Quarter tone

A quarter tone is a note halfway between two adjacent notes of a 12-tone chromatic scale. In typical music, there is an interval of 100 cents between two adjacent pitches. However, quarter tones have only 50 cents between each note in a chromatic scale. Now, instead of 12 notes in an octave, there are 24.
Quarter tones have their roots in the music of the Middle East, particularly Persian traditional music.Hormoz Farhat (2004). The Dastgah Concept in Persian Music. Cambridge University Press.  However, the first mention of the equally-tempered quarter tone scale, or 24 equal temperament, was made by 19th-century music theorists Heinrich Richter and Mikhail Mishaqa. The scale is called 24 equal temperament because it has 24 notes, each of which are the same distance away from each other (50 cents)Touma, Habib Hassan (1996). The Music of the Arabs, p. 16. Translator: Laurie Schwartz. Portland, Oregon: Amadeus Press. . Composers who have written music using this scale include: Pierre Boulez, Julián Carrillo, Mildred Couper, George Enescu, Alberto Ginastera, Gérard Grisey, Alois Hába, Thomas Heberer Ljubica Marić, Charles Ives, Tristan Murail, Krzysztof Penderecki, Giacinto Scelsi, Ammar El Sherei, Karlheinz Stockhausen, Tui St. George Tucker, Ivan Wyschnegradsky, Iannis Xenakis, and Seppe Gebruers.

 Types 

 Tuning systems with equal temperament 

The term quarter tone can refer to several different intervals, all very close in size. For example, some 17th- and 18th-century music theorists used the term to describe the distance between a sharp and a flat in certain music systems where they aren't the same pitch (e.g., D and E being different pitches). In the quarter-tone scale, also called 24-tone equal temperament (24-TET), the quarter tone is 50 cents. The frequency of each note in an octave is a multiple of   , or approximately 1.0293, higher than the previous note in that octave. This divides the octave into 24 equal steps (equal temperament). In this scale, the quarter tone is the smallest step. A semitone is thus made of two steps, and three steps make a three-quarter tone or neutral second, which is half the size of a minor third. The 8-TET scale is composed of three-quarter tones. Four steps make a whole tone.