Maths and Music & Dance

THIS EBOOK WAS PREPAREDAS A PART OF THE COMENIUS PROJECT

WHY MATHS?WHY MATHS?by the students and the teachers from:

BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS ( BELGIUM)

EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY)

GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA (PORTUGAL)

IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)

This project has been funded with support from the European Commission.This publication reflects the views only of the author, and the

Commission cannot be held responsible for any use which may be made of theinformation contained therein.

I.I. MATHS AND MUSIC – THEIR EVOLUTION THROUGH THEMATHS AND MUSIC – THEIR EVOLUTION THROUGH THE YEARSYEARS

II.II. MATHS , PHYSICS AND MUSICMATHS , PHYSICS AND MUSICIII.III. MATHS AND DANCEMATHS AND DANCE

Maths and Music -Their evolution through the years"Music is a science that must have certain rules: they must be extracted from a self-evident principle, which can not be known without the help of Mathematics. I must admit that, despite all the experience I have acquired with a long musical practice, it is only with the help of Mathematics that my ideas are arranged, and that the light has dispelled the darkness "(Jean-Philippe Rameau, Treatise on Harmony reduced to its basic principles (1722))

Maths and Music have always had very close relation: Pythagoras and the Greek musicMusic played a key role in the transition from whole numbers to rational ones: despite Pythagoras had based his philosophy on Integers and, in particular, on the first 4 numbers (tetraktýs), practicing the music for cathartic purposes, he discovered that the heights of the sounds were linked to each other by precise numerical ratios: a fundamental discovery, enough to be immortalized in the saying that "all is number (rational)".The legend says that the discovery was made by striking a jar filled with water, which then, further filled, issued the same note but more acute.

"Tetraktis" is the sum of ten identical objects, arranged as an equilateral triangle was the figure most sacred to the Pythagoreans, the triangle had four points on each side and a point at the center (or could be seen as a point on the highest level, just below two, then three, and finally four). This is a quote by Pythagoras: “There is Geometry in the humming of the strings. There is Music in the spacing of the spheres.”Later on, Pythagoras built a primordial guitar (evolution of the monochord) and studied the sounds produced by bungee cords made from ox nerves strained by different weights: he discovered that the consonance between pairs of sounds was repeated when these tensions were linked by a relationship of 1:4 or 9:4: the distance between these notes was the interval of an octave.

An interval, more than a distance is then the ratio between the frequencies of the notes that are considered: the Pythagoreans discovered that, by pressing a rope in the middle of its length and pinching one of its halves, they obtained the same note an octave higher: the lyre was invented. It was a

stringed musical instrument played by the ancient Greeks and was probably the most important and well-known instrument in the Greek world. It usually had two fixed upright arms (pecheis) or horns (kerata) and a crossbar (zygos) with tuning pegs (kollopes) were made by bronze, wood, ivory, or bone. Between the crossbar and the chordotonon, a fixed tailpiece seven strings (neurai or chordai) of equal length but varying thickness (usually made from sheep gut) stretched and were stretched. It was played by strumming or plucking by hand, usually using a plectrum made of wood, ivory, or metal. The lyre was played either alone or as an accompaniment to singing or lyric poetry in many different occasions such as official banquets, private drinking parties (symposia), religious ceremonies, funerals, and in musical competitions.Pythagoras thought that sounds came from the planets and he supposed that the distance between each planet could correspond to specific sounds, the notes. He also calculated the relation of the distance of each planet from the Earth and attributed the result to an hypothetical sound.

The Middle-agesIn the 13th century choirmasters used to apply the golden section in compositions because they aspired to the perfection: the audience perceived the music like Mathematics perfection.

Starting from 1320, the Ars Nova developed the concept of mensural notation, adding new durations of the notes to those used until then, and extending the applicability of the binary division of the values. These musicians also accentuated the musical aspects of the compositions (by multiplying the voices of the singers and introducing such the shape of the motet) compared to textual aspects.

In the 15th century musicians were looking for a proportional ratio connecting each part of the composition with the whole structure in order to achieve the highest internal coherence.

The first solution was isorhytm according to which the lengthof the tenor of the composition is divided into equal time slots with the same rhythmic characteristics.

Fourtheenth-century mottetto’s compositionThe "tenor" was the basic pattern structure that was divided into equal parts, which were called “talea".The repetition of the tenor in the composition was called “color”.

Tenor, from the Latin "tenere" (‘to hold’), originally meant a sustaining part, through a series of derivations but later on it also come to mean a high male voice.In the Middle Ages and Renaissance, polyphonic pieces were usually based on a "cantus firmus" or given melody, which was normally assigned to the tenor part.

In an isorhythmic mottet, the tenor repeats the same sequence of rhythmic values several times: each repetition is called Talea. Isorhythm differs from the repetitive application of the same rhythm mode, not only in the length and complexity of Taleas, but also in the independent coexistence of melodic modules.The talea is a rhytmic palindrome (which is possible to read from right and from left)

In a color the 2/4 pause between two talea is the centre of symmetry.

A color of 28 notes is arranged with a four-note talea pattern which repeats seven times.Usually between the color and the talea there was a rational arithmetic relation of 3:1

This is the structural plan of the tenor of a late medieval isorhythmic motet with threefold diminution, called "Sub Arturo plebs" by Johannes Alanus. We can notice a color of 24 longae (48 bars in modern notation), divided in three taleae. The color is repeated three times, each in a different mensuration. Its length is subsequently diminished by the factors of 9:6:4.

Since symmetry was highly appreciated in the Middle Ages the palindromic structure offered a second solution. A palindrome is a word or phrase that reads the same backwards as forwards.

In the palindrome musical structure, the two voices play the same musical motif at the same time, but the notes of the follower are in reverse order.

A third solution was dividing the composition into equal sections: the same music was repeated twice so that the second repetition was lasted a fraction of the original length of time.

Other solutions were the golden section and Filbonacci's series.

The CanonMusic obeys rules of construction sometimes use symmetry. A structure is symmetrical so it is invariant when applying certain changes to its different parts. In broader sense, it is the presence of elements that meet, are pending. It is this sense that we must consider in music. We meet reflections ("mirror") and translations in time, the heights of sounds or tones (translation of a given interval). Reflective symmetry in time (demotion) can formally exist but not be heard in the ear if a short sequence and its retrograde (do mi sol / sol mi do) will be perceived as balanced, it is not true the same for a longer sequence.

In music a canon is a contrapuntal compositional technique that employs a melody with one or more imitations of the melody played after a given duration. The initial melody is called the leader (or dux), while the imitative melody is called the follower (or comes). In this case we have a musical phrase that is repeated successively by different voices or in unison (which may be an octave) is shifted by an interval. The sentence can be changed by reversing and / or demotion: the follower must imitate the

leader, either as an exact replication of its rhythms and intervals or some transformation thereof. The two melodies must overlap according to the harmonic rules, so that simultaneous notes “play well” together.

At the base of counterpoint include the process of imitation: repetition of a pattern between the different voices or instruments. Imitation may be described as a time translation symmetry and scale the heights of sound, ie, an interval gap between different voices. More complex figures (reflections or mirrors) symmetry also occur: inverted pattern (the ascending intervals become descendants) or retrograde ("musical palindrome") or combination of both.

Who has not sung the barrel "Frère Jacques" in his childhood? "Canon table" are so named because the partition, unique, was placed on a table between the musicians which requires two "mirror" up / down (inversion intervals) and start / end (demotion).

If the axis of symmetry is the abscissa, we obtain a copy of the theme in which each ascending interval becomes a descending interval. If we combine this inversion with the translation in time of the theme, we obtain an inversion canon. If the axis of symmetry is the ordinate, we obtain a time inversion of the theme, a retrograde canon also known as “crab canon”, because the theme, like the crab- in Latin “cancer”- in the follower goes backwards in time.

Bach: a musician and mathematicianJ.S. Bach (1685-1750) can be considered as a "mathematician" because of the intricate structures and symmetries present in his music. Symmetrical arrangements and repetitions were typical of compositions in Bach's time, but no one else approached his innovation and mastery of these forms. While symmetrical elements can be found throughout Bach's body of work, these elements ar e most apparent in his later pieces, particularly his canons.

The "crab canon" is a a single melodic line which is played forward and backward simultaneously: the two voices would consist of the first voice and its mirror image. The challenge is in constructing a melody which perfectly fits with its inverse. The canon in contrary motion has bilateral symmetry: the two voices progress by the same intervals, but move in opposite directions.

Probably the most well-known example of Bach's use of an underlying meaning in any of his music is the appearance of his name in what he had planned as the next-to-last fugue of The Artof Fugue (Kupferberg, 107).

Maths, Physics and MusicThere are many connections between Maths, Physics and Music.Everybody is aware that Music is the art of organization of the sound in the time and in the space.So, when we study the sound, we study a lot about Music.

The sound and its intensityThe sound is a particular type of longitudinal wave: its direction of propagation is the same as the direction of propagation of the wave itself. The wave, that travels necessarily in a medium and does not propagate in the vacuum, is given by a compression and rarefaction of the medium itself: from this it follows that the pressure of the medium varies when wave passes.As a wave, sound can't spread but through matter and has both features common to the other waves (frequency, width, wavelenght, etc.) and own (timber, pitch, intensity, etc.).The intensity of the sound is the relation between the medium power of the wave and the area on which it spreads; concretely, it's how our hearing perceives sound, if low or loud.As the sound propagates in all directions, then we can outline the propagation of sound from a single source with spherical wavefronts.

The waves carry energy, for example, makes to vibrate the tympanum of our ear. The amount of energy transported in one second by a wave is called power (measured, then in W = J / s). When we move away from the source the surfaces identified by the rays (always perpendicular to wavefronts) the surface increases: this surface is inversely proportional to the distance from the source. Since we define the sound intensity I as the ratio between the average sound power that crosses

perpendicularly a given surface area of the same surface I = P̄A , then, in the case of spherical wave

(assumed the power is uniform on the same wavefront) we have: I = P4 π r2 . So if we double the

distance from the source of the sound intensity becomes a quarter.

Given what we have just said, pressure's variations are directly proportional to intensity. In the measurement of the intensity of acoustic waves we can have an example of a particular use of logarithmic scale.As the unit atm is too big for pressure variations related to sound, that are very small, we use a smaller unit derived from it called Pascal (Pa); this is where logarithmic scale comes out: for a better graphic representation, decibel (dB) is used, a mathematical expression we reach multiplying logarithm by 10. In fact, the interval between the the lowest audible intensity and the loudest consists into 1 million pressure variations, really hard to represent through a linear scale.

Why do we use logarithm?The logarithm of a number is that exponent to which a certain base must be raised to obtain the number itself.

Logarithmic scale is a means to represent positive, real numbers alternative to linear scale, as it is unsuitable for data with very different orders of magnitude. It applies, for example, to electromagnetic spectrum, Richter scale and magnitudes (stars' luminosity).There is a value under which we can't perceive sounds: it's called minimum threshold of amplitude and equals to I= 1∙10-12 W/m2 and a pressure variation of 20 microPa.There's also a value beyond which sound causes hearing damages, in particular it : it's called threshold of pain and equals to 10 W/m2, that is 20 Pa.

The beatsA beat is an interference between two sound waves with slightly different frequencies and it is perceived as periodic variations in volumeThis phenomenon occurs, for example, when we superimpose the sound waves of two tuning forks with sightly different frequencies. In this way the difference in frequency generates a peculiar effect: the beating. In this effects are formed two zones of different interference: the first "constructive" in which the volume is higher, the second "destructive" in which is lower.We can demonstrate that the successive maximal and minimal values create a wave whose frequency equals the difference between the frequencies of the two starting waves. We analyze the simplest case, between two sine waves of unit amplitude.

The role of beats in MusicMusicians use interference beats to check tuning at the unison, perfect fifth, or other simple harmonic intervals. Piano and organ tuners even use a method involving counting beats, aiming at a particular number for a specific interval.

Let's demonstrate it!The two starting frequencies are quite close, so the frequency of the cosine of the right side of the expression above, that is (f1−f2)/2, is often too slow to be perceived as a pitch. Instead, it is perceived

as a periodic variation of the first in the expression above, whose frequency is ( f 1+ f 2)

2, that is, the

average of the two frequencies. Therefore the frequency of the envelope has twice the frequency of the cosine, which means that the audible beat frequency is: f (beat)= f 1− f 2

cos (2 π f 1 t)+cos(2 π f 2 t)=2cos(2 π( f 1+ f 2)

2t)⋅2 cos(2 π

( f 1− f 2)2

t)When 2 cos(2 π

( f 1− f 2)2

t) is equal to one, the two waves are in phase and they interfere

constructively. When it is zero, they are out of phase and interfere destructively. Beats occur also in more complex sounds, or in sounds of different volumes, though calculating them mathematically is not so easy.

The sound of the guitarAs the other kind of waves, sound waves too are measured with frequency (that determinates the highness of a note in the musical scale) and intensity (that represents the volume of the sound); but moreover they have the characteristic that probably makes music so beautiful, complex and charming to human being: the timbre, by which we can understand the diversity of the various musical instruments and get a sense of their particular sound. So, thanks to the timbre, the possible combinations of sounds rise steeply.

Simple and complex soundsBefore we have checked the physic peculiarities of a wave; now let’s focus on his Mathematics development.

In the figure the sinusoidal graphic of a simple wave is represented. The wave is a pure sound; in fact there are no timbre variations between the peaks, so it is unitary. But, for scientists’ misfortune and

musicians’ luck, there are less then 3 or maybe 4 pure sounds in the world; there aren’t instruments which can create a pure sound, or which are made for creating it (except diapason, a bifurcate iron piece that produces a pure “A” (=la) sound). As we can see in the next figure, from the meeting of different timbres results a very various and complex wave: and, as said in the first paragraph, this characteristic permits to the human’s ear to discriminate the different instruments, and to the instruments to have a particular and rich sound.

The guitar The guitar is a six-corded instrument which can be sounded with fingers, nails or pick. It was born in the XIII century in Italian courts, but the actual version is the one made by Antonio De Torres in 1869. The strings can be 6, 7, 8, or perhaps 12; there are three types of guitar: electric, acoustic and classical. There are plenty of varieties to the classical guitar, such as ukulele, banjo, mandolin.

Here in the photo we have a list of the parts of a guitar.

There are mainly three methods of picking the strings of a guitar: the note, that makes a single note sing, with the picking of one string, pressed with the other hand in a position of the fingerboard; the chord, that is the sound of more strings together; then the arpeggio, or rather the sound of more strings one after the other.

In the guitar, for us more than in other instruments, the complexity of sound is a fundamental component: and each part of the guitar contributes to the creation of a charming, sensual sound different from all the others. Paradoxically the sound made by a guitar is not so difficult to hear, actually it is simple, but the process because of which it is generated is fascinating and very long and delicate. In fact in guitar the sounds ring again from all over its body: for example, when we play a chord, the different frequencies of the various strings make different part of the guitar vibrate. But also when we play a single note the quality of the materials that are in the guitar permits to the sound to be massive and clear, and to resound in a unitary way even though it comes from different parts of a guitar. To this phenomenon is absolutely important one element of the guitar: the sound board. Set inside the guitar, the sound board is the internal structure of the instrument ribbed in sectors that resounds in different ways depending on the note (or the notes) which are played by the musician. Timber is very important in the fabrication of a guitar, because more the timber is certified, thin and dense, more the sound results rich, delicate and precise.

In conclusion, we can say that guitar is maybe the more charming instrument ever created by man: simple in hearing, difficult in creating sound, beautiful in playing.

Maths and Dance

We worked together about Maths and Dance and presented this work at the “Festival della Scienza” event in Genova on October 24th.

This is the copy of the local newspaper article:

This is the link to our presentation: http://www.slideshare.net/enricamaragliano1/maths-and-dance