There is research claiming positive effects of binaural beats, but many suffer from methodological weaknesses. I'll present two examples of weak studies, and one that in my opinion may show a real effect (but failing to rule out at least one confounding factor).
Owens et al. 1998
(Owens et al. 1998) tested on 29 subjects. Each subject performed a 30-minute visual vigilence task on three different days while listening to noise while listening to either simple tones (no binaural beats), or binaural beats in the beta (16 and 24 Hz) or theta/delta (1.5 and 4 Hz) range. Participants were blinded as to the presence of binaural beats.
Each participant participated in three 30-minute experimental sessions across three days.
Session 1 was intended as training and establishment of a baseline. The participant was presented the control stimulus (with no binaural beats) for this session.
Sessions 2 and 3 were treatment session, where the participants were presented with the beta and theta/delta binaural beats, one in each session, with the order counterbalanced across subjects.
They used a single-tailed t-test (rather than a two-tailed t-test) to test the hypothesis that the beta frequency beats would improve performance more than the theta/delta frequency beats would. They found a statistically significant increase in the number of correct targets and a statistically significant decrease in the number of false positives when using the beta frequency beats as compared to the theta/delta frequency beats. They also tested self-reported mood differences.
There are various methodological weaknesses to this research:
They did not confirm that the binaural beats were not noticed by the participants. They were audible to the experimenters when listening carefully, but only relied on the fact that no participant reported hearing the beats.
They used a single-tailed t-test with the justification that their hypothesis was that beta frequency beats would improve upon theta frequency beats. This is a nice story, but they provided no prior declaration of this hypothesis, or any rationale for why they would not have simply selected a null-hypothesis of no-difference. The single-tailed t-test is a more powerful test, and thus reports statistical significance for smaller differences. This really seems like an inappropriate use of a one-tailed t-test, as it removed the possibility of finding the opposite result.
They mention that session 1 was used for both training and establishment of a baseline, yet present no comparisons with a baseline. Were either of the treatment conditions better than baseline? This paper doesn't answer that question.
In my opinion, these are symptoms consistent with the researchers having tried various comparisons and analysis strategies (perhaps comparing against baseline, trying a two-tailed t-test, etc.) and only reporting those that turned out showing some effect.
Wahbeh et al. 2007
This was an uncontrolled pilot study. It was an unblinded, non-randomized trial. Participants were simply given a CD containing binaural beats and they listened to it over a 60 day period. This study admits that it is simply exploratory. It doesn't draw any conclusions concerning causation, because it was not designed to do so. It simply identified a few possible correlations (which may not even exist due to the methodological weakness described next) for future research to verify in a controlled experiment.
- They performed 35 separate hypothesis tests and found 4 statistically significant differences from the null hypothesis at the p=0.05 level. When using p=0.05, these tests are expected to produce 1 false positive for every 20 hypotheses tested. That means in 35 hypotheses tests, you should expect to find about 1.75 statistically significant results just by chance. They found 4, so maybe there's something there, but this is anomaly hunting, and the exact situation in which one should apply a correction factor to control for the family-wise error rate. They did not apply a correction factor.
Reedijk et al. 2013
This study is more rigorous. It asks the question, do binaural beats affect creativity. They used binaural beats that were noticeable by the participants, calling the binaural beats a form of "cognitive entertainment", and they added white noise in such a way that it "enhanced the clarity of the beats".
They compared a control (a non-beating tone), to alpha frequency, to gamma frequency and tested performance on a divergent thinking task (naming as many uses as possible for a common household item) and a convergent thinking task (given three seemingly unrelated words, find a single word that could be prepended to each to make three new words: market, hero, and star become supermarket, superhero, and superstar).
There were 24 subjects each exposed to each condition (control, alpha, and gamma), and had to perform each task (divergent and convergent) in each of the conditions. The order of condition and task was counterbalanced across subjects using a latin square design.
Eye blink rate was also recorded as a proxy for striatum dopamine levels.
In my opinion, they use appropriate statistical tests to conclude that the convergent thinking task is unaffected by the binaural beats and that the divergent thinking task is affected by the binaural beats (no matter the type), but mediated by the subject's baseline eye-blink rate (striatal dopamine levels). Binaural beats increased the performance of those with low striatal dopamine levels. Binaural beats did not increase, and perhaps slightly inhibited the performance of those with higher striatal dopamine levels.
In summary, this study found that audible binaural beats (no matter the type: alpha or gamma) resulted in improved performance at divergent thinking tasks for people with low levels of striatal dopamine.
The proposed mechanism is that binaural beats could be assisting in setting up synchronized neural states. That assistance may not be needed in people with more optimal levels of striatal dopamine.
They don't report a frequency-specific effect as stated in some of the quoted claims. They also note that the effect is individualized, as opposed to the broad claims in the question.
One weakness is that they didn't compare against other sounds with beats. Perhaps any audio with a consistent beat would be sufficient to produce this effect.
Owens, James D., E. Kasian Justine, and Gail R. Marsh. "Binaural auditory beats affect vigilance performance and mood." Physiology & behavior 63, no. 2 (1998): 249-252.
Reedijk, Susan A., Anne Bolders, and Bernhard Hommel. "The impact of binaural beats on creativity." Frontiers in human neuroscience 7 (2013): 786.
Wahbeh, Helane, Carlo Calabrese, and Heather Zwickey. "Binaural beat technology in humans: a pilot study to assess psychologic and physiologic effects." The Journal of Alternative and Complementary Medicine 13, no. 1 (2007): 25-32.