Boethius in the context of "Scolica enchiriadis"

The problem of universals is an ancient question from metaphysics that has inspired a range of philosophical topics and disputes: "Should the properties an object has in common with other objects, such as color and shape, be considered to exist beyond those objects? And if a property exists separately from objects, what is the nature of that existence?"

The problem of universals relates to various inquiries closely related to metaphysics, logic, and epistemology, as far back as Plato and Aristotle, in efforts to define the mental connections humans make when understanding a property such as shape or color to be the same in nonidentical objects.

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."

Musica enchiriadis is an anonymous musical treatise authored during the 9th century. It is the first surviving attempt to set up a system of rules for polyphony in western art music. The treatise was once attributed to Hucbald, but this is no longer accepted. Some historians once attributed it to Odo of Cluny (879–942). It has also been attributed to Abbot Hoger (d. 906).

This music theory treatise, along with its companion text, Scolica enchiriadis, was widely circulated in medieval manuscripts, often in association with Boethius' De institutione musica. It consists of nineteen chapters; the first nine are devoted to notation, modes, and monophonic plainchant.

From the time of Plato through the Middle Ages, the quadrivium (plural: quadrivia, Latin for "four ways") was a grouping of four subjects or arts—arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in the trivium, consisting of grammar, logic, and rhetoric. Together, the trivium and the quadrivium comprised the seven liberal arts, and formed the basis of a liberal arts education in Western society until gradually displaced as a curricular structure by the studia humanitatis and its later offshoots, beginning with Petrarch in the 14th century. The seven classical arts were considered "thinking skills" and were distinguished from practical arts, such as medicine and architecture.

The four mathematical arts were recognized by Pythagoreans such as Nicomachus of Gerasa, but the use of quadrivium as a term for these four subjects has been attributed to Boethius, when he affirmed that the height of philosophy can be attained only following "a sort of fourfold path" (quodam quasi quadruvio). It was considered the foundation for the study of philosophy (sometimes called the "liberal art par excellence") and theology. The quadrivium was the upper division of medieval educational provision in the liberal arts, which comprised arithmetic (absolute number), music (relative number), geometry (magnitude at rest), and astronomy (magnitude in motion).

Augustinianism is the philosophical and theological system of Augustine of Hippo and its subsequent development by other thinkers, notably Boethius, Anselm of Canterbury and Bonaventure. Among Augustine's most important works are The City of God, De doctrina Christiana, and Confessions.

Originally, Augustinianism developed in opposition to Pelagianism; it was widespread in medieval western philosophy until the arrival of Thomism and Aristotelianism.

Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist. This positional notation system has largely superseded earlier calculation systems that used a different set of symbols for each numerical magnitude, such as Roman numerals, and in some cases required a device such as an abacus.

Fortuna (Latin: Fortūna, equivalent to the Greek goddess Tyche), historically anglicized as Fortune, is the goddess of fortune and the personification of luck in Roman religion who, largely thanks to the Late Antique author Boethius, remained popular through the Middle Ages until at least the Renaissance. The blindfolded depiction of her is still an important figure in many aspects of today's Italian culture, where the dichotomy fortuna / sfortuna (luck / unluck) plays a prominent role in everyday social life, also represented by the very common refrain "La [dea] fortuna è cieca" (Latin Fortuna caeca est; "Luck [goddess] is blind").

Fortuna is often depicted with a gubernaculum (ship's rudder), a ball or Rota Fortunae (wheel of fortune, first mentioned by Cicero) and a cornucopia (horn of plenty). She might bring good or bad luck: she could be represented as veiled and blind, as in modern depictions of Lady Justice, except that Fortuna does not hold a balance. Fortuna came to represent life's capriciousness. She was also a goddess of fate: as Atrox Fortuna, she claimed the young lives of the princeps Augustus' grandsons Gaius and Lucius, prospective heirs to the Empire. (In antiquity she was also known as Automatia.)

In philosophy (particularly the theory of categories), the Porphyrian tree (also spelled Porphyrean tree) or Tree of Porphyry is a classic device for illustrating a "scale of being" (Latin: scala praedicamentalis), attributed to the 3rd-century CE Greek neoplatonist philosopher and logician Porphyry, and revived through the translations of Boethius.

Porphyry suggests the tree in his introduction ("Isagoge") to Aristotle's Categories. Porphyry presented Aristotle's classification of categories in a way that was later adopted into tree-like diagrams of two-way divisions, which indicate that a species is defined by a genus and a differentia and that this logical process continues until the lowest species is reached, which can no longer be so defined. No illustrations or diagrams occur in editions of Porphyry's original work; diagrams were eventually made, and became associated with the scheme that Porphyry describes, following Aristotle.