Pythagoras was quoted as saying, “there is geometry within the humming of the strings, there's music within the spacing of the spheres.” As poetic as this sounds, the famous Greek mathematician was actually making an immediate statement about the connection between mathematics and music. In times , math may be a subject generally related to left brained individuals and music may be a subject generally related to right brained individuals. What most don’t realize is that the themes go hand in hand, and areintertwined as early because the times of Greek antiquity; “great minds took such pains to incorporate music in their worldview and indeed saw music because the organizing principle of the universe” [rogers]. Viewing music and science as profoundly linked was the dominant and accepted way of thinking in Western and non-Western philosophers of the past.
Mathematics and music are both subjects that need an abstract way of thinking and contemplation. Both subjects require recognizing and establishing patterns. it'simportant to notice that although the themeson an entire are more similar than usually given credit for during this day and age, they're also interdisciplinary. Math has historically been wont to describe and teach music, and the other way around . Mathematics are often found etched in common musical concepts like scales, intervals, wave frequencies, and tones. J. Ph. Rameau, a French musicologist of the eighteenth century, said it best in his Traitd de l’harmonie rdduite d ses principes naturels (1722): “Music may be a science which must have determined rules. These rules must be drawn from a principle which should be evident, and this principle can't be known without the assistance of mathematics. i need to confess that in spite of all the experience which I even have acquired in music by practising it for a reasonably long period, it's nevertheless only with the assistance of mathematics that my ideas became disentangled which light has succeeded to a particular darkness of which i used to be not aware before.” [Papadopoulos]
In Greek antiquity, it had been public knowledge that the faculties of Aristotle, Plato, and Pythagoras deemed music a neighborhood of mathematics. A Greek mathematical treatise would typically be comprised of 4 topics: Number Theory, Geometry, Music, and Astronomy. Mathematics and music were strongly linked until the Renaissance, when the 2 subjects diverged – theoretical music becoming an independent field. Pythagoras is recognized because the first music theorist. His greatest discovery addressed the relation of musical intervals with ratios of integers. The story is: on a visit through a brazier’s shop, Pythagoras took note of the various sounds being produced by the hammers on anvils. He realized that the pitch he was hearing depended only on the load of the hammer. The place the hammer hit the anvil, the angle it had been hit, the magnitude of the stroke – none of those factored into the pitch. This lead him to ponder about the connectionbetween two notes produced by two different hammers. In Attic music there have been intervals of the octave, fifth, and fourth. He recognized that the consonant musical intervals the hammers were creating corresponded, in terms of weights, to the numerical fraction 2/1, 3/2, and 4/3, respectively.
Pythagoras discovered that musical intervals, and hence all harmony, are supportedmathematical ratios, ratios that also, amazingly, appear in astronomy [rogers]
Thus, Pythagoras thought that the relative weights of two hammers producing an octave is 2/1, and so on. As soon as this concept occurred to him, Pythagoras went home and performed several experiments using different sorts of instruments, which confirmed the connection between musical intervals and numerical fractions Papadopoulos
musical theory constructed by Pythagoras. Two sounds from an equivalent taut string are said to be consonant once they are pleasing to concentrate to simultaneously. within the Greek cultural arena of that period such sounds are produced by lengths of string that are inversely proportional to the numbers 1, 2, 3, and 4. These compose the famous Tetraktys (1 + 2 + 3 + 4 = 10), a diagram of figured numbers symbolising pure harmony, the “vertical hierarchy of relation between Unity and emerging multiplic Perrine
Pythagoreans considered a set of vases, filled partially with different quantities of an equivalent liquid, and observed on them the “rapidity and therefore the slowness of the movements of air vibrations.” By hitting these vases in pairs and taking note of the harmonies produced, they were ready toassociate numbers to consonances. The result's again that the octaves, fifths, and fourths correspond respectively to the fractions 2/1, 3/2 and 4/3, in terms of the quotients of levels of the liquid. Papadopoulos
rich musical evolution flowing from the Greek roots into the Latin world and right up to the fourteenth century of our era. In St. Augustine’s De Musica, written at the top of the fourth century, rhythms also are classified consistent with their proportions (the proportional notation used today came much later). Then within the ninth century, Perrine
Carolingian policy in educational and ecclesiastical matters defined new practices. It encouraged the utilization of neumes that indicate the inflexions of the voice, but not the pitch of the sounds. The names Do, Re, Mi, Fa, Sol, etc., appeared with Guido d’Arezzo within the eleventh century, derix4ng from the syllables at the start of the stanzas (voces) of a hymn addressed to St John the Baptist, written around 770 A.D. The notes (claves) also are designated by letters, a practice that's still in use today in Englishspeaking countries (La = A, Ti = B, Do = C, . . . ) and in Germany (with some specificities). Finally, polyphony created new needs for harmonic mastery, the response coming from Philippe de Vitry within thefourteenth century together with his Ars Nova: during this work he defined new musical notations also as new ways of mixingrhythms. However, this culmination of the pythagorean musical base that had developed over many centuries eventually degenerated within the following century because it proved to be inadequate for responding to the new aesthetic trends that were appearing also because the practical needs of musicians Perrine
Music makes use of a symbolic language, along side an upscale system of notation, including diagrams which, ranging from the eleventh century (in the case of Western European music), are almost likemathematical graphs of discrete functions in two-dimensional cartesian coordinates (the x-coordinate representing time and therefore the y-coordinate representing pitch). Music theorists used these “cartesian” diagrams long before they were introduced in geometry. Musical scores from the 20 thcentury have a spread of forms which are on the brink of all kinds of diagrams utilized inmathematics. Besides abstract language and notation, mathematical notions like symmetry, periodicity, proportion, discreteness, and continuity, among others, are omnipresent in music. Lengths of musical intervals, rhythm, duration, tempi, and a number of other other musical notions are naturally expressed by numbers. {Papadopoulos}
Logarithms The arithmetic of musical intervals involves during a very natural way the idea of logarithms.Pythagoras defined the tone because the difference between the intervals of fifth and of fourth. the purposenow's that the fraction associated to the tone interval isn't the difference 3/2 – 4/3, but the quotient (3/2)/(4/3) = 9/8. it's natural to define the compass of a interval because thenumber (or the fractions of) octaves it contains. Thus, once we say that two notes are n octaves apart, the fraction associated to the interval that they define is 2 n. The definition of the compass are often made in terms of frequency, and actually one usually defines the pitch because the logarithm in base 2 of the frequency. (Of course, the notion of frequency didn't exist intrinsically in antiquity, but it's clear that the traditionalGreek musicologists were aware that the lowness or the highness of pitch depends on the slowness or rapidity of the air vibration that produces it, as explained in Theon’s treatise [12], Chapter XIII.) The relation of musical intervals with logarithms also can be seen by considering the lengths of strings (which actually are inversely proportional to the frequency). as an example , if a violinist (or a lyre player in antiquity) wants to supplya note which is an octave above the note produced by a particular string, he must divide the length of the string by two. Thus, music theorists dealt intuitively with logarithms long before these were defined as an abstract mathematical notion. (It was only within the seventeenth century that logarithms were formally introduced in music theory, by Newton , then by Leonhard Euier and Jacques Lambert.) the idea of musical intervals may be a natural example of the sensible use of logarithms, an example easily explained to children, provided they needsome acquaintance with musical intervals. {Papadopoulos}
Today the Music of corpuscles and solitons is taking the place of the Music of spheres and mermaids. Considerations of the multiple infinitely small (chaos?) are replacing those on the only infinitely great (the cosmos?). The bifurcation happened at the top of the eighteenth century, at the very moment when musicians were being pushed into the category of artists, whose role was to supplypleasure for this , and mathematicians into the category of scientists, building the society of the longer term . Perrine