Geometrical Music Ratios Found in Platonic Solids

Geometrical Music Ratios Found in Platonic Solids. What musical intervals were considered the most pleasing or consonant? The answer is the same today as it was thousands of years ago.

When a musical instrument is tuned using a just intonation tuning system, the size of the main intervals can be expressed by small-integer ratios.

The  1:1 ratio forms a unison.
The 2:1 makes the octave.
The 3:2 creates a perfect fifth.
The 4:3 makes a perfect fourth.
A 5:4 ratio makes a major third.

The smallest intervals are called just intervals, or pure intervals. In music theory, an interval is the difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones. This includes  (1) two adjacent pitches in a melody and vertical. Also included are harmonic intervals. They pertain to simultaneously sounding tones.

The smallest three intervals include the unison, octave and perfect fifth.  Now let’s consider how the smallest of the 1st thee regular polyhedrons are like the 1st three musical intervals. Size is determined by the number of degrees in each polygon, Then the ratios of the total of the  degrees of the 1st three solids by size will be compared.

 

So What About the Geometrical Music ratios of the 1st 3 Platonic Solids?

Ancient computing was based on the Platonic Solids have geometrical music ratios
The tetrahedron becomes the unison with the 1:1 ratio.

Ancient Computing Utilized the Platonic Solids

Let’s total the degrees in the “unison”, octahedron and cube. for our geometrical music ratios.

  • A triangle has 180°.  As you can see the tetrahedron (above) has 4 triangulated faces. Each of the Platonic solids posses a special quality called duality. Make a point at the center of each of the four triangles on the tetrahedron,  Then internally connect the points. You end up sketching a smaller inverted tetrahedron. This figure is dual to itself. This parallels the 1:1 ratio of the unison.

We are now in a position to see how  some other Platonic solid ratios by degrees harmonize in the same way as perfect and harmonious musical intervals. This gives rise to our post title: geometrical music ratios:

  • The next largest regular polyhedron by degrees is the octahedron. It is composed of 8 triangles. Its total degrees would be double that of the tetrahedron as 1440° 70 720°. That is 2:1 just like the musical octave.
  • The next regular polyhedron in size by degrees is the cube, It contains a total of 2160°. With its 6 squares, 6 x 360° = 2160. Compare this total to the octahedron’s. We have 2160/1440 in the ratio of 3/2. These two solids become the ratio of the numbers of the perfect fifth. Duality exists between the octahedron and cube.  If you draw the mid points in each of the squares on the cube and connect them internally, you get the six vertices of the octahedron.

Here is the neatly laid out parallel which gives rise to the same geometrical music ratios:

  • tetrahedron/tetrahedron 1:1 unison interval is also 1:1
  • octahedron /tetrahedron is 2:1. Octave is also 2:1.
  • cube to octahedron is 3;2. The musical fifth is 3:2.
  • Image result for picture of the five regular polyhedrons
    The concept of the music of the spheres is here.

Feel free to sample another link on antiquities revisited

Auspicious Eight in Arithmetic, Geometry and Antiquity

 

 

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