Tag Archives: resonance

Why resonance also works with imprecision

22. February 2022 Hans Rudolf Straub Leave a comment

When does resonance occur?

Resonance between two physical media depends on the frequency ratio of their natural vibrations. When the two frequencies constitute a simple fraction such as 2/1 or 3/2, resonance can occur. In an earlier post, I explained how the ten simplest frequency ratios lead with mathematical necessity to the ten tones which appear in our musical scales, no matter whether they are major scales, the various minor scales, the ecclesiastical modes, the major pentatonic scales, the minor pentatonic scales, the blues scale, etc. Ten tones are sufficient to build all these different scales.

Pure intonation and temperament

However, does resonance also work in the equal temperament? In the post about the tempered scale, we saw how the two distributions differ. Fig. 1 shows the pure intonation in blue – i.e. the ten most resonant intervals plus the two fillers C sharp and F sharp – and below, the twelve intervals of the equal temperament in red.

Fig. 1: Pure intonation (blue) and equal temperament (red) with fundamental tone C (logarithmical representation)

Obviously, the frequency ratios of the equal temperament differ from those of the pure intonation and thus no longer correspond to the simple frequency ratios which originally resulted in our pure musical scales. Nonetheless, the impure intonation works, and we distinguish between minor and major thirds, fifths and sixths although they are not pure any longer. Are the tempered, i.e. impure intervals still resonant?

The answer is an unequivocal yes.

Why the impure temperament is still resonant

Fig. 2 below shows resonance as dependent on frequency relation and damping. The greater the damping, the lower the resonance.

What is interesting is how the frequency ratios – in Fig. 2 marked on the horizontal from 0.0 to 3.0 – impact on the emergence of resonance. Resonance is strongest at 1.0, i.e. when both media, the stimulating and the stimulated, have an identical frequency. Yet resonance occurs even if the frequency ratio is not precisely 1. This is the reason why we experience the tempered scale as resonant, too.

Fig. 2: Resonance as a function of the precision of frequency ratios
[Source: https://en.wikipedia.org/wiki/Resonance (22/Feb/2022)]

As we can see in Fig. 2, the tempered scale with its not quite precise fractions still leads to resonances between the intervals, albeit to slightly weaker ones. Since we practically only hear music that is based on the tempered scale, we have become accustomed to it. Pure intonation can only be produced by instruments which can alter the pitches . It does not work on keyboard instruments. Pure string ensembles or unaccompanied singers are able to make music with pure intonation, and good ensembles actually do so.

Additional effects of the tempered scale

The key benefit of the temperament is the immense extension of compositional possibilities.

There are, however, further additional effects: the fact that the intervals are slightly “out of tune” leads to interferences (beats), with resonance ebbing and surging. The friction between two impurely tuned tones can result in the emergence of a third tone, which overlays the other two. Such effects can also be consciously sought in pure intonation in that a singer or instrumentalist slightly alters the pitch, thus creating a conscious musical effect with which they can play.

However, I don’t want to expand on these effects, nor on the interesting effects which piano tuners have to take into consideration, such as the so-called stretching across the whole range of tones. Tuning, for instance of a piano, has to keep several goals in view at the same time. Here, too, the three worlds have a simultaneous impact: the mathematics of the pure figures, the physics of the real piano strings and our subjective impression.

It is for two reasons, however, that I won’t pursue these considerations here. Firstly, the above-mentioned acoustic phenomena have been described very well, and secondly by experts who have specialised in them and know substantially more than a computer scientist and amateur musician like me. To me, the equal temperament is simply an ingenious and practical invention which I readily accept because it distinctly extends the harmonic possibilities of music.

I will therefore continue this series with the extensions of the compositional possibilities that result from the equal temperament.

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

Theory of the Three Worlds

The equal temperament

14. February 2022 Hans Rudolf Straub Leave a comment

Initial wish: changing the fundamental tone during a piece of music

In the preceding post, we saw that the just intonation is not pure any more when the fundamental tone is changed since certain intervals change. The further removed the key, the more tones fail to accord with the calculated, i.e. resonant tones.

If the frequencies of the scale tones are very slightly shifted – i.e. tempered – then we can also change over into neighbouring keys, i.e. we can modulate. In the equal temperament, we can actually change over to any fundamental tone whatever, and this temperament has successfully prevailed in Europe since the Baroque.

How the equal temperament works

In Fig. 1, you can see the resonant, i.e. pure intervals between a fundamental tone and its octave.

Fig. 1: Resonant intervals above the fundamental tone C in a logarithmical representation

In the above figure, I also included the indirectly resonant tones C# and F#, thus closing the gaps in the band of the most resonant intervals. What is conspicuous in Figure 1 is the fact that the twelve tones are not regularly, but almost equally distributed across the octave. Could this be exploited musically?

Fig. 2: Pure (blue) and equal (red) distribution of the 12 scale tones

Figure 2 shows the comparison between a natural, i.e. pure distribution of scale tones with a completely equal distribution. As you can see, the displacements are visible, but not all that big. The irregular, pure tones are slightly shifted, which results in a completely equal distribution of the tones. This slightly changed, but now equal distribution is called the equal temperament.

Since the distances between the twelve tones are precisely equal, it does not matter on which fundamental tone we establish the musical scales:

C major: C – D – E – F – G – A – B – C

E-flat minor: Eb – F – G – Ab – Bb – C – D – Eb

As you can easily check, the relative distances between the individual tones are now precisely equal in the tempered frequencies, no matter whether in C major, E-flat major or any other key.

The equal temperament is a radical solution and as such the end of a historical development that underwent several interim solutions (such as the Werckmeister temperament and many more). This historical development and the details of the practical configuration are extensively documented in the internet and the literature and are easy to find. What interests us here, however, are two completely different questions:

Why does tempering work although we do not have any precise fractions for the resonances any longer?
What are the compositional consequences?

Fractions and Resonances

9. April 2021 Hans Rudolf Straub Leave a comment

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.

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.
The interval must be able to resonate: numerator and denominator must be integers.
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.