A constant

timbre In music, timbre ( ), also known as tone color or tone quality (from psychoacoustics), is the perceived sound quality of a musical note, sound or tone. Timbre distinguishes different types of sound production, such as choir voices and music ...

at a constant pitch is characterized by a

spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...

. Along a piece of music, the spectrum measured within a narrow time window varies with the melody and the possible effects of instruments. Therefore, it may seem paradoxical that a constant spectrum can be perceived as a melody rather than a stamp. The paradoxA. Chaigne (1988), “Psychoacoustique”, ENST, 114 pages. is that the ear is not an abstract

spectrograph An optical spectrometer (spectrophotometer, spectrograph or spectroscope) is an instrument used to measure properties of light over a specific portion of the electromagnetic spectrum, typically used in spectroscopic analysis to identify mat ...

: it "calculates" the

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

of the

audio signal An audio signal is a representation of sound, typically using either a changing level of electrical voltage for analog signals, or a series of binary numbers for digital signals. Audio signals have frequencies in the audio frequency range of ro ...

in a narrow time window, but the slower variations are seen as temporal evolution and not as pitch. However, the example of paradoxical melody above contains no infrasound (i.e. pure tone of period slower than the time window). The second paradox is that when two pitches are very close, they create a beat. If the period of this beat is longer than the integration window, it is seen as a sinusoidal variation in the average rating: sin(2π(f+ε)t) + sin(2π(f-ε)t) = sin(2πft)cos(2πεt), where 1/ε is the slow period. The present spectrum is made of multiple frequencies beating together, resulting in a superimposition of various pitches fading in and out at different moments and pace, thus forming the melody.

MATLAB/Scilab/Octave code

Here is the program used to generate the paradoxical melody: n=10; length=20; harmon=10; df=0.1; t=(1:length*44100)/44100; y=0; for i = 0:n, for j = 1:harmon, y=y+sin(2*3.1415927*(55+i*df)*j*t); end; end; sound(y/(n*harmon),44100);

MATLAB/Scilab/Octave code

References

See also