In 1973, biophysicist Dr. Gerald Oster published a famous article in Scientific American titled “Auditory Beats in the Brain”, which found that when two pure tones of varying frequencies were combined, a third rhythmic beat was created which he called binaural or monaural beats. According to Oster, monaural beats occur when two tones are combined and sent through a loudspeaker, while binaural beats occur when stereo headphones are used to deliver each tone separately to each ear. Oster concluded that monaural beats were a more effective form of brainwave entrainment.
By exposing an individual to repeatedly external stimulus which can be flashing lights and/or recurring sounds, the entrainment process consists in pushing the brain to adjust to match the frequency provided. Unlike other forms of neurotherapy which require the individual to actively respond to stimulus, entrainment creates an immediate neurophysical response which is instinctive and effortless for the person being provided with the light and/or sound combinations. The key outcome of the sensory stimulation is known as the ‘Frequency Following Response’.

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Generally speaking, the brain will usually entrain to the strongest stimulus which would be isochronic tones over binaural beats. So when you see people add binaural beats at a different frequency to the isochronic tones, that would not produce additional brainwave entrainment at another frequency. If they are both at the same frequency I haven’t seen any research to indicate whether that would be beneficial or not, but my belief is that it would weaken the potential for entrainment. When you look at the waveform of an isochronic tone there is a distinct empty space between each beat, making it very pronounced and effective. When you add binaural beats at the same frequency it looks like this: http://www.mindamend.com/wp-content/uploads/2017/11/isochronic-tones-binaural-beats-combined-waveform.jpg. The depth of the waveform is now half as deep and less effective. This is before the binaural beats are formed inside your head, where the waveform is hard to determine and measure. From listening to that type of combination the beats sound much less pronounced, which has to make them much less effective in terms of a brainwave entrainment stimulus, compared to isochronic tones on their own.
Above is a nice, simple and short chart of the various brainwave ranges and the types of effects they generally have and what sorts of activities they may be able to enhance. There simply is nothing more detailed than what this chart contains about entrainment frequencies and what they are useful for. Those other elaborate lists to be found on the internet are nothing but the product of overactive imaginations.
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Subsequently, the term 'entrainment' has been used to describe a shared tendency of many physical and biological systems to synchronize their periodicity and rhythm through interaction. This tendency has been identified as specifically pertinent to the study of sound and music generally, and acoustic rhythms specifically. The most ubiquitous and familiar examples of neuromotor entrainment to acoustic stimuli is observable in spontaneous foot or finger tapping to the rhythmic beat of a song.
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