mathematics in music

In music, 1, 2, and 3 are not numbers, but special signs, and are sung as do, re, mi, which come from first syllable of each of first seven lines of a medieval Italian hymn. However, history of music is as long as history of language, and its origin cannot be verified. But what is amazing is that we can use mathematical knowledge to explain many of rules of music, including basic elements of music - principles of musical sound, that is, 1, 2, 3 ... These symbols really have a numerical value. or math background.

Learning music always starts with scales, and our usual scales are made up of 7 basic tones:

1, 2, 3, 4, 5, 6, 7

or in roll call

do, re, mi, fa, so, la, si

Various combinations of 7 tones and tones one or more octaves above and one or more octaves below them are "melodies".

Renowned American music theorist Percy Goetschius (1853-1943) said: “For music students and music lovers who are eager to learn, there is no such musical element as “scale” that arises directly and arouses their curiosity and wonder During a long time".

A scale of 7 notes is arranged from lowest to highest according to its "pitch". To understand principle of scale, you must first know what is "pitch" of a sound? What is "altitude" difference between sound and sound?

When an object vibrates, it makes a sound. The strength of vibration (the size of energy) is reflected in size of sound.

In physics, speed of vibration is represented by frequency. Frequency is defined as number of vibrations of an object per second. The unit of vibration per second is hertz. A sound with a frequency of 261.63 Hz is denoted in music by letter c1. Accordingly, scale is expressed as

c, d, e, f, g, a, b

When C is sung like "do", it's called key of C.

The human ear cannot perceive or feel unpleasant sounds that are too high or too low in frequency. The frequency range commonly used in music is around 16–4000 Hz, and most expressive frequency range of human voice and instrumental music is around 60 Hz. ~1000 Hz.

On a stringed instrument, pull an empty string and it will make a sound of a certain frequency. If you are asked to sing this note, how do you know that vibration frequency of your vocal cords is exactly equal to vibration frequency of empty string? This requires "principle of resonance": when frequencies of two vibrations are equal, combined effect is maximized without any attenuation. Therefore, you must adjust vibration frequency of vocal cords through experience and perception so that vibration of vocal cords resonates with vibration of empty string. At this time, vibration frequency of vocal cords is equal to vibration frequency of empty string.

It has long been found that sound produced by an empty string has a very harmonious effect, or close to "resonance", with sound producedwith same empty string, but twice as short, this is called ratio with octaves. We can use "like a shadow to follow" to describe a pair of octaves, unless frequency of two tones is same, an octave is most closely related sound in terms of auditory harmony.

At beginning of 18th century, British mathematician Taylor (1685-1731) obtained a formula for calculating frequency of vibration of a string f:

l represents length of string, T represents tension of string, and ρ represents density of string.

This shows that for same string (same material, same thickness) frequency is inversely proportional to length of string, and frequency ratio of a pair of octaves is 2:1.

Now we can describe difference in pitch between notes: suppose an empty string produces a C sound, a half string produces a C an octave higher; a string of length 8/9 produces up to re, 64/81. a string of length 3/4 gives fa, a string of length 2/3 gives so, a string of length 16/27 gives la, a string of length 128/243 gives si, and so on. For example, this is how a string would sound an octave higher, length of which is half length of 2/3, that is, 1/3.

To make it convenient to count frequency of sound "c" by one unit, frequency of sound "c" an octave higher is equal to two units, and frequency of sound "re" is 9/8 units. The names of sounds and their corresponding frequencies are listed in following table:

Table 1:

Sound name C D E F G A B C

Frequency 1 9/8 81/64 4/3 3/2 27/16 243/128 2

two

Having learned numerical ratio of do, re, mi, fa, so, la, si, a new question arises: why are tones with these frequencies used to form scale? In fact, question that should be answered in first place is why use 7 notes to form a scale?

This is an eternal mystery. Since it cannot be confirmed from past history, there are all kinds of inferences and conjectures in ancient and modern China and abroad. The 7 major planets (this was in an era when only 7 planets were known) make different sounds, forming musical scales. We will solve riddle mathematically.

We use different tones to create melodies. Of course, we must consider whether these tones are harmonious together. As mentioned earlier, octaves are closest tones in terms of auditory harmony, but only octaves are used. form a beautiful melody - at least there are too few of them, for example, in musical frequency range of octaves c1 and c1 there are only following 8: Do2 (16.35 Hz), Do1 (32.7 Hz), Do (65.4 Hz) , C (65.4 Hz), c (130.8 Hz), c1 (261.6 Hz), c2 (523.2 Hz), c3 (1046.4 Hz), c4 (2092.8 Hz), for human voice has only 4 tones C, c, c1 and c2.

To produce new harmonics, remember that cause of harmony of pair of octaves mentioned above is approach to resonance. Mathematical theory tells us that each sound can be decomposed into a superposition of a harmonic and a series of harmonics with a frequency multiplied by an integer. still withreading that frequency of sound c is 1, then it is decomposed into a superposition of harmonics with frequencies 1, 2, 4, 8, ..., and frequency of high-octave sound c is 2, which is decomposed into frequencies 2, 4, 8, 16,. .. Superposition of harmonics, frequency of these two series of harmonics is almost same, which is mathematical explanation for approximate resonance of a pair of octaves. The principle can be deduced from this: if frequency ratio of two tones is a simple integer ratio, then two tones have a harmonic ratio, because each tone can be decomposed into a superposition of a harmonic and a number of integer multiple harmonics. , two tones The simpler integer frequency ratio, more corresponding two harmonic columns contain harmonics of same frequency.

The next simple integer ratio to 2:1 is 3:2. Try it, an open string note (assumed to be C in Table 1, and used as do) will work in harmony with a 2/3 string note, either sequentially or simultaneously. It can be concluded that ancients must have been very excited when they discovered this phenomenon. In fact, we have more reason to be excited than ancients because we understand mathematical principles involved. Then playing 3/2 string produces a note that is also in harmony. Its frequency is 2/3 of C frequency, which is already lower than C frequency. To make it easier to study in an octave, it is replaced by a higher octave of 4/3 of C frequency. Obviously, we got G(so) and F(fa) in Table 1.

The problem is, we can't do this forever, otherwise we'll end up with infinite polyphony instead of 7 tones!

If a new tone is played sequentially from C with a frequency ratio of 3:2, and new tone is a few octaves higher than C, then new tone will not be played later. Unfortunately, mathematics can prove that this is impossible, because there are no natural numbers m and n that make following formula true:

(3/2)m = 2n

Can natural development of rational thought now establish an approximate equation? After calculating (3/2)5 = 7.594 ≈ 23 = 8, therefore, ratio to 1 is considered to be 23, that is, ratio three octaves higher is considered as same note, and (3/2) 6 and (3 /2 The ratio )1 is also equal to 23, that is, ratio is three octaves higher and so on is also considered to be same sound. In sense of "one octave" there are only 5 notes, and their frequencies are:

1, (3/2), (3/2)2, (3/2)3, (3/2)4 (1)

Equivalent to an octave precision:

1, 9/8, 81/64, 3/2, 27/16

Comparing Table 1, we know that these are 5 sounds: Do (do), Re (re), E (mi), Sol (co), A (la). This is so-called pentatonic scale. not widely used in musical culture, but usual "corrupt newspaper songs" consist of pentatonic scales.

Further, according to (3/2)7 = 17.09 ≈ 24 = 16, there should be 7 notes in total to form scale. Based on (1), we use frequency ratio 3:2 go up once and down once to get a sequence of 7 tones whose frequency is

(2/3), 1, (3/2), (3/2)2, (3/2)3, (3/2)4, (3/2)5

Equivalent to an octave precision:

1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128

An ordinary pentatonic seven-tone major scale is obtained as shown in Table 1.

Study ratio of frequencies of two adjacent tones in scale. Thanks to calculations, we know that there are only two cases: frequency ratio of do-re, re-mi, fa-so, so-la and la-si is 9:8, called full-tone ratio, mi-fa, frequency ratio of si-do is 256:243, called halftone ratio.

The law of harmonics that occurs when ratio of frequencies is 2:1 and 3:2 is called law of fifths. In China, earliest written references to law of five degrees can be found in "Diyuan pian" of classical "Guanzi". to 3rd century BC Western scholars believe that Pythagorean school of ancient Greece in VI century BC. first proposed law of five powers.

According to approximate equation (3/2)12 = 129.7 ≈ 27 = 128 and method described above, pentatonic twelve-tone scale can be composed as follows:

Table 2:

Sound name C #C D #D E F #F

Frequency 1 (37)/(211) (32)/(23) (39)/(214) (34)/(26) (22)/(3) (36)/(29)

Sound name G #G A #A B C

Frequency 3/2 (38)/(212) (33)/(24) (310)/(215) (35)/(27) 2

There are two frequency ratios between two adjacent tones in twelve-tone pentatonic scale: 256:243 and 2187:2048, which are called natural semitones and modified semitones, respectively. The table shows that relationship between two tones with different sound names, such as #C-D, is a natural semitone, and relationship between two tones with same sound names, such as C-#C, is a semitone. change.

In course of human history, emergence of a particular musical culture cannot be limited to one time or one place, but rhythm of five steps appeared in East and West almost simultaneously, because this shows connection between artistic talent of a person.

Three

Among various temperaments, apart from fifth temperament, most widely used are twelve equal temperaments and pure temperament.

Twelve Equal Temperaments - People have noticed that two semitones in pentatonic twelve-tone scale have little difference. If this difference is eliminated, it will be very convenient for keyboard modulation, because each of keyboards has a pitch. pitch is fixed, unlike pitch of a plucked or strung instrument, which is determined by position of finger. The way to eliminate difference between two semitones is to make frequency ratios of adjacent tones equal. This is mathatic problem for high school students: insert 11 numbers between 1 and 2 so that they form a geometric sequence. Obviously, overall ratio is, AND following inequalities hold

1.05350 = 256 / 243 <= 1.05946 < 2187 / 2048 = 1.06787

Thus there are twelve equal temperaments, and frequency ratio of any two adjacent tones is same, and there is no difference between natural semitones and changing semitones.

The heptatonic scale, consisting of twelve equal temperaments, looks like this:

Table 3:

Sound name C D E F G A B C

Frequency 1 ()2 ()4 ()5 ()7 ()9 ()11 2

Same as pentatonic heptatonic scale, C-D, D-E, F-G, G-A, A-B are ratios of whole tones, E-F, B-C are ratios of semitones, but whole tone is exactly two semitones.

The Twelve Equal Laws are a reference to Fifth Law and a rebellion against Fifth Law.

The appearance of twelve equal temperaments shows that irrational numbers have entered music, which is an amazing phenomenon. Irrational numbers are big monster in mathematics. Today, a non-mathematics college student still does not understand what irrational numbers are after graduating from a mathematics college. Mathematicians have been using irrational numbers for over 2500 years and did not understand irrational numbers until late 19th century. Musicians don't seem to care about complexity of irrational numbers, and it's easy to call elegant music irrational numbers.

The appearance of twelve equal temperaments also makes principle of harmony we introduced earlier - simpler integer ratio between frequency ratio of two tones, more harmonious two tones - no longer valid. But don't let that discourage you, because in essence artistic behavior does not have to obey scientific principles. Just as painting that follows golden ratio is art, painting that does opposite is also art.

The inventor of historically recorded 12-tone equal law is Dutchman Steffen (Stevin circa 1548 - circa 1620) in Europe. it was Zhu Zaiyu (1536-1612), a Ming Dynasty scholar and musician. He expressed twelve equal temperaments and even calculated each degree to 24 decimal places (completed before 1581). The establishment of twelve equal temperaments is another amazing manifestation of connection between artistic talent of a person and musical culture.

Pure temperament - ratio of frequencies 1, 3, 5 (do, mi, co) of three tones of pentatonic seven-tone scale is 1:81 / 64:3 / 2, that is, 64:81: 96, Pure Temperament changes this ratio to 1:5/4:3/2, which is equal to 64:80:96 or 4:5:6, simplifying frequency ratio between 1-3-5 tritones majorfoot triad. Then press frequency ratio 1:5/4:3/2 to copy two tones 7, from 5 (so) tones, copy two tones, , from 1 (before) tone to descending, namely,,, 1, 3, 5, 7, frequency ratio

(2/3):(5/4)(2/3):1:(5/4):3/2:(5/4)(3/2):(3/2)2< /p>

A total of 7 tones are converted into octaves to form a pure heptatonic temperament scale:

Table 4:

Sound name C D E F G A B C

Frequency 1 9/8 5/4 4/3 3/2 5/3 15/8 2

Compared to seven-tone pentatonic scale (Table 1), 4 tones C, D, F and G are same, but 3 tones E, A and B are different.

According to frequency ratio of two adjacent tones in a purely tempera heptatonic scale, there are 3 ratios: 9:8, 10:9, 16:15. From a numerical point of view, it is simpler than heptatonic pentatonic, but there are more types than heptatonic pentatonic (the heptatonic pentatonic has only 2 frequency ratios of two adjacent tones). What is good and what is bad in art is no longer judged by mathematics.