beschreiben timbre

Often people try to characterize types of timbre quality as fat, deep or in terms of having colour.

but calling the sound of 3 overlapping oscillators "Fatter" just because there are more sound sources is a bit inaccurate. It seems color is naturally suited to describe differences of timbre between instruments.

A friend once observed that "colour" was a pretty natural choice of metaphor when describing the timbre of a sound, refering to the fact that the presence of multiple frequencies is at play both in the visible light waves we call colour and in the waves of musical tones.

In general, the timbre of a given musical sound (indeed any sound) is determined by the harmonic content of the accoustic pressure wave. OK statements like this are deceptively simple (save perhaps for the term harmonic to some readers) but they contain within a great deal of information. Let's simplify this statement and explore the concepts it encapsulates.

For starters, if the term "harmonic" sounds confusing we can do without it for now, because in any sound wave there are two components (among others) that make up / describe the qualities and characteristics of the wave: Amplitude (the loudness of the wave over time), and its frequency (the number of times the periodic waveform is repeated in a given unit of time). The latter , frequency, or rather a multitude of frequencies - combined together (added, interfering) to form the Waveform of the sound , ie the way the wave's amplitude looks during the time of a single cycle of the wave - make up the harmonic content of a wave.

I would also like to add that when I speak of waves I shouldn't only be speaking of accoustic or sound waves, which because they travel through air contracting and expanding it, I like to call by their other physical name, pressure waves.
There are also electric waves. And much of what gets said of the sound produced by material accoustic instruments , also applies to waves of electrical current, voltage waves , or electrical signals, streams of them. These are usually produced not by material accoustic instruments but by electric circuits that generate waves of different types and that modify them in different ways, (this is called synthesizing) and then drive loud speakers which convert the electric current waves into actual sounds that we hear (this is called reproduction). The electric waves themselves could also be analogs of wave information initially produced as binary data that describe all the properties of the waves produced, from waveforms to full performances. This binary data, called digital, is converted to an analog electrical wave or signal or current (stream) , using a device called DAC (Digital to Analog converter), and eventually the "analog" electric current is itself converted into accoustic sound waves or pressure waves, which are audible analogs of the electric current waves, and that travel in the air to reach our ears.

Let's simplify the earlier statement and then proceed to discuss the waves and their timbres.

In general, the timbre of a given sound is determined by the content of its wave.

The wave itself is a composite of several other waves, all added together - That is, they produced pretty much at the same time, and their intensities (loudness) combine to give the total loudness of the wave. These several other waves have different loudness values but are all considerably lower than the loudest composite wave, which has a frequency equal to the pitch of the musical note being sounded.
This loudest wave is called the fundamental, and the other less intens waves are called the overtones of the waves. The fundamental and the remaining overtones are also known as the harmonics of the wave. Each one of the overtones (aka harmonics) has a frequency that is a multiple of the frequency of hte fundamental, that is a multiple of the musical note's own pitch. I won't get into the arithmetic and trigonometrical relations between the harmonics here now because I'd like to focus on timbre now. But I'll say the easiest representation of the relationship (ratios) between the harmonics (and their frequencies) is the famous one using the length of a guitar string.

Because the overtones have frequencies that are multiples of the fundamental harmonic of the musical note (or electric signal), the harmonics of any sounds are distributed over the same values of frequency, namely, (f, 2f, 3f, 4f, 5f, ... and so on ).
What differs from sound to sound (among other things) is the relative loudnesses of the different harmonics.

There are other factors affecting timbre, such as the relative phases of the harmonics of the wave, as well as the presence or absence of other waves combined each with its own set of harmonics. But I believe the relative loudness of the overtones is the crucial factor in a sound's timbre or colour.

The harmonics of an accoustic instrument depend on the materials of which the sound generating components are made as well as the shape of the sound box of the instrument. The shape and materials of a violin's sound board, the hairs of its bow, the make of the strings, affect the harmonics of the instrument, and so despite the kinship different violin instruments sound different from one another. The same is true for most other instruments. Even playing techniques affect the tones' timbre.

Now when we want to look at a representation of the harmonics of a given wave, we have to use a diagram that plots Amplitude against (not time as in the more familiar wave diagram - called time domain diagrams but against) Frequency.
Based on the aforementioned, we should expect to see that at the the frequency of the fundamental the amplitude has the highest value, and that the other overtones appear in the graph as spikes in amplitude right where each overtone's frequency is on the horizontal axis. Here is a sample diagram of the harmonics of a wave produced on a VST synth and plotted by the FREE fre(a)koscope 0.8 spectrum analyzer plugin,



The tools that let us look at a wave's harmonics in this way are called spectrum analyzer, which are desendants or cousins of the oscilloscope which plots the Amplitude of a wave (vertical axis) against the time axis (horizontal axis).

Note the name spectrum here is not used in vain. Because as you see in the picture, the graph actually shows us a spectrum of frequencies, with spikes where overtones occur in the wave. This is very much like the spectra we get when analyze chemical elements and compounds to get to see which frequencies of light they emit (or absorb i don't remember) in the visible light band of the electromagnetic spectrum. Very much in the same vain, the color of an object is also those portions of visible light that are reflected or emitted by it.
Each object has its own spectrum (ie, collection of frequencies that make up its color) In the case of material objects , the frequency is that of electromagnetic waves, in teh case of sounds, the frequencies are those of accoustic pressure waves in the air, and in the case of synthesized sounds, they are frequencies of an electrical current signal.

Therefore, as my friend has noted, it does make perfect sense to use colour when describing a musical sound.

The images that follows as the one above, are of the same synth preset, but the harmonics look different , the peaks more or less are the same but the all sorts of amplitude changes occur to different frequencies,






This is because, the harmonics , and therefore the timbre of a sound vary with time, as the instrument is used to play a melody.

The images below show the frequency spectrum for a software electric piano playing C3 and C4 respectively,



In the latter figure the resolution of the sampling window was increased adding sharpness to the frequency spectrum graph, (chalk one up for the software designers and engineers),



This image is of a software organ playing a C4 (Do 4) note ,



The images used here are screenshots of the free fre(a)koscope 0.8 FFT-based realtime spectrum analyzer plugin, available on the FFT tools page at smartelectronix.com , a collective of quality-friendly audio and music software developers.