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.

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

The bit is the basis of IT

Our information technology is based on the bit. Everything that happens in our computers is based on this smallest basic element of information. If someone asks you what a single bit means, you may well answer that the bit can assume two states, of which one means 0 and the other means 1. As is generally known, this enables us to write numbers of any size; all we have to do is to line up a sufficient number of bits.

But is this really true? Does the one state in the bit really mean 0 and the other 1? Can these two states not also assume completely different meanings?

A bit can be attributed arbitrary meanings

In fact, the two states of the bit can assume any meaning. Besides 0/1, true/false, yes/no and positive/negative are also popular; but in principle and in practice, a bit can be attributed any meanings from the outside. Of course, inversions are also possible, i.e. 0/1 and 1/0.

The attribution of the meaning of the bit comes from the outside

Whether the specific bit in the computer programme means 0/1 or 1/0 or something else, does of course play a crucial part. However, the meaning is not in the bit itself, for the bit is a most radical abstraction. It only says that two states exist and which is currently active. What the two mean, however, is a completely different story, which goes far beyond the single bit. In a computer program, it can be declared, for instance, that the bit corresponds to the TRUE/FALSE pair of values; but the same bit, together with another bit, can also be interpreted as part of a number or a letter code – very different meanings, then, depending on the program context.

Digital and analogue context

The software program is the digital context, and of course it consists of further bits. The bits from the surroundings can be used to determine the meaning of a bit. Let’s assume that our bit and other bits are involved in defining the letter ‘f’. Our program is also organised in such a manner that this letter will appear in a table, in a column which is headed ‘Gender’. All this is clearly set out in the software. Now, does the software determine the meaning of the bit? You will doubtless not be surprised if the ‘f’ means ‘female” and the table probably lists various people who can be male (m) or female (f). But what do male and female mean? It is only in the analogue world that these expressions receive a meaning.

The bit, the perfect abstraction

In fact, the bit represents the final point of a radical abstraction of information. In a single bit, information is reduced to what is absolutely elementary in that the information about the meaning has been completely removed from the bit. The bit merely says that two states exist that have been described outside it and which of the two is active at a specific point in time.

This radical abstraction is intentional and makes a great deal of sense in a software, for in this way, the same physical bit in the chip of the computer can be put to a new use again and again, once as a TRUE/FALSE pair, once as 0/1, once as YES/NO, etc. This is very practical and enables the computer to solve any task whatsoever. The perfect abstraction that has thus been achieved, however, simultaneously deprives the single bit of its individual meaning, which can and must be attributed to it anew for every application.

The endless regress

When the meaning of the bit is given from the outside, then of course other bits can take on this task and define the meaning of a bit. For this purpose, however, these outside bits must have the necessary effective power, which of course they cannot have without their own meaning. And naturally, the meanings of the bits of this outer circle are not in these bits themselves – for the same reason as above – but have to be given from the outside, i.e. by a further circle of bits. The bits of this second outer circle must be explained in a further circle, and the meanings of the bits of this further circle in another outer circle… Of course this process of attributing meanings never reaches an end in a world of bits: the regress is endless.

The endless regress only ends in the analogue world

Only when we step out of the program into the real world are we really able to attribute a meaning to the information from the computer.

Selective and descriptive information content

If we recapitulate the above, we can make the following distinction in the bit:

The descriptive information content says what the bit means; it describes the two states of the bit but does not say which state has currently been selected. The selective information content says which of the two states is currently active but does not know anything about the properties of the two states and thus about their individual meanings.

The distinction between the selective and descriptive information content was coined by the British radar pioneer and information scientist Donald McKay in the 1940s, practically at the same time as the first mention and description of the classic bit by the American Shannon. McKay also clearly recognised that Shannon’s bit only carries a selective information content and that the descriptive information content must be given from the outside.

Surprisingly, this insight of McKay’s has almost fallen into oblivion today.

Conclusion:

1. The bit supplies the selective information content.
   2. The descriptive information content is not located in the bit.
   3. Thus the bit on its own does not have any meaning.
   4. The meaning of the bit is always given from the outside.
   5. This initiates an endless regress.
   6. Only in the analogue world does the endless regress end.

Scales are patterns

When you hear a melody, it is based on a musical scale, i.e. a number of a few distinct notes which may occur in the melody. In a linear sequence, these tones constitute the scale. Most melodies that can be heard in our cultural area can be traced back to one single scale, the Ionian or major scale, which is made up of seven notes in very specific scale steps.

Thousands of scales

However, there are thousands of different scales. Presumably you are familiar with the minor as well as with major scale and may have heard something about pentatonic scales or about whole tone scales, Lydian and Phrygian scales, Indian ragas, Japanese and African scales. All these scales differ from each other.

As we will see, however, they have some astounding similarities. Why should people all over the world, in all cultures and with all their differences, comply voluntarily and strictly with these similarities? The reasons for this are easy to explain if we don’t merely look at one world, but at the interaction of all three worlds.

In which of the three worlds do the scales exist?

Scales are part of our reality, no matter how we define reality – unless we define reality as that which we call matter. In that case, the scales are not part of matter. They may manifest themselves in the physical world, for instance if a human being sings or plays them, but they have an identity which is independent of the individual way they are performed. In this sense, scales are non-local, as is typically the case with entities of the Platonic world. Between the scale and its performance, there is thus the relationship of an abstract, i.e. Platonic pattern with its material instance. This is always a 1/n relationship, for the pattern is unique but can be the source of any number of instances.

As a pattern, scales are part of the Platonic world, even though they manifest themselves in the material world. Mathematics, in particular, has much to do with the form of scales, which can be demonstrated easily, yet you don’t need to know anything at all about this kind of mathematics in order to recognise the scales correctly or to sing them. Your mental world in which you experience these scales has no need of figures and formulae.

Scales thus exist in all three worlds:

Platonic world: here, a scale exists as an entity, i.e. as a unity and as a whole. Here, every scale exists only once.

Physical world: here, a scale exists as any number of occurrences – whenever melodies are produced on the basis of it.

Mental world: here, i.e. in your head, you recognise the melodies and the scales.

Of course, each world is organised in its very own way. Now, how do the three worlds interact?

This is a text in the series about the theory of the three worlds.

Translation: Tony Häfliger and Vivien Blandford

The world of the natural sciences

The physical world is what is explored by the natural sciences and by physics in particular. The successes of the natural science programme are obvious, both with regard to its insights into how the world works, i.e. theory, and with regard to the technologies this enables, i.e. practice. The natural sciences have changed our world fundamentally ever since Galileo Galilei.

Objectivity

The success became possible because from the Renaissance onwards, the thinkers and explorers in Europe did not solely refer to what had been handed down since ancient times, but conducted unprejudiced research and looked for results themselves. Whereas the monks in the monasteries interpreted and compiled old manuscripts (scholasticism), the free spirits dared to believe what they themselves could see in nature – even if it was in contradiction to the monastic authorities.

However, the loss of the old authorities called for a new guiding principle to prevent a situation whereby everyone would be able to claim anything. Therefore researcher’s proposition should be independently verifiable, and solely what was discernible by everybody beyond any doubt should in future be true and applicable. Thus the ideal of objectivity was born.

Measurability

But the world should not only be able to be described objectively, but also measured with the highest possible precision. This has two advantages: a) A proposition is all the more credible the more precise its predictions are. The more precisely we are able to measure things, the more significant the observations. b) Besides more precise insights, precise measurements also enable us to build increasingly precise instruments and machines.

The world from the outside

The natural sciences thus focus on what can be seen and measured from the outside. This is what Penrose describes as the physical world. Not my internal view, my feeling or my belief is what is called for, but what can be observed and measured from the outside beyond any doubt. This is the physical world.

The physical world in interaction

We could consider the physical world to be the sole true reality, but our view of the world becomes more inclusive if we add the other two worlds. How do the three worlds interact? – There are bridges from one to the other, there are crossovers, and there are effects from one world on the others. I would like to illustrate how these interactions work with examples from the field of music – a field in which obviously all three worlds are involved.

This is a text in the series about the theory of the three worlds.

Translation: Tony Häfliger and Vivien Blandford