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Maths and Music
Transcript of Maths and Music
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 length
of the tenor of the composition
is divided into equal time slots
with the same rhythmic
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.
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.
Ex. ½ , ¼,
The composers of the
15th century used all
these criteria to achieve
a unity .
Other solutions were the golden section and Filbonacci's series.
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).
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.
"Frere Jacques" is a widely known example of repeating canons in which all voices are musically identical.
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
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 unity based on proportions
and corresponding lands.
Elena Abd El and Anna Spirito
school year : 2012-13
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 cancrizans, because the theme, like the crab- in Latin “cancer”- in the follower goes backwards in time.
Their evolution during the History
Maths and Music have always had very close relations
Music 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)
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.
So the lyre was invented...
A an octave higher
...that worked according to these rules:
"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))
In 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.
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.
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
The "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”.
Fourtheenth-century mottetto’s composition
But the history goes on...
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.
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.
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.
Bach: a musician and mathematician
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 Art
of Fugue (Kupferberg, 107).
J.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 are 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.
There is Geometry in the humming of the strings.
There is Music in the spacing of the spheres.
Pythagoras (6th C. B.C.)
This is an example of direct canon in two voices (translation along the x-axis): the note "e" voice of the second harmonic with the note "b" and "c" of the first, the note "f" with the second notes "b" and "a" of the first and so on ...
A video about J.S. Bach's Crab Canon