Aristoxenus of Tarentum (Greek : Ἀριστόξενος ὁ Ταραντῖνος; born c. 375, fl. 335 BC) was a Greek Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been lost, but one musical treatise, Elements of Harmony (Greek: Ἁρμονικὰ στοιχεῖα;Latin: Elementa harmonica ), survives incomplete, as well as some fragments concerning rhythm and meter. The Elements is the chief source of our knowledge of ancient Greek music. 
Aristoxenus was born at Tarentum (in modern-day Apulia, southern Italy), and was the son of a learned musician named Spintharus (otherwise Mnesias).  He learned music from his father, and having then been instructed by Lamprus of Erythrae and Xenophilus the Pythagorean, he finally became a pupil of Aristotle,  whom he appears to have rivaled in the variety of his studies. According to the Suda ,  he heaped insults on Aristotle after his death, because Aristotle had designated Theophrastus as the next head of the Peripatetic school, a position which Aristoxenus himself had coveted having achieved great distinction as a pupil of Aristotle. This story is, however, contradicted by Aristocles, who asserts that he only ever mentioned Aristotle with the greatest respect. Nothing is known of his life after the time of Aristotle's departure, apart from a comment in Elementa Harmonica concerning his works.  
His writings were said to have consisted of four hundred and fifty-three books,  and dealt with philosophy, ethics and music. Although his final years were in the Peripatetic school, and he hoped to succeed Aristotle on his death, Aristoxenus was strongly influenced by Pythagoreanism, and was only a follower of Aristotle in so far as Aristotle was a follower of Plato and Pythagoras. Thus, as Sophie Gibson tells us,  “the various philosophical influences” on Aristoxenus included growing up in the profoundly Pythagorean city of Taras (Tarentum), home also of the two Pythagoreans Archytas and Philolaus, and his father's (Pythagorean) musical background, which he inculcated into his son. Gibson tells us that, after the influence of his father:
The second important influence on Aristoxenos’ development was Pythagoreanism. Born in Tarentum, the city in which both Archytas and Philolaos had lived, it can be seen that the extended period of time that Aristoxenus spent in a Pythagorean environment made an indelible impact on the subject matter of his writings. Such titles as "Pythagorou bios", "Peri Pythaorou kai ton guorimon autou" and "Peri tou Pythagorikou biou" indicate Aristoxenus’ interest in the society. Furthermore, his works on education show evidence of Pythagorean influence, particularly in their tendency towards conservatism. Most importantly, speculation on the structure of music had its origin in a Pythagorean environment. Its focus was on the numerical relationship between notes and, at its furthest stretch, developed into a comparison between musical, mathematical and cosmological structures. 
However, Aristoxenus disagreed with earlier Pythagorean musical theory in several respects, building on their work with ideas of his own. The only work of his that has come down to us is the three books of the Elements of Harmony, an incomplete musical treatise. Aristoxenus' theory had an empirical tendency; in music he held that the notes of the scale are to be judged, not as earlier Pythagoreans had believed, by mathematical ratio, but by the ear.  Vitruvius in his De architectura  paraphrases the writings of Aristoxenus on music. His ideas were responded to and developed by some later theorists such as Archestratus, and his place in the methodological debate between rationalists and empiricists was commented upon by such writers as Ptolemais of Cyrene.
The Pythagorean theory that the soul is a 'harmony' of the four elements composing the body, and therefore mortal ("nothing at all," in the words of Cicero  ), was ascribed to Aristoxenus (fr. 118–121 Wehrli) and Dicaearchus. This theory is comparable to the one offered by Simmias in Plato's Phaedo.
In his Elements of Harmony (also Harmonics), Aristoxenus attempted a complete and systematic exposition of music. The first book contains an explanation of the genera of Greek music, and also of their species; this is followed by some general definitions of terms, particularly those of sound, interval, and system.  In the second book Aristoxenus divides music into seven parts, which he takes to be: the genera, intervals, sounds, systems, tones or modes, mutations, and melopoeia.  The remainder of the work is taken up with a discussion of the many parts of music according to the order which he had himself prescribed. 
While it is often held among modern scholars that Aristoxenus rejected the opinion of the Pythagoreans that arithmetic rules were the ultimate judge of intervals and that in every system there must be found a mathematical coincidence before such a system can be said to be harmonic,  Aristoxenus made extensive use of arithmetic terminology, notably to define varieties of semitones and dieses in his descriptions of the various genera. 
In his second book he asserted that "by the hearing we judge of the magnitude of an interval, and by the understanding we consider its many powers."  And further he wrote, "that the nature of melody is best discovered by the perception of sense, and is retained by memory; and that there is no other way of arriving at the knowledge of music;" and though, he wrote, "others affirm that it is by the study of instruments that we attain this knowledge;" this, he wrote, is talking wildly, "for just as it is not necessary for him who writes an Iambic to attend to the arithmetical proportions of the feet of which it is composed, so it is not necessary for him who writes a Phrygian song to attend to the ratios of the sounds proper thereto."  However, this should not be construed as meaning that he postulated a simplistic system of harmony resembling that of modern twelve tone theory, and especially not an equally tempered system. As he urges us to consider, "(a)fter all, with which of the people who argue about the shades of the genera should one agree? Not everyone looks to the same division when tuning the chromatic or the enharmonic, so why should the note a ditone from mesé be called lichanos rather than a small amount higher?" 
It is sometimes claimed that the nature of Aristoxenus' scales and genera deviated sharply from his predecessors. That Aristoxenus used a model for creating scales based upon the notion of a topos, or range of pitch location,  is fact, however there is no reason to believe that he alone set this precedent, as he himself does not make this claim. Indeed, the idea of unfixed pitch locations that cover certain ranges, the limits of which may be defined by fixed points, is a notion that was popular until the modern fixation upon fixed pitch systems, as is indicated by Baroque theoretical systems of pitch and intonation. Another way of stating this, however perhaps less accurate, is that instead of using discrete ratios to place intervals, he used continuously variable quantities.
The postulation that this resulted in the structuring of his tetrachords and the resulting scales having 'other' qualities of consonance  is one that can only be accounted for by the recourse to often repeated inconsistencies amongst his interpreters and modern confirmation bias in favour of simplified twelve tone theories. Aristoxenus himself held that "(...) two things must not be overlooked: first, that many people have mistakenly supposed us to be saying that a tone can be divided into three equal parts in a melody. They made this mistake because they did not realise that it is one thing to employ the third part of a tone, and another to divide a tone into three parts and sing all three. Secondly we accept that from a purely abstract point of view there is no least interval." 
In book three Aristoxenus goes on to describe twenty eight laws of melodic succession, which are of great interest to those concerned with classical Greek melodic structure. 
Part of the second book of a work on rhythmics and metrics, Elementa rhythmica, is preserved in medieval manuscript tradition.
Aristoxenus was also the author of a work On the Primary Duration (chronos).
A five-column fragment of a treatise on meter (P. Oxy. 9/2687) was published in Grenfell and Hunt's Oxyrhynchus Papyri , vol. 1 (1898) and is probably by Aristoxenus.
The edition of Wehrli presents the surviving evidence for works with the following titles (not including several fragments of uncertain origin):
Pythagoras of Samos was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, the West in general. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a gem-engraver on the island of Samos. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to Croton in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although modern scholars doubt that he ever advocated complete vegetarianism.
Dicaearchus of Messana, also written Dikaiarchos, was a Greek philosopher, geographer and author. Dicaearchus was a student of Aristotle in the Lyceum. Very little of his work remains extant. He wrote on geography and the history of Greece, of which his most important work was his Life of Greece. Although modern scholars often consider him a pioneer in the field of cartography, this is based on a misinterpretation of a reference in Cicero to Dicaearchus' tabulae, which does not refer to any maps made by Dicaearchus but is a pun on account books and refers to Dicaearchus' Descent into the Sanctuary of Trophonius. He also wrote books on ancient Greek poets, philosophy and politics.
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.
Archytas was an Ancient Greek philosopher, mathematician, music theorist, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder of mathematical mechanics, as well as a good friend of Plato.
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, in modern Calabria (Italy). Early Pythagorean communities spread throughout Magna Graecia.
In Western music theory, a major second is a second spanning two semitones. A second is a musical interval encompassing two adjacent staff positions. For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff positions. Diminished, minor and augmented seconds are notated on adjacent staff positions as well, but consist of a different number of semitones.
The intervals from the tonic (keynote) in an upward direction to the second, to the third, to the sixth, and to the seventh scale degrees (of a major scale are called major.
Nicomachus of Gerasa(Greek: Νικόμαχος; c. 60 – c. 120 AD) was an Ancient Greek Neopythagorean philosopher from Gerasa, in the Roman province of Syria. Like many pythagoreans, Nicomachus wrote about the mystical properties of numbers, best known for his works Introduction to Arithmetic and Manual of Harmonics, which are an important resource on Ancient Greek mathematics and Ancient Greek music in the Roman period. Nicomachus' work on arithmetic became a standard text for Neoplatonic education in Late antiquity, with philosophers such as Iamblichus and John Philoponus writing commentaries on it. A Latin paraphrase by Boethius of Nicomachus's works on arithmetic and music became standard textbooks in medieval education.
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".
Ocellus Lucanus was allegedly a Pythagorean philosopher, born in Lucania in the 6th century BC. Aristoxenus cites him along with another Lucanian by the name of Ocillo, in a work preserved by Iamblichus that lists 218 supposed Pythagoreans, which nonetheless contained some inventions, wrong attributions to non-Pythagoreans, and some names derived from earlier pseudopythagoric traditions.
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music.
Eudemus of Rhodes was an ancient Greek philosopher, considered the first historian of science, who lived from c. 370 BCE until c. 300 BCE. He was one of Aristotle's most important pupils, editing his teacher's work and making it more easily accessible. Eudemus' nephew, Pasicles, was also credited with editing Aristotle's works.
Music was almost universally present in ancient Greek society, from marriages, funerals, and religious ceremonies to theatre, folk music, and the ballad-like reciting of epic poetry. It thus played an integral role in the lives of ancient Greeks. There are some fragments of actual Greek musical notation, many literary references, depictions on ceramics and relevant archaeological remains, such that some things can be known—or reasonably surmised—about what the music sounded like, the general role of music in society, the economics of music, the importance of a professional caste of musicians, etc.
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.
Adrastus of Aphrodisias was a Peripatetic philosopher who lived in the 2nd century AD. He was the author of a treatise on the arrangement of Aristotle's writings and his system of philosophy, quoted by Simplicius, and by Achilles Tatius. Some commentaries of his on the Timaeus of Plato are also quoted by Porphyry, and a treatise on the Categories of Aristotle by Galen. None of these have survived. He was a competent mathematician, whose writings on harmonics are frequently cited by Theon of Smyrna in the surviving sections of his On Mathematics Useful for the Understanding of Plato. In the 17th century, a work by Adrastus on harmonics, Περὶ Ἁρμονικῶν, was said by Gerhard Johann Vossius to have been preserved, in manuscript, in the Vatican Library, although the manuscript appears to be no longer extant, if indeed this was not an error on Vossius' part.
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.
A septimal 1/3-tone is an interval with the ratio of 28:27, which is the difference between the perfect fourth and the supermajor third. It is about 62.96 cents wide. The septimal 1/3-tone can be viewed either as a musical interval in its own right, or as a comma; if it is tempered out in a given tuning system, the distinction between these two intervals is lost. The septimal 1/3-tone may be derived from the harmonic series as the interval between the twenty-seventh and twenty-eighth harmonics. It may be considered a diesis.
Ptolemais of Cyrene was a music theorist, author of Pythagorean Principles of Music. She lived perhaps in the 3rd century BC, and "certainly not after the first century AD." She is the only known female music theorist of antiquity.
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.
Elementa harmonica is a treatise on the subject of musical scales by Aristoxenus, of which considerable amounts are extant. The work dates to the second half of the 4th century BC. It is the oldest substantially surviving work written on the subject of music theory.