A Gregorian mode (or church mode) is one of the eight systems of pitch organization used in Gregorian chant.
The name of Pope Gregory I was attached to the variety of chant that was to become the dominant variety in medieval western and central Europe (the diocese of Milan was the sole significant exception) by the Frankish cantors reworking Roman ecclesiastical song during the Carolingian period.  The theoretical framework of modes arose later to describe the tonal structure of this chant repertory, and is not necessarily applicable to the other European chant dialects (Old Roman, Mozarabic, Ambrosian, etc.).
The repertory of Western plainchant acquired its basic forms between the sixth and early ninth centuries, but there are neither theoretical sources nor notated music from this period. By the late eighth century, a system of eight modal categories, for which there was no precedent in Ancient Greek theory, came to be associated with the repertory of Gregorian chant. This system likely originated from the early Byzantine oktōēchos , as indicated by the non-Hellenistic Greek names used in the earliest Western sources from about 800. 
In the traditional system of eight modes (in use mainly between the 8th and 16th centuries) there are four pairs, each pair comprising an authentic mode and a plagal mode.
The authentic modes were the odd-numbered modes 1, 3, 5, 7, and this distinction was extended to the Aeolian and Ionian modes when they were added to the original eight Gregorian modes in 1547 by Glareanus in his Dodecachordon.  The final of an authentic mode is the tonic, though the range of modes 1, 2, and 7 may occasionally descend one step further. This added degree is called the "subfinal" which, since it lies a whole tone below the final, is also the "subtonium" of the mode. The range of mode 5 (Lydian) does not employ a subfinal, and so always maintains F as its lower limit.  These four modes correspond to the modern modal scales starting on re (Dorian), mi (Phrygian), fa (Lydian), and so (Mixolydian).  The tenor, or dominant (corresponding to the "reciting tone" of the psalm tones), is a fifth above the final of the scale, with the exception of mode 3 (Phrygian), where it is a sixth above the final. This is because a fifth above the tonic of mode 3 is the "unstable" ti (in modern solfège), which may be flattened to ta.
The older Byzantine system still retains eight echoi (sing. ἦχος – echos), each consisting of a small family of closely related modes that, if rounded to their diatonic equivalents, would be the eight modes of Gregorian chant. However, they are numbered differently, the authentic modes being 1, 2, 3, 4.  Other Eastern Christian rites use similar systems of eight modes; see Syriac usage of Octoechos and Armenian usage of Octoechos.
A plagal mode (from Greek πλάγιος 'oblique, sideways, athwart')   has a range that includes the octave from the fourth below the final to the fifth above. The plagal modes are the even-numbered modes 2, 4, 6 and 8, and each takes its name from the corresponding odd-numbered authentic mode with the addition of the prefix "hypo-": Hypodorian, Hypophrygian, Hypolydian, and Hypomixolydian. 
The earliest definition of plagal mode is found in Hucbald's treatise De harmonica (c. 880), who specifies the range as running from the fourth below the final to the fifth above. Later writers extend this general rule to include the sixth above the final and the fifth below, except for the Hypolydian mode, which would have a diminished fifth below the final and so the fourth below, C, remained the lower limit.  In addition to the range, the tenor (cofinal, or dominant, corresponding to the "reciting tone" of the psalm tones) differs. In the plagal modes, the tenor is a third lower than the tenor of the corresponding authentic mode, except in mode 8 (Hypomixolydian), where it is raised to a 4th above the finalis (a second below the tenor of the authentic mode 7) in order to avoid the "unstable" degree ti, which may be flattened (in the authentic mode 3, the tenor is similarly raised to the sixth above the finalis, and the tenor of plagal mode 4—Hypophrygian—is therefore also a fourth above the finalis).
In Byzantine modal theory ( octoechos ), the word "plagal" ("plagios") refers to the four lower-lying echoi, or modes.  Thus plagal first mode (also known as "tone 5" in the Russian naming system  ) represents a somewhat more developed and widened in range version of the first mode. The plagal second mode ("tone 6" in the Russian system) has a similar relation to the second mode, and the plagal fourth mode—respectively to the fourth mode. Though there is no "plagal third mode", the mode that one would expect ("tone 7") is called the "grave tone". 
Two characteristic notes or pitches in a modal melody are the final and cofinal (tenor, dominant, or reciting tone ). These are the primary degrees (often the 1st and 5th) on which the melody is conceived and on which it most often comes to rest, in graduated stages of finality.  [ page needed ] The final is the pitch in which the chant usually ends; it may be approximately regarded as analogous (but not identical) to the tonic in the Western classical tradition. Likewise the cofinal is an additional resting point in the chant; it may be regarded as having some analogy to the more recent dominant, but its interval from the tonic is not necessarily a fifth. In addition to the final and cofinal, every mode is distinguished by scale degrees called the mediant and the participant. The mediant is named from its position—in the authentic modes—between the final and cofinal. In the authentic modes it is the third degree of the scale, unless that note should happen to be B, in which case C substitutes for it. In the plagal modes, its position is somewhat irregular. The participant is an auxiliary note, generally adjacent to the mediant in authentic modes and, in the plagal forms, coincident with the cofinal of the corresponding authentic mode (some modes have a second participant). 
Given the confusion between ancient, medieval, and modern terminology, "today it is more consistent and practical to use the traditional designation of the modes with numbers one to eight". 
In music theory, the term mode or modus is used in a number of distinct senses, depending on context.
Gregorian chant is the central tradition of Western plainchant, a form of monophonic, unaccompanied sacred song in Latin of the Roman Catholic Church. Gregorian chant developed mainly in western and central Europe during the 9th and 10th centuries, with later additions and redactions. Although popular legend credits Pope Gregory I with inventing Gregorian chant, scholars believe that it arose from a later Carolingian synthesis of the Old Roman chant and Gallican chant.
In Western musical theory, a cadence is the end of a phrase in which the melody or harmony creates a sense of resolution. A harmonic cadence is a progression of two or more chords that concludes a phrase, section, or piece of music. A rhythmic cadence is a characteristic rhythmic pattern that indicates the end of a phrase. A cadence is labeled like or less "weak" or "strong" depending on the impression of finality it gives. While cadences are usually classified by specific chord or melodic progressions, the use of such progressions does not necessarily constitute a cadence—there must be a sense of closure, as at the end of a phrase. Harmonic rhythm plays an important part in determining where a cadence occurs.
Dorian mode or Doric mode can refer to three very different but interrelated subjects: one of the Ancient Greek harmoniai ; one of the medieval musical modes; or—most commonly—one of the modern modal diatonic scales, corresponding to the piano keyboard's white notes from D to D, or any transposition of itself.
Mixolydian mode may refer to one of three things: the name applied to one of the ancient Greek harmoniai or tonoi, based on a particular octave species or scale; one of the medieval church modes; or a modern musical mode or diatonic scale, related to the medieval mode.
The Aeolian mode is a musical mode or, in modern usage, a diatonic scale also called the natural minor scale. On the white piano keys, it is the scale that starts with A. Its ascending interval form consists of a key note, whole step, half step, whole step, whole step, half step, whole step, whole step. That means that, in A aeolian, you would play A, move up a whole step to B, move up a half step to C, then up a whole step to D, a whole step to E, a half step to F, a whole step to G, and a final whole step to a high A.
The Locrian mode is the seventh mode of the major scale. It is either a musical mode or simply a diatonic scale. On the piano, it is the scale that starts with B and only uses the white keys from there. Its ascending form consists of the key note, then: half step, whole step, whole step, half step, whole step, whole step, whole step.
The Hypolydian mode, literally meaning "below Lydian", is the common name for the sixth of the eight church modes of medieval music theory. The name is taken from Ptolemy of Alexandria's term for one of his seven tonoi, or transposition keys. This mode is the plagal counterpart of the authentic fifth mode.
The Hypophrygian mode, literally meaning "below Phrygian ", is a musical mode or diatonic scale in medieval chant theory, the fourth mode of church music. This mode is the plagal counterpart of the authentic third mode, which was called Phrygian. In the Middle Ages and Renaissance this mode was described in two ways: the diatonic scale from B to B an octave above, divided at the mode final E ; and as a mode with final E and ambitus from the A below to the C above. The note A above the final had an important melodic function. The melodic range of the ecclesiastical Hypophrygian mode therefore goes from the perfect fourth or fifth below the tonic to the perfect fifth or minor sixth above.
The Hypodorian mode, a musical term literally meaning 'below Dorian', derives its name from a tonos or octave species of ancient Greece which, in its diatonic genus, is built from a tetrachord consisting of a semitone followed by two whole tones. The rising scale for the octave is a single tone followed by two conjoint tetrachords of this type. This is roughly the same as playing all the white notes of a piano from A to A: A | B C D E | (E) F G A. Although this scale in medieval theory was employed in Dorian and Hypodorian, from the mid-sixteenth century and in modern music theory they came to be known as the Aeolian and Hypoaeolian modes.
In chant, a reciting tone can refer to either a repeated musical pitch or to the entire melodic formula for which that pitch is a structural note. In Gregorian chant, the first is also called tenor, dominant or tuba, while the second includes psalm tones as well as simpler formulae for other readings and for prayers.
Phthora nenano is the name of one of the two "extra" modes in the Byzantine Octoechos—an eight mode system, which was proclaimed by a synod of 792. The phthorai nenano and nana were favoured by composers at the Monastery Agios Sabas, near Jerusalem, while hymnographers at the Stoudiou-Monastery obviously preferred the diatonic mele.
Phthora nana is one of the ten modes of the Hagiopolitan Octoechos consisting of 8 diatonic echoi and two additional phthorai. It is used in different traditions of Orthodox chant until today. The name "nana" is taken from the syllables sung during the intonation which precedes a melody composed in this mode. The name "phthora" derived from the verb φθείρω and means "destroy" or "corrupt". It was usually referred to the diatonic genus of the eight mode system and as a sign used in Byzantine chant notation it indicated a "change to another genus", in the particular case of phthora nana a change to the enharmonic genus. Today the "nana" intonation has become the standard name of the third authentic mode which is called "echos tritos" in Greek and "third glas" in Old Church Slavonic.
Falsobordone is a style of recitation found in music from the 15th to the 18th centuries. Most often associated with the harmonization of Gregorian psalm tones, it is based on root position triads and is first known to have appeared in southern Europe in the 1480s.
Ionian mode is a musical mode or, in modern usage, a diatonic scale also called the major scale.
The Octoechos is a liturgical book containing a repertoire of hymns ordered in eight parts according to eight echoi. Originally created in the Monastery of Stoudios during the 9th century as a hymnal complete with musical notation, it is still used in many rites of Eastern Christianity. The book with similar function in the Western Church is the tonary, and both contain the melodic models of an octoechos system; however, while the tonary serves simply for a modal classification, the octoechos is organized as a cycle of eight weeks of services. The word itself can also refer to the repertoire of hymns sung during the celebrations of the Sunday Office.
The Hypoaeolian mode, literally meaning "below Aeolian", is the name assigned by Henricus Glareanus in his Dodecachordon (1547) to the musical plagal mode on A, which uses the diatonic octave species from E to the E an octave above, divided by the final into a second-species fourth (semitone–tone–tone) plus a first-species fifth (tone–semitone–tone–tone): E F G A + A B C D E. The tenor or reciting tone is C, mediant B, the participants are the low and high Es, the conceded modulations are G and D, and the absolute initials are E, G, A, B, and C.
The Hypolocrian mode is an almost entirely theoretical mode, introduced into chant theory in the 19th century by the editors of the Pustet-Ratisbon, Mechlin, and Rheims-Cambrai Office-Books, who designated it mode 12. It is the plagal counterpart to the authentic Locrian mode, mode 11 in that system of numbering, in which the Ionian and Hypoionian become modes 13 and 14. The ambitus of the mode lies between F and the F an octave higher, divided at the final, B. Its reciting tone, is E, and its mediant is D. It has two participants, G and C. Although a few plainchant melodies, as well as polyphonic compositions, have been attributed to this mode by some writers, it will generally be found that they are really derived, by transposition, from some other tonality.
Oktōēchos is the name of the eight mode system used for the composition of religious chant in most Christian churches during the Middle Ages. In a modified form the octoechos is still regarded as the foundation of the tradition of monodic Orthodox chant today.
Oktōēchos is the name of the eight mode system used for the composition of religious chant in Byzantine, Syriac, Armenian, Georgian, Latin and Slavic churches since the Middle Ages. In a modified form the octoechos is still regarded as the foundation of the tradition of monodic Orthodox chant today.