## Geometry, music and mathematics.

Geometry, music and mathematics are more connected than you might think. Geometric shapes found in Platonic solids mirror nice sounding musical intervals and small whole numbers, while shapes not found in them mirror bad sounding intervals and irrational numbers. To really understand this the best place to start is cymatics, this is the game of vibrating water or particles with sound to see what shapes emerge.

The following video is the holy grail of cymatics - in my eyes. It shows how each harmonic in the harmonic series can be physically connected to each regular polygon, with the same amount of sides as that harmonic's sequence number in the series. The same shapes can be created with water in a bowl on a speaker, and in various other ways, but this is the best example I have found. I guess having no container removes some factors that would otherwise make the results less clear.

IMPORTANT if you are using a phone and the images are on a dark background and are all above each other, scroll to the bottom of the page and select "mobile" view instead of "web".

The following video is the holy grail of cymatics - in my eyes. It shows how each harmonic in the harmonic series can be physically connected to each regular polygon, with the same amount of sides as that harmonic's sequence number in the series. The same shapes can be created with water in a bowl on a speaker, and in various other ways, but this is the best example I have found. I guess having no container removes some factors that would otherwise make the results less clear.

IMPORTANT if you are using a phone and the images are on a dark background and are all above each other, scroll to the bottom of the page and select "mobile" view instead of "web".

The harmonic series is infinite, and does not end where the video ends. The spaces between these harmonics are where musical intervals come from. The octave, fifth, fourth and all intervals in normal equal temp tuning are based on, and sound similar to, these intervals. The actual harmonic intervals are written as ratios. A perfect fourth, for example, has a ratio of 4/3 because it is found between harmonics 4 and 3, while a perfect fifth with a ratio of 3/2 is found between harmonics 3 and 2.

Not all intervals come from the spaces between successive harmonics. Ratios like 5/3 are also used. The 7/6 Septimal minor third however sounds odd and is not similar to anything on a normal piano.

Here is an example of a nice sounding ratio based scale called "Ptolemy's intense diatonic scale". It sounds almost exactly like a normal equal temp C major scale. The pure ratios do make it sound smoother, though. If you don't know how ratios work in scales it is simple. C = 1/1 because it is the root note of the scale (perfect prime), and all of the other notes are measured against this note using the ratios. For example: D = 9/8 because C multiplied by 9 and divided by 8 = D.

Here is an example of a nice sounding ratio based scale called "Ptolemy's intense diatonic scale". It sounds almost exactly like a normal equal temp C major scale. The pure ratios do make it sound smoother, though. If you don't know how ratios work in scales it is simple. C = 1/1 because it is the root note of the scale (perfect prime), and all of the other notes are measured against this note using the ratios. For example: D = 9/8 because C multiplied by 9 and divided by 8 = D.

According to cymatics, the 5/4 major third contains a pentagon and a square, while a 4/3 perfect fourth contains a triangle and a square. Putting a triangle over a square, or a pentagon over a square in a 2D space, does not reveal much. You could just as well put a 7 sided septagon over the square and it will not look much worse. If, however, you think in 3D and see what you can build with these shapes, everything changes and things became really interesting. To understand this you need to look at the Platonic and Archimedean solids. The Platonic solids are named after Plato, but seem to have been around even before his time.

A Platonic solid is a regular, convex polyhedron. They are made from identical-in-shape-and-size, regular (all angles equal and all sides equal) polygonal faces, with the same number of faces meeting at each vertex. Basically, the Platonic solids can only be made from triangles, squares or pentagons. Only five solids can be made using these rules; they are the simplest 3D shapes that exist:

Dodecahedron

All of the Platonic solids are made from only triangles, squares and pentagons, and each one is made of only one of them. If you use more than one regular polygon together in one shape, you can make the Archimedean solids. Obviously, these are named after Archimedes and only 13 of them exist. They have very long names, so I am not adding them here.

Archimedean solids contain triangles, squares, pentagons, hexagons, octagons and decagons. In the levitated drop of water, a hexagon is an octave of a triangle, which is why it contains 2 triangles. In the same way an octagon is an octave of a square and a decagon is an octave of a pentagon. In music and vibration, all octaves of a frequency have the same note name (C, D, E etc). So, musically speaking, the building blocks of both the Platonic and Archimedean solids are really just triangles, squares and pentagons (see the 3 images below).

7, 11 and 13 sided polygons below don't work for building nice solids, and they don't contain any smaller regular polygons. You won't find these in any Platonic or Archimedean solids.

If you look at the degrees of the angles in the corners of the regular polygons, you will see that 7, 11, 13 and 14 sided polygons have long irrational numbers while the rest have whole numbers. 14 is an octave of 7 so they have the similar properties mathematically. just as they do in music and geometry.

A regular triangle (3 sides) has 60 degrees in one corner.

A regular square (4 sides) has 90.

A regular pentagon (5 sides) has 108.

A regular hexagon (6 sides) has 120.

A regular heptagon (7 sides) has

A regular octagon (8 sides) has 135.

A regular nonagon (9 sides) has 140.

A regular decagon (10 sides) has 144

A regular hendecagon (11 sides) has

A regular dodecagon (12 sides) has 150.

A regular tridecagon (13 sides) has

A regular tetradecagon (14 sides) has

A regular pentadecagon (15 sides) has 156.

A regular hexadecagon (16 sides) has 157.5.

A regular triangle (3 sides) has 60 degrees in one corner.

A regular square (4 sides) has 90.

A regular pentagon (5 sides) has 108.

A regular hexagon (6 sides) has 120.

A regular heptagon (7 sides) has

**128.57142857142857142857142857143...**A regular octagon (8 sides) has 135.

A regular nonagon (9 sides) has 140.

A regular decagon (10 sides) has 144

A regular hendecagon (11 sides) has

**147.27272727272727272727272727273...**A regular dodecagon (12 sides) has 150.

A regular tridecagon (13 sides) has

**152.30769230769230769230769230769...**A regular tetradecagon (14 sides) has

**154.28571428571428571428571428571...**A regular pentadecagon (15 sides) has 156.

A regular hexadecagon (16 sides) has 157.5.

Somehow the mathematics behind the polygons mirrors the way they work in a physical space. But that is not all. The same thing happens when you translate all of this to audio. If you listen to the ratio based intervals below in the Bandcamp players and look at the cymatic shapes I have added, you will see that ones containing triangles, squares and pentagons sound the best, while ones that contain 7, 11 or 13 sided polygons sound bad.

3/2 Fifth is the next best interval.
Three lines make a triangle which is interesting, but a fifth can also be 6/4, then you get a more sensible hexagon and a square. Hexagons contain two regular triangles because a hexagon is an octave above a triangle (3 x 2 = 6). Hexagons, triangles and squares are good for building solids. |

This is all quite amazing to me. 7, 11 and 13 sided polygons have irrational numbers in their angles, they are no good for building 3D shapes, and they make bad sounding intervals in music. I don't know why all of this works so well, but I no longer have a doubt that mathematics, geometry and sound are connected in a really deep way. Maybe 3D sound waves behave in a similar way to 3D geometry, with some numbers of waves just not fitting well with others? While numbers themselves have similar mathematical properties to the way they sound in ratios and work in 3D geometry ? It certainly seems that way to me.

If you want to learn more about this, you might enjoy my book "Mathemagical Music Production". You can read the first few pages for free on Amazon (link below).

If you want to learn more about this, you might enjoy my book "Mathemagical Music Production". You can read the first few pages for free on Amazon (link below).

Web page made on 10 May 2018