Applying Binaural Beats and Tones to the Music

We used standard frequency bands:

Delta       .5Hz to 4Hz, the frequencies of deep sleep

Theta       4Hz to 7.5Hz, the frequencies of relaxation, meditation, drowsiness

Alpha       7.5Hz to 12Hz, alert without executive or analytic thinking

Beta         12Hz to 38Hz, alert and focused, attentive, active analytical and executive thinking

Gamma    38Hz to 90Hz. Where thought patterns are widely structured and static (in a rut), gamma disrupts the coordination between brain areas, allowing new thinking and creativity.

Binaural beats, Isochronic tones

Binaural beats, at left, are the audio illusion created by the brain when two tones, close in pitch but different, are played, at the same volume, one in each ear. The difference in the frequency of the two tones is the frequency of the beat.

This shows the nature of a binaural beat, and how the tones alternately reinforce each other (louder, near the ends) and cancel each other out (softer, in the middle).

In these recordings, the tone that matches the key of the music plays in the left ear, and the interference tone plays in the right ear.

Imaginary middle tone?

Normally, when creating binaural beats, the carrier tone is considered a tone between the higher and lower of the two constituent tones. For instance, if one uses 100Hz and 90Hz to create a 10Hz binaural beat, the theoretical carrier is 95Hz, or halfway between them. The perceived middle tone is anecdotal at best, and the perception of it is not consistent, since it depends on the relative pitches and the distances between them.

For SongRest’s purposes, we discard the idea of a carrier tone being theoretically anywhere. Instead, the carrier tone is normally equivalent to the key or the perfect fifth of the key of the music. Since the tones will be perceived in a music bed, the key of the music will take precedence over any theoretical middle tone. For this collection, to design a consistent experience, we arbitrarily chose to serve the carrier tone to the left ear and the interference tone to the right ear.

Isochronic tones, shown at left, are pure tones that pulse in volume, growing louder and softer in a consistent tempo. At left is an example of an isochronic tone.

The number of wave peaks per second is the frequency; the height of the waves is the loudness.

The isochronic tone gets louder and softer at the same frequency as the binaural beats’ interference pulses.

SongRest’s approach differs from some experimental standards in significant ways. Where we find the experimental standard is based on its own internal logic, and doesn’t translate to music in a practical way, we have replaced it with our own music-based standards.

The SongRest algorithm, developed for SongRest’s research into music for chronic pain, helped us to tune brainwave frequencies exactly to the key and notes in the music. We began with the key of the music, and assessed the major tones in that particular piece, and the desired brainwave band. The next steps:

1. Calculated the note frequencies and harmonics that fell in the frequency band, or were simple fractions or multiples of values in the frequency band.

2. Determined the minimum note frequency required to create the binaural beat, using Oster’s Curve as a guide to proportions. We do not rely on Oster’s absolute frequency values, as they are often impractical in music.

3. Created the carrier tone and the interference tone for the beat.

4. Used the binaural beat carrier tone as a potential carrier for the isochronic beat.

5. Tested and refined as needed.

The SongRest Algorithm

SongRest has developed an algorithm for matching brainwaves to music for pain, and applied the same algorithm to this music, in order to tie the brainwave frequencies tightly in with the music. The mathematical relationships between the brainwaves and the music give this music great potential in brain entrainment to the brainwave frequencies assigned to each piece of music.

The frequency bands used in the music are generally balanced, with at least four pieces in each band. This makes the collection a comprehensive and powerful brainwave collection.

For those interested in the technical aspects of the project, the SongRest algorithm is described below.

K = key frequency (root note hz)
Ka= K adjusted to a harmonic of f, namely an integer multiple or fraction, harmonic or subharmonic).
T = tempo in Hz = BPM/60
Ta = tempo adjusted to allow brainwave mapping
f = target entrainment frequency
l= multiplier based on key frequency
BBc= binaural beat carrier frequency
BBi= binaural beat interference frequency
e = isochronic carrier

Formula: Where f a Ka a Ta, BBc = Ka ( l), BBi  = BBc - f,  e = Ka x l a (f)

Description of the formula: where the target frequency is proportional to the adjusted frequency and to the adjusted tempo, the binaural beat carrier is the adjusted frequency times a multiplier. The multiplier could be a whole number, in which case the carrier is a mathematical harmonic of the key, or a fraction, in which case the carrier is a lower tone, a subharmonic. Subtract f, the target frequency, from the carrier tone to get the interference tone frequency. The complications come in matching notes of the chromatic scale with frequencies in the brainwave bands.

Below is an image from music editing software, modified to show how  closely the volume of the tones follows the music. You’re seeing only the last 25% of the music, so you can imagine the work that went into matching the tones and adjusting them throughout each piece.

In this image, the original music is in teal green. The “carrier” tone of the binaural beat is in purple. This is the foundation tone, matching the key of the music.

The volume of the tones is calibrated to follow every nuance of the volume of the music, keeping the tones always just in the audible range. Some instruments and some frequencies sound louder at a given volume than others, and that is one of the factors in our sound adjustments. This was not an automated process; pitch, tonal quality, and other aspects of the music as well as the sound qualities of the beats and tones themselves were factors in adjusting volume. 

SongRest used both the music editor interface and also direct editing of XML code to obtain precision of volume matching. Together the Gordons and SongRest went through many rounds of edits to get them just right.

Of course people differ, so some tones will be more audible than others to different individuals, but we have done our best to make this music into a valuable tool for relaxation and transformation for everyone.

Tools and methods

Our primary tool for sound design was Adobe Audition CC 2018, a professional DAW (digital audio workstation) package designed for editing music. It can change tempo and frequency of music tracks independently with precision (to 1/100Hz) without degrading sound quality, and can generate pure tones for binaural beats. We used Audition to create the binaural beat tones as separate files, and, also in Audition, imported them and aligned them with the music.

To generate isochronic tones, we used Audacity 2.13, a free program, with a plugin called Isomod. Both Audition and Audacity run on both Macintosh and Windows operating systems. Everything was generated as wav or aiff files, so as not to lose any quality in editing.

Our software for this project ran on Macintosh OSX 10.13.2 (High Sierra) on a MacBook Pro with a 2.9GHz Intel i7 processor (four cores) and 16GB RAM. The drive is a 500GB flash drive.

Using sound files at full uncompressed quality meant larger music files. Since the music averaged 30 minutes per selection, they were in the vicinity of 1GB per final mixed music file.

We got around a minor technical limit of the Audacity Isomod plugin by creating pure tones and isochronic beat profiles at fractions of the intended frequencies, within an accuracy of .001Hz to reduce errors when scaled to 2-3 times the original frequency.

We increased the frequencies in Audition by scaling the duration of the clips to match the exact requirements of the music. This required careful calculations and exacting placement of the tones with the music, often zooming in to extreme levels as shown, and editing by fractions of a cycle to match generated tones exactly with the music. 

The waveform image shows the manual alignment we accomplished to match wave peaks of the binaural beat carrier tone (293.67Hz, the D just above middle C), the BB low tone of 244.725Hz (almost but not quite a half-step lower), with the center of the isochronic pulse. The isochronic pulse for this piece pulsed at 16.33Hz on a baseline tone of 587.33Hz (D an octave above the binaural beat carrier). The 16.33Hz frequency is 1/36 of its carrier tone, and if we could hear a sound that low, it would also be the note D.

Dealing with any sound, we work with two fundamental factors: pitch and amplitude. Pitch is how high it is, based on the frequency (the number of waves per second). Amplitude is the loudness (the relative height of the waves to each other).

SongRest diverges from previous practice in isochronic tone production. In many research conditions, the isochronic tone pulses between silence and on, with a small ramping up and down to keep it from producing a “click” effect.

SongRest sets the baseline level amplitude for the tone, still audible, and pulses to a higher amplitude, ramping up and down at the beginning and end of each pulse, so that the peak amplitude lasts for 33% of the time between pulses where possible. This design is shown below

Listen in headphones, ear buds or speakers

We use binaural beats plus isochronic tones for two reasons. One, the two modes enhance each other, making a music experience of greater power. Two, when listening in speakers, the isochronic tone pulses may be more audible than the binaural beats, providing benefit even when listening through speakers. Even with stereo separation, binaural beats function in speakers as monaural beats, in which both tones are heard by both ears.

We have tested the music thoroughly, using high quality speakers, laptop speakers, studio headphones, low-cost headphones and ear buds. The selected tones and beats are based on the standards illustrated.

Historical note on Brainwave Bands

You would think the brainwave frequencies would be listed in the order of the Greek alphabet (alpha, beta, gamma, delta, etc.). But because they were discovered at different times, the order of frequency names is not in the nicely ascending order of the Greek alphabet.

Instead they were named as they were discovered:

1. alpha (7-12Hz) and beta (12-38Hz), discovered by Hans Berger, the inventor of the electroencephalograph, or EEG,  in 1924

2. Delta waves (.5-4Hz), discovered by W. Grey Walter, 1930s, who improved the EEG.

3. Theta (4Hz-7.5Hz), identified by Jung and Kornmüller in the 1930s and clarified by John D. Green and Arnaldo Arduini

4. Gamma  (38Hz-90Hz), identified by Pheiffer and Smythies in 1964.

The order from lowest to highest frequencies is: Delta, Theta, Alpha, Beta, Gamma.

Around 200Hz there are also Lambda waves, which, unlike most brainwaves, are saw-toothed triangular waves around 200 Hz (most other waves are smoothly-curving sine waves). Lambda waves are associated with activity of eyes open, visually scanning bright complex views . Gamers produce a lot of lambda waves.

Lambda waves being primarily visual, we have not included them in the music.

Hz is named for Heinrich Herz, who first proved the existence of electromagnetic waves such as radio waves, which are measured in cycles per second.

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Pfeiffer, Carl Curt, and John R Smythies. International Review of Neurobiology. Vol. Volume 7 /. International Review of Neurobiology, V. 7. Place of Publication Not Identified: Academic Press, 1964.  https://www.ncbi.nlm.nih.gov/pubmed/21809747 (abstract only)