Theon of Smyrna (Greek : Θέων ὁ ΣμυρναῖοςTheon ho Smyrnaios, gen. Θέωνος Theonos; fl. 100 CE) was a Greek philosopher and mathematician, whose works were strongly influenced by the Pythagorean school of thought. His surviving On Mathematics Useful for the Understanding of Plato is an introductory survey of Greek mathematics.
Little is known about the life of Theon of Smyrna. A bust created at his death, and dedicated by his son, was discovered at Smyrna, and art historians date it to around 135 CE. Ptolemy refers several times in his Almagest to a Theon who made observations at Alexandria, but it is uncertain whether he is referring to Theon of Smyrna.The lunar impact crater Theon Senior is named for him.
Theon wrote several commentaries on the works of mathematicians and philosophers of the time, including works on the philosophy of Plato. Most of these works are lost. The one major survivor is his On Mathematics Useful for the Understanding of Plato. A second work concerning the order in which to study Plato's works has recently been discovered in an Arabic translation.
His On Mathematics Useful for the Understanding of Plato is not a commentary on Plato's writings but rather a general handbook for a student of mathematics. It is not so much a groundbreaking work as a reference work of ideas already known at the time. Its status as a compilation of already-established knowledge and its thorough citation of earlier sources is part of what makes it valuable.
The first part of this work is divided into two parts, the first covering the subjects of numbers and the second dealing with music and harmony. The first section, on mathematics, is most focused on what today is most commonly known as number theory: odd numbers, even numbers, prime numbers, perfect numbers, abundant numbers, and other such properties. It contains an account of 'side and diameter numbers', the Pythagorean method for a sequence of best rational approximations to the square root of 2,the denominators of which are Pell numbers. It is also one of the sources of our knowledge of the origins of the classical problem of Doubling the cube.
The second section, on music, is split into three parts: music of numbers (hē en arithmois mousikē), instrumental music (hē en organois mousikē), and "music of the spheres" (hē en kosmō harmonia kai hē en toutō harmonia). The "music of numbers" is a treatment of temperament and harmony using ratios, proportions, and means; the sections on instrumental music concerns itself not with melody but rather with intervals and consonances in the manner of Pythagoras' work. Theon considers intervals by their degree of consonance: that is, by how simple their ratios are. (For example, the octave is first, with the simple 2:1 ratio of the octave to the fundamental.) He also considers them by their distance from one another.
The third section, on the music of the cosmos, he considered most important, and ordered it so as to come after the necessary background given in the earlier parts. Theon quotes a poem by Alexander of Ephesus assigning specific pitches in the chromatic scale to each planet, an idea that would retain its popularity for a millennium thereafter.
The second book is on astronomy. Here Theon affirms the spherical shape and large size of the Earth; he also describes the occultations, transits, conjunctions, and eclipses. However, the quality of the work led Otto Neugebauer to criticize him for not fully understanding the material he attempted to present.
Theon was a great philosopher of harmony and he discusses semitones in his treatise. There are several semitones used in Greek music, but of this variety, there are two that are very common. The “diatonic semitone” with a value of 16/15 and the “chromatic semitone” with a value of 25/24 are the two more commonly used semitones (Papadopoulos, 2002). In these times, Pythagoreans did not rely on irrational numbers for understanding of harmonies and the logarithm for these semitones did not match with their philosophy. Their logarithms did not lead to irrational numbers, however Theon tackled this discussion head on. He acknowledged that “one can prove that” the tone of value 9/8 cannot be divided into equal parts and so it is a number in itself. Many Pythagoreans believed in the existence of irrational numbers, but did not believe in using them because they were unnatural and not positive integers. Theon also does an amazing job of relating quotients of integers and musical intervals. He illustrates this idea in his writings and through experiments. He discusses the Pythagoreans method of looking at harmonies and consonances through half-filling vases and explains these experiments on a deeper level focusing on the fact that the octaves, fifths, and fourths correspond respectively with the fractions 2/1, 3/2, and 4/3. His contributions greatly contributed to the fields of music and physics (Papadopoulos, 2002).
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. 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, which is ≈702 cents wide.
In music theory, an interval is the difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.
Aristoxenus of Tarentum 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, survives incomplete, as well as some fragments concerning rhythm and meter. The Elements is the chief source of our knowledge of ancient Greek music.
In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.
Philolaus was a Greek Pythagorean and pre-Socratic philosopher. He argued that at the foundation of everything is the part played by the limiting and limitless, which combine together in a harmony. He is also credited with originating heliocentrism, the theory that the Earth was not the center of the Universe. According to August Böckh (1819), who cites Nicomachus, Philolaus was the successor of Pythagoras.
Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone, Italy. Early Pythagorean communities spread throughout Magna Graecia.
The musica universalis, also called music of the spheres or harmony of the spheres, is an ancient philosophical concept that regards proportions in the movements of celestial bodies – the Sun, Moon, and planets – as a form of music. This "music" is not thought to be audible, but rather a harmonic, mathematical or religious resonance. The idea continued to appeal to scholars until the end of the Renaissance, influencing many schools of thought, including humanists. Further scientific exploration discovered orbital resonance in specific proportions in some orbital motion.
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale. For example, C is adjacent to C♯; the interval between them is a semitone.
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".
Hippasus of Metapontum was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods for divulging this. However, the few ancient sources which describe this story either do not mention Hippasus by name or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer.
The twelfth root of two or is an algebraic irrational number. It is most important in Western music theory, where it represents the frequency ratio of a semitone in twelve-tone equal temperament. This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals as consisting of different numbers of a single interval, the equal tempered semitone. A semitone itself is divided into 100 cents.
In music theory and musical tuning the Holdrian comma, also called Holder's comma, and rarely the Arabian comma, is a small musical interval of approximately 22.6415 cents, equal to one step of 53 equal temperament, or . The name comma is misleading, since this interval is an irrational number and does not describe the compromise between intervals of any tuning system; it assumes this name because it is an approximation of the syntonic comma (play ), which was widely used as a measurement of tuning in William Holder's time.
Greek mathematics refers to mathematics texts written during and ideas stemming from the Archaic through the Hellenistic periods, extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by Greek culture and the Greek language. The word "mathematics" itself derives from the Ancient Greek: μάθημα, romanized: máthēmaAttic Greek: [má.tʰɛː.ma]Koine Greek: [ˈma.θi.ma], meaning "subject of instruction". The study of mathematics for its own sake and the use of generalized mathematical theories and proofs is an important difference between Greek mathematics and those of preceding civilizations.
According to legend, Pythagoras discovered the foundations of musical tuning by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously. According to Nicomachus in his 2nd century CE Enchiridion harmonices Pythagoras noticed that hammer A produced consonance with hammer B when they were struck together, and hammer C produced consonance with hammer A, but hammers B and C produced dissonance with each other. Hammer D produced such perfect consonance with hammer A that they seemed to be "singing" the same note. Pythagoras rushed into the blacksmith shop to discover why, and found that the explanation was in the weight ratios. The hammers weighed 12, 9, 8, and 6 pounds respectively. Hammers A and D were in a ratio of 2:1, which is the ratio of the octave. Hammers B and C weighed 9 and 8 pounds. Their ratios with hammer A were and. The space between B and C is a ratio of 9:8, which is equal to the musical whole tone, or whole step interval.
In musical tuning theory, a Pythagorean interval is a musical interval with frequency ratio equal to a power of two divided by a power of three, or vice versa. For instance, the perfect fifth with ratio 3/2 (equivalent to 31/21) and the perfect fourth with ratio 4/3 (equivalent to 22/31) are Pythagorean intervals.
Music theory has no axiomatic foundation in modern mathematics, although some interesting work has recently been done in this direction, yet the basis of musical sound can be described mathematically and exhibits "a remarkable array of number properties". Elements of music such as its form, rhythm and metre, the pitches of its notes and the tempo of its pulse can be related to the measurement of time and frequency, offering ready analogies in geometry.
Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900.
Robert Lawlor is a mythographer, symbologist and New Age author of several books.
Education for Greek people was vastly "democratized" in the 5th century B.C., influenced by the Sophists, Plato, and Isocrates. Later, in the Hellenistic period of Ancient Greece, education in a gymnasium school was considered essential for participation in Greek culture. The value of physical education to the ancient Greeks and Romans has been historically unique. There were two forms of education in ancient Greece: formal and informal. Formal education was attained through attendance to a public school or was provided by a hired tutor. Informal education was provided by an unpaid teacher and occurred in a non-public setting. Education was an essential component of a person's identity.
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, to The Perfect Immutable System, encompassing a span of fifteen pitch keys