Musical Geometry

Music gives a soul to the universe, wings to the mind, flight to the imagination and life to everything. …Geometry will draw the soul toward truth and create the spirit of philosophy.  ― Plato

Music is the universal language of mankind.  ―Henry Wadsworth Longfellow

Did you know you’re a mathe-magical musician with an inherent understanding of music theory?  All living organisms are. It’s no coincidence.  Just sing a note, any note.

Laaaaaa

Now, hum another note that’s higher than the last, but make it sound harmonic, the best you can.

You just sung the opening two notes to Star Wars™, and learned the perfect fifth in the process.  Two notes with a frequency ratio of 2:3 are perfect fifths.

Dunnn (2) DUNNNNN (3) …… Dun Dun dah Duhhn duhn….

The perfect octave can be found in the song “Some Where” (over the rainbow)
Some (1) Where (2)

The perfect fourth if found in every wedding. Here Comes (the bride)
Here (3) Comes (4). Listen to how the pitch of the note goes up one third, from 3 to 4, this is the perfect fourth.

The perfect fifth is the most natural and pleasing sound there is. Pythagoras and his buddies focused on the Octave, Fifth, and Fourth, probably because these can be constructed within the tetractys.

by Chad Adams

The perfect fifth is the most natural sounding resonance possible. The music of the spheres takes it literal with the orbital resonance of Pluto and Neptune, upon making themselves around the Sun.

The perfect fifth is like a template that embeds space and time.  As soon as unity (one note) manifests, duality, in the form of the most pleasing sound possible, the perfect fifth, will always emerge.

This GIF shows the geometry of the major scales around the Circle of Fifths.

A man should hear a little music, read a little poetry, and see a fine picture every day of his life, in order that worldly cares may not obliterate the sense of the beautiful which God has implanted in the human soul.
― Johann Wolfgang von Goethe

Some more geo-musical and mathe-magical relationships.

Music is … A higher revelation than all Wisdom & Philosophy.
― Ludwig van Beethoven

The Tetractys Reveals Perfect Musical Intervals

Semi-minor axis

The orbital distances of the planets creates musical relationships, at least according to NASA.  (Keplerstern verlag)

There are two ways to square the circle, that is, draw a square and circle with commensurate perimeters and areas.  Squaring the circle with equal area creates the whole step, whole tone, or major second, with a ratio of 9:8.

The Pythagorean comma is the small difference between 12 perfect fifths and 7 octaves.  In other words if you take any note, and scale it up twelve fifths, you’ll land almost exactly on a note that is seven octaves above it.  The difference is called the Pythagorean comma.  Earth’s yearly cycle echoes this phenomenon, as compared to the degrees in a circle.

The comma can be found in the Eye of Horus (or Ra)

An ancient myth describes a battle between Horus and Set in which Horus´ right eye was torn out.  Thoth magically restored Horus´ eye, at which point it was given the name “Wadjet” (“whole” or “healthy”).

The Eye of Horus(Thoth, Ra) is broken up into 6 sections.  Comprising these six sections are fractions that sum to 63/64ths.  There are many explanations as to why Ra’s eye doesn’t add up to a whole.  One is the pythagorean comma, yet it is only 99.78% accurate.  The other, more interesting explanation lies in the Babylonian Talmud as well as the book of Isaiah in the Old Testament.  This will be discussed much more in a coming article. 

Eye of Horus  = 64/63   = 1.01587302…

Pythagorean Comma = 531441/524288 = 1.01364326…

^Image courtesy of John(Chad) Adams 

Geometry seemed to me to be a knowledge which man appeared to have produced but which had, nevertheless, a significance quite independent of man. Here one is permitted to know something which the mind alone, through its own power, experiences. In this feeling I found the justification for the spiritual world.  -Rudolf Steiner

Plato’s solids, in and of themselves, display the musical ratios in their total angular sum.  The perfect fourth isn’t as pretty as we have to use two octahedra, but then again, it doesn’t sound nearly as pleasant as the perfect fifth or octave.

The music is not in the notes but in the silence between them.
~ Mozart

Circles of Fourths in 3-D.  (concept by Marshall Lefferts) I wanted to see what synchronicities would arise when I used all flats, instead of both flats and sharps most musicians are familiar with when using the Circle of Fifths.

The Circle of Fifths in 3 dimensions on the cuboctahedron

Music is liquid architecture; Architecture is frozen music.
― Johann Wolfgang von Goethe

Now getting back to the first picture illustrated in this article….

Musical Geometry: Visualizing Sound

The intervals in music theory, the perfect octave, fifth, and fourth, are shown in various sized circles. The diameter of the red circle is six, the first perfect number (1+2+3=1x2x3). Inside it, all the perfect intervals can be found. They’re “perfect” because they sound so nice, and can be formed using the nodes of the tetractys(lower left).

The perfect octave is represented by the green(3) and red(6) circle, or the yellow(1) and blue(2) circle. This doubling is found in cellular mitosis, vortex math, and in the song “SOME WHERE over the rainbow”.

The perfect fifth, the most pleasing sound imaginable, is a 2 to 3 ratio found in blue and green. The opening line to Star Wars™ (Dun, Dun…) is the perfect fifth, as well as TWINKLE TWINKLE little star. The higher the pitch you sing, the bigger the circle.

The perfect fourth is a 3:4 ratio, shown in green and orange. “HERE COMES the bride” where “here” is green, and “comes” is orange. “Comes” is exactly a perfect fourth above “here”. You also here a lot of fourths in Super Mario Bros.

The circle of fifths wraps around the red circle clockwise. Each following note is a fifth above the previous. The circle of fourths opposes the fifths, rotating counter clockwise.

It’s really weird how these ratios, audibly displayed upon Earth, are reflected temporally in the cosmos.

As above, so below. 

On Earth as it is in heaven.

We are the music makers, and we are the dreamers of dreams.
― Arthur O’Shaughnessy

Even More:

https://www.youtube.com/watch?v=LJVO-FcxDio

http://www.megageometry.com/2015_01_01_archive.html

http://www.earmaster.com/products/free-tools/interval-song-chart-generator.html

https://www.youtube.com/watch?v=8_1mA8XFp4c

https://vimeo.com/129891456

https://sites.google.com/site/432octaves/geomusic

http://cosmometry.net/basics-of-the-music-system

http://www.cymascope.com/cyma_research/astrophysics.html

https://www.sciencedaily.com/releases/2008/04/080417142454.htm

http://www.cymaticmusic.co.uk/overview.htm

http://www.esotericonline.net/profiles/blogs/cymatics-precession-time-and-the-platonic-solids

http://www.roelhollander.eu/en/blog-music/zodiac-signs-and-tonality/

http://www.schillerinstitute.org/fid_91-96/fid_911_jbt_tune.html

http://mathemagicalmusic.weebly.com/

http://ray.tomes.biz/alex.htm

http://ddurham.weebly.com/geomusic.html

http://www.harmonisphere.com/SacredGeometryOfMusic.htm

http://whatmusicreallyis.com/research/physics/

http://www.cocreatorsworld.com/science-of-harmony/the-forty-nine-octaves-of-sound-and-light/

https://www.youtube.com/watch?v=Wja0WBLgaiQ

http://www.svpvril.com/musicuni.html

http://www.artofclaytaylor.com/single-post/2016/05/28/HARMONIC-GEOMETRY-AND-TONAL-MAGNETISM-1