Pythagoras (of Samos, c. 570 BC – c. 495 BC) worked out the notes and intervals of a musical scale. This was the result of experiments with vibrating strings. For example, he worked out that halving the length of the string would raise the pitch of a note by an octave; dividing it in the ratio 2/3 would result in a harmonious note and other simple ratios would also be notes with a satisfying harmonious quality, such as a string divided 3/4.
The philosophical basis was the belief that simple whole numbers were the key to how the universe worked; simple - and beautifully harmonious - laws.
Pythagoras's musical and mathematical theories of cosmic application were accepted for 2000 years until the Renaissance and the Age of Enlightenment, in other words, until recently.
By convention, to define an interval between two notes, both the starting and ending notes are included. For example, the interval from notes A to B is a second; from A to C is a third, etc.
Every sound can be analysed as a series of single pitches in combination. What gives the sound its particular quality, or "timbre" is mainly dependent on the varying proportions (loudness) of these individual notes.
These notes are called harmonics (sometimes called partials) and together form the natural harmonic series and are: the key or bass note, the octave above this, the interval of a fifth above this, the interval of a fourth above this, and so on.
In terms of Pythagoras's string ratios, they are: 1, 1/2, 1/3, 1/4, etc. A natural pattern based on simple whole numbers emerges.
Intervals and Chords
The Greeks weren't particularly interested in applying harmony in music because their music was a monody, a single line, usually played to accompany poetry or a recitative. It is only later that harmony was explored in any detail.
The basis of western harmony theory is the triad. This is a group of three notes, commonly referred to as a chord. The lowest note is called the root. Add to this a note which is an interval of a third above and then another note an interval of a third above this (a fifth above the root) and you have a simple chord. For example, the notes A, C and E.
The order of the notes can be varied, e.g., C, E and the A above,. More notes can be added to give the sound a different flavour and effect.
Some notes in a chord can be deliberate discords which then drop into harmony later in the music. This creates a process of tension and release which drives the music forward. A device called a suspension was and is frequently used to achieve just this effect.
One particularly discordant interval called "The Devil's Interval" (an "augmented fourth") was banned in the Middle Ages by church authorities.