mathematics classics

Chapter 4 Mathematics in Music
problem
1, 2, and 3 in music are not numbers but special marks. They are sung as do, re, and mi, which come from the first syllable of each of the first seven lines of a medieval Italian hymn.However, the history of music is as long as the history of language, and its origin cannot be verified.But what is amazing is that we can use mathematical knowledge to explain many of the rules of music, including the basic elements of music - the composition of musical tones, that is, 1, 2, 3... These symbols do have a numerical or mathematical background.

Learning music always starts from the scale, our common scale is composed of 7 basic tones: 1, 2, 3, 4, 5, 6, 7 or do, re, mi, fa, so, la , si uses 7 tones and one or several octaves higher than them, and one or several octaves lower than them to make various combinations is "tune".

Why is music composed of seven scales
The famous American music theorist Percy Gaicius said: "For music learners and music lovers who are eager for knowledge, there is no musical element like the 'scale', which immediately and persistently arouses their curiosity and surprise. gone."

The 7-tone scale is arranged from low to high according to the "height". To understand the principle of the scale, you must first know what is the "height" of the sound?What is the "height" difference between the sound and the sound?
When an object vibrates, it produces a sound. The strength of the vibration (the size of the energy) is reflected in the size of the sound. The vibration of different objects is reflected in the different timbre of the sound, and the speed of the vibration is reflected in the height of the sound.

The speed of vibration is represented by frequency in physics. Frequency is defined as the number of times an object vibrates per second. The unit of vibration per second is called Hertz.The sound with a frequency of 1 Hz is represented by the letter c261.63 in music.Correspondingly, the scale is expressed as: c, d, e, f, g, a, b, which is called C key when the C sound is sung as "do".

The sound with too high or too low frequency cannot be perceived or felt uncomfortable by the human ear. The frequency range commonly used in music is about 16-4000 Hz, and the most expressive frequency range in vocal and instrumental music is about 60-1000 Hz. .

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

It was discovered very early that the sound produced by an empty string has a very harmonious effect, or close to "resonance", with the sound produced by the same empty string but the length is halved. octave relationship.We can use "like a shadow to follow" to describe a pair of octaves, unless the frequency of the two tones is exactly the same, the octave is the most closely related sound in terms of auditory harmony.

At the beginning of the 18th century, British mathematician Taylor obtained the calculation formula of string vibration frequency f: f=12LTρl represents the length of the string, T represents the degree of tension of the string, and ρ represents the density of the string.

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

现在我们可以描述音与音之间的高度差了：假定一根空弦发出的音是do，则二分之一长度的弦发出高八度的do；8/9长度的弦发出re，64/81长度的弦发出mi，3/4长度的弦发出fa，2/3长度的弦发出so，16/27长度的弦发出la，128/243长度的弦发出si等等类推。例如高八度的so应由2/3长度的弦的一半就是1/3长度的弦发出。

In order to conveniently count the frequency of the c sound as one unit, the frequency of the octave-higher c sound is two units, and the frequency of the re sound is 9/8 units. The sound names and their respective frequencies are listed in the following table: Table The sound name is CDEF GAB.

Frequency 198816443322716243128.

After knowing the numerical relationship of do, re, mi, fa, so, la, si, the new question is why use the tones with these frequencies to form the scale?In fact, the question that should be answered first is why use 7 tones to form a scale?
This is a mystery of the ages. Since it is impossible to verify it from the past history, there are all kinds of inferences and conjectures in ancient and modern times. For example, a saying in Western culture is based on the mysterious color of the number "7". The planets (this was in the days when only 7 planets were known) made different sounds to form musical scales.We'll solve the mystery mathematically.

我们用不同的音组合成曲调，当然要考虑这些音放在一起是不是很和谐，前面已谈到八度音是在听觉和谐效果上关系最密切的音，但是仅用八度音不能构成动听的曲调——至少它们太少了，例如在音乐频率范围内c1与c1的八度音只有如下的8个：C2（16.35赫兹）、C1（32.7赫兹）、C（65.4赫兹）、c（130.8赫兹）、c1（261.6赫兹）、c2（523.2赫兹）、c3（1046.4赫兹）、c4（2092.8赫兹），对于人声就只有C、c、c1、c2这4个音了。

In order to produce new harmonics, recall that the reason for the harmony of a pair of octaves mentioned earlier is to approximate resonance.Mathematical theory tells us that each sound can be decomposed into the superposition of a harmonic and a series of harmonics of integer times frequency.Still assuming that the frequency of c is 1, then it is decomposed into the superposition of harmonics with frequencies of 1, 2, 4, 8, ..., and the frequency of the high-octave c sound is 2, and it is decomposed into frequencies of 2, 4, 8 , 16,... The superposition of the harmonics, the frequency of these two series of harmonics is almost the same, which is a mathematical explanation for the approximate resonance of a pair of octaves.From this, a principle can be deduced: if the frequency ratio of two tones is a simple integer relationship, then the two tones have a harmonious relationship, because each tone can be decomposed into the superposition of a harmonic and a series of integer multiple harmonics, the two tones The simpler the integer relationship of the frequency ratio means that the corresponding two harmonic columns contain more harmonics of the same frequency.

The next simple integer ratio to 2:1 is 3:2.Try it, the sound from an empty string (assumed to be C in Table 1, and used as do) and the sound from the 2/3 length string are very harmonious no matter whether they are played successively or played at the same time.It can be inferred that the ancients must have been very excited when they discovered this phenomenon. In fact, we have more reasons to be excited than the ancients, because we understand the mathematical principles involved.Next, the sound from the 3/2 length string is also harmonious.Its frequency is 2/3 of the C frequency, which is already lower than the frequency of the C sound. In order to facilitate the investigation in the octave, it is replaced by its higher octave, that is, the frequency is 4/3 of the C sound.Obviously we have got G(so) and F(fa) in Table 1.

The problem is that we can't keep doing this, otherwise we will get infinite polyphony instead of 7 tones!

如果从C开始依次用频率比3∶2制出新的音，在某一次新的音恰好是C的高若干个八度音，那么再往后就不会产生新的音了。很可惜，数学可以证明这是不可能的，因为没有自然数m、n会使下式成立：（3/2）m=2n此时，理性思维的自然发展是可不可以成立近似等式？经过计算有（3/2）5=7.594≈23=8，因此认为与1之比是23即高三个八度关系算作是同一音，而（3/2）6与（3/2）1之比也是23即高三个八度关系等等也算作是同一音。在“八度相同”的意义上说，总共只有5个音，他们的频率是：1，3/2，（3/2）2，（3/2）3，（3/2）4。

(1) Converted to within an octave is:

1，9/8，81/64，3/2，27/16
Comparing Table 1, we know that these five tones are C (do), D (re), E (mi), G (so), A (la). This is the so-called pentatonic scale. It is not widely used, but the familiar "selling newspaper songs" are composed of pentatonic scales.

接下来根据（3/2）7=17.09≈24=16，总共应由7个音组成音阶，我们在（1）的基础上用3∶2的频率比上行一次、下行一次得到由7个音组成的音列，其频率是：2/3，1，3/2，（3/2）2，（3/2）3，（3/2）4，（3/2）5折合到八度之内就是：1，9/8，81/64，4/3，3/2，27/16，243/128得到常见的五度律七声音阶大调式如表一。

Examine the ratio of the frequencies of two adjacent tones in the scale. Through calculation, we know that there are only two cases: the frequency ratio of do-re, re-mi, fa-so, so-la, and la-si is 9:8, which is called Whole tone relationship; mi-fa, si-do frequency ratio is 256:243, known as semitone relationship.

以2∶1与3∶2的频率比关系产生和谐音的法则称为五度律。在中国，五度律最早的文字记载见于典籍《管子》的《地员篇》，由于《管子》的成书时间跨度很大，学术界一般认为五度律产生于公元前7世纪至公元前3世纪。西方学者认为是公元前6世纪古希腊的毕达哥拉斯学派最早提出了五度律。

根据近似等式（3/2）12=129.7≈27=128并仿照以上方法又可制出五度律十二声音阶如下：表二音名C＃CD＃DEF＃F
频率1（37）/（211）（32）/（23）（39）/（214）（34）/（26）（22）/3（36）/（29）。

The sound name is G#GA#ABC.

频率3/22（38）/（212）（33）/（24）（310）/（215）（35）（27）2五度律十二声音阶相邻两音的频率之比有两种：256∶243与2187∶2048，分别称为自然半音与变化半音。从表中可看到，音名不同的两音例如#C-D的关系是自然半音，音名相同的两音例如C-#C的关系是变化半音。

In the course of human history, the emergence of a certain music culture cannot be limited to one time or one place, but the rhythm of five degrees appeared in the East and the West almost at the same time, after all, it shows the connection of human artistic endowment.

Among the various temperaments other than the fifth temperament, the twelve equal temperament and pure temperament are the most widely used.

Twelve equal temperament - People have noticed that the two semitones in the pentatonic twelve-tone scale have little difference. If this difference is eliminated, it will be very convenient for the modulation of keyboard instruments, because the tone of each key of keyboard instruments The pitch is fixed, unlike the pitch of a plucked or drawn instrument that is determined by finger position.The way to eliminate the difference between two semitones is to make the frequency ratio of adjacent tones equal. This is a math problem for middle school students - insert 1 numbers between 2 and 11 to make them form a geometric sequence. Obviously, the common ratio is, And there is the following inequality: 1.05350=256/243<122=1.05946<2187/2048=1.06787.

What is obtained in this way is the twelve equal temperament, and the ratio of any adjacent two tones of it is the same, and there is no distinction between natural semitones and changing semitones.

The seven-tone scale composed of twelve equal temperament is as follows:
音名CDEFGABC频率16232（122）5（122）7（42）3（122）112。

Like the pentatonic seven-tone scale, CD, DE, FG, GA, and AB are whole-tone relationships, and EF and BC are half-tone relationships, but its whole tone is exactly equal to two semitones.

The twelve equal laws are not only a reference to the law of five degrees, but also a rebellion against the law of five degrees.

The advent of equal temperament shows that irrational numbers entered music, which is an amazing thing.Irrational numbers are a big monster in mathematics. Today, a non-mathematics major college student still does not understand what irrational numbers are after finishing college mathematics. Mathematicians have used irrational numbers for more than 2500 years and did not really understand irrational numbers until the end of the 19th century.Musicians don't seem to care about the difficulty of irrational numbers, and easily label elegant music as irrational numbers.

The emergence of the twelve equal temperament also makes the principle of harmony that we introduced earlier - the simpler the integer relationship between the frequency ratio of two tones, the more harmonious the two tones are - no longer holds true.But don't be discouraged by this, because in essence, artistic behavior does not have to obey scientific principles.Just as painting that conforms to the principle of the golden section is art, painting that does the opposite is also art.

The inventor of the twelve equal temperaments in the historical records is the Dutchman Steffen in Europe, who strictly established the twelve equal temperaments by using the frequency ratio of the two tones around 1600; , the twelve equal laws he expressed even calculated 122 and each power to 24 decimal places (completed about 1581 years ago).The establishment of twelve equal temperaments is another amazing manifestation of the connectivity of human artistic endowment in music culture.

纯律——五度律七声音阶的1、3、5（do、mi、so）三音的频率之比是1∶81/64∶3/2，即64∶81∶96，纯律将这修改为1∶5/4∶3/2，即64∶80∶96或4∶5∶6，使大三和弦1-3-5三音间的频率之比更显简单。然后按1∶5/4∶3/2的频率比从5（so）音上行复制两音7、2·，从1（do）音下行复制两音6·、4·，即4·、6·、1、3、5、7、2·的频率之比是：（2/3）∶（5/4）（2/3）∶1∶（5/4）∶3/2∶（5/4）（3/2）∶（3/2）2。

共得7个音折合到八度之内构成纯律七声音阶：表四音名CDEFGABC频率19/85/44/33/25/315/82。

Compared with the pentatonic seven-tone scale (Table 4), there are 3 tones C, D, F, and G that are the same, and [-] tones E, A, and B are different.

在相邻两音的频率比方面，纯律七声音阶有3种关系：9∶8、10∶9、16∶15。从数字看，它比五度律七声音阶简单，然而种类却比五度律七声音阶多（五度律七声音阶只有2种相邻两音的频率比）。在艺术上孰好孰坏，已不是数学所能判断的了。

Pure law originated in ancient Greece and was formally established by the British Odintang at the end of the 13th century.

(End of this chapter)