LEXIS


(おと)(ひび)
楽音(がくおん)音響(おんきょう)

When energy acts on a body, the body deforms. The absorbed energy is passed on in the body. This creates waves. Their direction of propagation and their speed depend on the properties of the medium through which they move.

Waves on a String

If pressure is exerted on a tensioned string, its parts move away from the “source of interference” and back again; the string begins to vibrate, creating a wave of “mountains” and “valleys”. The distance between two mountains or valleys is called wavelength. This wave spreads over the entire free-running length of the string. At each vibration, it sets one wavelength. Such a perpendicularly propagating wave is called transversal. Its speed (frequency, measured in hertz = vibrations per second) depends on how strongly the string is stretched and how heavy or dense the string is. The faster the oscillation takes place (the higher the frequency is), the shorter the wavelength. The vibration ends when the absorbed kinetic energy has been distributed over the entire string. The vibration becomes audible because it triggers sound waves in the surrounding air.

Sound Waves

In air (and all other aeriform or liquid media) waves propagate longitudinally: The medium is first compressed in the direction of propagation of the wave and then diluted again. This changes the air pressure, depending on how high the temperature of the air is and how dense it is. The higher the air temperature, the faster waves will spread. At room temperature (20° C), its speed is about 340 m/s, at 33° C already 350 m/s. The air molecules that are caught by a wave get into harmonic vibration. Again, the higher the frequency, the shorter the wavelength and vice versa.

Waves in the air are caught by membranes in the human ear and perceived as sound. How loud sound is perceived depends on the strength of its intensity (power per area, measured in W/m²), but also on the frequency of the sound. Humans can register sound waves with a frequency between 20 and 20,000 Hz.

Tone

Tone is a periodic (i.e., periodically recurring) vibration perceivable by humans as sound, which can be represented as a sinusoid.

Overtones

The strings of a ukulele are fixed between bridge and nut. If you pluck an open string, it will vibrate at a certain frequency called natural frequency, which will trigger a wave whose wave field has exactly one “belly” (loop); this wave field is called first harmonic; the wave itself is called fundamental. A frequency twice as high as the natural frequency produces a two-legged wave; between them lies an area called node whose center does not move (also the two end points of the string are nodes). This wave field is the second harmonic. Its wavelength is half the size of the natural frequency. These effects are repeated at all integer multiples of the natural frequency. They are called overtones.

Multiples of natural frequency
= number of loops
Wave fieldTone
11st harmonic (fundamental)key tone
22nd harmonic1st overtone
33rd harmonic2nd overtone
44th harmonic3rd overtone
55th harmonic4th overtone
66th harmonic5th overtone
77th harmonic6th overtone

Standing Waves

The nodes (i.e., stationary points) of these wave fields do not change their position; therefore they are called standing waves. Oscillation always goes from one node to the next and vice versa back (this corresponds to its wavelength, in other words: the distance from one node to the next loop is one quarter of the wavelength).

Resonance Spectrum

The totality of all frequencies of a string at which such standing waves can arise is called resonance spectrum. When playing a string, most of the kinetic energy is connected to the fundamental wave; however, small parts combine with the overtone waves of the resonance spectrum. The superimposition of all these waves forms the tone color of a string.

To make only the overtones of a string audible, the technique of the flageolet is applied. To shift the spectrum of audible overtones, the technique of registration is applied.

Tones

The tones on an ukulele are created by vibrating a string at a certain point. These strings are marked by the frets. If the string is pressed at these points, a vibration node is created: At the pressed position, the string does not move, before and behind it oscillates wavy. This creates certain pitch ratios of the string, which are defined as musical intervals (distances to the root of the string). The integer division ratios are the most important for the creation of harmony, because their attack in turn produces certain overtones.

FretRatioInterval
122:1Octave
73:2Fifth
54:3Fourth
35:4Third

Resonance

The ukulele strings are attached to a resonant or sounding body (see body). When a vibrating string transports its energy to this resonator, it begins to vibrate itself. Its vibrations in turn trigger a wave and overlap (interfere) with the vibration of the string. When a standing wave is created, the intensity of the vibration of the string is amplified. This effect is called resonance. It also contributes to the timbre of an instrument.

In addition, the resonant body decisively amplifies the volume because it has much more contact with the surrounding and trapped air than the strings and therefore causes stronger sound.

Moreover, strings that are not struck are also made to vibrate when their root note or their overtones are struck on a neighboring string. This too is a resonance phenomenon (sympathetic or string resonance).

Beat

If the vibrations of the string and the resonator are the same, but there is some offset, a tone is created that swells (beat). This effect is perceived as vibrato or tremolo.

Sound

Tones created on a ukulele become sounds by adding overtones and resonances. These are complex tones and unlike pure tones no longer sinusoidal. This is the reason why an ukulele sounds different from other instruments.

Timbre

The determining factors for the specific sound of an ukulele (its timbre) are therefore above all:

  • the length, density and tension of the strings,
  • the selected register,
  • the type of attack,
  • the resonance characteristics of the tonewood.

Ambient Sound

On the other hand, how well a ukulele can be heard depends on the conditions under which the sound waves it emits can spread and how much they are broken (reflected). For the farther the listeners are from the sound source, the less direct (unreflected) sound they hear. The reflections are inevitably shifted in time; this creates reverberation. After a certain time delay (which depends on the volume), this reverb is perceived as a disturbing echo. Another problem is that low notes are always reflected more than high ones.

Depending on purpose and size, a room should reflect sound differently. Ideally, a room should not be round, elliptical, or parallel (the optimal shape is a trapezoid), and the walls and ceilings should not be curved inward. If a room produces too much reverb, the reflections can be damped by absorbers. Upholstered furniture, carpets, curtains, blankets, bookshelves and foam boards (but not cardboard egg cartons!) do a good job here. If a room does not return the sound sufficiently, standing waves may arise, at whose knot there is barely anything to hear, while at their loops it may become too loud. In this case reflectors can be used. Since sound waves propagate faster at higher temperatures, room temperature also plays a role.

Reverberation Time

The most important measure of the acoustic quality of a room is the reverberation time. Reverberation time is measured by how many seconds it takes for a sound signal to be 60 dB quieter (RT60).

Proven values are:

Size (m²)HallStudio
(rehearsal room)
300,70,4
1001,00,6
3001,20,8
1,0001,41,0
2,0001,71,1
5,0002,21,2

For measuring reverberation time, a measuring microphone and software, e.g. the free Room EQ Wizard, can be used.

Sound perception

With physical conditions alone, the effect of tones and sounds can not be explained. Sound perception rather takes place in the head of the listener and is therefore the subject of psychoacoustics.

To an inner, spiritual something correspondes an outer, physical something: tone. Music happens when both are “attuned” to each other. Tone is a psycho-physical fact.1)

References

  • Tipler, Paul A.; Mosca, Gene: Physik für Wissenschaftler und Ingenieure. Heidelberg: Springer, 6. Aufl. 2012
  • Hoffmann, Jerry: String Theory
  • Taylor, Charles: Der Ton macht die Physik: Die Wissenschaft von Klängen und Instrumenten. Heidelberg: Springer 2013

1) Ernst Levy: A Theory of Harmony. 1985, S. 3