<P> Tempus perfectum was indicated by a circle, while tempus imperfectum was denoted by a half - circle (the current symbol, used as an alternative for the 4 4 time signature, is actually a holdover of this symbol, not a letter C as an abbreviation for "common time", as popularly believed). While many of these innovations are ascribed to Vitry, and somewhat present in the Ars Nova treatise, it was a contemporary--and personal acquaintance--of de Vitry, named Johannes de Muris (Jehan des Mars) who offered the most comprehensive and systematic treatment of the new mensural innovations of the Ars Nova (for a brief explanation of the mensural notation in general, see the article Renaissance music). Many scholars, citing a lack of positive attributory evidence, now consider "Vitry's" treatise to be anonymous, but this does not diminish its importance for the history of rhythmic notation . However, this makes the first definitely identifiable scholar to accept and explain the mensural system to be de Muris, who can be said to have done for it what Garlandia did for the rhythmic modes . </P> <P> For the duration of the medieval period, most music would be composed primarily in perfect tempus, with special effects created by sections of imperfect tempus; there is a great current controversy among musicologists as to whether such sections were performed with a breve of equal length or whether it changed, and if so, at what proportion . This Ars Nova style remained the primary rhythmical system until the highly syncopated works of the Ars subtilior at the end of the 14th century, characterized by extremes of notational and rhythmic complexity . This sub-genera pushed the rhythmic freedom provided by Ars Nova to its limits, with some compositions having different voices written in different mensurations simultaneously . The rhythmic complexity that was realized in this music is comparable to that in the 20th century . </P> <P> Of equal importance to the overall history of western music theory were the textural changes that came with the advent of polyphony . This practice shaped western music into the harmonically dominated music that we know today . The first accounts of this textural development were found in two anonymous yet widely circulated treatises on music, the Musica and the Scolica enchiriadis . These texts are dated to sometime within the last half of the ninth century . The treatises describe a technique that seemed already to be well established in practice . This early polyphony is based on three simple and three compound intervals . The first group comprises fourths, fifths, and octaves; while the second group has octave - plus - fourths, octave - plus - fifths, and double octaves . This new practice is given the name organum by the author of the treatises . Organum can further be classified depending on the time period in which it was written . The early organum as described in the enchiriadis can be termed "strict organum" Strict organum can, in turn, be subdivided into two types: diapente (organum at the interval of a fifth) and diatesseron (organum at the interval of a fourth). However, both of these kinds of strict organum had problems with the musical rules of the time . If either of them paralleled an original chant for too long (depending on the mode) a tritone would result . </P> <P> This problem was somewhat overcome with the use of a second type of organum . This second style of organum was called "free organum". Its distinguishing factor is that the parts did not have to move only in parallel motion, but could also move in oblique, or contrary motion . This made it much easier to avoid the dreaded tritone . The final style of organum that developed was known as "melismatic organum", which was a rather dramatic departure from the rest of the polyphonic music up to this point . This new style was not note against note, but was rather one sustained line accompanied by a florid melismatic line . This final kind of organum was also incorporated by the most famous polyphonic composer of this time--Léonin . He united this style with measured discant passages, which used the rhythmic modes to create the pinnacle of organum composition . This final stage of organum is sometimes referred to as Notre Dame school of polyphony, since that was where Léonin (and his student Pérotin) were stationed . Furthermore, this kind of polyphony influenced all subsequent styles, with the later polyphonic genera of motets starting as a trope of existing Notre Dame organums . </P>

When singers were not available a tradition began in the middle ages of using