In early Greek music theory, an octave species (εἶδος τοῦ διὰ πασῶν, or σχῆμα τοῦ διὰ πασῶν) is a sequence of incomposite intervals (ditones, minor thirds, whole tones, semitones of various sizes, or quarter tones) making up a complete octave (Barbera 1984, 231–32). The concept was also important in Medieval and Renaissance music theory.
Ancient Greek theory 
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" (Aristoxenus, 92.7–8 & 92.9–11 (da Rios), translated in Barbera 1984, 230). The basis of 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" (Gombosi 1951, 22). The species defined by the different positioning of the intervals within the tetrachord in turn depend upon genus first being established (Barbera 1984, 229). Incomposite in this context refers to intervals which are 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 (Barbera 1984, 229). 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 (Barker 1984–87, 2:15):
Mixolydian ¼ ¼ 2 ¼ ¼ 2 1 Lydian ¼ 2 ¼ ¼ 2 1 ¼ Phrygian 2 ¼ ¼ 2 1 ¼ ¼ Dorian ¼ ¼ 2 1 ¼ ¼ 2 Hypolydian ¼ 2 1 ¼ ¼ 2 ¼ Hypophrygian 2 1 ¼ ¼ 2 ¼ ¼ Hypodorian 1 ¼ ¼ 2 ¼ ¼ 2
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 (Cleonides 1965, 41) and, similarly, the lower interval of the pyknon must be smaller or equal to the higher one (Barbera 1984, 229–30). The whole tone added to created 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 (Cleonides 1965, 41; Barbera 1984, 233). 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 (Barbera 1984, 232). 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 (Gombosi 1951, 23–24). 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 (1930, D. 49–53; Barbera 1984, 235) labels the resulting fourteen pitches with the (Greek) letters from Α ( Alpha α) to Ο ( Omega Ω).(A diagram is available at System 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" (Barbera 1984, 237–39). Those octave species that cannot be mapped onto the system are therefore to be rejected (Barbera 1984, 240).
Medieval theory 
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 continued to be used as a basis of the theory of modes, in combination with other elements, particularly the system of octoechos borrowed from the Byzantine Church (Powers 2001).
- Aristoxenus. 1954. Aristoxeni elementa harmonica, edited by Rosetta da Rios. Rome: Typis Publicae Officinae Polygraphicae.
- Barker, Andrew (ed.) (1984–89). Greek Musical Writings. 2 vols. Cambridge & New York: Cambridge University Press. ISBN 0-521-23593-6 (v. 1) ISBN 0-521-30220-X (v. 2).
- Barbera, André. 1984. "Octave Species". The Journal of Musicology 3, no. 3 (Summer): 229–41.
- Cleonides. 1965. "Harmonic Introduction," translated by Oliver Strunk. In Source Readings in Music History, vol. 1 (Antiquity and the Middle Ages), edited by Oliver Strunk, 34–46. New York: W. W. Norton.
- Gombosi, Otto. 1951. "Mode, Species". Journal of the American Musicological Society 4, no. 1 (Spring): 20–26.
- Powers, Harold S. 2001. "Mode §II: Medieval Modal Theory". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.
- Ptolemy. 1930. Die Harmonielehre des Klaudios Ptolemaios, edited by Ingemar Düring. Göteborgs högskolas årsskrift 36, 1930:1. Göteborg: Elanders boktr. aktiebolag. Reprint, New York: Garland Publishing, 1980.
- Ptolemy. 2000. Harmonics, translated and commentary by Jon Solomon. Mnemosyne, Bibliotheca Classica Batava, Supplementum, 0169-8958, 203. Leiden and Boston: Brill. ISBN 90-04-11591-9
- Solomon, Jon. 1984. "Towards a History of Tonoi". The Journal of Musicology 3, no. 3 (Summer): 242–51.
Further reading 
- Chalmers, John H. Jr. 1993. Divisions of the Tetrachord. Hanover, NH: Frog Peak Music. ISBN 0-945996-04-7 available on-line http://eamusic.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/index.html