The Physical Basis of Resonance
Resonance is always based on the natural oscillations of two physical media and their mutual coupling
Coupling of Natural Oscillations of Physical Objects
The natural vibrations are standing waves whose frequency is determined by the properties of the physical medium (size, shape, material, etc.).
Two such media can enter into resonance via their oscillations. The resonance is created by coupling the two oscillations so that the two physical media form a coupled unit in their oscillating behaviour.
The coupling takes place via a physical exchange of energy, either directly or indirectly, e.g. via the air. The prerequisite for the coupling to occur is that the frequencies of the natural vibrations of the two physical media involved are in a suitable mathematical ratio.
Stability of Resonance Over Time
Once the resonance state has been established, it remains stable for a certain period of time, i.e. the coupled oscillation state remains stationary, often over a longer time period. This astonishing behaviour has to do with the energy situation, which is particularly favourable when the two objects are coupled
Internal Oscillation
The internal oscillation of an individual physical object can also be described as resonance. For example, an electron around the nucleus of an atom resonates with itself during its cycles and can therefore only swing into very specific orbital frequencies that allow it to resonate with itself on its orbital trajectory. This leads to the specific frequency of the natural oscillation of the electron in the atom. The same applies to the vibrational behaviour of a string of a musical instrument.
Resonance as Abstraction
Although the physical material determines the inherent frequency of the vibrating media, the frequency of the coupled resonance, however, results from the appropriate ratio of the natural frequencies of the two involved media. The frequency of the evolving resonance follows mathematical rules. Amazingly simple mathematics is enough to discern which reonance frequencies will evolve and how strong the resonance between the two vibrating physical media will be.
Once the internal oscillations of the involved objects are set, it is only the abstract mathematical ratio of the two frequencies which allow the development of a resonance and determine their strength.
Three Worlds, According to Roger Penrose
The emergence of resonance impressively demonstrates the interplay between two of the Three Worlds that, according to Nobel Prize winner Roger Penrose, form our reality, namely the physical and the platonic one. The term platonic refers to the abstract world of ideas, to which mathematics belongs. By using this term for the world of mathematics, Sir Roger alludes to the European cultural history. Here, the discussion about the reality of ideas is not only part of Plato’s philosophy, but also characterises large parts of philosophical discourse in the Middle Ages, known as the problem of universals.
The question has lost none of its relevance since then: How real are ideas? Why does the abstract prevail in the material world? What is the relationship between abstract ideas and the concrete, i.e. physical world?
About a year ago, I thought that the emergence of resonance in music would be a good example to explore the relationship between physics, maths and the third world, i.e. our subjective perception. I was surprised at how amazingly clear the relationship between the three worlds can be presented here and how astonishingly simple, logical and far-reaching the maths in the harmonies of our music is.
Resonance in Music
In a piece of music, the resulting resonances between the notes change again and again, creating a fascinating change of colour. We can experience it intuitively, but we can also explain it rationally, as a play of resonances between the notes.
Only the Overtone Series?
At school I learnt that the overtone series determines our scales. But that is a gross simplification. The phenomenon of resonance can explain our scales much more simply and directly than the overtone series on its own can. The overtone series only describes the vibration behaviour within one physical medium – the resonance in music, however, always arises between at least two different media (tones). For the considerations regarding the resonance of two tones, we must consequently also compare two overtone series. Only the juxtaposition of the two series explains what is happening – a fact that is usually ignored in textbooks.
Chords consist of three or more tones. Here too, the resonance analysis of the three or more tones in question can explain the chord effect with astonishing simplicity. Only this time, it is not the frequencies of two but of a bunch of tones that have to be taken into account simultaneously.
Pure and Tempered Tuning
In Europe, equal temperament tuning became established in the Baroque period, which expanded the compositional possibilities in many ways. The first thing the layman finds on the theory of scales is therefore a precise description of the deviations of the tempered tuning from the pure tuning – but these deviations are only of marginal importance for the development of resonances. Pure tuning is not a condition for resonance; the mathematics of resonance presented here explains the phenomenon precisely even with tempered tuning.
This is a contribution to Penrose’s Three-World Theory and the Origin of Scales.
Translation: Juan Utzinger