Modality is not a difficult concept to grasp, when one understands that it is practical in nature. Since the early centuries of the Church, a very common format for Gregorian chant was:
Antiphon—Psalm verse—Antiphon—Psalm verse—Antiphon, etc.
Needless to say, it was very important that the Antiphon connected nicely with the Psalm verse, and vice versa. Choosing a Psalm tone that did not connect well with the Antiphon was just asking for trouble. They needed a system to facilitate this process, and that is the origin of the Modal system.
At first, they started classifying the Antiphons based on the beginning pitch. This is totally logical, based on the aforementioned pattern of Antiphon and Psalm verses in succession. However, they eventually decided it would be better to classify Antiphons based on the ending pitch. Either way would probably work, but in the end, the ending pitch prevailed (pardon the pun). The ending pitch is called the final or finalis.
Therefore, simply locate the final, and that tells you the Mode of the piece:
Pretty simple, right? However, you are not finished, because you have to decide whether it is authentic or plagal. Odd numbers (1, 3, 5, 7) are authentic. Even numbers (2, 4, 6, 8) are plagal. Authentic Modes have a higher range. Plagal Modes have a lower range.
Have you ever wondered what those little numbers are? The ones listed in the Index of the chant books? They tell you the mode:
The Mode is also listed at the very beginning of each chant:
But what if a piece of Gregorian chant has a range (“ambitus”) that extends both high and low? Obviously, it could be authentic or plagal, and I suspect that the editors simply flipped a coin in such circumstances. What if the piece has a very limited range, say, only three notes? The editors either flip a coin, or (sometimes) they will not assign a mode. Sometimes, rather than flipping a coin, the editors consulted ancient MSS of the Antiphonale or Graduale to help them decide. However, the ancient MSS themselves frequently disagreed about Modal assignment. I have seen 10th century Antiphonals where the same exact chant is classified differently. We must not lose sleep over this, because the assignment was purely practical: as long as the Psalm verses connected nicely, nothing else mattered.
The astute reader is probably asking the following question:
If Modality is purely practical, then why do we classify all chants? For instance, the Alleluia Verse was never sung as Antiphon—Psalm verse—Antiphon—Psalm verse—Antiphon? Why, then, does it have a Modal classification?
The answer is quite simple: musicians became accustomed to classifying the chants, and as the centuries rolled along it seemed natural to classify all the chants according to Mode. Also, a common practice was to organize each Antiphonale or Gradual according to Mode, and I feel this practice must have had an influence, as well.
Sometimes the final of the Mode plays a big role in the chant. Sometimes it does not. Some authors consider the dominant (a.k.a. “co-final” or “tenor”) to be the second most important note of the Mode. Sometimes the dominant plays a big role in the chant. Sometimes it does not. Looking at the the Introit Intret orátio mea, we notice that throughout the entire piece, the final is only sung three times, yet the piece still has to be called Mode 3 because it ends on MI:
At first glance, Benedícta et venerábilis es (PDF) appears to end on RE, so why is it Mode 4? A careful examination shows that while the Verse ends on RE, the actual Gradual ends on MI. The Church allows the first section of the Gradual to be repeated, so (perhaps) there is an argument to be made that this Gradual should always be sung repeating the first section.
As time went on, “Modal theory” began to exert and influence on melodies that had existed for centuries, and (frankly) began to mangle some. However, even to this day, there are still some chants that “defy” Modal theory, like Kyrie IV, which ends on LA. Many Graduals also end on LA, by the way. For theories on how such a thing came to be, see Willi Apel’s book, Gregorian chant.
Once you know the Mode, you can easily transition between Antiphons and Psalm verses. The easiest way to learn about the different “Simple” Psalm tones is to visit Psalmi in Notis (URL), which has numerous books wherein each verse is carefully written out for every single Psalm tone. To learn about the “Solemn” Psalm tones, download Versus Psalmorum et Canticorum (PDF), or visit Communio (URL) and scroll to the bottom.
Simple Psalm tones are the more common ones, and are generally “accentual” — that is, based on the tonic accent of the Latin words. Solemn Psalm tones are slightly less common, and are often “cursive” rather than “accentual.” Sometimes it is said that Latin Psalm tones are based “solely on the Latin tonic accent.” This is not technically true, since many are cursive rather than accentual.