Music theory in the context of "Counterpoint"

Eratosthenes of Cyrene (/ɛrəˈtɒsθəniːz/; Ancient Greek: Ἐρατοσθένης [eratostʰénɛːs]; c. 276 BC – c. 195/194 BC) was an Ancient Greek philosopher, polymath and scholar. He was known as a mathematician, geographer, poet, astronomer, and music theorist. Eratosthenes became the chief librarian at the Library of Alexandria. His work was the precursor to the modern discipline of geography, and he introduced some of its terminology, coining the terms geography and geographer.

He is best remembered as the first known person to calculate the Earth's circumference. He was also the first to calculate Earth's axial tilt, which similarly proved to have remarkable accuracy. He created the first global projection of the world incorporating parallels and meridians based on the available geographic knowledge of his era. Eratosthenes was the founder of scientific chronology; he used Egyptian and Persian records to estimate the dates of the main events of the Trojan War, dating the sack of Troy to 1184 BC. In number theory, he introduced the sieve of Eratosthenes, an efficient method of identifying prime numbers and composite numbers.

China has been the source of many innovations, scientific discoveries and inventions. This includes the Four Great Inventions: papermaking, the compass, gunpowder, and early printing (both woodblock and movable type). The list below contains these and other inventions in ancient and modern China attested by archaeological or historical evidence, including prehistoric inventions of Neolithic and early Bronze Age China.

The historical region now known as China experienced a history involving mechanics, hydraulics and mathematics applied to horology, metallurgy, astronomy, agriculture, engineering, music theory, craftsmanship, naval architecture and warfare. Use of the plow during the Neolithic period Longshan culture (c. 3000 – c. 2000 BC) allowed for high agricultural production yields and rise of Chinese civilization during the Shang dynasty (c. 1600 – c. 1050 BC). Later inventions such as the multiple-tube seed drill and the heavy moldboard iron plow enabled China to sustain a much larger population through improvements in agricultural output.

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) circa 530 BC. Early Pythagorean communities spread throughout Magna Graecia.

Already during Pythagoras's life it is likely that the distinction between the akousmatikoi ("those who listen"), who is conventionally regarded as more concerned with religious, and ritual elements, and associated with the oral tradition, and the mathematikoi ("those who learn") existed. The ancient biographers of Pythagoras, Iamblichus (c. 245 – c. AD 325) and his master Porphyry (c. 234 – c. AD 305) seem to make the distinction of the two as that of 'beginner' and 'advanced'. As the Pythagorean cenobites practiced an esoteric path, like the mystery schools of antiquity, the adherents, akousmatikoi, following initiation became mathematikoi. It is wrong to say that the Pythagoreans were superseded by the Cynics in the 4th century BC, but it seems to be a distinction mark of the Cynics to disregard the hierarchy and protocol, ways of initiatory proceedings significant for the Pythagorean community; subsequently did the Greek philosophical traditions become more diverse. The Platonic Academy was arguably a Pythagorean cenobitic institution, outside the city walls of Athens in the 4th century BC. As a sacred grove dedicated to Athena, and Hecademos (Academos). The academy, the sacred grove of Academos, may have existed, as the contemporaries seem to have believed, since the Bronze Age, even pre-existing the Trojan War. Yet according to Plutarch it was the Athenian strategos (general) Kimon Milkiadou (c. 510 – c. 450 BC) who converted this, "waterless and arid spot into a well watered grove, which he provided with clear running-tracks and shady walks". Plato lived almost a hundred years later, circa 427 to 348 BC. On the other hand, it seems likely that this was a part of the re-building of Athens led by Kimon Milkiadou and Themistocles, following the Achaemenid destruction of Athens in 480–479 BC during the war with Persia. Kimon is at least associated with the building of the southern Wall of Themistocles, the city walls of ancient Athens. It seems likely that the Athenians saw this as a rejuvenation of the sacred grove of Academos.

Porphyry (/ˈpɔːrfɪri/; Koine Greek: Πορφύριος, romanized: Porphýrios; c. 234 – c. AD 305) was a Phoenician Neoplatonic philosopher born in Tyre, Roman Phoenicia during Roman rule. He edited and published the Enneads, the only collection of the work of Plotinus, his teacher.

He wrote original works in the Greek language on a wide variety of topics, ranging from music theory to Homer to vegetarianism. His Isagoge or Introduction, an introduction to logic and philosophy, was the standard textbook on logic throughout the Middle Ages in its Latin and Arabic translations. Porphyry was, and still is, also well known for his anti-Christian polemics. Through works such as Philosophy from Oracles and Against the Christians (which was banned by Constantine the Great), he was involved in a controversy with early Christians.

In music theory, the term period refers to forms of repetition and contrast between adjacent small-scale formal structures such as phrases. In twentieth-century music scholarship, the term is usually used similarly to the definition in the Oxford Companion to Music: "a period consists of two phrases, antecedent and consequent, each of which begins with the same basic motif." Earlier and later usages vary somewhat, but usually refer to notions of symmetry, difference, and an open section followed by a closure. The concept of a musical period originates in comparisons between music structure and rhetoric at least as early as the 16th century.

In Western music theory, a chord is a group of notes played together for their harmonic consonance or dissonance. The most basic type of chord is a triad, so called because it consists of three distinct notes: the root note along with intervals of a third and a fifth above the root note. Chords with more than three notes include added tone chords, extended chords and tone clusters, which are used in contemporary classical music, jazz, and other genres.

Chords are the building blocks of harmony and form the harmonic foundation of a piece of music. They provide the harmonic support and coloration that accompany melodies and contribute to the overall sound and mood of a musical composition. The factors, or component notes, of a chord are often sounded simultaneously but can instead be sounded consecutively, as in an arpeggio.

In music theory, a phrase (Greek: φράση) is a unit of musical meter that has a complete musical sense of its own, built from figures, motifs, and cells, and combining to form melodies, periods and larger sections.

Terms such as sentence and verse have been adopted into the vocabulary of music from linguistic syntax. Though the analogy between the musical and the linguistic phrase is often made, still the term "is one of the most ambiguous in music....there is no consistency in applying these terms nor can there be...only with melodies of a very simple type, especially those of some dances, can the terms be used with some consistency."