Tag Archives: Three worlds

The mental world

What is the mental world?

The mental world is the world in our heads. It is the way in which we perceive the world; it consists of our sensations, feelings and thoughts. It is a completely subjective world.

In the theory of the three worlds, the mental world is the third world besides the physical and the Platonic worlds.

Difference from the physical world

Whereas the physical world is objectively comprehensible, the mental world remains subjective.

In other words: whereas we can observe the objects of the physical world from the outside, this is impossible with the objects of the mental world.

The example of colour

Objectively (physically), colours can be measured as wavelengths of light waves. Thus yellow and blue, for instance, have specific wavelengths, which can be objectively measured in nanometres. What we perceive, however, is not the wavelength. We have a subjective impression of yellow or blue, which is actually triggered off by the physical phenomenon of the light wave, yet what we sense is not the wavelength, but a very subjective impression of a yellow or blue colour. Thus we perceive green, for example, as a very specific colour which corresponds to a certain wavelength. As we know from our art lessons, however, green can be mixed from blue and yellow. This means that what hits our eyes physically is a combination of photons with a “blue” and “yellow” wavelength. However, we do not perceive these two objectively existing wavelengths, but we apprehend the combination as green, i.e. as a completely different wavelength. This subjective impression is called qualia in the literature.


Does the mental world really exist?

Or is it simply an emanation of the physical world? Many people believe that it is. The subjective impression that we sense is generated in the brain by electrical currents triggered by photons on our retina. In these terms, the mental world does not really exist but is an emanation of the physical world, a mere effect of physics which makes us believe that we see a colour.

At the other end of the spectrum, we find the solipsists and the radical constructivists such as Ernst von Glasersfeld. For solipsists, the mental world – i.e. their own imagination – is the only world that exists with certainty. Everything else can be deception, a dream; only one’s own imagination is certain.

Thus we have two extremes
  1. a) Physicalists: only the physical world exists; the mental world is wholly constructed out of the physical world.
  2. b) Solipsists: only the mental world exists; it fakes the existence of an external physical world.

What is more interesting than these two extremes are the opinions in between. Roger Penrose, for instance, advocated with his theory of the three worlds that none of the three worlds should be excluded as non-existing. Rather, he wanted to clarify the interrelations of the three worlds.

Coexistence

This is also my stance: although it seems plausible to see mental sensations and processes as mere effects of the physical world, it strikes me as sensible to view the mental world as a world of its own – not because it could not have emanated from the physical world but because it can be described better in this way. To return to the example of the colours: it is irrelevant to human behaviour whether green is generated by its own green wavelength or by a combination of yellow and blue wavelengths; I always see the same colour and also behave accordingly. The description of human thought, perception and behaviour becomes at once simpler and more precise if we tackle the processes in the mental world directly. This is possible, but only from the inside, if I imagine the thoughts, colours, etc. of the mental world myself.

Communication about mental objects (thoughts, colours, etc.) is possible, too, but also requires a subjective basis of experience; this time, one that the interlocutors have experienced in a similar way.


Where does the mental world play a part?

Wherever inner perceptions and processes are involved, we are in the mental world.

The following areas can hardly be described without accepting the existence of the mental world:

  • psychology,
  • culture,
  • values, morality,
  • politics,
  • art.

Thus the mental world is not quite irrelevant.

Semantics

In my own field, semantics, a clear dividing line between the objective and the subjective worlds can be discerned. Whereas words and sentences are part of the objective world, the concepts, i.e. the meanings of the words, and the thoughts that are expressed by means of the sentences, are part of the subjective, i.e. mental world.


This is a post about the theory of the three worlds.

What are the advantages of the equal temperament?

The equal temperament has prevailed in our occidental music culture – despite the obvious shortcoming that its intervals are not pure any longer. This was only possible because some substantial advantages offset the flaw of impurity:

1.  One single tuning serves all keys: the fundamental tone is freely selectable.

In pure intonation, instruments basically have to be retuned for each key and each fundamental tone. With a harpsichord, this concerns a few strings, but with an organ, this is really a great undertaking in view of the vast number of registers and pipes. The further the keys are apart from each other – i.e. the more crosses and Bs they have – the worse the detuning is. The equal temperament changes all this. Although it is never quite pure, it is never so detuned as it would be in a distant key, either. With the equal temperament, instruments can be used to play in a different key immediately without retuning. Organs, string, wind and any other instruments can now play together in all keys.

2. Free modulation

This advantage is particularly obvious and effective. In the same piece of music, we are now able to change from one key into any other (modulate) – without any break and retuning. With pure intonation, this is only possible in a limited way and for closely related keys. The equal temperament, however, removes all the limits to modulation.

3. “Bonus keys”

The seven tones of the major key do not merely serve to play major scales. On the white keys of the piano, you can play all the seven ecclesiastical modes, depending on which of the seven tones you choose as the fundamental tone:

C: Ionian = major
D: Dorian
E: Phrygian
F: Lydian
G: Mixolydian
A: Aeolian
B: Locrian

The same applies to the pentatonic scales. If you only press the black keys on a piano, for example, you will automatically play pentatonically (penta = five → five black keys). Depending on which tone you choose as the fundamental tone, you will play a different pentatonic scale, for instance the major pentatonic scale (with F sharp as the fundamental tone) or the minor pentatonic scale (with E flat as the fundamental tone).

This principle of generating new scales with the tones of an existing scale also applies to the ‘melodic minor’ scale in jazz. As the major scale, the melodic minor scale is a selection of seven tones, but in intervals that cannot be played with the white keys of the piano alone. With these seven tones of the melodic minor scale, we are again able to construct seven different musical scales which differ greatly in terms of character, merely depending on which tones we choose as the fundamental tone.

4. Polytonality

This stylistic device emerged in the 20th century with Stravinsky and other composers, and it is also used in modern jazz. To achieve this, several keys are mixed; in practice, it is usually two (= bitonality). This appears to be slightly risky at first sight, but if we take into account the resonances (!) correspondingly, it actually sounds quite catchy.

Conclusion from the perspective of the theory of the three worlds

The mathematical purity (Platonic world) of the interval resonances is violated by the temperament, but in such a minimal way that the resulting vibration phenomena (physical world) are produced all the same and the listening experience (mental world) is hardly diminished at all.

On the other hand, free modulation extends the number of possible combinations of the still only twelve tones immensely. This mathematical extension (Platonic world) is audible and exciting (mental world). Through minimal impurity, music gains in variants and richness.


This is a post about the theory of the three worlds.

How the tempered scales came into being

Musical scales before tempering

Natural musical scales

The musical scales of human cultures developed naturally, i.e. without any conscious mathematical considerations whatsoever, in the course of millennia. The fact that there is a great deal of mathematics behind them nonetheless has something to do with the resonances between the scale tones and the fundamental tone. These resonances strike us as attractive, and music that is based on such resonances is capable of uniting human communities.

Mathematically, resonances can be traced back to fractions with as low numbers as possible, and we were able to deduce mathematically which nine intervals have to display the most distinctive resonances. It is not by chance that all the musical scales commonly used worldwide – i.e. the standard pentatonic scales, our major and our minor scales, the ecclesiastical modes and many more – exclusively consist of a selection of five or seven of these nine tones.

The fundamental tone and the tension

What can also be observed in all cultures, and what we cannot imagine being otherwise, is the fact that all musical scales have a clearly defined basis, i.e. a fundamental tone, to which the other tones are always related. This is the fundamental tone in relation to which each scale tone builds up its resonance; the stronger it is, the more harmonious the tone sounds beside the fundamental tone, and also inside a melody. On the other hand, the higher the numbers in the fractions, i.e. the poorer the resonance between a tone and the fundamental tone, the tenser the melody tone appears to us. The sharpest tone in our major scale is the major seventh (for instance H in C major), which is a semitone below the octave. This tone has the strongest tension of all tones in the major scale; it calls for its resolution to the octave, thus leading the melody from tension to relief. This tension is only possible because the fundamental tone resonates audibly or inaudibly in the course of a melody and the resonance of the major seventh to it is tense.

All this, however, still has nothing to do with tempering and works in pure major.


Why were the musical scales tempered?

Two yardsticks: linear and exponential

Tempering means that the frequencies of the scale tones are slightly altered, reduced or increased – i.e. tempered. At first sight, this slightly weakens the resonances; nevertheless, tempering has prevailed in Europe’s musical culture as a matter of course.

To understand tempering, it is helpful to understand that we measure intervals against two different yardsticks: one of them is linear, the other is exponential. A precise explanation of the reasons for and consequences of these two yardsticks can be found here. In simple terms, this is about the fact that all the intervals are relative. Thus if I build a musical scale on the basis of C, the scale tones are related to C; if I choose another fundamental tone, for instance D, then the frequency of the E in the key of C is different from that of the E in the key of D. Let’s have a closer look at this surprising fact:

An example of the relativity of two yardsticks

In C major, the tone E is a major third above the fundamental tone C; the frequency of E amounts to 5/4 of the fundamental tone C. A tone D is a major second above the C and thus has 9/8 of the frequency of the C. If we now choose this D (=9/8) as the fundamental tone, then E also occurs in D major, but this time as a major second. The crucial point is that this E is not precisely as high as was the E of C major previously:

Tonality  (key) Frequency in relation to
the fundamental scale tone
Frequency
in relation to C major
Function of the tone
in C major and D major
C major C = 1 1 Fundamental tone
D = 9/8 9/8 Major second
E = 5/4 5/4 Major third
D major D = 1 9/8 Fundamental tone
E = 9/8 9/8 x 9/8 = 81/64 Major second

Table 1: Relativity of the frequencies in relation to the fundamental tone

In Table 1 you see that the seemingly identical E has different pitches in the two musical scales:

E in C major = 5/4     = 1.25
E in D major = 81/64 = 1.266

When I tune a string to the E of C major, the string will not completely accord with the expected E in D major. The difference is slight, but it can be measured and is perfectly audible to keen ears.

The pure intonation only works for a defined fundamental tone 

As soon as the fundamental tone changes, all the scale tones relate to the new fundamental tone with regard to resonance, and the pitches of the earlier key are not all the same any longer. As outlined before, the reason for this is the two different ways of measuring the intervals: once in a linear way (auditory perception) and once exponentially (physical frequencies).


The three worlds

Again, this is about the three worlds according to Penrose: the auditory perception takes place in the mental world, the frequencies are part of the physical world, and the ideal world is that of the mathematics of the integer fractions. In music, all three worlds interact with each other in an extremely interesting way.


The objective of tempering

The point of tempering is the fact that the fundamental tone can thus be freely exchanged without the resonant musical scales known from before having to be given up. Tempering is an ingeniously chosen compromise, which is really capable of uniting both goals.


Historical development

The fascination with resonant acoustics is typical of all human cultures. This is how the highly resonant musical scales came into being: the standard pentatonic scales and the major scales, various minor scales and the Doric mode frequently used in Gregorian chant. The music which previously used to be played with these musical scales always had a constant tonality, i.e. one fundamental tone that is not changed as long as the melody is being played. All the tones of the melody resonate with the fundamental tone, and the fact of how strongly the tone of the melody resonates with the fundamental tone indicates the tension which the melody has with the individual tones.

To support the fundamental tones, polyphonic instruments from earlier cultural epochs often have additional strings (drone strings) or pipes (bagpipe) whose pitch cannot be changed to ensure that precisely this tension between the melody tone and the fundamental tone is emphasised. Whereas the tones of the melody vary, the fundamental tone (drone) is sounded throughout the piece and provides it with a solid base.

However, certain novelties gained ground in the Renaissance. Thus people began to change the fundamental tone during the piece. This allows for greater diversity in music. As long as the keys were related to each other, only slight frequency deviations were caused; in the case of really closely related keys, they also merely concerned one tone. The further the keys were apart from each other, however, the more difficult things became. Thus it sounded very unpleasant, for example, if someone tried to play F sharp major on an organ tuned to C. This marked the beginning of a period in Europe of a wide variety of experiments with slightly altered temperaments which attempted to reconcile the incompatibility of the fundamental tone shift with the purity of the intervals in various compromises. What ultimately prevailed in the late Baroque period was the equal temperament, which constitutes a really convincing compromise and enabled the rich development of harmonics in classical music and modern jazz.

More about this in the next post


This is a post about the theory of the three worlds.

Two Less Resonant Intervals for the Gaps

Starting point: two gaps

In the previous post, we saw that in the sequence of the ten scales tones found so far, there are two gaps. Can we find resonant tones there, too? We already know the following:

  • We already know the ten most resonant intervals in the octave.
  • These ten intervals serve to constitute the five standard pentatonic scales and our major and minor scales. There, the gaps are not obtrusive; they are only conspicuous in the distribution of all the ten potential scale tones.
  • Intervals do not occur on their own, either in a chord or in a melody. Thus if we have a resonant interval, we add another one to it and calculate the resulting sum interval. Or we look at the distance between two intervals and calculate the distance between both intervals, i.e. we subtract one from the other.
  • Owing to the exponential progression of the frequencies, and contrary to our intuitive expectation, when we calculate the frequency of the two intervals added together, we do not have to build the sum of the two interval fractions, but have to multiply them, and with regard to the distance between the two intervals, we must not subtract one interval from the other, but divide one by the other.

Since we have already found all the resonant intervals within the octave, we cannot expect any more highly resonant intervals for the two gaps. Yet although, if we are successful, the intervals are not all that resonant any longer in terms of their resonance with the fundamental tone, they may still have direct and thus very resonant relations to other scale tones. In musical configurations, this also makes them interesting in terms of resonance, depending on the situation.

Here are my attempts to fill in tones for the gaps. To illustrate this, I will first show the distribution of the ten most resonant intervals from the previous post:

Fig. 1: Distribution of the ten most resonant scale tones in an octave (logarithmic representation)

The minor seconds

Regarding the first gap, we cannot find a tone with good resonance with the fundamental tone. Because the two tones of the interval must be close together, the fraction of their frequencies must be close to 1 and therefore its numerator and denominator cannot be small numbers – as would be preferred for a good resonance. We therefore check as candidates fractions that are still created by relatively small numerators and denominators:

Octave – major seventh = 2 : 15/8 = 16/15 = 1.067
Fourth – major third = 4/3 : 5/4 = 16/15 = 1.067
Minor sixth – fifth = 8/5 : 3/2 = 16/15 = 1.067
Minor third – major second = 6/5 : 9/8 = 48/45 = 16/15 = 1.067
Major third – minor third = 5/4 : 6/5 = 25/24 = 1.042
Major seventh – minor seventh = 15/8 : 9/5 = 75/72 = 25/24 = 1.042
Major sixth – minor sixth = 5/3 : 8/5 = 25/24 = 1.042
Minor seventh – major sixth = 9/5 : 5/3 = 27/25 = 1.08

In this way, we can find several intervals which satisfy the requirements and fit into the first gap:

  • 16/15 = 1.042
    – major third – minor third
    – major sixth – minor sixth
    – major seventh – minor seventh
  • 25/24 = 1.067
    – minor third – major second
    – minor sixth – fifth
    – fourth – major third
    – octave – major seventh
  • 27/25 = 1.08
    – minor seventh – major sixth

These three intervals all sound rather twangy. We call them minor seconds. In the pure temperament, there are at least three of them.

The tritones

The second gap is right in the middle of the musical scale. We are now going to try to fill this gap with a combination of two known resonant intervals:

Major seventh – fourth = 15/8 : 4/3 = 45/32 = 1.406
Minor seventh – major third = 9/5 : 5/4 = 36/25 = 1.44
Major sixth – minor third = 5/3 : 6/5 = 25/18 = 1.389
Octave + minor third – major sixth = 2 x 6/5 : 5/3 = 12/5 : 5/3 = 36/25 = 1.44

Once more, this results in three intervals that are close together:

  • 25/18 = 1.389
  • 45/32 = 1.406
  • 36/25 = 1.440

Insertion of the minor seconds and the tritones into the sequence of the scale tones

Our calculations so far have concerned frequency ratios. As explained in the previous post, we have to convert them logarithmically to ensure that their distances correspond to what we perceive in our mental world. In a logarithmic representation (on basis 2), we arrive at the following distribution:

Fig. 2: The three minor seconds (“semitones”) and three tritons fill the gaps of Fig. 1

Although we can see that the proposals fill their respective gaps well, we have three proposals for each gap! Which of them is the best? We could argue in favour of that with the smallest figures in the numerator and denominator, or in favour of that with the greatest frequencies among the variants, or in favour of that with the closest relation to a musical scale we already know. However, it is for a good reason that we defer this issue until we deal with the well-tempered scales.


This is a post about the theory of the three worlds.

The Distribution of Tones within the Octave

The ten most resonant tones within the octave

In this series of texts, we examine musical scales from the perspective of the three worlds. All three worlds are involved, as we saw, for example, when we answered the question as to why the musical scales of all musical cultures always cover precisely one octave. This cannot be explained in purely mathematical or physical terms. It is only through the involvement of the third world, namely our mental world, that the significance of the octave becomes evident.

The selection of the tones used in a musical scale is determined by all three worlds through the phenomenon of resonance, as we saw in previous posts. Let us now have a look at how the ten most resonant tones are distributed within the span of the scale octave. We will see that there are gaps in this distribution, and then think about what conclusions we can draw from this.

Here are these ten tones once again. (The table lists eleven tones, but the fundamental tone and the octave count as one tone, since the octave is both the highest tone in the present octave and simultaneously the fundamental tone of the next higher octave, in which the musical scale repeats itself.)

Table 1: The ten most resonant tones in an octave 

The middle column of Table 1 lists the fractions which indicate the ratio between the scale tone and the fundamental tone; with the fifth, for instance, the frequency is 3/2 times the fundamental frequency. On the far right, I have expressed these fractions as decimal numbers to make comparisons easier. Of course, the numbers range from 1 (fundamental tone) to 2 (octave).

 The distribution of the ten tones

To see how the ten tones are distributed within the octave, we take the frequencies of the tones and compare them with the frequency of the fundamental tone. These frequency ratios are in the right-hand column of Table 1. I have transposed these numbers into Figure 1, and you can see how the frequency ratios are arranged linearly.

This is what the arrangement of the intervals of Table 1 looks like:

Fig. 1: Frequency ratios of the potential scale tone (Table 1) in linear representation. Here, the intervals are regarded as starting from C, i. e. C = fundamental tone.

In Fig. 1, we are immediately struck by how irregular the distribution is. It displays four major gaps, namely between C and D, F and G, A and B, and H and C. Subjectively, the distribution does not appear to be , either; for example, the distance between the fundamental tone and the fourth (C-F) is much shorter than that between the fifth and the octave (G-C). However, we perceive both intervals as the same, namely as two fourths, for the distance between G and C is also a fourth, just like that between C and F. Nonetheless, the distance between G and C in Figure 1 is much longer than that between C and F. Why does the perceived distance not correspond to the real frequency ratio? – The answer is again in the exponential progression of the frequencies (physical world), which does not correspond to our linear perception (mental world). We therefore have to notate the frequencies logarithmically, thus arriving at a representation which corresponds to our subjective perception:

Fig. 2: The frequency ratios of Fig. 1 in a logarithmic representation.

We can see that in Fig. 2, the distances correspond to our subjective perception as being truer to scale. Thus, in contrast to Fig. 1, the distance between C and F is the same as between G and C, which we perceive as correct, namely as a fourth each time. The other distances in Fig. 2 also correspond to our perception.

The distribution still looks irregular, though, and there still are gaps. However, these have also shifted now. The gap between C and D has become bigger, whereas that between H and C has become smaller. What is really conspicuous are the two gaps C-D and F-G. Will we be able to do something here? Will we once more be able to resort to our reflections on resonance in order to fill the gaps? I will explain how this works in the next post.


This is a post about the theory of the three worlds.

How does the pythagorean comma come about?

The Pythagorean comma

The Pythagorean comma demonstrates that our tonal system is not perfectly consistent but has a gap whose form and cause I will describe in this post. The comma is relevant in terms of both ourpythagorean comma musical practice, since it has very specific effects, and of philosophy and science, since it is typical of the problems that we observe in the interplay of our  three worlds (according to Penrose). Thus it is a topic that is not solely relevant to musicians but also to people who are interested in the question as to how mathematics (ideal world), physics (physical world) and our experience (mental world) relate to each other.

To begin with, I’ll explain why this comma occurs.


Adding intervals

What happens if we add two intervals, for instance a fifth and a fourth? We will see that such an addition works perfectly in some cases but leads to problems in others. This is where the origin of the comma lies and the reason which led to the tempered mood. Why do problems occur here at all? This is what I would like to have a closer look at now, later also at the solution to the problem.

For intervals, adding means multiplying

Can we add intervals just like that? The problem is that we sense the intervals as behaving linearly but the frequencies do not increase linearly, but exponentially.

Fig. 1: Exponential increase of the frequencies

Between tone A (110Hz) and tone a (220 Hz), there is a distance of 110 Hz. This distance corresponds to an octave – but only there! If we now measure the distance from tone a to tone a’, which again corresponds to an octave, we do not get 110Hz, but 200Hz. And from a’ to a”, we already get 880Hz. This is because the frequencies do not increase in a linear fashion, but exponentially, as depicted in Fig. 1.

This exponential behaviour is where the root of the problem lies and which has led European musicians to try out various tempered moods, with the equal temperament ultimately prevailing. The exponential behaviour has the following unexpected effect: When we add intervals, we cannot add, but have to multiply their frequency ratios, and when we subtract them, we have to divide them. The operations shift from addition/subtraction to multiplication/division.

This shift of operations is known in other domains, too: Before pocket calculators and computers, technicians used so-called logarithmic tables and slide rules, which are based on precisely this shift. This effect can also be observed in the combinatorial explosion and in the progress of epidemics.


Examples of interval additions

 What always interests us when we look at intervals is the ratio of the frequencies of the two tones of the interval, i.e. the fraction between the frequency of the higher tone divided by the frequency of the lower tone: X = f2/f1. This fraction determines the intervals which we perceive.

Octave and octave

Cf. Fig. 1: we add two octaves, for example A-a and a-a.

The first octave (A-a) measures 220/110 = 2.
The second octave (a-a’) measures 440/220 = 2.
The two octaves together make 2×2 = 4.

The multiplication is correct here since a’ is 4 times as fast as A (440Hz / 110Hz). Thus two octaves result in a quadruplication of the basic frequency.

Fifth and fourth

The fifth corresponds to a frequency acceleration to 3/2,
the fourth to an acceleration to 4/3.

The fifth and the fourth together result in 3/2 * 4/3 = 12/6 = 2.

2 corresponds to an octave. Thus mathematically, a fifth and a fourth together result in precisely one octave. Here, too, the mathematics is in line with what we expect to hear.

Further additions to an octave

Minor third plus major sixth = 6/5 * 5/3 = 30 /15 = 2.
Major third plus minor sixth = 5/4 * 8/5 = 40 /20 = 2.


Above an octave

Two fifths: 3/2 * 3/2 = 9/4 = 2.25.

2.25 is bigger than 2, i.e. we have left the range of an octave. Where have we ended up? With a ninth. A ninth is an octave plus a major second. Will this work out? We calculate:

Octave plus major second = 2 * 9/8 = 18/8 = 9/4 = 2.25.

Yes, it works! When we add two intervals, we must multiply their fractions.

Subtraction becomes division

Of course, it also works the other way round: we can subtract one interval from another. Then, we have to divide:

An octave minus a fifth = 2 : 3/2 = 4/3 = a fourth.
A fifth minus a fourth = 3/2 : 4/3 = 9/8 = a major second.


Why do these calculations work out?

I was surprised to see that despite the shift in arithmetic operations, the calculation provides precisely the results which a musician expects. How can this be? After all, a multiplication is something completely different from an addition. Why can we calculate like this all the same?

The fact that the calculations work out so perfectly is not a matter of course, and they do not always work out perfectly, either, as we are about to see. In the above examples, though, they work out, and this is only the case because we made a clever choice of intervals. The reason for this is that in the course of many millennia, countless musical people developed a tonal system which allows for precisely this.

As represented in earlier posts, the choice was everything but random, but made according to the criteria for resonant musical scales. And we noticed that intervals are resonant when they represent fractions with integral numerators and denominators and that it is necessary for a simultaneous resonance of several intervals that both numbers are as low as possible and, in particular, do not contain a prime higher than 5 in the prime factorisation.

These restrictive conditions enable us to reduce the fractions when we compare the intervals with each other. High numbers, and high primes in particular, do not lend themselves to reduction. As we have seen in the above examples, however, reduction is a great help for us.

This is most important for scales and chords which both have more than just two tones. Let us look at a combination of two intervals into a bigger interval. If we calculate the combined interval, we have to multiply numerators and denominators, which will soon result in very high numbers. High numerators and denominators, however, mean low resonance. If we are now able to reduce the resulting fraction, the reduced numbers of numerator and denominator will show that resonance still is going to take place. Therefore combinations of several intervals – a property of scales and chords – may lead to new intervals that still are resonant – if it is possible to reduce. And only in this way will a combination of two resonant intervals result in another resonant interval. And only then will we remain within our resonant musical scale.

Therefore, reducing is quite practical, especially if we combine several intervals.

– Only, it does not always work out.


It would be too nice…

The shift from linear to exponential growth leads to problems faster than we expect. Even the simplest calculations do not work out:

Major second plus minor seventh = 9/8 * 9/5 = 81 / 40 = 2.025.
We do not actually expect 2.025, but 2, i.e. a pure octave.

Major third minus major second = 5/4 : 9/8 = 40/36 = 10/9 = 1.111.
We actually expect a major second, i.e. 9/8 = 1.125.

A fourth and a fourth = 4/3 * 4/3 = 16/9 = 1.777.
We actually expect a minor seventh = 9/5 = 1.800.

As you can see, our expectation that calculations with intervals work out has been disappointed. Only very few interval combinations allow for “pure” calculations, and then only because we have chosen our scale system so well. All the other interval combinations do not work out. Usually, the discrepancies are not very big, but they still exist distinctly. The phenomenon that interval combinations do not work out has become famous under the name of Pythagorean comma.


The Pythagorean comma

Pythagoras already knew that natural, i.e. well-sounding intervals can be traced back to simple fractions with low numbers. As we know, the simplest intervals with the lowest figures are the octave (2/1), the fifth (3/2) and the fourth (4/3).

As we have seen above, two fifths together result in a ninth. How many fifths does it take to return to the fundamental tone? Let’s look at this from the C and examine how many fifths it takes to reach another C:

C – G:  first fifth
G – D:  second fifth

The whole sequence is as follows:
C – G – D – A – E – H – F# – C# – Ab – Eb – B – F – C

We thus have twelve fifths. The low and the high Cs are seven octaves apart. Thus the upper C, calculated through the fifths is 3/212 = 129.746 and calculated through the octaves is 27 = 128. The discrepancy between the two calculations is 129.746 : 128 =  1.0136.

This small discrepancy of 0.0136 is the Pythagorean comma.


Placing the comma into a larger context

The Pythagorean comma is inevitable and ultimately derives from the fact that mathematically speaking, we are sailing under two different flags here, namely one which adds and another which multiplies. Our thinking, which takes its primary orientation from matter and space, is accustomed to the linear calculations with which lengths are measured. We also look at the intervals in this manner. However, these work through frequencies and their mutual ratios, and those are not linear, but exponential.

Incidentally, this is not the only field where our habit of thinking in a linear way becomes a trap. Many processes progress exponentially; examples are the combinatorial explosion, the observation of a collective of several objects, and the progress of an epidemic, social trends, etc. As soon as what happens becomes complex, exponential conditions should not surprise us.

This leads us back to the fundamental issue of this series, namely the relations between the three worlds. What is the role of mathematics for physics and our minds? I will leave this question unanswered here and remain in the field of music for the time being. In the next post, I will explain the advantages provided by our solution to the comma problem, namely the equal temperament.


Conclusion

 Calculations with intervals are made by multiplying and dividing the fractions of their frequencies.

  1. This is in contradiction to our intuitive idea that this involves additions and subtractions.
  2. For this reason, most “additions” and “subtractions” of two intervals do not result in the pure intervals we expect.
  3. Only very few additions/subtractions of intervals result in other pure intervals. This is only possible if the fractions of the intervals involved can be reduced.
  4. This is subject to the same rules as those applying to the determination of resonant scale tones: numerators and denominators must be low numbers; higher primes, in particular, hamper resonance and calculability.
  5. The Pythagorean comma is an expression of this fundamental mathematical incompatibility of linearity (addition) and exponentiality (multiplication).
  6. Thus the Pythagorean comma places a limitation on the pure intonation.

We will soon see how the problem of the Pythagorean comma is solved by means of the equal temperament. But before that, we will have a look at the distribution of the tones within the octave in just intonation.


This is a post about the theory of the three worlds.

The major scale introduces tension to the resonances

The major scale

The major scale (Ionian mode) is the most widespread musical scale both in Europe and globally. It is a heptatonic scale, i.e. a musical scale with seven tones. It is characterised by very special resonance ratios, which serve well to explain its worldwide appreciation.

Below, I have listed the tones of the major scale of C, ascending from the bottom to the top, together with the intervals between each tone and the fundamental tone. Of course, it is these intervals that constitute the musical scale. We could also start the musical scale with any other tone and only speak of the intervals (second, third, etc.) since we only need the distances between the tones to describe a musical scale. However, I am using the tone of the C major scale, simply because it is clearer and also enables you to reproduce it more easily on a piano or other instrument.

The interval designates the ratio of the frequency of a scale tone to the frequency of the fundamental tone. This interval always lies between 1 (fundamental tone) and 2 (octave) in every musical scale. We express it in the form of a fraction.

C      2
B      15/8
A      5/3
G      3/2
F      4/3
E      5/4
D      9/8
C      1

Table 1: The C major scale

The fractions enable us to recognise what is typical of the major scale. It can be shown very well that what we hear subjectively (mental world) is perfectly in parallel with what we are able to represent mathematically in simple fractions (Platonic world). It turns out again that the three worlds (according to Penrose) interact perfectly in the field of music.

All the tones resonate with the fundamental tone

In a previous post, I established resonance criteria for scale tones. These criteria yield ten tones which strongly resonate with the fundamental tone. Like the standard pentatonic scales, the major heptatonic scale consists of a selection of these ten most resonant tones. Thus we may assume that the major scale is generally strongly resonant in itself. Yet not every tone is equally resonant with the fundamental tone. And among themselves, in particular, tones resonate very differently. This is where the story becomes interesting. To begin with, we’ll have a look at the difference between the major heptatonic scale and the major pentatonic scale.

The major heptatonic scale as an extension of the major pentatonic scale

The standard pentatonic scales are the most strongly resonant musical scales in general, and the most resonant of them is the major pentatonic scale. The major heptatonic scale can be regarded as an extension of the major pentatonic scale. Both musical scales are subsets of the ten most resonant tones.

Table 2: Comparison between the major pentatonic and the major heptatonic scales

The tones which are added to the heptatonic scale in comparison with the pentatonic scale explain the difference. Whereas the pentatonic scale is resonant throughout and all tones can be mixed at random without any tensions occurring, this is no longer the case with the heptatonic scale. The two additional tones, F and H, introduce the necessary tension for the matter to become interesting.

The first thing that strikes us is the fact that with H, that tone is added which among the ten most resonant tones has the highest digits in the numerator and the denominator. Among the ten most resonant tones, it is therefore the tone that resonates most poorly with the fundamental tone, the tone that creates most tension. This concerns its relation to the fundamental tone.

However, the intervals between different scale tones apart from the fundamental tone play a role in musical scales, too. We calculate this by dividing the frequency ratio of the upper tone by that of the lower tone (for the reasons for dividing, cf. Appendix). The two new additional tones in the heptatonic scale, i.e. F and B, generate a tension both mathematically and audibly such as has not occurred in the pentatonic scales so far. If, for instance, we sound F together with E, then this results in a frequency ratio of 4/3 : 5/4 = 16/15, a fraction which indicates a resonance that is difficult to achieve. H and the upper C sounded together have a similar result; here, the ratio is 2/1 : 15/8 = 16/8 : 15/8  = 16/15. Thus the interval between H and C is as full of tension as that between E and F. The interval between F and H is even more critical; here, the fraction is 15/8 : 4/3 = 45/32. These are two high numbers indeed, compared to what we have seen before, and they offer no possibility of reducing.

Tension and easing

Poor resonance means tension since both tones do not easily go together. We apprehend this subjectively (mental world), as you can easily find out by striking an E and an F at the same time on the piano and comparing the sound with the simultaneous sounding of, say, E and G. E and F produce more friction. Physically, the mathematics of the frequency ratios express a lesser or higher degree of readiness to enter into a resonant relationship, and we can hear this.

Musically, however, tension is not without interest. The major pentatonic scale without F and H may strike us as quiet and harmonious, but also as a bit dull. Conversely, the major heptatonic scale contains little peppercorns which introduce an exciting pungency, similarly to the hotness of chilli peppers in a dish. Yet whereas you can still feel this hotness as a long aftertaste, this pungency in music can be switched on and off with precision by simply replacing the tone that creates tension with a quiet one that resonates without any problem. This interplay of tension and easing is extensively used in music.

The major heptatonic scale as a subset of the ten most resonant tones

As illustrated in Table 2, the major scale is a selection of seven tones from the list of the ten most resonant tones. This selection packs a punch. I will shortly deal with the mathematical details which result from it. You will probably not be surprised that these mathematical details are again in parallel with our listening experience. Let’s first look at which of the ten tones are missing in the major scale: they are Eb, Ab and Bb. As always, we look at the fractions of these three intervals: 6/5, 8/5 and 9/5. We quickly notice that all these fractions have the denominator 5. Conversely, none of the tones in the major heptatonic scale do have the denominator 5.

Reducing the denominator

This fact of the missing denominator 5 facilitates the resonances within the major heptatonic scale. If two tones have the same denominator, this is cancelled out when we sound the two tones simultaneously. If, conversely, there are different denominators, resonance will be made more difficult. Different denominators, however, can be reduced too, if they can be divided by each other.

For this purpose, we conduct an integer factorisation. In the major scale, for example, the major seventh and the major third resonate perfectly with each other: in order to calculate the resonance mathematically, we divide the major seventh by the major third, which results in 15/8 : 5/4 = 3/2. These 3/2 luckily indicate a perfectly resonant interval, namely the fifth.

The cancelling in this case is possible because in integer factorisation, both the denominator 8 and the denominator 4 contain the prime number 2 twice (8=2×2×2 und 4=2×2). Thus the factor 2 can be cancelled out (even twice, since both denominators contain it twice). This cancelling (reducing) can be done whenever two numerators contain the same factors.

For this reason, it is “clever” of the major scale to omit precisely all the tones with the denominator 5. In this way, the 5 never intrudes, and reductions are more easily possible. And reduced fractions with regard to the frequency ratios mean better resonance both physically and mentally.

The resonance of the totality of all scale tones together

We can roughly estimate the reduction behaviour of the totality of all the tones in a musical scale by calculating the lcm (least common multiple) of all the denominators, as we already did with the pentatonic scales. The tones of the major heptatonic scales have an almost unbeatable lcm of 24; it is even the same as that of the major pentatonic scale with two fewer tones.

This low lcm is of course the result of the fact that there aren’t any tones with the denominator 5 in the scale; otherwise we would have to multiply the lcm by 5, which would result in 120.

The lcm is useful but does not say everything

However, the lcm doesn’t show the whole resonance behaviour of the musical scale. It is only a measure for the resonance of all the scale tones with the fundamental tone, but does not say anything about how the scale tones resonate with each other.  For the above-mentioned interval of the tones F and B, for instance, we obtain a frequency ratio of 45/32, which means that even if the tones resonate well with the fundamental tone, they can be full of tension among themselves.

This is not a weak point, though; rather, it makes the musical scale interesting. In this respect, the major heptatonic scale is clearly more interesting than the major pentatonic scale although both of them have the same lcm.

Nonetheless, the lcm is a valuable yardstick against which the basic resonance potential in the musical scale can be roughly measured, since with a high lcm, i.e. when the denominators cannot be reduced, the dissonances are sharper in any case.

Triads in the major scale

We now turn our resonance considerations to three tones that are sounded simultaneously. Let us analyse the triad of C, E and G. The frequencies (cf. Table 1) are: 1 – 5/4 – 3/2. To calculate the ratio of all three tones to each other, we must place all three of them on the same denominator. It is precisely for this that we need the lcm again, which in this case is 4. Thus 1 – 5/4 – 4/2 results in a ratio of 4/4 – 5/4 – 6/4. The common denominator can be cancelled out, and the ratio of the frequencies of C-E-G is 4 – 5 – 6.

This is absolutely the most resonant ratio which is possible when three different tones are sounded together. The C-E-G triad is the major triad with which everyone is familiar. On the piano, it is slightly distorted due to temperament, but even so you can easily try out for yourself how catchy and attractive this triad is. No wonder it plays such a dominant role in pop and folk music.

Three major triads in the major heptatonic scale

However, the major heptatonic scale does not contain the major triad only once, but three times. Look at the tones F – A – C, in fractions 4/3 – 5/3 – 2, or all the tones placed on the common denominator 3: 4/3 – 5/3 – 6/3, i.e. again 4 – 5 – 6. This demonstrates once more the benefit which the common denominator (here: 3) represents for the resonances. Thus the major heptatonic scale did well to include the tone of F, which allows for a second major triad.

But H is also well chosen, for it enables a third major triad in the major heptatonic scale. This time, we start with G and add H. As a third note, we take D, an octave higher than usual, i.e. just above the upper C. For this purpose, we have to multiply the 9/8 of D by 2, which results in 9/4 (cf. calculation rules).  This fraction is higher than 2, i.e. is already above the octave. Let us now look at the tones G – H – D; the frequencies are 3/2 – 15/8 – 9/4. With the lcm=8, we obtain 12/8 – 15/8 – 18/8. We can now reduce the numerators and denominators, which again results in 4 – 5 – 6, i.e. the same ratio as above, the same perfectly resonant major triad as those with the roots C or F.

Thus the major heptatonic scale contains the major triad no fewer than three times, for three tones of the heptatonic scale can be combined in this most resonant manner three times. What is also remarkable, however, is the fact that the three triads do not mix very well with each other. This is easily heard; again, the mathematics perfectly corresponds to our subjective mental experience (sorry, I have to mention this thing with the three worlds again and again; I myself am surprised how well they converge in musical scales).

Of course, the major scale was not “invented”, and certainly not by a mathematician. Rather, this musical scale was found by people who themselves actively made music and thus became aware of the particularly interesting resonance relationships which result from the combination of tones.

As we just have seen, there are three independent collections of tones inside the major scale which resonate well within themselves but harmonise less well with the other selections. This results in three different colours or harmonies inside the major scale which can be used separately and whose sequence can be planned in a piece of music and thus tells us a story. The three colours are defined by the root of each triad, namely by the fundamental tone of the musical scale (C), its fourth (F) and its fifth (G). The three tones are called tonic (fundamental tone), subdominant (fourth) and dominant (fifth). The possibility of playing with such colour far exceeds the possibilities of the major pentatonic scale and was studied and perfected in Europe throughout the course of the centuries.

Minor triads

These triads, too, have a special resonance ratio, namely 10 – 12 – 15. The numbers are slightly higher than in the major triad, which makes the minor triad slightly less resonant. For three different tones, however, the ratio is still extremely simple and thus resonant, and minor triads are certainly not dissonances.

In the minor triad, the minor third is the first tone with a frequency ratio with the denominator 5, whereas the major scale does not have this and prefers denominators based on the prime number 2. This produces a distinctly different colour. With the denominator 5, we have already reached the highest “permitted” prime number, much higher than 2 and its easily divisible multiples found in the major scale. The minor scale therefore sounds softer, more special and not as as the major scale.

The minor triads cannot only be found in the minor heptatonic scale but also in the major heptatonic scale, simply based on less prominent scale tones, specifically on D, E and A. In principle, minor colours can also be produced in the major scale, even though only on less prominent tones.

Conclusion

All in all, the seven tones of the major scale constitute an almost inexhaustible source of combinations. The major scale combines a maximum degree of resonance with the possibility of generating tension and different colours. All this can be easily reconstructed by means of simple mathematical fractions – in complete harmony with what we hear subjectively.

To continue, we will have a look at the Pythagorean comma. This is particularly interesting because it shows how the mathematical world reaches its limits in the physical world. This fact has resulted in an evenly tempered intonation, an “impure” intonation. So have a look at how the Pythagorean comma emerges.


This is a post about the theory of the three worlds.

Standard pentatonic scales

As we have seen in the previous post, the tones C – D – E – G – A – C constitute the standard major pentatonic scale.

All in all, another four pentatonic scales can be created with the simple criteria for resonant pentatonic scales. These five pentatonic scales are the five musical scales which according to our mathematical criteria allow for resonances among all their tones.

We will see later on that we are able to create all the musical scales traditionally used in Europe with our pool of the nine most resonant tones. In the heptatonic scales, however, for instance in our diatonic major scale, certain tones resonate poorly with each other, which is actually more interesting in musical terms, since this creates a natural structure within the scale tones.

Conversely, pentatonic scales do not have any “wrong” tones however we mix them. It appears as if there would be no resistance, no matter which tones we sound together.

The five pentatonic scales are:
(on the basis of C)

C –  D  –  E  –  G  –  A  –  C

C –  D  –  F –  G  –  A  –   C

C – Eb – F –  G   – Bb  – C

C – Eb – F –  Ab – Bb  – C

C – D  –  F –  G   – Bb  – C

The same tones in fractions:
(fundamental tone = 1)

1 – 9/8 – 5/4 – 3/2 – 5/3 – 2

1 – 9/8 – 4/3 – 3/2 – 5/3 – 2

1 – 6/5 – 4/3 – 3/2 – 9/5 – 2

1 – 6/5 – 4/3 – 8/5 – 9/5 – 2

1 – 9/8 – 4/3 – 3/2 – 9/5 – 2

For the purpose of the calculation, I assume that all the tones are sounded simultaneously and within an octave with the fundamental tone as the lowest tone. Thus I calculate the intervals between each tone and the fundamental tone. Then I look at the denominators of all the intervals and look for the least common multiple (lcm) of these denominators. This shows how easily resonances can occur between all the tones:

Table 1: Vibration ratios of 5 standard pentatonic scales
How do you read Table 1?

I have placed the same intervals of the different pentatonic scales in the same column. This results in holes in the table. Thus the major pentatonic scale does not have a fourth and a seventh. Furthermore, I have given tones with the same denominator in the fraction the same colour.

What do the colours show? – As we know, identical denominators mean that the two tones are particularly resonant because in the calculation of their frequency ratio, the two denominators “reduce themselves away” when the tones sound together. This leads to particularly simple, i.e. particularly resonant ratios between tones with the same denominator. For seconds, major thirds and fifths, I have chosen three different shades of green. The denominators are not identical but are always based on the prime number 2, which means that a reduction by two is always possible. Therefore the different greens always mix very well.

In mathematical terms, the major second has the highest denominator of all intervals in the pool – 8 – but this is not a problem. 8 is not a prime number, but 23, just as the denominator of the major third, 4, equals 22. If we sound the fifth together with the major second, the ratio of the two tones is 3/2 : 9/8 = 3×8 / 2×9 = 4/3, i.e. a fourth. The fourth is the third most resonant interval that is possible within an octave; a major second and a fifth are therefore perfectly resonant. Reduction proves effective.

Furthermore, the colours in Table 1 also show the different forms of the third and the sixth. Thus the 5/4 third is the major third and the 6/5 third is the minor third.

Musical scales are about the interval between scale tones and the fundamental tone, but also about the ratio of scales tones among each other. The lcm is the indicator of how close the resonances are in this respect. The lower the lcm, the more resonant the musical scale as a whole. However, critical tones can also be omitted or inserted into a melody as a special accent, but this is hardly possible in the standard pentatonic scales with their low lcms.

How do the five pentatonic scales differ from each other?

Major and minor scales

In functional harmony, a genuine and trailblazing European invention, the third plays an important part. Whether major or minor is a question that is always being asked. However, we are not going to deal with functional harmony (yet), but we can nevertheless look at our five pentatonic scales from the perspective of the third. We then see that we have a major pentatonic scale (with a major third) and two minor pentatonic scales (with a minor third).

The two minor pentatonic scales differ from each other in that one of them does not have a fifth. Although the lcm is the same and low in both cases, the lack of a fifth is a great handicap musically (and with regard to resonance), which is why the minor pentatonic scale without the fifth is hardly ever used. Our customary minor pentatonic scale is the one with the fifth.

When you look at Table 1, you can see the difference at once by the colours: the minor pentatonic scales has tones that are reddish in colour (denominator 5) whereas the major pentatonic scale does not. In each case, the third attracts further intervals with the same denominator. The reason for this is to be found in resonance again: identical denominators guarantee strong resonance.

Sus pentatonic scales

The term “sus” derives from “suspended fourth”. Where does the expression come from? – In classical European music, i.e. in functional harmony, the third is the crucial factor. A chord which does not have a third but has a fourth instead, is suspended, i.e. it must first be resolved; the fourth is regarded as a suspension and must be resolved into the major third. In other styles such as jazz or modern pop music, the sus chord is a chord like any other, a colour like minor and major. There are also sus musical scales without thirds, whether minor nor major, both in world music and in jazz. After the octave and the fifth, the fourth (4/3) is the most resonant interval.

Again, there are two forms of sus pentatonic scales. The high lcm of 120 of one pentatonic scale results from the fact that it has both a major second (denominator = 9) and a minor seventh (denominator = 5), which means that reduction is not possible (as you can work out for yourself if you calculate the lcm yourself). The sus pentatonic scale with the sixth has another problem: the fourth and the sixth combined with the upper fundamental tone constitute a major chord (4/3 – 5/3 – 6/3 →4-5-6). This major chord on the fourth is extremely resonant and thus becomes so dominant that the musical scale is easily misinterpreted as a major pentatonic scale.

Which pentatonic scales are commonly used?

As a consequence of the above-mentioned weaknesses of the one minor pentatonic scale and the two sus pentatonic scales, only the major pentatonic scale and the minor pentatonic scale with the fifth are commonly used. However, these two pentatonic scales are ubiquitous and very easy to sing. They can also be combined with other musical scales/chords in a melodious way, which is of particular musical interest. On their own, they sound somewhat commonplace, but show their whole strength in combinations. They are perfect building blocks for musicians.


To continue, we’ll have a look at the major musical scale. How resonant is it?


You can find an overview of the texts about the theory of the three worlds here.

Expressions around waves and sine waves

Sine waves play a crucial part for our considerations of resonance. On this page, I would like to explain the terms that I use in this respect.

Sinusschwingung

Wave

A wave is a motion in time which oscillates around a baseline.

A wave can have different shapes. For our considerations of resonance, we use pure sine waves; such a wave is shown in the graph above.

Amplitude

The amplitude is the deviation of a wave from the baseline. It does not play any primary role in our considerations.

Period

A period lasts as long as the wave takes to arrive in the same position to repeat itself again in precisely the same way. Depending on the shape of the wave, it crosses the baseline two or more times; in the case of the sine wave, it crosses it twice in opposite directions.

Wavelength

The wavelength is the length of a period.

Frequency

Frequency denotes the number periods per unit of time. It is the crucial value for our considerations of resonance, for whereas amplitudes and wavelengths change depending on the carrier medium of the wave, the frequency remains the same if the wave is transferred from one medium to another.

“Belly”

In each period, the sine wave has one belly in the positive direction and one belly in the negative direction. When we count the number of bellies per unit of time, we thus measure the frequency, i.e. counting bellies measures the frequency. This slangy expression has the advantage of being graphically illustrative. At the same time, the term “belly” emphasises indivisibility (mathematically: whole numbers!), whereas the frequency can be expressed in any real number.

With respect to resonance ratios, we always compare whole numbers, i.e. the number of completed bellies. Whether we only count the positive bellies or both the positive and the negative ones is irrelevant, since in a frequency comparison, only the relations of the frequencies matter.

Natural frequency

Certain physical media (strings, air columns in pipes, etc.) have the property of oscillating (vibrating) in a very specific frequency, their natural frequency.

Fundamental tone and overtones

Besides the natural frequency as the fundamental frequency (fundamental tone), the medium can also oscillate with a multiple integer of the fundamental frequency. Multiple integer means that the crucial factor is the number of whole periods (bellies).

Calculating with frequencies and intervals

On this page, I will explain some rules which are applicable when we calculate with intervals and their frequencies.

Intervals are fractions

An interval ranges from a lower tone to a higher one. The fraction of the interval is calculated by dividing the frequency of the higher tone by the frequency of the lower tone, for instance

E  =  330 Hz
A  =  440 Hz

440/330 = 4/3. This is a fourth. The interval of the fourth is always 4/3: in the fourth, the higher tone is precisely 4/3 times as fast as the lower tone.

What counts here are only the relative values, not the absolute ones. Whether I start the fourth with the E (E-A) or with the C (C-F) is immaterial; the relative frequency ratio is always 4/3. In other words, intervals are always relative.

The exponential progression of the frequencies

When we compare the intervals with each other, there is a crucial feature: the progression of the frequencies is exponential.

Fig. 1: The progression of the frequencies is exponential

In Fig. 1, you can see the frequencies of various A tones, from the great A (capital letter) to the treble a” (small letter with two inverted commas). On the piano keyboard, it looks as if the distances between all four As are the same, but if we look at the frequencies, the distances become increasingly bigger. In other words: the frequencies increase faster than the intervals. In mathematical terms, the intervals progress in a linear fashion whereas the frequencies progress exponentially. This has some consequences for calculations with intervals.

Adding intervals

When we add two intervals, then this is a multiplication with regard to frequencies. Thus the tone of the (great) A has the frequency of 110 Hz. When we move up an octave, then the (small) a has twice that frequency, namely 220 Hz. The distance between 110 Hz and 220 Hz is 110 Hz. Yet this 110 Hz is only an octave if we start with the capital A. If we move up another octave from the small a, we must not add the 110 Hz of the lower octave (which would result in 330 Hz), but have to add 220 Hz, thus getting from 220 Hz to 440 Hz.

Our spontaneous idea that an octave corresponds to a value in Hz is incorrect. An octave means that the lower frequency is multiplied by 2 (with 2 because the octave always doubles). The addition of the intervals becomes a multiplication. At first sight, this change in arithmetic operations may be confusing, but once we’re aware of it, the matter is not so difficult. We must therefore remember:

the addition (of intervals) becomes a multiplication (of frequencies)

Subtracting intervals

Unsurprisingly, subtractions work in perfect analogy. Subtractions become divisions.

Example

We are looking for the distance between a major third and the fifth above it. Our knowledge of music tells us that the distance between the two intervals is a minor third. Can we also calculate this?

In this comparison, we subtract the major third from the fifth. But instead of subtracting, we divide:

Fifth                 =  3/2
Major third  =  5/4

Fifth – major third → 3/2 : 5/4 = 3×4 / 2×5 = 12/10 = 6/5

As we know, 6/5 is the minor third. This method always works, for any interval:

To check, we can add the two thirds again and the result – of course with a multiplication – is:

5/4 x 6/5 = 30 / 20 = 3/2

The major and minor thirds again result in the fifth (3/2) in this way.

The advantage: we are able to reduce!

The shift from addition and subtraction to multiplication and division has the advantage that with fractions we are able to reduce, as shown in the above examples, we are able to reduce.

This has a direct impact on the resonances: whenever we are able to reduce, the numbers in the fractions become smaller – and small numerators and denominators in the fractions are an advantage for resonance. This also explains why we prefer not to have any prime numbers higher than 5 in the intervals. Non-prime numbers such as 6, 8, 9, 10, etc. can be reduced, which is why we can find a major second (9/8) in common musical scales, but no intervals with the fractions of 7/4 or 8/7.

Relating scale tones to each other

When we compare two tones within a musical scale in order to decide whether they resonate with each other, we always relate them to their shared fundamental tone. This is essentially connected with the character of musical scales (and chords). All tones of a musical scale are based on this one fundamental tone (tonality).

Since the intervals are always relative, the crucial factor is not the absolute frequency of a tone, but its relation to the fundamental tone. When we set the

fundamental tone = 1,

all scale tones are described as relations of their frequency to the frequency of the fundamental tone.

Distance between two scale tones

What, for example, is the relationship between the fourth and the fifth? The fourth is 4/3 above the fundamental tone, and the fifth is 3/2. If we want to compare these two tones, we can calculate the distance between them. The distance is a subtraction, and with frequencies, a subtraction is a division. We divide the higher tone by the lower one, and the result is: 3/2 : 4/3 = 9/8.

9/8 is a major second; the distance between the fourth and the fifth is a major second.

Resonance between the fundamental tone and two scale tones

Whenever we look at two scale tones, they always have the fundamental tone in their foundation. The fundamental tone determines the tonality, and the tonality places the musical scale on an absolute basis.

But how do the three tones mix – fundamental tone, fourth and fifth?

For resonance to be generated, all three tones must stand on a shared base, or mathematically speaking, they require something like a shared counting time for all three frequencies, i.e. for the fundamental tone (1), the fourth (4/3) and the fifth (3/2).

For this purpose, we look for the smallest denominator which tallies for all three figures. In the example, this is the denominator 6:

Fundamental tone    1     =   6/6
Fourth                            4/3  =   8/6
Fifth                                 3/2  =   9/6

This common denominator is always the lcm, the least common multiple of all the individual denominators involved.

A further example

Fundamental tone    1    =  15/15
Minor third                 6/5  =  18/15
Major sixth                 5/3  =  25/15

The lcm of 1, 5 and 3 is 15.

What does this common denominator of several tones mean?

I propose the hypothesis that resonance occurs more easily, the smaller this common denominator is. For our consideration of resonance, the following applies on the basis of this hypothesis:

Tones which have a low common denominator mix easily.

The higher the common denominator, the smaller the internal resonance of the tones. 

The consequences of these conclusions are dealt with in the text about specific musical scales and chords.


This is where you will find an overview of the texts about the theory of the three worlds.

First musical scales

Starting position

Do the criteria which we have postulated so far already enable us to create musical scales which are so attractive that they occur in reality? After all, the criteria look rather artificial and theoretical at first sight – can they nonetheless serve to explain musical scales that have grown naturally?

Indeed they can. The mathematical criteria for resonance have obviously had an impact on the human ear and have for millennia prompted people again and again to invent music which is fundamentally structured by precisely those musical scales that we are able to derive mathematically with the help of our criteria.

Pool of resonant tones

With our resonance criteria alone, we established a first pool which contains those tones of which we expect the strongest resonance with the fundamental tone. I list these nine potential scale tones again below:

Of course this is merely a pool of many tones, and not a meaningful musical scale. The problem is that all these tones can easily and quickly resonate with the fundamental tone – but do they also resonate among each other?

Two scale tones and the fundamental tone

Thus this is not solely about the resonance of a tone with the fundamental tone, but additionally about its resonance with other tones. This has a mathematical basis: we have a look at the lcm (least common multiple) of the denominators involved. How this works, and why this is the case, is explained on the page of calculation principles for resonances.

The lcm (least common multiple) of the denominators

The resonance criteria indicate a good resonance if the lcm of the denominators involved is as small as possible. The fundamental tone has the denominator 1 and therefore fits every lcm; therefore it will fit every combination well. But how do the scale tones fit each other?

Example 1

Fourth and fifth: The denominators are 3 and 2; the lcm is 6, i.e. low. The numerators are also low. By way of a further indication of resonance, we can calculate the distance between the two tones. This is 3/2 : 4/3 = 9/8, i.e. a major second. Although 8 is a relatively high denominator, which may interfere with the resonance with other scale tones, both the 8 of the second and 2 of the fifth fit the very important octave very well. Also, both the fourth and the fifth have unbeatably low numerators and denominators, which has a favourable impact on the mixing ratios for further scale tones.

In other words: the fourth and the fifth are a perfect pair for resonance – at least mathematically. But does this also sound good?

How does the combination sound in our subjective perception?

Naturally, this is not solely about mathematics. The mental world, our subjective perception, determines whether we like a certain type of music and how we receive it. If you sound the fundamental tone along with the fourth, the fifth and the octave, you can hear what we have calculated: the resonance of the four tones is pure. Indeed, the combination even sounds rather banal, and we may miss the pep that dissonances introduce into the music to which we are accustomed. We also miss the sweetness of the thirds (denominators 5 and 6).

Example 2

We mathematically combine the major second with the minor third, i.e. 9/8 with 6/5: the lcm is 40, the interval between the two tones is 6/5 : 9/8 = 48/45 = 16/15. With the high lcm and the short distance, this pair is a bit more problematical – at least if we pay attention to good resonance and want to avoid any stridency in music.

The first two musical scales found

If you feel like it, you can calculate the lcm and the distance between all the above-mentioned yourself and thus compile a pool of tones in which both aspects are optimised and as much resonance as possible can develop. Of course you also want to select more than just three of four tones for a musical scale. What about five?

The two musical scales with the strongest resonance that can be found in this way are remarkable – both of them are very well-known musical scales:

1 – 9/8 – 5/4 – 3/2 – 5/3 – 2
1 – 6/5 – 4/3 – 3/2 – 9/5 – 2

Or with the fundamental tone C:

C – D  – E – G –  A  –  C
C – Es – F – G – Bb – C

Pentatonic scales

These are two pentatonic scales, i.e. two musical scales with five tones (the C occurs twice but is only counted once). It is not really surprising that both musical scales are good old friends – they are none other than the major and minor pentatonics.

Nor does it come as a surprise that both these musical scales occur in practically all the world’s cultures from the rain forest to all the advanced civilisations, either in their pure form or else as the basic structure of more sophisticated musical scales.

Theoretical and mathematical considerations have led us to postulate these two musical scales which are not only well-known worldwide but are easy to understand for everybody and felt to be catchy by everyone, including small children.

In my view, this is not an accident. It looks as if considerations so far are highly compatible with observed reality.


In the following post, you can find a resonance analysis of the five standard pentatonics.

This is a post about the theory of the three worlds.

Criteria for attractive musical scales (overview)

What is this about?

According to the theory of the three worlds, mathematics (ideal world) plays a part in physical processes (physical world). Without our subjective perception (mental world), however, we would not be able to notice any of this. I examine the way in which these very different worlds converge in reality with the example of musical scales. There are some riddles here, for instance why human cultures have created thousands of different musical scales, but every one of them uses the octave.

This constancy of the octave can easily be explained by resonance, which in the case of the octave is particularly striking because it corresponds to a mathematical ratio of 2:1. The higher tone of the octave vibrates with twice the frequency of the fundamental tone – an obvious example of how physics, mathematics and subjective perception come together.

In the preceding post I examined whether, with the help of mathematical considerations, we can find further intervals which occur in real musical scales. With only a few mathematical criteria, we found a first pool of intervals which have a high degree of affinity to resonance, i.e. the musical scale tone and the fundamental tone are in a frequency ratio which very easily produces a strong resonance.

Criteria for musical scale tones

Tone range

All musical scales known to me are situated within the range of an octave. This alone is amazing, and we have seen that all three worlds are involved in their creation.The crucial factor for the musical scale tones is their frequencies (f). What is decisive is not the absolute height of the frequency, but only its relative height in comparison with other tones. Below, the number 1 will denote the frequency of the fundamental tone and the number 2 the (doubled) frequency of the octave. All the musical scale tones must be situated between the fundamental tone and the octave; in mathematical terms, this means for the relative frequency f(ST)of the scale tone (ST) in relation to the fundamental tone:

    1 ≤ f(ST) ≤  2

2.  Resonance

The physical precondition for resonance to occur is that two frequencies are in a “rational” ratio. In number theory, “rational” describes a number which can be written as a fraction of two integers. In other words: f2 = m/n f(m andn are integers, f1 and f2 are the two frequencies). We now hypothesise that a musical scale tone (ST) is preferably in a ratio (M/N) to the fundamental tone (FT) in which a resonance can develop. In mathematical terms, this is expressed as follows:

    f(ST)  =  m/n   f(FT)

This precondition is part of the physical and mathematical worlds and, independently of subjective and cultural perception, a generally valid physical precondition for resonance, also in the non-acoustic sphere.If calculating with interval fractions is new to you, I would like to recommend the annexes, in which the calculation processes and their backgrounds are described in detail.

3.  Strong resonance

A resonance does not always occur equally quickly and equally strongly. Some intervals can therefore develop a resonance more quickly and more strongly than others. I hypothesise that a resonance is the stronger, the smaller m and n are, and I have examined, starting from number 1, what intervals we obtain with as small values as possible for m and n.Surprisingly, it turns out that all the small denominators up to 5 result in musical scale tones which we already know as the most important ones in our traditional European musical scales.The search also reveals that the absolute height of m and n is not the sole determining factor; rather, it comes down to the prime numbers which result from the integer factorisation of m and n. Here it turns out again that among the prime numbers, 5 is also the limit for the numerator.  With this in mind 6, 8 and 9, which are not prime numbers, can be factorised (6=2×3, 8=2×2×2 und 9=3×3) and used for resonances. 7, however, which is a prime number, cannot be factorised and is therefore already too high for m and n in practice.

  m and n should be as small as possible,

Or more precisely: In the integer factorisation of m and n, 5 should be the highest number.

Why integer factorisation? – The musical scale tone does not exist on its own, i.e. it should not only have a resonance in relation to the fundamental tone, but also in relation to the other musical scale tones. This results in compound resonances, mathematically speaking in fractions, in which numerator and denominator can often be reduced. This quickly turns an 8 into a 4, a 15 into a 3, and we again have very small m and n values. Examples will follow.

4. Further criteria

Resonance is not the only requirement for attractive musical scales. Musicians and listeners must also be able to distinguish between the tones. This is easier if there are not too many tones and if they are not too close together. These are not mathematical requirements, nor have they anything to do with resonance. The origin of this demand on a musical scale has something to do with our reception, i.e. with what we are able to perceive – that is, our mental world. With regard to this, our perception depend on the physics of our inner ear and on the physics of our brain. These criteria are thus a demand from the mental world, on the basis of the physical world. Evidently, all three worlds are involved in the creation of the many thousand different musical scales. The following are criteria that are not related to resonance:

4a)
 The number of the musical scale tones must not be too high.

4b) The musical scale tones must not be too close to each other.

Neither the number of tones nor their minimum distance from each other are the same in every culture. Besides resonance, there are further principles in music, namely that of interference and the higher/lower principle. If the tones are close to each other, music-related training provided by cultural experience may make precisely such tones interesting. However, this does not happen across the whole octave range of the musical scale, for if tones are very close to each other, the overall number of tones in the scale must still not become excessive. The distance between other tones must occasionally be wider again. A musical scale with quarter tones across the whole octave is conceivable but is unlikely to become prevalent in practice.Besides interferences and the higher/lower principle, there is a further principle, the variety principle. This has neither a mathematical nor a physical basis, but a mental one. If everything is the same, things become boring – and this also applies to the distances within a musical scale. This is why we do not experience the whole tone musical scale, which solely consists of identical intervals, as particularly exciting. Yet in combination with other scales with more variants, it is precisely that same whole tone musical scale which can provide an interesting contrast. Pieces of music that would only use whole tone musical scales would not be interesting for long. We thus have a further criterion which is not related to resonance:

4c) The distances in the musical scale should not all be identical.

4d) Fundamental tone principle:

The fundamental tone is the tone which holds the musical scale together. We will have a closer look at this in the section about ecclesiastical modes. As a consequence of the fundamental tone principle, we will always relate the musical scale tones to the fundamental tone.

We compare all the musical scale tones through the fundamental tone.


With the help of these conditions, we will examine our pool of intervals from the preceding post and filter out some first musical scales. Not surprisingly, they are musical scales which do not only have a simple mathematical structure, but also exist worldwide. They are certain pentatonic scales which are not only in use in their pure forms around the globe, but are also contained as a basic structure in more complex musical scales, such as our standard major scale. Firstly, we will have a look at the most resonant musical scales.


Here you can find an overview of the texts about the theory of the three worlds.


 

Fractions and Resonances

Resonance works through shared overtones

Resonance occurs when two vibratory physical objects vibrate together. What matters is the two objects’ natural frequency:

  • 1st degree resonance: both objects vibrate in the same frequency (f2 = f1).
  • 2nd degree resonance: one object vibrates in an overtonefrequency of the other
    (f2 = n * f1).
  • 3rd degree resonance: both objects vibrate in a shared overtone frequency
    (f2 = n/m * f1).

The 3rd degree resonance reveals itself by the fact that the ratio of the two frequencies corresponds to a fraction of integers (n/m). This 3rd degree resonance is what interests us, for it has an effect on musical scales and chords rather than the 2nd degree frequency, as is often assumed (for this, cf. previous post).

Example: the fifth

One string a' has a basic frequency of 440 Hz and a string e" the frequency of 660 Hz. In this case, the second overtone on string a' vibrates with 3 × 440 = 1320 Hz and the first overtone on string e" vibrates with 2 × 660 = 1320 Hz. The overtones on the two strings have the same frequency and string a' can stimulate string e" through this common overtone. The ratio of the two basic frequencies is 3/2.

Proposition

For resonance to be generated, the frequencies must have a ratio of a fraction with integers.

 Why integers?

When a vibration medium (string, basilar membrane) vibrates, standing waves are produced. These are characterised by the fact that the string does not vibrate at either end, but only in the middle, with one or several antinodes (bellies). The number of the vibration bellies in the middle must be an integer since otherwise the standing wave would not be zero at either end.

Assessing musical scale tones with the help of resonances

 As we can hear, octaves and fifths can easily be made to resonate (fifth experiment) and are characterised by very simple resonance ratios (f2/f1), namely 2/1 for the octave and 3/2 for the fifth. With very few mathematical preconditions, we can now find further musical scale tones:

Criteria for the tones of a musical scale

In the cases below, we look at the interval between the tone and its fundamental tone.

  1. The interval must be situatedwithin an octave: this means that the fraction of the two frequencies (musical scale tone to fundamental tone) must be ≥ 1 and ≤ 2.
  2. The interval must be able to resonate: numerator and denominator must be integers.
  3. The resonance should be as vibrant as possible: the denominator of the fraction should be as small as possible.

The last two criteria are crucial but require some explanation, and I will illustrate the reasons for this later. For the time being, however, I will take all three mathematical criteria as given and see whether they enable us to find further well-known intervals.

Generation of a pool of possible musical scale tones with the help of the three criteria

 We begin with the denominators, starting with denominators 1 and 2. Our musical scale already includes:

All the other tones with denominators 1 or 2 are situated outside our octave range (1-2). Therefore we look whether further well-known intervals result from denominator 3:

All the other tones with denominator 3 are situated outside our octave range. We therefore continue with denominator 4:

All the other tones with denominator 4 are situated outside our octave range. We therefore continue with denominator 5:

Any further fractions with denominator 5 are situated outside our octave range.

A thought in between  

What is striking here is the fact that fractions with the numerator 7 do not exist as intervals here in Europe. My proposition is that this has something to do with the fact that seven is a prime number. This can explain why 8/5 and 9/5 are intervals that we are accustomed to although the denominator and numerators in these fractions are higher than with 7/5. Eight is 2×2×2 and nine is 3×3. We will see later that we can also compare several intervals with each other to assess the resonance. In frequency analysis, a comparison of two intervals means that the fraction of one interval is divided by the fraction of the other interval. In this situation it is of advantage that we can reduce the fraction. A prime number does not provide us with this option, but with figures such as 8 or 9, a reduction is often possible, particularly if, with fifths and fours, we have 2 or 3 in the denominator. Examples will follow.

In the case of our generation of resonant musical scale tones, however, it follows that we ignore 7/4 and 7/5 because the 7 in the numerator is too high a prime figure.

We continue with denominator 6:

Denominator 6 thus does not produce any new tones.

We omit denominator 7 as a high prime number and go directly to denominator  8:

Thus denominator 8 produces two new intervals, namely the major second (9/8 and the major seventh (15/8). The numerator and the denominator may be quite high in both fractions, but they can be divided by 2, 3 and 5. Thus the fractions of the intervals in combination with musical scales and chords become reducible, and the intervals prove to be resonant.

This is the end of our search for tones, and we can list our findings in the order of ascending frequencies:

Pool of the strongly resonating intervals

The following is striking in this pool:

a) Most intervals used by us have very simple frequency ratios.

b) The fourth, with a low denominator and numerator, is the “fourth most logical” musical scale tone. However, this tone is never an overtone. Nonetheless it makes sense, both in the mathematical world (simple fraction), in the physical world (simple resonance ratios) and mentally (subjective musical hearing experience).

c) The minor second and the tritone are missing from our pool. Mathematically speaking, these tones are not simple fractions and sound correspondingly sharper.

In music, such sharpness is interesting, because it brings change and diversity, but these intervals definitively fail to reach the ideal of a smooth resonance.

We will see later how the tritone nonetheless fits well into the resonances in certain conditions, whereas the minor second always sounds sharp and thus becomes the actual leading tone in European music.

For the time being, however, we are leaving it at our pool of ten tones. This is sufficient for many musical scales, particularly those which are used most frequently.

Further thoughts in between

As is well known, mathematicians like prime numbers. In this respect, we can define the above-mentioned criterion 3 (as small as possible numbers for numerator and denominator) more precisely:

The 3rd criterion in more precise terms:

For an interval to be able to resonate well in combination with other intervals, the prime factorisation of the numerator and the denominator should result in the smallest prime numbers possible:

2 is better than 3
3 is better than 5
5 is better than 7
7 is already too high for practical purposes.

Intervals and rhythms

This is different from rhythms, where fraction ratios also play a part. Why this is different there, and why rhythms with 7 or 11 beats sound good, can be plausibly explained with the theory of the three worlds. More about this later.

From a pool to musical scales

We now look for further criteria for attractive musical scales. The pool we have found is not a musical scale yet, but merely our starting point from which tones for different musical scales can be compiled. Here, further criteria are applicable.

Further criteria

4th criterion: preferably, the musical scale tones should also to be able to enter into resonances with each other.

5th criterion: the musical scale has a fundamental tone (a so-called tonality), which has a very special function in the musical scale.

6th criterion: the musical scale tones must not be too close to each other; if they are, we as human beings (lay people) cannot distinguish between them any longer. This is a practical constraint from the mental world. In a purely mathematical world, any number of differentiations would be conceivable; in a real situation, this is not the case. More about the criteria for musicl scales on the following page.


This is a post about the theory of the three worlds.

“Breaking down” the Fifth

The fifth

Let us first have a look at the fifth. It is a feature of  practically all the musical scales of human cultures. Musical scales without this pure fifth do exist, but these musical scales strike me either as artificial and deliberately designed like the whole tone scales or rather uncommon like the Locrian mode. The blues scale makes use of the “blues note” – the “flat five”, a note close to the fifth known as the diminished fifth – but also uses the perfectly normal fifth. After the octave, the fifth is certainly the interval that occurs most frequently in all the thousands of musical scales on this earth.

Fifth and twelfth

Can this normal fifth be generated by resonance like the octave? Although it is not an overtone, it can still be reached through the overtones. Below, I will show how this works, namely with a short detour through the twelfth, the third overtone.

The graph with the vibrating overtones will serve to illustrate this:

Fig. 1: A vibrating string with the fundamental tone and the first four harmonics (overtones)

In Fig. 1 I have even described the third overtone as a fifth, which is actually wrong, for in reality it is a twelfth. Nonetheless, this tone immediately strikes us as a fifth when we hear it. In Fig. 2, you can see an example of fifths and twelfths on a piano keyboard:

Fig. 2: Octave and twelfth on the piano

In our example, the fundamental tone is a (capital) C. The first overtone, the octave, is a (small) c and the second overtone, the twelfth, a (small) g. In relation to the fundamental tone C, the small g is a twelfth, but in relation to the first overtone, i.e. the small c, the g is a fifth.

The frequency of the fifth

What about this c-g interval in terms of frequency? To find out, let us compare Figures 1 and 2: Tone 3 (g) is three times the fundamental vibration (C) and tone 2 (c) twice the fundamental vibration. Thus tone 3 (g) vibrates 3/2 times as fast as tone 2 (c). If we take the small c rather than the capital C as the fundamental tone, then the small g is the fifth. And in the fifth, the upper tone (g) vibrates 3/2 times as fast as the lower tone (c). This is generally applicable:

A tone which vibrates 3/2 times as fast as another sounds a fifth higher to us.

The three worlds in the fifth

The fraction 3/2 is the mathematical aspect of the fifth. We have deduced it through the physics of the string vibrations. At the same time, we complied with the previously mentioned conditions (constraints) from the mental world: if the quint is meant to be a tone of a musical scale, it must not be too far away from the fundamental tone. This applies to every tone of a musical scale; it must be within an octave. In mathematical terms, this means that the ratio between its frequency and the frequency of the fundamental tone must be between 1 (= fundamental tone) and 2 (= octave). The fifth satisfies this requirement with the frequency ratio of 3/2 = 1.5. In the case of the twelfth, the frequency ratio is 3, i.e. greater than 2, and thus the twelfth is not a note of the musical scale. We perceive it as a fifth, but as mentioned before, in the mental world we perceive the octave as the “same” tone.

The resonance experiment for the fifth

To illustrate the relationship between the fundamental tone, the fifth and the twelfth, I propose a further resonance experiment on the piano:

Fig. 3: Resonance experiment for the fifth. In comparison to Fig. 2, the key on which we investigate the resonance is now the fifth – the capital G – and not the small g.

To begin with, we again test the twelfth and press the twelfth (the small g) with the right hand like in the octave experiment . The string should not make a sound, but the key must be kept depressed. With the left hand, we briefly and vigorously hit the C, i.e. the fundamental tone. Like in the octave experiment, the depressed string (g) should make a sound although it was not hit. This is pure resonance; the string makes a sound because it was stimulated by sound waves. This works because the small g is an overtone of the capital C.

But what about the capital G, i.e. the fifth? To test this, keep the capital G mutely depressed while vigorously hitting the fundamental tone, i.e. the capital C. You can now hear a high tone. If you listen carefully, you will notice that this is not the fifth, i.e. the capital G, but the twelfth, i.e. the small g. How come this tone sounds although you don’t keep the key of the small g depressed at all?

In fact the sound of the small g is produced on the string of the capital G! This means that the string does not vibrate in its fundamental vibration but in its first upper vibration, the octave. This works well because the string can vibrate almost as well with two antinodes (bellies) as with one. This is the so-called flageolet-note.

In other words: you have stimulated a harmonic on the string whose frequency is twice the speed of the fundamental frequency of the string. But where did the stimulus for this frequency come from? – It is the capital C string which initiated the overtone. The vibration of this C string contains the small g as a harmonic, namely the second overtone. This second overtone stimulates the (capital) G string to resonate, but not with its fundamental vibration (G), but with its first overtone, the small g, for only this can be stimulated as a harmonic of the (capital) G string. You can hear this tone (g) on the G string as long as you keep the G key depressed.

Table 1: The resonance of the fifth

Thus the resonance of the fifth is generated by the detour of the overtones. Neither string is involved in its fundamental vibration, but both strings are involved through their harmonics. The fact that this works has been demonstrated with the fifth experiment.

The fifth, a simple fraction

In Table 1, the fifth is represented as a fraction: 3/2.

As we have seen, all the tones of a musical scale must be within the range of an octave, i.e. their frequency must be between the single value and the double value of the frequency of the fundamental note. This has been achieved with the frequency ratio of 3/2 = 1.5. We have thus found the first tone within the octave range which has a very simple interval relation to the fundamental note. Whereas the octave vibrates at twice the speed of the fundamental note, the fifth vibrates 3/2 times as fast.

With the exception of the octave, the overtones do not fulfil the conditions for musical scales. Nonetheless, they play a role as transmitters of resonance. We have produced the fifth by simply breaking down the twelfth (the second overtone) by one octave. This breaking down by an octave manifests itself as the 2 in the denominator. The 3 in the numerator is the “inheritance” of the second overtone, the twelfth, which vibrates at three times the rate of the fundamental tone.

Summary

With the fraction of 3/2, which defines the fifth, we have obtained a strikingly simple ratio. This is no accident. We will see how these simple ratios (ideal world!) also play a part for the other notes of a musical scale.

The fact, however, that we are no longer able to use ratios of single integers, like with the overtones, for the musical scales and that instead, fractions (of two integers) are used, is due to the constraint of the octave limitation, which is a constraint of the physical/mental world.

The fifth is not the sole fraction among our musical scale intervals. Integer fractions define the most important tone intervals. We can find them with a simple design rule. On the following page, I will show how this design rule works.


This is a post about the theory of the three worlds.

The overtone series is not a muscal scale

The series of overtones

As we know, the octave is the first overtone. The physical phenomenon of resonance has helped us understand the role played by the overtones: in the physical world, vibrating media such as strings or pipes can vibrate with their basic frequency, but also with an integral multiple of this frequency. In this way the octave results as the first integral multiple of the basic frequency – i.e. by the fact that its frequency is doubled.

It would now be obvious to add the further overtones, which are also integral multiples of the basic frequency, and to explain the single notes of the musical scales in this way.

Fig. 1: The fundamental tone and the first four harmonics

Fig. 1 shows how a string vibrates and how the octave and further overtones are added. Whereas the fundamental tone vibrates with precisely one “belly”, the overtones vibrate with two, three, four and more bellies. If we start with the fundamental tone C, this will result in the following series of overtones:

Fig. 2: Overtone series ranging from the fundamental tone C (note 1) to c“‘ (note 16)

The musical scale works in a closed frequency range

Resonance explains the overtone series, including the octave. How, though, does a musical scale come about, which is meant to fill the narrow range between the fundamental tone and the first overtone, i.e. the octave? As mentioned in the previous post, the notes of the musical scales are subject to this constraint whereby they all have to be within the range of an octave. The overtone series, however, leads far beyond an octave.

In addition, Fig 2 also reveals that the distances between the overtones vary a great deal. Whereas initially, they are very far apart, they get closer and closer as the series proceeds. This would be highly impractical for a musical scale to be applied in reality.

We see, however, that the overtones from note 4 onwards, and even more between notes 8 and 16, constitute something like a kind of major scale: c, d, e, (f), g (a), (h), h, c: nearly our major scale, but not exactly, since the tones in brackets (11, 13, 14) don’t quite fit.

The natural “musical scale” of the alphorn

In fact, the musical scale between notes 8 and 16 corresponds to the natural musical scale of the alphorn – including, however, the “off” notes 11, 13 and 14, and without the actually important fourth, namely the f. Nonetheless, the natural tone series of the alphorn – but also only between notes 8 and 16 – is almost something like a natural musical scale in that it packs a reasonable number of notes into the range of an octave, and this even on the basis of resonance.

Yet the overtone series that can be played on the alphorn is still not a real musical scale. Only the musical scale between notes 8 and 16 can be played on this instrument; most notes of the musical scale below this range are missing while above it, there is a confusing array of many more tones, which are increasingly closer to each other. This is not tantamount to a musical scale which repeats itself octave after octave. Additionally, the instrument is somewhat impractical. In order for the notes to be played at a normal pitch, the horn must of necessity be very long. This is different with a violin or a flute, i.e. substantially more practical. With these smaller instruments, however, we cannot achieve the tones of the musical scale as overtones (which would be flageolet tones with the violin, and with the flute pure overblows like with the alphorn), but by means of deliberate mechanical manipulation of the physical object that vibrates, namely the strings of the violin and the air column of the flute.

But do the overtones and resonances play a role in the musical scales nonetheless? – They do! You can read about this in the next post, which explains how a fifth comes about.

This is a post about the theory of the three worlds.