Music and mathematics

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 **are**intertwined 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's**important **to notice** that although **the themes**on **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 connection**between 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 **supported**mathematical 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 to**associate 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 the**fourteenth century **together with his** Ars Nova: **during this** work he defined new musical notations **also** as new ways **of mixing**rhythms. 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 like**mathematical 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 th**century have **a spread** of forms which are **on the brink of** all **kinds of** diagrams **utilized in**mathematics. 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 the**number (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 traditional**Greek 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 supply**a 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 need**some 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 supply**pleasure for **this** , and mathematicians into the category of scientists, building the society of **the longer term** . Perrine