So to summarize their claim, they're saying that entrainment means that a binaural beat will cause your brain's electroencephalogram to match the pattern of the phantom beat. Well, if it did, entrainment certainly doesn't apply and would not be part of the equation, so we can scratch that off the list. But it doesn't make the claimed observation wrong. We do know that certain electroencephalogram waveforms are often associated with certain kinds of activity. For example, physical activity or REM sleep often produces an electroencephalogram with a sine wave of between 4 and 8 Hz, which we term a theta pattern. Waking relaxation with eyes closed often produces a pattern from 8 to 12 Hz, which is called an alpha pattern. There are only a few characterized patterns, and pretty general descriptions of what kinds of activities go with them. The claim made by the binaural beat sellers depends on much more granular and specific matches. For example, the claim that a binaural beat with a frequency of X produces the same effect in your brain as Vicodin is wholly implausible. Such claims presume that we know the exact frequency of the electroencephalogram in each of these desired conditions, and the fact is that brain waves don't work that way. It is wholly and absolutely implausible to say that desired brain condition X will occur if we get your EEG to read exactly X Hz.
The composer Alvin Lucier has written many pieces that feature interference beats as their main focus. Italian composer Giacinto Scelsi, whose style is grounded on microtonal oscillations of unisons, extensively explored the textural effects of interference beats, particularly in his late works such as the violin solos Xnoybis (1964) and L'âme ailée / L'âme ouverte (1973), which feature them prominently (note that Scelsi treated and notated each string of the instrument as a separate part, so that his violin solos are effectively quartets of one-strings, where different strings of the violin may be simultaneously playing the same note with microtonal shifts, so that the interference patterns are generated). Composer Phill Niblock's music is entirely based on beating caused by microtonal differences.
Theta brainwaves in the frequency range of 6 to 9 hertz are known as thalpha waves because of their proximity to alpha waves. They are seen during states of high suggestibility, during hypnosis, and during paranormal experience and are also connected to an increase in human growth hormone (HGH) levels and higher blood flow to the brain. Theta brainwaves sit between the realms of the subconscious and conscious mind, making them an especially useful brainwave state for developing creative thinking, working through emotional problems, and integrating subconscious and conscious experience.
The activity of neurons generate electric currents; and the synchronous action of neural ensembles in the cerebral cortex, comprising large numbers of neurons, produce macroscopic oscillations. These phenomena can be monitored and graphically documented by an electroencephalogram (EEG). The electroencephalographic representations of those oscillations are typically denoted by the term 'brainwaves' in common parlance.