geometryinmusic - Tumblr blog

[geometryinmusic]

MUSIC

As Dmirti Tymoczko quoted once

 "Music is as if we had maps of many small neighborhoods in a city, but we didn't have a sense of how those maps fit together." It is generally the arrangement of art and sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. There have been lots of geometrical representations of music.

This blog explains some concepts of the music theory which corelate with geometry.

[geometryinmusic]

SEMITONES AND TONES

The semitones and tones are mostly prominent in keyboard and piano.

Semitones are of two types:

Chromatic - is the interval separating a note with the same note raised by a sharp or lowered by a flat. A chromatic semitone has 5 commas. Ex. G-G# or E-Eb

Diatonic - is the smallest interval separating two conjunct notes. It measures only 4 commas. The diatonic semitone always involves two notes with different and contiguous names. Ex. G-Ab or E-D#

Whereas a Tone is formed of a chromatic semitone and a diatonic semitone. It therefore has a total of 9 commas. Ex. C-D or Eb-F.

Here a comma is a slight difference in frequency occurring when a note of a scale, say E in the scale of C, is derived according to different systems of tuning. And these commas are called the Pythagorean commas which were first discovered and introduced by Pythagoras which were later studied by Euclid and some other Chinese mathematicians.

[geometryinmusic]

STAFF NOTATION

It is a way of expressing musical notes in five parallel lines which have equal gaps between them known as staffs.

The staffs when combined with musical notes is together called the staff notation.

[geometryinmusic]

NOTES IN STAFF NOTATION

Quarter Note

The quarter note lasts for one count. Quarter notes, or quarter note crotchet, are probably the most common note type in music. 

Half Note

Half notes last for 2 counts, or twice as long as a quarter note. Half notes are probably the second most common type of rhythm after quarter notes.

Whole Note

The whole note last the same time as four quarter notes.

Dotted Half Note

A Dotted Half Note gets three counts. Two counts for the half note, plus one count for the dot.

[geometryinmusic]

CIRCLE OF FIFTH

The circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths.

It deals with the sharpening and flattening of the music notes. In the circle of fifth, there are Pythagorean commas which is an interval of quarter of a semitone which shortens the width of one of the 12 perfect fifths.

It was known in antiquity by Pythagoras which was later calculated by Euclid and some other mathematicians.

The circle of fifth is also called the Pythagorean circle.

[geometryinmusic]

TIME SIGNATURE

The time signature is defined as the number of notes per a beat. and, these beats are symmetrical in nature. It help us divide music into phrases and make it easier to memorize and understand more complex passages and songs. Without them, measures in notated music would run on forever, much like a sentence without any sort of punctuation. There are different types of time signature:

Simple-3/4,4/4....

Compound-9/8,12/8...

Complex-5/4,7/8...

Additive-3+2+3/8

Irrational-3/10,5/24....

[geometryinmusic]

CIRCLE OF FOURTH

The circle of fourths is the clock of music, showing the relationship between the 12 key signatures. There are 12 notes in music and therefore 12 key signatures, each with a varying number of sharps and flats. These 12 keys each contain seven notes in the form of a diatonic scale. These diatonic scales are connected to the chords that belong in each of these keys too. If you used all 12 notes to construct a scale, you would merely have what’s called a chromatic scale, which is sometimes used in music, but isn’t in any key signature specifically. Since music is always played in one of the 12 keys, it's important that you understand how each key differs.

[geometryinmusic]

SACRED GEOMETRY IN MUSIC

Music and Sacred Geometry explores patterns, structures, and forms that exist in both music and nature. Taking a page from ancient Greek philosophy, it connects music not only to math, astronomy, and our physical world, but also to a kind of ethos, or soul - something that transcends physicality and brings us into the realm of spirit.

we can identify symbols by drawing harmony from 'sacred' geometry, and understanding the many ways music has been used throughout the ages in ritual, ceremony, and healing modalities, the class uncovers the intrinsic intelligence and power music provides us at many levels of our existence.

The below image shows how symmetry is followed in the Harmonisphere:

[geometryinmusic]

ORBIFOLDS

According to Dmitri Tymoczko, music is a lattice of points folded up in a symmetrical space known as orbifolds.

Orbifolds capture the multidimensionality in music: how chords are connected with each other, even how notes are arranged "to minimize the amount of effort that musicians have to make when moving from chord to chord." This clearly relates to one of the principles of geometry stating everything is connected.

He also explained how a tetrahedron represents chords to form a Chopin piano and how chords have a non-Euclidean geometry.

[geometryinmusic]

THE FIBONACCI SEQUENCE AND GOLDEN SPIRAL IN MUSIC

The Fibonacci Sequence has been nicknamed ‘nature’s code’, ‘the divine proportion’, ‘the golden ratio’, ‘Fibonacci’s Spiral’ amongst others.

it’s a series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

The next number in the sequence is found by adding up the two numbers before it. The ratio for this sequence is 1.618. This is what some people call ‘The Divine Proportion’ or ‘The Golden Ratio’. When you make squares out of those values, it makes a nice-looking spiral called the golden spiral.

Now, if we compare this to notes in music we find out that it looks similar to the bass clef in the staff notation:

The Fibonacci Sequence plays a big part in Western harmony and musical scales:

An octave on the piano consists of 13 notes. Eight are white keys and five are black keys. A scale is composed of eight notes, of which the third and fifth notes create the foundation of a basic chord In a scale, the dominant note is the fifth note, which is also the eighth note of all 13 notes that make up the octave. Eight divided by 13 equals 0.61538... the approximate Golden Ratio.

Also hailed as the master of violin making, Antonio Stradivari has made some of the most beautiful and sonorous violins in existence.

The Golden Ratio can be found throughout the violin by dividing lengths of specific parts of the violin. Some people think this is one of the reasons it sounds so good.

As well as being used to craft violins, the Golden Ratio that comes from the Fibonacci Sequence is also used for saxophone mouthpieces, in speaker wires, and even in the acoustic design of some cathedrals.

Even the most famous music artists like Lady Gaga,  Beethoven, Bartok and  Debussy have used the Fibonacci sequence to compose their music.

Hence, we can see how geometry and music are connected in every small detail.