Sound Healing Articles
Converting Sound into Shapesby Jill Mattson
The Chart, Room Shape, and Frequency reveals the frequencies of a square and rectangles. Barbara Hero calculated the frequencies of curved and angular shapes. She did this based on the work of Jules Antoine (1822- 1880), a physicist, who also connected sound and shapes. Antoine developed the Lissajous apparatus that bounced a beam of light off a mirror attached to a vibrating tuning fork. This then reflected it off a second mirror that connected to a perpendicular vibrating tuning fork (usually of a different pitch, creating a harmonic interval) and onto a wall. The two sounds were displayed at right angles to each other, creating shapes named Lissajous figures.
By varying pitch and rhythm, one could create shapes. For example:
- Two sine waves (sounds) of equal frequency and in-phase, produce a diagonal line to the right.
- Two sine waves of the same frequency and 180 degrees out-of-phase create a diagonal line to the left.
- Two sine waves of equal frequency and 90 degrees out-of-phase draw a circle.
Ratios of string lengths create musical intervals. Hero used such ratios and multiplied them by the arctangent (the x, y model of trigonometry). She then matched this math to the math that created the Lissajous figures from sound. In other words, she reduces a frequency to the interval ratio that created it, and multiplied it by the arctangent, then matched this number with frequency numbers that created curves and angles in matter. In this way, she mathematically could see the curved and angular shapes of musical intervals/ratios
Sounds & Shapes
Mathematics by Greg Opalinski
Right Angle Trigonometry
Sine and cosine ratios generate the Lissajous Patterns
The Math of Sine and Cosine Ratios
16 x 16 Lambdoma - fundamental frequency of 32 Hz
- 32 hertz frequency has been inserted into the 1/1 position and the math completed for this chart
- The 32 Hz diagonal has ratios of 1/1, 2/2, 3/3, …, 16/16
- The top row of frequencies (32, 64, 128, …) have ratios of 1/1, 2/1, 3/1, …, 16/1
- The first column of frequencies (32, 16, 10.66667, …) have ratios of 1/1, 1/2, 1/3, …, 1/16
- Further ratios in the matrix are formed by taking the top row numbers (in red) and putting them over the denominators in the blue column.
Creating Lissajous Patterns from Musical Intervals
Lissajous patterns can be constructed from the following set of equations:
where a is the harmonic number and b the subharmonic number
See gold arrow: 64 Hz – the ratio is 4/2 (simplified 1/2)
Lissajous for 64 Hz 4/2
Note that this is the same pattern as for the ratio of 2/1, or any equivalent. Thusly, this pattern will reoccur throughout the matrix.
See the red arrow on earlier page: 24 Hz – the ratio is 6/8 (simplified 3/4)
Lissajous for 24 Hz 6/8
Note that this is the same pattern as for the ratio of 3/4, or any equivalent. Thusly, this pattern will reoccur throughout the matrix.
The Ratio is 4/3
Lissajous for 42.66667 Hz 4/3
The number of times each ratio occurs in the 16 X 16 Lambdoma matrix.
The coloring is irrelevant and only intended to sort.
Major 7th 243:128
Minor 7th 16:9
Major 6th 27:16
Minor 6th 128:81
Perfect 5th 3:2
Perfect 4th 4:3
Major Tone 9:8
Minor Tone 10:9
The angles of whole number ratios are translated into frequencies. This creates a (musical) interval for any angle. These angles can then be applied to Platonic Solids. (The tangent trigonometric function relates the two legs of a right triangle.)
Again, the 16 X 16 matrix with a fundamental of 32 Hz The following will demonstrate how the arctan function is applied:
Any ratio of frequencies, or ratios, will yield the same degree measure.
In general, the angles can be determined by:
In my estimation, it is easier to apply ratios, for they will generate equivalent angle measurements.
Keep in mind that frequencies need to be applied to the angle measurement. Frequencies that I assume are based on specific conditions.
An example provided by Hero using a tetrahedron: By using the encoding of angles described, the tetrahedron angles can be shown to be equivalent to a note of <G. That is, the dihedral angle of this Platonic solid is 70°:31^':41''. We would round to 70°. This is similar to the arctan(14:5), or arctan(11:4). These ratios correspond to
Notes, Ratios, Angles and Musical Intervals of the Lambdoma Matrix
All the ratios in the Lambdoma Matrix were translated into angles using the arctan of the given ratio, based upon 1/1 being 256 hertz. (Barbara Hero), *Source Doug Benjamin