To test the hypotheses that (i). electroencephalograms (EEGs) are largely made up of oscillations at many frequencies and (ii). that the peaks in the power spectra represent oscillations, we applied a new method, called the period specific average (PSA) to a wide sample of EEGs. Both hypotheses can be rejected. Although the principal peaks in the two spectra agree most of the time, quite often a peak in the power spectrum accompanies no periodicity peak and some periodicity peaks have no power spectral peak. The Fourier spectrum is not a reliable indication of rhythms. EEG samples from patients during waking, sleeping and seizure states, and volunteer healthy subjects doing cognitive tasks quite often show no significant rhythms, on an arbitrary, common sense definition. When clear rhythms are seen, they involve one or two, rarely up to four or five simultaneous non-harmonically related frequencies. Rhythms are special cases; most of the power spectrum most of the time is nonrhythmic. "Good" rhythms usually have quite narrow peaks, with frequency modulation of <5%, strengths of >2.5 up to >10 times the expectation from chance, and they often show fine structure by being quite local and brief. Most rhythms are quasisinusoidal but others are sharp-cornered recurrent events with <50% duty cycle. In the face of wide variability, we do not report any systematic differences in periodicity among EEGs from different parts of the brain or different brain states or species; it will take many more exemplars of each state, species or brain part to establish characteristic features. The PSA method may be the best so far proposed to demonstrate and quantify periodicity in wide-band time series with noise, but it has serious limitations. Discussion leads to the conclusion that it is time for a new paradigm or metaphor for brain waves.