This article may be too technical for most readers to understand.(May 2021)
In the musical system of ancient Greece, an octave species (εἶδος τοῦ διὰ πασῶν, or σχῆμα τοῦ διὰ πασῶν) 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.
The concept of octave species is very close to tonoi and akin to musical scale and mode, and was invoked in Medieval and Renaissance theory of Gregorian mode and Byzantine Octoechos.
Greek theorists used two terms interchangeably to describe what we call species: eidos (εἶδος) and skhēma (σχῆμα), defined as "a change in the arrangement of incomposite [intervals] making up a compound magnitude while the number and size of the intervals remains the same".  Cleonides, working in the Aristoxenian tradition, describes three species of diatessaron, four of diapente, and seven of diapason in the diatonic genus. Ptolemy in his Harmonics calls them all generally "species of primary consonances" (εἴδη τῶν πρώτων συμφωνιῶν). In the Latin West, Boethius, in his Fundamentals of Music, calls them "species primarum consonantiarum".  Boethius and Martianus, in his De Nuptiis Philologiae et Mercurii, further expanded on Greek sources and introduced their own modifications to Greek theories. 
The most important of all the consonant species was the octave species, because "from the species of the consonance of the diapason arise what are called modes ".  The basis of the octave species was the smaller category of species of the perfect fourth, or diatessaron; when filled in with two intermediary notes, the resulting four notes and three consecutive intervals constitute a "tetrachord".  The species defined by the different positioning of the intervals within the tetrachord in turn depend upon genus first being established.  Incomposite in this context refers to intervals not composed of smaller intervals.
Most Greek theorists distinguish three genera of the tetrachord: enharmonic, chromatic, and diatonic. The enharmonic and chromatic genera are defined by the size of their largest incomposite interval (major third and minor third, respectively), which leaves a composite interval of two smaller parts, together referred to as a pyknon; in the diatonic genus, no single interval is larger than the other two combined.  The earliest theorists to attempt a systematic treatment of octave species, the harmonicists (or school of Eratocles) of the late fifth century BC, confined their attention to the enharmonic genus, with the intervals in the resulting seven octave species being: 
Species of the perfect fifth (diapente) are then created by the addition of a whole tone to the intervals of the tetrachord. The first, or original species in both cases has the pyknon or, in the diatonic genus, the semitone, at the bottom  and, similarly, the lower interval of the pyknon must be smaller or equal to the higher one.  The whole tone added to create the species of fifth (the "tone of disjunction") is at the top in the first species; the remaining two species of fourth and three species of fifth are regular rotations of the constituent intervals, in which the lowest interval of each species becomes the highest of the next.(   Because of these constraints, tetrachords containing three different incomposite intervals (compared with those in which two of the intervals are of the same size, such as two whole tones) still have only three species, rather than the six possible permutations of the three elements.  Similar considerations apply to the species of fifth.
The species of fourth and fifth are then combined into larger constructions called "systems". The older, central "characteristic octave", is made up of two first-species tetrachords separated by a tone of disjunction, and is called the Lesser Perfect System.  It therefore includes a lower, first-species fifth and an upper, fourth-species fifth. To this central octave are added two flanking conjuct tetrachords (that is, they share the lower and upper tones of the central octave). This constitutes the Greater Perfect System, with six fixed bounding tones of the four tetrachords, within each of which are two movable pitches. Ptolemy   labels the resulting fourteen pitches with the (Greek) letters from Α (Alpha α) to Ο (Omega Ω). (A diagram is shown at systema ametabolon)
The Lesser and Greater Perfect Systems exercise constraints on the possible octave species. Some early theorists, such as Gaudentius in his Harmonic Introduction, recognized that, if the various available intervals could be combined in any order, even restricting species to just the diatonic genus would result in twelve ways of dividing the octave (and his 17th-century editor, Marcus Meibom, pointed out that the actual number is 21), but "only seven species or forms are melodic and symphonic".  Those octave species that cannot be mapped onto the system are therefore rejected. 
In chant theory beginning in the 9th century, the New Exposition of the composite treatise called Alia musica developed an eightfold modal system from the seven diatonic octave species of ancient Greek theory, transmitted to the West through the Latin writings of Martianus Capella, Cassiodorus, Isidore of Seville, and, most importantly, Boethius. Together with the species of fourth and fifth, the octave species remained in use as a basis of the theory of modes, in combination with other elements, particularly the system of octoechos borrowed from the Eastern Orthodox Church. 
Species theory in general (not just the octave species) remained an important theoretical concept throughout Middle Ages. The following appreciation of species as a structural basis of a mode, found in the Lucidarium (XI, 3) of Marchetto (ca. 1317), can be seen as typical:
We declare that those who judge the mode of a melody exclusively with regard to ascent and descent cannot be called musicians, but rather blind men, singers of mistake... for, as Bernard said, "species are dishes at a musical banquet; they create modes." 
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.
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈702 cents wide.
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.
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. The term is derived from Latin enharmonicus, from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος (enarmónios), from ἐν (en) and ἁρμονία (harmonía).
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale. For example, C is adjacent to C♯; the interval between them is a semitone.
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 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".
Nicola Vicentino was an Italian music theorist and composer of the Renaissance. He was one of the most progressive musicians of the age, inventing, among other things, a microtonal keyboard.
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 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.
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.
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.
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.
An incomposite interval is a concept in the Ancient Greek theory of music concerning melodic musical intervals between neighbouring notes in a tetrachord or scale which, for that reason, do not encompass smaller intervals. Aristoxenus defines melodically incomposite intervals in the following context:
Let us assume that given a systēma, whether pyknon or non-pyknon, no interval less than the remainder of the first concord can be placed next above it, and no interval less than a tone next below it. Let us also assume that each of the notes which are melodically successive in each genus will either form with the fourth note in order from it the concord of a fourth, or will form with the fifth note from it in order the concord of a fifth, or both, and that any note of which none of these things is true is unmelodic relative to those with which it forms no concord. Let us further assume that given that there are four intervals in the fifth, of which two are usually equal and two unequal, the unequal ones are placed next to the equal ones in the opposite order above and below. Let us assume that notes standing at the same concordant interval from successive notes are in succession with one another. Let us assume that in each genus an interval is melodically incomposite if the voice, in singing a melody, cannot divide it into 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.