A composer who wrote the most famous music in the world.
Most of the creations are actually spent in a state of deafness.
therefore
How did he create it?
What about such intricate and wonderful music?
answer
What is hidden behind these wonderful notes?
In the laws of mathematics ...
? 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 50 bars as an example.
The first half consists of three notes in D major (D, F-sharp, A).
Add them together.
That is, a harmonious melody-triad.
They represent the mathematical relationship between different note frequencies.
These frequencies form geometric series.
If we start with a note A3 with a frequency of 20hz.
This series can be expressed by the function f = ar n.
Where n stands for consecutive keys on the keyboard.
Moonlight Sonata Trio in D Major
The values of n are 5/9/ 12 respectively.
N-valued introduction function
You can draw a sine curve of each note.
Draw three functions
At the same distance
D completed two cycles, F rose two and a half, and A was three cycles.
This melody is called chord interval.
Nature is beautiful, sweet and moving.
Beethoven's use of dissonance
It is also full of magic and fascinating.
In section 52 -54
The main triplet contains b and c sounds.
Their sinusoidal surfaces fluctuate.
Stronger than chords
It's extremely difficult to match, almost nothing.
Discordant interval
Compare the triad in D major above.
Beethoven in mathematical certainty
Added unquantifiable elements.
Namely, emotion and creativity.
Beethoven's real musical talent lies not only in his ability to recognize melodies without listening to music, but also in his perception of musical effects. 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 separated from Wan Li, but there are many magical intersections. ...
Pythagoras was the first person to discover the connection between music and mathematics.
One day, Pythagoras passed by a blacksmith's shop and was attracted by the rhythm of high and low while the iron was being struck. He found that the harmony of sound is related to the volume ratio of the vocal body, so he did many experiments on the strings to find the harmonious and beautiful sound law of the strings and finally found the number of music.
The pitch of music depends on the length of the vocal body (such as strings). When playing the piano, your fingers move on the strings and constantly change the length of the strings, and the piano will emit a kind of 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 most harmonious and beautiful, so people call 3, 4 and 6 "music numbers".
At the same time, he further found that a vibrating string can produce pleasant intervals as long as it is divided in proportion: for example, 1∶2 produces octaves, 2∶3 produces fifth degrees, and 3∶4 produces fourth degrees. Then it is found that every harmony combination of strings can be expressed as an integer ratio, and the whole scale can be produced by increasing the length of strings according to the integer ratio.
From this, he thinks: "Music is sacred and sublime, because it reflects the relationship between numbers as the essence of the universe."
Symphonic poems of mathematics and music have been sung since then, and have fascinated countless people for thousands of years. For example, on the keyboard of the piano of the King of Musical Instruments, from one key C to the next key C is an octave in music, in which * * * contains 13 keys, 8 white keys, 5 black keys and 5 black keys, which are divided into two groups, one with 2 black keys and the other with 3 black keys.
It 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 of any two adjacent numbers is approximately equal to the golden ratio (0.6 18).
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, Dream, the whole song is divided into 6 sentences and 24 verses. The high tide will appear in the section 14, which is calculated according to the golden ratio: 24×0.6 18≈ 14.83.
For example, Chopin's Serenade in D flat major has 76 bars. Theoretically, the golden section is in 46 bars, which is where the climax of the whole song appears!
Beethoven's pathetique sonata Op. 13, second movement, complete ***73. Theoretically, the golden section should be in 45 bars, and the climax of the whole song is formed in 43 bars. With the change of mode and tonality, the climax is basically consistent with the golden section.
A typical example is Mozart's sonata in D major. The first movement is 160. If you multiply the number of bars by the golden section ratio, 160×0.6 18=98.88, the reproduction part of the music is located in the 99th bar, just at the golden section! 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, look for the golden section in music!
"Bach would be a mathematician if he were not a musician", which classical lovers must be familiar with. The crab cannon of BWV 1079 is a "mathematical work" that sounds strange, but it is even more amazing after watching the soundtrack.
Bach's contribution to music includes ten canon (referring to counterpoint), and the copying of the theme realizes many techniques such as dislocation, reflection, pull-ups, transposition and so on. Music score has sliding 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 the19th 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 1920s, Joseph Hillinger, an American professor of mathematics and music, drew a commercial curve with paper on the The New York Times, converted each basic segment of the curve into music at appropriate intervals, and then played it on musical instruments. It turns out that this is a beautiful music, which is very similar to Bach's music works. He therefore believes that:
All musical masterpieces can be converted into mathematical formulas.
His student george gershwin tried and boldly created a system of composing music with mathematics. It is said that he used this system to create the famous opera Bogey and Beth. ...
in the final analysis
Mathematics and music are a perfect match.
The abstract beauty of mathematics
Artistic beauty of music
Has stood the test of time.
infiltration
Maybe when composers create "nice" music,
There is an "invisible hand of mathematics"
Play a role in the dark