Liturgical Modern Pitches in ascending order
I - Authentic modes (tones in medieval paralance)
Dorian Dorian |D| E F G (A) B C D
Phrygian Phrygian |E| F G A B (C) D E
Lydian Lydian |F| G A B (C) D E F
Mixolydian Mixolydian |G| A B C (D) E F G
II - Plagal modes (derived tones)
Hypodorian Aeolian A B C |D| E (F) G A
Hypophrygian Locrian B C D |E| F G (A) B
Hypolydian Ionian C D E |F| G (A) B C
Hypomixolydian Dorian D E F |G| A B (C) D
[Grout 1973]
The enclosed pitch |X| is the 'finalis' of the mode, what we might today call the tonic note; the pitch enclosed (X) is the 'cofinalis' or what we would call a dominant. In chants sung in a given mode, the cofinalis was often used as a secondary tonal center. In modern harmonic theory, the tonic going up to the dominant always form the interval of the fifth.
Among the authentic modes, the Phrygian is the transgressor to this convention. These designated pitches must not be construed in terms of absolute pitch; never, as that concept and standard was not yet developed.
A liturgical mode should rather be understood as specified by a sequence of intervals (not well tempered), a range, and by its finalis and cofinalis. It must also be recognized that a mode is a theoretical construct derived by abstraction from practice. The medieval construct of mode relied also very heavily on the misreading of the available ancient Greek theory.
In the modern modal sense, the finalis is always the first note of the mode's 'scale', wwhile the dominant (analog of the cofinalis) is always gotten by counting up four "white steps" from the finalis (tonic).
In all cases but the Locrian mode, this gives an interval of a well tempered fifth between tonic and dominant; in the locrian mode, it gives a tritone. If we use an accidental to correct the tritone to a fifth, we have to replace F with F#; doing this, however, gives the G Phrygian mode; so one is really stuck with the tritone relation, the intervallus diabolus.
Writing in an essentially C major key, but making the resting tone B (which is classically considered the leading tone - that is leading to C) requires a considerable amount fancy footwork, and a placing of a great emphasis on the returning to B as a resting tone. The Locrian mode was not and is not much used for exactly that reasoning.
With regard to use of the accidental operators (#, b), only Bb was in use. An older German notation calls "B" what we now call "Bb" and calls "H" what we call "B": thus the possibility of motivizing the name of BACH, exploited by J. S. Bach himself in the final and uncompleted (B A C H) fugue from his The Art of the Fugue, as well as by numerous other composers in homage to the master; most notably, in works for the organ by Liszt and Reger.
Since it is possible to think of a keyboard tuned to a just tempered C major tonality including the Bb, the above notation is a justifiable one, and ones sees either eight or nine (with the Bb) pitches, with different sets of finalis and cofinalis. The expression of these distinguished pitches within the chant was by way of certain characteristic phrases or cadential formulas for the mode to terminate a phrase or semiphrase on the finalis or cofinalis respectively.
It is probably worth mentioning that in addition to modes of pitch in medieval church music, there were also modes of rhythm [Grout 1973]. For some other musical definitions Eric's Treasure Trove: music is helpful.
If you can serialize pitch intervals (à la Schoenberg), then you can also serialize rhythms à la Milton Babbit. This idea, clearly, has a long history.
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