The evolution of music theory and notation is almost more amazing than the evolution of human writing. When you really think about it, modern music notation is like Esperanto that lots of people can actually speak. People all over the Western world, and much of the Eastern world as well, know how to communicate with each other effectively through sheet music, chord theory, and the Circle of Fifths. Here are five music theorists who have helped define how musicians look at music or have changed the view of music entirely.

Pythagoras: 582–507 B.C.

Anybody who's ever taken a geometry class has heard of Pythagoras. Obsessed with the idea that everything in the world could be broken down into a mathematical formula, and that numbers themselves were the ultimate reality, Pythagoras came up with equations that could theoretically calculate everything from determining the size of a mountain by measuring its shadow to his famous Pythagorean Theorem.

According to legend, Pythagoras took a piece of string from a lyre (the most popular instrument of the day), plucked it, measured its tone and vibration rate, and then cut that string in half and made a new set of measurements. He named the difference between the rate of vibration of the first length of string and the second an octave, then went to work breaking the octave up into 12 evenly divided units, with each unit equal to 100 "cents." Every point around his Pythagorean Circle (which would evolve into the Circle of Fifths) was assigned a pitch value, with each pitch exactly 1/12 octave higher or lower than the note next to it.

Pythagoras' circle wasn't perfect, at least to a musician's ears, and for the next 2000 years, musicians and theorists concentrated on "tempering" this circle, with its 12 spots and shapes left intact, but creating a circle that was much more musician- and audience-friendly.

Nicola Vicentino: 1511–1576

Nicola Vicentino was an Italian music theorist of the Renaissance period whose experiments with keyboard design and equal-temperament tuning rival those of many 20th-century theorists. He served briefly as a music tutor for the Duke of Este to support himself while writing treatises on the relevance of ancient Greek music theory in contemporary music, and why, in his opinion, the whole Pythagorean system could be thrown out the window. He was both adored and reviled by contemporaries for his disdain for the diatonic (12-tone) system and was invited to speak at international music conferences on his beliefs.

Vicentino amazed the music world even more when, to further prove the inadequacies of the diatonic scale, he designed and built his own microtonal keyboard that matched a music scale of his own devising, called the archicembalo. On the archicembalo, each octave contained 36 keys, making it possible to play acoustically satisfactory intervals in any key — predating the well-tempered meantone keyboard in use today by nearly 200 years. Unfortunately, he built only a few of the instruments, and before his work could catch on, he died of the plague.

Harry Partch: 1901–1974

At age 29, Harry Partch gathered up 14 years of music he had written, based on what he called the "tyranny of the piano" and the 12-tone scale, and burned it all in a big iron stove. He devoted the next four-and-a-half decades to producing sounds found only in microtonal scales — the tones found between the notes sounded by the piano keys. By the time he died in 1974, he had built around 30 instruments and had devised complex theories of intonation and performances to accompany them, including a 46-tone scale, with which he built most of his compositions.

Karlheinz Stockhausen: 1928–2007

Karlheinz Stockhausen's greatest influence as a theorist can best be felt in the genres of music that came directly out of his teachings. During the 1950s, he helped develop the genres of minimalism and serialism. Much of the 1970s "krautrock" scene was created by his former students at the National Conservatory of Cologne, Germany, while his teachings and compositions greatly influenced the musical renaissance of 1970s West Berlin (notable characters include David Bowie and Brian Eno). In the long run, Stockhausen can be seen as the father of ambient music and the concept of variable form, in which the performance space and the instrumentalists themselves are considered part of the composition, and changing even one element of a performance changes the entire performance.

Stockhausen is also responsible for polyvalent form music, in which a piece of music can be read upside-down, from left to right or right to left, or, if multiple pages are incorporated, the pages can be played in any order the performer wishes.

Robert Moog: 1934–2005

Although no one really knows who built the first fretted guitar, or who truly designed the first real keyboard, we do know who created the first pitch-proper, commercially available synthesizer. Robert Moog is widely recognized as the father of the synthesizer keyboard, and his instrument revolutionized the sound of pop and classical music from the day it hit the streets in 1966. Unfortunately, Moog wasn't the greatest businessperson — or perhaps he was just very, very generous with his ideas — and the only synthesizer-related patent he ever filed was for something called a low-pass filter.

When he first began building synthesizers, his goal was to create a musical instrument that played sounds completely different from any instrument that came before. However, as people began to use synthesizers to re-create "real" instrument sounds (putting some musicians out of business), he became disillusioned with the instrument and decided that the only way to get people to work with the "new" sounds was to break away from the antiquated keyboard interface altogether. His North Carolina-based company, Big Briar, began working on Leon Theremin's theremin design to create a MIDI theremin, which was designed to eliminate the interval steps between each note but still keep the tonal color of each individual instrument's MIDI patch.