Tetrachord in the context of "Tone row"

Gregorian chant is the central tradition of Western plainchant, a form of monophonic, unaccompanied sacred song in Latin (and occasionally Greek) 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 he only ordered a compilation of melodies throughout the whole Christian world, after having instructed his emissaries in the Schola Cantorum, where the neumatical notation was perfected, with the result of most of those melodies being a later Carolingian synthesis of the Old Roman chant and Gallican chant.

Gregorian chants were organized initially into four, then eight, and finally 12 modes. Typical melodic features include a characteristic ambitus, and also characteristic intervallic patterns relative to a referential mode final, incipits and cadences, the use of reciting tones at a particular distance from the final, around which the other notes of the melody revolve, and a vocabulary of musical motifs woven together through a process called centonization to create families of related chants. The scale patterns are organized against a background pattern formed of conjunct and disjunct tetrachords, producing a larger pitch system called the gamut. The chants can be sung by using six-note patterns called hexachords. Gregorian melodies are traditionally written using neumes, an early form of musical notation from which the modern four-line and five-line staff developed. Multi-voice elaborations of Gregorian chant, known as organum, were an early stage in the development of Western polyphony.

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifths which are "pure" or perfect, with ratio ${\displaystyle 3:2}$. This is chosen because it is the next harmonic of a vibrating string, after the octave (which is the ratio ${\displaystyle 2:1}$), and hence is the next most consonant "pure" interval, and the easiest to tune by ear. 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 (i.e., the untempered perfect fifth), which is ≈ 702 cents wide.

The system dates back to Ancient Mesopotamia;. (See Music of Mesopotamia § Music theory.) It is named, and has been widely misattributed, to Ancient Greeks, notably Pythagoras (sixth century BC) by modern authors of music theory. Ptolemy, and later Boethius, ascribed the division of the tetrachord by only two intervals, called "semitonium" and "tonus" in Latin (256:243 × 9:8 × 9:8), to Eratosthenes. The so-called "Pythagorean tuning" was used by musicians up to the beginning of the 16th century. "The Pythagorean system would appear to be ideal because of the purity of the fifths, but some consider other intervals, particularly the major third, to be so badly out of tune that major chords [may be considered] a dissonance."

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.

Any discussion of the music of ancient Greece, theoretical, philosophical or aesthetic, is fraught with two problems: there are few examples of written music, and there are many, sometimes fragmentary, theoretical and philosophical accounts. The empirical research of scholars like Richard Crocker, C. André Barbera, and John Chalmers has made it possible to look at the ancient Greek systems as a whole without regard to the tastes of any one ancient theorist. The primary genera they examine are those of Pythagoras and the Pythagorean school, Archytas, Aristoxenos, and Ptolemy (including his versions of the genera of Didymos and Eratosthenes).

In the musical system of ancient Greece, an octave species (Greek: 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.