The Hypophrygian (deuterus plagalis) mode, literally meaning "below Phrygian (plagal second)", 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 (B–C–D–E + E–F–G–A–B); and as a mode with final E and ambitus from the A below to the C above. The note A above the final (the tenor of the corresponding fourth psalm tone) 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 name Hypophrygian originates in an octave species of ancient Greek music theory. According to Aristoxenus, this octave species was originally described around the year 400 BC by the Harmonicist school of Eratocles in terms of the enharmonic genus of the tetrachord: a series of rising intervals of two quarter tones followed by a ditone, together spanning a perfect fourth. The Dorian octave species begins with this tetrachord, which is followed by a whole tone and another tetrachord to complete the octave with a pattern of ¼, ¼, 2, 1, ¼, ¼, and 2 tones. This pattern is rotated downward one degree for the Hypolydian, and one more for the Hypophrygian, for an octave species of 2, 1, ¼, ¼, 2, ¼, and ¼ tones. 
The name was appropriated by Ptolemy of Alexandria for one of his seven tonoi, or transposition keys. Ptolemy's system differed from the earlier Aristoxenian model, which had thirteen transpositional levels each a semitone from its neighbours. Ptolemy substituted a diatonic sequence of seven transpositions pitched either a whole tone or a semitone apart. The entire double-octave scale system was then transposed onto each of these relative pitch levels, requiring (in modern terms) a different key signature in each case, and therefore a different sequence of whole and half steps in the fixed central octave span. The Hypophrygian transposition was the second-lowest of these, a whole tone above the Hypodorian. A whole tone higher was the Hypolydian, followed a semitone higher still by the Dorian, then after another whole tone by the Phrygian, and so on.   Four centuries later, the term was taken from Ptolemy in exactly the same sense by Boethius, who described these seven names as "toni, tropi, vel modi" (tones, tropes or modes) in the fourth book of his De institutione musica. In the late 9th century, in the Carolingian treatises Alia musica and in a commentary on it called the Nova expositio, this set of seven terms, supplemented by an eighth name, "Hypermixolydian", was given a new sense, designating a set of diatonic octave species, described as the tonal embodiments of the eight modes of Gregorian chant. 
Missa Mi-mi (Missa quarti toni) by Johannes Ockeghem is a well-known example of a work written in the Hypophrygian mode.[ citation needed ]
In music theory, a diatonic scale is any heptatonic scale that includes five whole steps and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, depending on their position in the scale. This pattern ensures that, in a diatonic scale spanning more than one octave, all the half steps are maximally separated from each other.
In music theory, the term mode or modus is used in a number of distinct senses, depending on context.
In music theory, a tetrachord is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion —but in modern use it means any four-note segment of a scale or tone row, not necessarily related to a particular tuning system.
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 Phrygian mode can refer to three different musical modes: the ancient Greek tonos or harmonia sometimes called Phrygian, formed on a particular set of octave species or scales; the Medieval Phrygian mode, and the modern conception of the Phrygian mode as a diatonic scale, based on the latter.
In the musical system of ancient Greece, genus is a term used to describe certain classes of intonations of the two movable notes within a tetrachord. The tetrachordal system was inherited by the Latin medieval theory of scales and by the modal theory of Byzantine music; it may have been one source of the later theory of the jins of Arabic music. In addition, Aristoxenus calls some patterns of rhythm "genera".
A heptatonic scale is a musical scale that has seven pitches, or tones, per octave. Examples include the major scale or minor scale; e.g., in C major: C D E F G A B C—and in the relative minor, A minor, natural minor: A B C D E F G A; the melodic minor scale, A B C D E F♯G♯A ascending, A G F E D C B A descending; the harmonic minor scale, A B C D E F G♯A; and a scale variously known as the Byzantine, and Hungarian, scale, C D E♭ F♯ G A♭ B C. Indian classical theory postulates seventy-two seven-tone scale types, collectively called thaat, whereas others postulate twelve or ten seven-tone scale types.
Aurelian of Réôme was a Frankish writer and music theorist. He is the author of the Musica disciplina, the earliest extant treatise on music from medieval Europe.
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 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 the musical system of ancient Greece, an octave species is a specific sequence of intervals within an octave. In Elementa harmonica, Aristoxenus classifies the species as three different genera, distinguished from each other by the largest intervals in each sequence: the diatonic, chromatic, and enharmonic genera, whose largest intervals are, respectively, a whole tone, a minor third, and a ditone; quarter tones and semitones complete the tetrachords.
Ionian mode is a musical mode or, in modern usage, a diatonic scale also called the major scale.
Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900.
A Gregorian mode is one of the eight systems of pitch organization used in Gregorian chant.
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 musical system of ancient Greece evolved over a period of more than 500 years from simple scales of tetrachords, or divisions of the perfect fourth, into several complex systems encompassing tetrachords and octaves, as well as octave scales divided into seven to thirteen intervals.
Pyknon, sometimes also transliterated as pycnon in the music theory of Antiquity is a structural property of any tetrachord in which a composite of two smaller intervals is less than the remaining (incomposite) interval. The makeup of the pyknon serves to identify the melodic genus and the octave species made by compounding two such tetrachords, and the rules governing the ways in which such compounds may be made centre on the relationships of the two pykna involved.
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.