In Modern music notation, all notes are binary, which means that any note duration (ie, half-note) will take the same time as two of the next-shorter note duration (ie, two quarternotes) and it will also occupy one-half of the time period of the next longer note duration (ie, two half-notes are required to fill the time of a whole note). If you wish to arrange for a note duration to be filled with a different number of lesser-duration notes, it is customary to mark this with a number over or under the note head (ie, triplets). In actual fact, this number is the denominator of the fraction that represents each note's part of the whole.
In renaissance notation, trinary division of note values was considered more proper. There were many possible reasons: the Holy Trinity represented perfection in a tripartite God; a three-part division gave the completeness of having a beginning, a middle, and an end.
Because of this perceived perfection of the triune division, notes so related were called `Perfect'. And if a note was to be divided in two, it was felt to have lost a part, and was therefor `Imperfect'.
Additionally, four levels of notes could be Perfect or Imperfect. (Interestingly, the smaller notes were automatically binary in common use.)
The Semibreve
could be perfect, containing
three minims
, or it could be imperfect, containing only two.
This was referred to as Prolation. Thus, in Prolatio Perfectus, = while in Prolatio Imperfectus, =. Perfection of the Breve
fell under the name Tempus.
Thus, Tempus Perfectus meant = while Tempus Imperfectus meant =. Likewise, the Long and Maxima could be perfect or imperfect. In actual practice, these longer duration note shapes were quite binary, there were times when they would be split in tre. The perfection of the Long would be discussed as `mood' or `mode', while the perfection of the Maxima is referred to by the term `maximodus'.
These, then, are the `levels' of mensuration: Maximodus, Mode, Tempus, and Prolatio. Any of them could be perfect or imperfect. The state of mensuration for a piece (or a portion of a piece) was notated with a time signature. Tempus Perfectus was notated with a circle,
, it's
wholeness being `perfect'. Tempus Imperfectus was noted with a broken circle, 
. Prolation was indicated by the presence
or absence of a concentric dot: With the dot was Prolatione perfecta 
,
without it was imperfecta. Because Tempus and Prolation were the two most common levels to be affected (remember, for the most part, the larger and smaller note-relations were simply binary unless specifically shown to be otherwise), these four symbols were the most common. Apel chose a shorthand for referring to them, consisting of two numbers in brackets. The first was 2 for Tempus imperfectum, 3 for Tempus Perfectum. The second, which was additionally italicized, was 2 for Prolatione Imperfecta, 3 for Prolatione Perfecta. Thus, Tempus Imperfectum cum prolatione perfecta
would be [2, 3].
Perfection and imperfection in the `larger' levels, Modus and Maximodus, were not indicated by time signatures. Instead, the relations between them were indicated in the form of rests which usually appeared in the tenor part, as the larger modes were used most commonly in sacred works based upon Cantus Firmus. The Long Rest would cover two spaces if the Mode was imperfect, three if the mode was perfect. Likewise, the Maxima rest would be a grouping of two long rests (which might be two or three spaces tall themselves, depending upon the mode) for Maximodus Imperfectus, or three long rests for Maximodus Perfectus.
The best way to think of this is to consider where the tactus falls. With the tactus on the semibreve, we can think of a breve as occupying a measure. In Maximodus Imperfectus, Modus Imperfectus, Tempus Imperfectus, Prolatione Imperfecta, we have a simple 4-bar phrase. The phrase is split into two sub-phrases, each containing two bars. Each bar has two major beats in it (the tacti), and two less accented beats. The four bar phrase would be the Maximima. The two sub phrases would be Longs, and each bar would be a breve. The half-notes would be semibreves, the quarternotes would be minims.
This is actually a very close correlation between our modern way of thinking about music, and that of the renaissance musician. We would tend to think of the music as being either in 2/2 or 4/4 time. In either case, there will be two tacti to a measure, and a binary division of the tacti (for quarternotes). We wouldn't tend to care too much on a conscious level about the fact that the music forms 2-bar sub-phrases, and would feel, rather than consciously look for the four-bar phrases.
Similarly, the renaissance musician would notate the Tempus and Prolation, using the broken circle with no central dot, and hide the mode and maximode in the form and grouping of the longer rests. After all... it gives the tenors something to do.
One last thing: 4/4 time is notated now-a-days as C. We are usually told in grade school that the C stands for `Common Time'. Here, however, we can see a much more clear reason for the
. That's all for now.
