waves - Does timbre consist in pitch and volume?

I read that the physical properties of a sound wave correspond to its audible qualities: pitch, volume, and timbre. However, an oscilloscope uses only two-dimensions to accurately depict the physical properties of a wave. Intuitively, pitch and volume seem more basic than timbre does, so I surmise that timbre must consist in those two properties.

1. Does timbre consist in those two properties?


2. How do two-dimensional waveform diagrams depict three properties of sound waves?

Answer

Actually I have read (although I can't find a reference) that the subjectively perceived psychological notion of pitch itself, although very nearly wholly set by the sound wave's frequencies, is also weakly dependent on the intensity of the sound: that is, a higher intensity sound wave does seem ever so slightly sharper (higher in subjective pitch) than one of the same frequency but lower intensity. Hopefully another answerer to this question can add some more information.

Actually pitch and timbre are closely related. Timbre is primarily the harmonic mix of the sound wave: i.e if you wrote out a sound wave as:

$$y(t) = \sum\limits_{k=1}^\infty \left(A_k\, \cos(k\,\omega_0\,t) + B_k\, \sin(k\,\omega_0\,t)\right)$$

then, approximately we can make the following equations between physical properties of the sound waves (on the left) and subjective qualities of the sound perception experience (on the right):

$$\begin{array}{llcl}\text{Fundamental frequency}&\omega_0 & \mapsto & \text{"pitch"}\\\text{Intensity}& I = \sum\limits_{k=1}^\infty \left(A_k^2+B_k^2\right) & \mapsto & \text{"loudness"}\\\text{Relative Harmonic Mix}& \left(\sqrt{\frac{A_k^2+B_k^2}{I}}\right)_{k=1}^\infty & \mapsto & \text{"Timbre"}\end{array}$$

in other words, think of finding the relative amplitudes of each harmonic component and normalising them in an vector whose head lies on the unit sphere in $\ell^2$. The normalised position on the unit sphere is the timbre, whilst the squared length of the unnormalised vecotr is the loudness.

AS I said, the above are only approximate equations and a detailed answer would need to study the relevant psycho-acoustics carefully. Some interesting bits of trivia I've picked up as a singer and also learning to tune pianos:

1. For many instruments - the piano's effervescent timbre is a good example - timbre has a great deal to do with nonlinear phenomenon of intermodulation, whereby sum and different frequencies produce significant components of the wave at frequencies that are away from the lowest frequency in a chord, so the Fourier series actually has a very low fundamental frequencies but only closely spaced, high harmonic number "clusters" of them around the harmonics of the perceived fundamental frequency are present;


2. Modern pianos are NOT tuned to harmonics in a Fourier series, the are tuned to equal temperament, an invention mainly of J. S. Bach, whereby every semitone interval on the piano has the same frequency ratio, namely $2^{\frac{1}{12}} $ so that, on a logarithmic frequency scale, the semitones are evenly spaced. The motivation is that frequency relationships between notes in a melody, chord and so forth are then covariant with respect to any modulation (change of diatonic key) in the music. This allows each key to be equally well in tune, which was the motivation for J.S.Bach's (or F. Chopin's) 24 preludes, each a variation on the same theme in each of the 12 diatonic keys in major and minor modes. They were, so to speak, "showing off" the possibilites openned up by realising, through equal temperament, the covariance of geometry of any musical pattern with respect to modulation. In contrast, flutes and clarinets, or any wind instrument that uses harmonics to realise several registers, are constrained by the instrument's physics to have the same fingering in different registers (e.g. $C_0$ and $G_1$ on a b-flat clarinet) harmonically related; most often a register jump corresponds to a tripling of frequency (corresponding to the first two harmonics of an open pipe). Therefore, not all instruments can play in all keys equally well. An upshot of all this is that it is certain that our subjective sense of pitch would be rather different from, say, Bach's contemporary Vivaldi, as we have learnt to ignore the "out of tunehood" of well tempered tuned notes (difference between them and those produced by a harmonic series);


3. When chords are played with notes repeated at the harmonics, it becomes highly subjectives as to what is pitch and what is timbre and how accustomed the listener's ear's is to various instruments and their harmonic content weighs on the perceived pitches are in the chord. Experience with different instruments will cause the listener to group harmonics together as a single note differently, depending on what "timbre" is present in that instrument;


4. In extreme cases, pitch and timbre become irresolvably tangled, as it does with in very big chords with equal spacings between them. If you play a dimished seventh chord or augmented chord (equal spacings of a minor and major third, respectively) spanning many octaves (you have to do this with a computer using something like Sibelius) and make an "upwards" or "downwards" progression by the equal step in the chord, it becomes about a 50% chance whether a listener will perceive the progression as upwards or downwards (this is because, if the series were truly infinite, there would be no difference in the change). I have a similar experience listening carefully to the bass line (left hand) in the third movement of Chopin's piano sonata #2 (Op 35): if you look at the score you see why: the notes in successive chords line up in endlessly rising "stripes".