even these relationships in natural phenomena can sometimes be perceived in another way, which has more connection with subject oriented, intuitive perception.   

The concept of Correspondence is perhaps best known in relation to the Dionysian Great Chain of Being, and its various adaptations in Christian Hermeticism. The Great Chain of Being is a 'chain' of beings, or levels of being, from Man to God, presented as an ascending hierarchy, sometimes pictorially depicted. The important hierarchical relationship between say, the beings presented as Cherubim, Dominations and Thrones, or between Archangels and Angels in the Chain, may be depicted in a number of ways. Robert Fludd provides graphical depictations of the relationships between the stages of the Great Chain, and explicitly integrates the system with musical intervals. 

Click here to see an example from Fludd's Tomus Secundus De Supenaturali, Naturali, Praeternaturali Et Contranaturali Microcosmi historia, in Tractatus tres distributa (1619). 

This is Correspondence directly connected with the perceived (philosophical) idea that everything comes from the One, or everything is an expression of God at a different level, and in particular that Man is capable of being the consciousness at any level, up to and including that depicted as the Seraphini (Seraphim). 

Correspondence, however, is also recognisable in another way without dependence on such an extensive  philosophical idea, in natural phenomena, and even in 'abstract' mathematical principles. For example, below (and at the top of this page) is Andrew Tacquet's model from Proposition 14 of Theorems of Archimedes by Tacquet (Hawkins, A General History of the Science and Practice of Music). The solid shapes have the same relationships as harmonic musical intervals. Hawkins did not really regard this as an example of correspondence at all, but rather, as a demonstration that musical intervals are determined by natural laws : 

In respect of both volume and surface area, the ratio of the cylinder to the sphere is the same as the harmonic ratio for the perfect fifth, i.e. 3:2, and the ratio of the cone to the cylinder is also 3:2. The three solids from the sphere to the cone correspond to two ascending perfect fifths. Hawkins describes another figure derived from Archimedes' 34th Theorem, shown below :  

In this case, the ratio of the surface area of the large cylinder to that of the small cylinder is 2:1, corresponding to the octave, and the ratio of the sphere to the small cylinder inside it, is 4:3, corresponding to the perfect fourth. The ratio of the large cylinder to the sphere is 3:2, corresponding to a perfect fifth. The three solids from the small cylinder to the large cylinder correspond to a rising perfect fourth, and a rising perfect fifth, completing the octave.

About this, there is a school of thought, which, roughly speaking, has this to say :

"Such correspondences exist because of the mathematical 'form' that the objects (whether geometric solids, musical intervals or portions of a musical string's length) inherit and extend. The mathematical 'form' itself is a set of abstract mathematical objects and their relationships. The other objects, geometric solids and musical intervals, etc, are objects that actually exist in the world of physical phenomena. The objects of 'pure' mathematical 'form', on the other hand, being 'abstract', do not have to be manifest in physical phenomena, in order to be, and to have the relationships that they have". 

This, basically, is the idea behind the supposed notion of the 'Platonic' mathematical 'Reality'. 

Plato says much that can be interpreted (by object oriented thinkers) as suggesting this so-called mathematical 'Reality' is the Reality behind all things. But such an interpretation of Plato is surely an error. Mathematical 'form' certainly exists, and where it manifests in the behaviour or properties of matter, that matter does extend the mathematical objects and their relationships that are inherent in mathematical 'form'. However, if abstract mathematical 'form' exists, we are compelled to ask where does it exists, and is there a prior matrix that produces it?

Before even speculating on this, it should be acknowledged that any mathematical 'form' exists in the mind, as soon as a mind finds it in mathematical abstraction, and this happens often before its appearance in material phenomena has been recognised, discovered, or utilised. So it is that the meson (sub-atomic particle), for example, was 'predicted' (by Pauli), which means it was discovered in mathematical 'form', before it was discovered in 'material' form.

We cannot, however, jump to the conclusion that abstract mathematical 'form' is the Reality behind all things, just because mathematical 'form' can be grasped in the conceptualising, object oriented mind, and because this mind can 'intuit' it to exist somewhere other than in the mind. We might jump to this conclusion, if we are entitled to jump to conclusions, but to suggest this is what Plato or Socrates was doing, on the basis that Plato uses Number as a symbol, is an unjustified leap.

Correspondences of the kind noted by Hawkins, are, arguably, little more than coincidental. Those who believed in the harmony of the spheres, but also believed it was about actual orbital speeds and distances of the planets, attached too much importance to correspondence of the object oriented kind, and failed to sufficiently differentiate it from the more esoteric kind of correspondence that, for example, Fludd so clearly illustrates.

The actual behaviour of the planets is 'inherited' from the mathematical 'form' that the physical celestial bodies, their properties and behaviors, inherit in their own realm of material actuality. This mathematical 'form' is not one of simple circles and whole number ratios. This mathematical 'form' is not 'ideal' - it does not consist of the mathematical 'form' for perfect circles and whole number ratios. 

The same sort of situation is true of the musical strings that the Pythagoreans were supposed to use, and from which Pythagoras is supposed to have discovered the 'divine' harmonic string length ratios. What is conventionally referred to in acoustics as an 'ideal' musical string, is really an abstract theoretical or mathematical 'form'. It has no material existence. It is, for example, perfectly flexible, and no material string is perfectly flexible. The 'ideal' string perfectly exhibits 'ideal' behaviour, in which the Pythagorean harmonic ratios are perfectly represented. Material strings do not perfectly represent the harmonic ratios in their behaviour, although they may do so very closely. So, to 'ideal' limits, Pythagoras can be said to have been 'wrong' in his discoveries and assertions about divine, harmonic whole number ratios.

But was he really 'wrong', because he perceived the ideal, rather than the physical actuality? In other words, cannot the 'ideal' remain pristinely true and 'simple' - completely symbolised by perfect whole number ratios - but in being translated into physical actuality become corrupted, complicated and less than ideal? In this case, the 'ideal form' is not the 'mathematical form', even though it may be expressed, apparently, mathematically. In this hierarchy, the 'corruption' has to take place between the ideal and the abstract mathematical 'form'. Scientific, or object oriented thinking, would of course see no point in such a 'speculation'.

Hawkins' illustrations above, express both kinds of 'form' - the mathematical and the ideal - which in these particular 'symbols', happen to coincide. There are two ways of looking at the same 'symbol'. If we see the mathematical 'form' in the relationships, we are seeing in an object oriented way. We are then 'scientific'. If we see the ideal 'form' appearing in matter, we are 'seeing through' both the physical and the mathematical 'form' to something that transcends both.

Statements about ideal 'form' have nothing to do with science, because they are not attempts to describe mathematical 'form' or laws. They are an attempt to describe something hierarchically 'above' mathematical 'form', which mathematical 'form' merely emulates, imperfectly. Even in the Hawkins figures, the emulation is actually imperfect, because the ideal 'form' is not really a form of numerical relationship at all, but is merely symbolised by numerical relationships. It is not so much that the ideal is 'corrupted' here on the way down to the abstract mathematical, as that it becomes 'represented' in the mathematical. Finally, it is 'represented' in the physical objects themselves. 

So it is in the description of the creation of the world, in Timeaus. So it is, in the idea of the harmony of the spheres.