This is a post about the theory of the three worlds and continues the post about the octave.
We generate a resonance
If you regard resonance as an abstract phenomenon – or as a musical phenomenon that you have not yet experienced – I recommend that you should conduct the following simple experiment: look for a piano (not a digital one) and for a tone on that piano that you can sing well. Press the key of this note and sing it. Of course, this already requires the resonance in your inner ear, otherwise you would not hit the tone. Then press the piano key again, but in such a manner that no sound is produced, and keep the mute key depressed. In this way, the string can vibrate freely. Now sing the tone again. If you have struck the pitch of the key, then the tone will sound in the piano without you pressing the key again. This works best when the piano is open; possibly, you may merely have to sing a bit louder. Now you can sing various tones, for instance a short melody, and you will see that the tone will sound in the piano whenever your voice has the same pitch as the key.
This experiment gives you a sensuous (= physico-mental) impression of the phenomenon of resonance. If you find it difficult to hit the note, there is a simpler way. Depress the piano pedal on the right. Now all the strings can vibrate freely. Now shout at the piano, preferably with the lid open. Again, you will hear how the strings vibrate as an echo of your voice.
Simple resonance
The “long-distance effect” in the above experiment is not magic but produced by sound waves. These waves resonate with the strings. The typical feature of this is the fact that resonance does not occur with any frequency but precisely when the sound wave hits the string’s natural frequency. Natural frequencies are properties of many physical systems; for example, a bridge can have its natural frequency, as can a glass, a piece of wood or a pot. String and wind instruments are perfected in such a way as to resonate particularly well, i.e. to ensure that their natural frequencies are particularly strong and sonorous.
Higher-level resonances
Again I propose a small experiment, and again you will need a piano, which should be tuned this time.
Fig. 1: Two Cs on the piano at the distance of an octave
Now depress the higher C key on the piano (the one on the right). Of course there are many such Cs on the piano; it would be best to take two neighbouring Cs in the middle of the keyboard, where the experiment can be heard most clearly. You can also take tones other than C; the experiment works with all tones, provided that the distance between the two tones is precisely an octave. You will now also realise why the octave is so called: the higher C is at a distance of eight (Latin octo) tones from the lower one (when you count the musical intervals, the starting tone is always counted as well).
Now keep the higher (right-hand) C key mutely depressed. Now hit the lower C key briefly and strongly. You will again hear a “long-distance effect”. Clearly, the string of the higher C started to vibrate when you hit the lower C. Now hit the keys immediately to the right and left of the lower C. You will not be able to get the higher C to vibrate with those keys; no resonance occurs.
Why resonance occurs precisely with an octave
Fig. 2: Possible vibrations of a string
In Fig. 2, you can see five possible vibration patterns for a stretched string. At the bottom (at 1), the string vibrates with precisely one wave peak. At 2, there are two peaks; at 5, there are five. Yellow represents the vibrating string; the black line shows the corresponding sound wave, i.e. the travelling sound wave which has the same frequency as the standing wave, which is represented by the resonant string.
State 1 in Figure 2 is the basic state, i. e. the tone which you hear when you press the piano key. State 2 is the next permitted state of the vibration. Here, the string vibrates with two peaks; at 3, there are three, etc. All states in which the string at its two fixed ends does not vibrate are states that permit the string to vibrate unimpeded. Thus it is not only the state of simple string vibration that is possible, but in principle every state that corresponds to a wavelength that fits integrally into the string length.
In state 2, the wavelength is half as long as in the basic state and thus the frequency is twice as fast/high. State 2, with its frequency twice as high, corresponds to the tone that sounds an octave higher; state 4 corresponds to the tone that sounds two octaves higher.
Now why does the higher C also resonate when you hit the lower C as in the experiment proposed above? The reason for this is the fact that the string of the lower C – like any other string – does not only generate sound in the basic state (state 1 in Fig. 2), but more or less in all the permitted vibrations. These vibrations are thus superimposed on each other. When the sound waves emitted from the lower string reach the string of the higher C, then they do not only contain the basic vibration, but a bit more faintly also the higher vibrations and therefore precisely also the vibration of the higher C string. Thus a resonance is generated.
Sine vibrations and overtones
In mathematical terms, the two black curves in Fig. 2 are sine curves. With a technical device, it is possible to generate such curves acoustically; we then speak of a sine vibration. In natural resonating bodies such as piano strings, the human voice or everywhere else in nature, such pure sine vibrations do not exist; rather, the sound waves produced by them always also contain higher vibrations (levels 2ff. in Fig. 2) in complex superimpositions. They are called overtones. The proportions of the individual overtones, i.e. the extent to which the vibrations of levels 2ff. resonate besides the fundamental tones in the sound mixture, are highly variable and are determined by the sound-producing medium. These mixtures account for the character of the sound of the individual instrument.
Interpretation of the string vibrations in the three worlds
Platonic → physical (from simple to complex)
The example of the vibrating string shows us how mathematical laws from the Platonic world determine the physical world. In the physical world, however, they manifest themselves in many different ways, and a great diversity emerges: several vibrations are simultaneously generated on the string; besides the fundamental tone, there are always many overtones at the same time. Each single one of these vibrations can be described very simply in mathematical terms. The mixture, however, is very complex.
What is very simple in the mathematical world, i.e. in the Platonic world, quickly becomes complex as soon as it takes effect in the physical world.
Penrose’s endless staircase and the “anti-Penrose” direction
In my view, the funnels in Penrose’s sketch represent only one direction of the relationships. Penrose emphasises in his explanation that a description of physics does not require the whole of mathematics and thus arrives at proportions such as those represented in the sketch with the funnels and which appear to contradict logic as in the endless staircase.
In my view, however, the funnels can also be seen in the opposite direction when we focus on the volume of information. This is larger in the physical world than in the Platonic one. When mathematics enters physics, new things emerge, namely the complex diversity of the mixing ratios. This concrete diversity in the physical world constitutes a piece of information which far exceeds the information of the original mathematical world. The volume of information increases in the direction from Platonic to physical. In terms of volume, this represents a countermovement to Penrose’s funnel. Thus when we take a closer look, the endless staircase of the three worlds loses some of its paradoxical terror.
Platonic → mental
Presumably you have heard technically generated sine vibrations. They were the beginning of electronic music and had the charm of something new and technical at the time. It was precisely their naked purity that was impressive. However, these tones also become very boring very quickly. The purity and the sterile banality of these technical sounds is caused by the lacking resonance of the overtones. As listeners, we perceive the rich information of these additional vibrations, and they account for the wealth of natural sounds. I wouldn’t want to have to do without them.
In the next post, I would like to explain why the octave is so important in the mental world and what contributes to the fact that musical scales in all cultures are always based on the octave.
This is a post about the theory of the three worlds.