The composer who composed the most famous music in the world
Most of his works were really deaf
So
How did he create
such intricate and wonderful music?
The answer
lies in the
mathematical law hidden behind these wonderful notes ...
? Take the famous piano sonata "Moonlight" No.14 as an example. The first paragraph is slow in rhythm and stable in scale, with three notes as a group. Each triplet contains an elegant and beautiful melody structure, revealing the extremely interesting relationship between music and mathematics ...
Take the first half of 5 bars as an example
The first half consists of three notes in D major (D, F, A)
superposition together
is a harmonious melody-triad
They represent the mathematical relationship between the frequencies of different notes
These frequencies form a geometric progression
If we start with the note A3 with a frequency of 2hz
we can use the function f = ar n
where n represents several keys on the keyboard
. The value of n is 5/9/12
If you bring the value of n into the function
, you can draw the sine curve of each note
Draw three functions
In the same distance,
D completes two cycles, F is raised for two and a half, and A is three cycles
This melody is called chord interval
It is natural and pleasant
. Fascinating
In verses 52 -54,
the main triplet contains the B sound and the C sound
Their sinusoidal surfaces fluctuate
more violently than the chords
It is extremely difficult to match, and there is almost no
Compare the dissonant interval
with the above triad in D major
Beethoven's mathematical certainty
. As james sylvester said:
Perhaps, music cannot be described as emotional mathematics. But maybe, mathematics can be rational music? Musicians can perceive mathematics and mathematicians can think about music.
These two disciplines seem to be thousands of miles apart, but there are many magical intersections ...
Pythagoras was the first to discover the connection between music and mathematics.
One day, Pythagoras passed by a blacksmith's shop and was attracted by the high and low rhythmic sound of iron striking from inside. He found that the harmony of the sound is related to the proportion of the volume of the vocal body, so he did many experiments on the strings to find the law of harmonious and beautiful sound of the strings, and finally found the number of music.
the pitch of the musical sound depends on the length of the vocal body (such as the strings). When you play the piano, your fingers move on the strings, changing the length of the strings constantly, and the piano will make a ups and downs and cadence sound. If three strings are pronounced at the same time, only when their length ratio is 3∶4∶6 can the sound be the most harmonious and beautiful, so people call 3, 4 and 6 "musical numbers".
At the same time, he further found that as long as a vibrating string is divided in proportion, it can produce pleasant intervals: for example, 1: 2 produces octaves, 2: 3 produces fifth degrees, and 3: 4 produces fourth degrees. Then it is found that each harmonious combination of strings can be expressed as an integer ratio, and increasing the length of strings according to the integer ratio can produce the whole scale.
Therefore, he thinks: "Music is sacred and lofty because it reflects the relationship between numbers as the essence of the universe."
The symphonic poem of mathematics and music has been sung since then, which has fascinated countless people for thousands of years. For example, on the keyboard of the piano, the king of musical instruments, from one key C to the next key C is an octave in music, in which * * * includes 13 keys, 8 white keys and 5 black keys, and 5 black keys are divided into two groups, one group has 2 black keys and the other group has 3 black keys.
just embodies the special property that the sum of the first two numbers of any three numbers in the famous Fibonacci sequence in the history of mathematics is equal to the third number.
another property of Fibonacci sequence is that the ratio between any two adjacent numbers is approximately equal to the golden section ratio (.618).
If we carefully study the structure of musical works, it is not difficult to find that the golden ratio can be seen almost everywhere in musical forms. In the works of different scales in classical music, the bar where the climax notes are located is almost exactly at the golden section of the whole song.
For example, in Dream, the whole song is divided into 6 sentences and 24 bars. The climax will appear in the 14th section, which accords with the calculation according to the golden ratio: 24×.618≈14.83.
For example, Chopin's "Serenade in D-flat Major" has 76 bars. Theoretically, the golden section is at 46 bars, which is exactly where the climax of the whole song appears!
Beethoven's pathetique sonata Op.13, the second movement, is complete with 73 sections. Theoretically, the golden section should be at 45 bars, and the climax of the whole song is formed at 43 bars. With the transformation of mode and tonality, the climax is basically consistent with the golden section.
a more typical example is Mozart's sonata in d major, the first movement of which is 16 bars in length. If you multiply the number of bars by the golden section ratio, it is 16×.618=98.88, and the reproduction part of the tune is located at the 99th bar, which is exactly at the golden section point! Further analysis of Mozart's works also shows that 94% of Mozart's piano concertos conform to this law.
Next time you listen to music, you might as well look for the golden section in music!
"Bach would be a mathematician if he wasn't a musician", which must be familiar to classical lovers. The crab cannon of BWV179 is a "mathematical work" that sounds strange, but even more amazing after reading the score.
Bach's Dedication to Music includes ten canon (referring to counterpoint), and the copy of the theme implements various techniques such as dislocation, reflection, pull-up and inversion. Music score has slip reflection symmetry.
What is even more amazing is that many people have summed up the beauty of "function" and even deduced the function formula.
As early as the 19th century, the French mathematician Fourier discovered that all musical sounds, whether instrumental or vocal, can be expressed by mathematical functions. Tone is related to the frequency of mathematical curve, volume is related to amplitude, and timbre is related to periodic shape.
in the 192s, Joseph Hillinger, an American professor of mathematics and music, described the last business curve in the new york Times on paper, converted each basic segment of the curve into music at appropriate intervals, and then played it on an instrument. As a result, it turned out to be a piece of music with beautiful tune and very similar to Bach's music works. From this, he thinks:
All musical masterpieces can be transformed into mathematical formulas.
george gershwin, his student, made an attempt and boldly created a system for composing music with mathematics. It is said that he used this system to compose the famous opera "Bogey and Beth" ...
In the final analysis,
Mathematics and music are a perfect match
The abstract beauty of mathematics
The artistic beauty of music
has stood the test of years
Mutual penetration
.