Prior to the innovations of Franco of Cologne
 in the mid-thirteenth century, the value of the longa was in common 
usage in both theoretical and practical sources but appeared primarily 
in pre-mensural notation ligatures,
 symbols representing two or more notes joined together. A ligature that
 began with a longa was said to lack "propriety", while ligatures ending
 with a longa possessed "perfection", since in the view of that era a 
"proper and perfect" rhythmic sequence was the succession of a brevis 
followed by a longa, justified by the fact that the ligature 
representing this rhythm is written the same way as a plainchant 
ligature (a different usage of the term from above). As a result, there 
were four possible ligature types: those beginning with a brevis and ending with a longa, which had both propriety and perfection; the reverse, which had neither; those both beginning and ending with a longa, which lacked propriety but had perfection; and those beginning and ending with a brevis, which were proper but not perfect.[1][2] Two longae, rarely three, had the combined value of a maxima. The theoretical value of a maximodus perfectus could only be written with three longae or a maxima plus a longa.[3]