A remarkable common feature
All the musical scales known to me encompass an octave. Even scales which tones unusual to us Europeans – Arabic, Indian, Japanese and African ones – encompass precisely an octave, i.e. the deepest and highest tones have a distance of precisely an octave, whatever scale this may be.
I find this extremely remarkable. This is as if all the world’s languages, which after all have very different words, used the same word for a certain concept, had always done so, and done so independently of each other. What is the reason for this?
The theory of the three worlds is capable of providing a plausible explanation of the unusual common feature of the musical scales of all human cultures.
The octave in Platonic terms
If you pluck a string on a violin, you will produce a tone. If you press a finger onto the fingerboard exactly in the middle of the string, the tone will be one octave higher. The same applies to pipes. A pipe that is half as long as another one sounds an octave higher. Obviously, the octave is based on a ratio of 1:2. This is the Platonic, i.e. mathematical side of the octave. Simple mathematical ratios (= fractions) also play a part in connection with other intervals, which will be discussed later.
These mathematical ratios of the relationships between the tones – i.e. the intervals – have long been known and were taught by the Greek Pythagoras, who founded an influential school in southern Italy before Socrates and Plato.
Fig. 1: A vibrating string. In the upper graph, the string is attached on the left and on the right (0 and 1) and is consequently incapable of vibrating there. The farther away it is from the attachment points, the more strongly it vibrates, most strongly in the middle. In the lower graph, a finger has been pressed onto the string, and it now vibrates in half its length and an octave higher. (With these descriptions, however, we have already left the Platonic world and entered the physical world.)
The simple ratio doesn’t yet explain the uniqueness of the common feature, the octave, across all human cultures. Why does the ratio play a part in musical scales at all?
To explain this, we’ll have to have a look at the other two worlds, namely the physical, in which the tones are produced, and the mental, in which we perceive them.
The octave in physical terms
Tones
Tones are material vibrations in a transmission medium such as air. As a rule, a tone contains a superimposition of several vibrations (fundamental tone plus overtones). At this moment, however, we are only looking at the fundamental vibration, which determines the recognisable pitch.
This fundamental vibration is a sine wave, and the pitch is indicated as a frequency, for instance 440 Hz. This frequency means that the sine wave vibrates at a rate of 440 movements per second. The same is done by the string.
When the string vibrates in a fixed place, we speak of a standing wave (cf. Fig. 1 above). Conversely, the vibration in the air moves away from this fixed place (travelling wave). The string is able to move the air by means of its stationary vibration and thus produces a vibration in the air, a sound wave. The string transmits the properties of its vibration, particularly its frequency, to the sound wave.
The wavelength in a travelling wave, i.e. a sound wave, but also a wave on a water surface, for example, is the distance between the antinodes (or wave peaks). In a standing wave such as the string in Fig. 1, the wavelength equals the (double) length of the vibrating string.
If the speed of a travelling wave is constant, then more antinodes must follow each other, the shorter the distances between them are. The distances between the wave peaks represent the wavelength, the number of peaks per time unit represent the frequency of the wave. The more peaks pass a specific place, the smaller their distances.
Thus there is an inversely proportionate ratio between wavelength and frequency, i.e. the shorter the wavelength, the higher the frequency must be. This is why the string that is half as long vibrates at twice the speed. This is the physical origin of the octave.
Tone generation
How does the vibration get into the string? This is the consequence of the fact that a stretched string has a tendency towards natural oscillation. The tension in the string leads to a situation whereby a stimulus, for instance the plucking of a string, triggers off a movement which does not stop at either end of the string but is pushed back. In this way, the standing wave is produced. The wavelength, i.e. the distance between the antinodes, is determined by the length of the string. The reason for this is the fact that no motion is possible any longer at either end of the string since it is attached there. The wave can only vibrate in between. The wavelength must therefore fit precisely into the length of the string.
The octave in mental terms
The inner ear
We perceive sound with our two inner ears. These are organs of an extremely refined design with the structure of a snail, which is why they are called cochlea (Greek for snail). The sound wave comes from outside and travels into the cochlea, which is filled with liquid, and by means of resonance generates a vibration in the so-called basilar membrane, which runs through the entire cochlea. Along the basilar membrane, so-called hair cells receive the vibrations of the basilar membrane and transmit them into the brain as electrical signals. The complex and refined structure of the cochlea, which is only cursorily described here, enables the acoustic signals to be analytically separated so that depending on the frequency, different hair cells are stimulated: the higher the frequency, the closer to the entrance of the cochlea; the lower the frequency, the deeper inside the cochlea.
Mental tone perception
Up to this point, tone perception through the inner ear has nothing to do with the mental world as yet; these are merely the anatomical requirements, i.e. the physical apparatus which specifically prepares the physical signals (the sound waves) for actual perception. This latter takes place in the brain and is a subjective process.
Subjective processes are characterised by the fact that they cannot be understood from the outside. I don’t know how you hear or feel something; this is entirely your own world. However, we have so many common properties as human beings that I may assume that you will experience many things very similarly to me. We have the same anatomy and the same living conditions. Why do many people perceive the same music as beautiful? If we are moved by the same music and apprehend it as cheerful, sad, comforting, rousing, etc. like other people, this demonstrates that our mental worlds are strongly connected to each other despite their subjectivity.
In this context, cultural aspects – learned habits – play a very important role. Ultimately, culture is also part of the mental world; it is the spirit, i.e. the subjectivity, that we share.
This individual and collective subjectivity, our mental world, is not least also based on the physical preconditions, i.e. the physical world.
Thus we are back with our topic: why do all human cultures have the octave in their musical scales, which are so different in other respects? The reason for this can now be explained in terms of physics and lies in resonance.
Resonance
Resonance is required for tones to arrive in the inner ear at all, for the basilar membrane in the inner ear receives the vibrations of the sound waves in a very specific manner. Not all the frequencies find the same resonance in the basilar membrane. The inner ear is structured in such a way that the basilar membrane resonates with high frequencies at its entry and with low frequencies in its depth. In this way, the ear analyses the various pitches. But resonance is responsible for much more, inter alia also for the fact that the octave is always present in the thousands of different musical scales.
This conspicuous observation will be the subject of the next post.
In a further post, I will then deal with the perception of the octave in the mental world, i.e. our subjective world.
This is a post about the theory of the three worlds.
The next post will deal with resonance in the three worlds.
Translation: Tony Häfliger and Vivien Blandford