<P> The Latin treatise De institutione musica by the Roman philosopher Boethius (written c. 500) was a touchstone for other writings on music in medieval Europe . Boethius represented Classical authority on music during the Middle Ages, as the Greek writings on which he based his work were not read or translated by later Europeans until the 15th century . This treatise carefully maintains distance from the actual practice of music, focusing mostly on the mathematical proportions involved in tuning systems and on the moral character of particular modes . Several centuries later, treatises began to appear which dealt with the actual composition of pieces of music in the plainchant tradition . At the end of the ninth century, Hucbald worked towards more precise pitch notation for the neumes used to record plainchant . </P> <P> Guido d'Arezzo' wrote in 1028 a letter to Michael of Pomposa, entitled Epistola de ignoto cantu, in which he introduced the practice of using syllables to describe notes and intervals . This was the source of the hexachordal solmization that was to be used until the end of the Middle Ages . Guido also wrote about emotional qualities of the modes, the phrase structure of plainchant, the temporal meaning of the neumes, etc.; his chapters on polyphony "come closer to describing and illustrating real music than any previous account" in the Western tradition . </P> <P> During the thirteenth century, a new rhythm system called mensural notation grew out of an earlier, more limited method of notating rhythms in terms of fixed repetitive patterns, the so - called rhythmic modes, which were developed in France around 1200 . An early form of mensural notation was first described and codified in the treatise Ars cantus mensurabilis ("The art of measured chant") by Franco of Cologne (c. 1280). Mensural notation used different note shapes to specify different durations, allowing scribes to capture rhythms which varied instead of repeating the same fixed pattern; it is a proportional notation, in the sense that each note value is equal to two or three times the shorter value, or half or a third of the longer value . This same notation, transformed through various extensions and improvements during the Renaissance, forms the basis for rhythmic notation in European classical music today . </P> <Ul> <Li> Bᾱqiyᾱ Nᾱyinῑ (Uzbekistan, 17th Century CE), Uzbek author and music theorist . Author of Zamzama e wahdat - i - mῡsῑqῑ ("The Chanting of Unity in Music"). </Li> <Li> Baron Francois Rodolphe d'Erlanger (Tunis, Tunisia, 1910--1932 CE), Arabic musicologist . Author of La musique arabe and Ta'rῑkh al - mῡsῑqᾱ al - arabiyya wa - usῡluha wa - tatawwurᾱtuha ("A History of Arabian Music, its principles and its Development") </Li> </Ul>

The basic ratio on which the notation of rhythm is built is