The pitch of a musical note (hereafter abbreviated as a note) is defined by its frequency measured in Herz (Hz), cycles per second.

The bigger the frequency, the higher the pitch.

The frequencies of all musical notes are the set of positive reals ℝ + . The interval between them is expressed by the ratio of their frequencies. As will be

shown below, in just intonation, D = 9 8 and A = 5 3 and the interval between them are calculated by division 5 3 / 9 8 = 40 27 . The reason for this is that use is

made of the logarithmic distance and the subtraction of logarithms is expressed as division of the anti-logs. In turn, the main reason why the logarithmic distance is used comes from the Weber-Fechner law (cf. e.g. [6] ) to the effect that stimuli are sensed logarithmically. For two notes a and b define

d ( a , b ) = | log a / b | = | log a − log b | ≥ 0.

Then this is a distance function. For d ( a , a ) = 0 , d ( b , a ) = d ( a , b ) and

d ( a , c ) = | log a − log b + log b − log c | ≤ | log a − log b | + | log b − log c | = d ( a , b ) + d ( b , c ) .

Hence, ( ℝ + , d ) is a metric space. Thus, we think of intervals between two notes as the logarithmic difference, which we substitute by the quotient, say b/a, of corresponding frequencies b > a and say that the interval between a and b is b/a (by which we understand the logarithmic interval log b − log a ).

We introduce the relation ∼ by

a ∼ b ⇔ a / b = 2 n   n ∈ ℤ . (2.1)

Then this is an equivalence relation. For a / a = 2 0 , and if a / b = 2 n , then b / a = 2 − n . Finally, if a / b = 2 m and b / c = 2 n , then a / c = 2 m + n . Hence ℝ + is classified into equivalence classes: ℝ + / ∼ .

This may be also viewed group-theoretically. The multiplicative group ℝ + or its subgroup ℚ + of all positive rationals contains the cyclic subgroup 〈 2 〉 = { 2 n | n ∈ ℤ } . And the quotient group ℚ + / 〈 2 〉 may be considered as the set of all musical notes up to octaves.

Two musical notes whose frequencies are different by powers of 2 are thought of as belonging to the same equivalence class, or those which sound alike. Musicians express the equivalence relation by using the same label for those notes in the same class. E.g. if the middle C has frequency 260 Hz, then the note three octaves higher than middle C has frequency 260 × 2 3 = 2080 Hz while the note two octaves below middle C has frequency 260 × 2 − 2 ≈ 65 Hz. They are denoted by the same symbol C and are referred to high C or low C, respectively. The concert pitch which is used today is A = 440 . Then with this value, in Pythagoras scale, the middle C = 261 while in just intonation it is C = 264 , cf. also (6.1).

Definition 1. Two intervals which combine to give an octave is called an inversion to each other.

After the octave, the next simplest is the perfect fifth 3:2 containing 7 semitones whose inversion is the perfect fourth 4:3 containing 5 semitones. The major third 5:4 is the interval containing 4 semitones whose inversion is the minor sixth 8:5 which contains 8 semitones (Figure 1 & Figure 2).

The sequence of three notes arranged in the order of the major third and the minor third is called a major triad.

The common major chord (do-mi-so-do) has the ratio 4 : 5 : 6 : 8 , while the common minor chord (do-mi^ç-so-do) has the ratio 10 : 12 : 15 : 20 (Figure 3).

Remark 2.1 For the meaning of “fifth’’, cf. Remark 5.1 below. There are three ratios with the prefix “perfect’’, which are perfect first (which is nothing but the unison), perfect fifth and perfect fourth. The major third is the interval from A ( f = 440 ) to C # ( f = 550 ) , which is a representative of the equivalence class consisting of the intervals 5/4, 5/2, 5/1, 10/1,….

Pythagoras was the first who raised the question “Why is consonance associated with the ratios of small numbers?’’ Here “numbers’’ means integers. This is referred to as Pythagoras’ law of small numbers to the effect that only the small integer multiples of the fundamental notes can create harmony and consonance. There is a variation of this law, the law of cyclotomic numbers, cf. §7.

In this section as a partial explanation of Pythagoras’ law of small numbers, we refer to the experiment of Helmholtz and Joahim stated in [3] and we refer to

their result as the Helmhotz-Joahim scale. H. von Helmholtz with the help of the renowned violinist J. Joachim, made an experiment and tabulated the notes which are the most pleasing to the ears (Table 1).

Table 2 is the basis of Coxeter’s speculation of the law of cyclotomic numbers in §7.

When we pile up the notes starting from a basic note, we obtain an increasing sequence of pitches. The way of piling up is not at one’s disposal and one has to struggle to achieve good consonants among them. There are several sequences constructed and they are called a scale or sometimes pitch with the inventor’s name or with the describing term (sometimes also called an intonation).

In the case of Pythagorean scale (§5), it is formed using only powers of 2 and 3.

Since one whole-tone consists of two semi-tones and there are two semi-tones and five whole-tones in one octave, it follows that one octave has 12 semitones, cf. (8.6) below. Therefore, piling up the notes on the basic one, the 12th power is

very important. In the case of the Pythagorean scale, what are piled up are powers of 3 2 , so that

( 3 2 ) 12 ≈ 129.7,

which is a little higher than 7 octaves: 2 7 = 128 . The interval

129.7 128

is known as the comma of Pythagoras. This discrepancy accounts for many difficulties in obtaining an organized system of pitches.

In the equal tempered system, one octave is equally divided into 12 semitones with ratio being the 12th root of unity

1   tempered   semitone = 2 1 / 12 = 1.05946 ⋯ (4.1)

The Pythagorean scale is made of 2 p 3 q , p , q ∈ ℤ , − 3 ≤ q ≤ 3 . As is explained above, starting from middle C, A (la) is obtained as ( 3 2 ) 3 ∼ 27 16 .

From middle C (doh) by piling up 3 2 , we get G (sol), then ( 3 2 ) 2 = 9 4 ∼ 9 8 , which is D (re). The next is ( 3 2 ) 3 = 27 8 ∼ 27 16 , which is A (la) with frequency 440 Hz. The 5th is ( 3 2 ) 4 = 81 16 ∼ 81 64 = ( 3 2 ) 2 , which is E (mi).

Definition 2. These 5 notes C, D, E, A, G form the pentatonic scale. It is the basis of many folk-songs in Scotland. Figure 4.

The 6th is ( 3 2 ) 5 = 243 32 ∼ 243 32 1 2 2 = 243 128 , which is B (si). Thus C (doh), G (sol), D (re), A (la), E (me), B (si). The 7th F (fa) is obtained by coming back from higher C. i.e. 2 ⋅ ( 3 2 ) − 1 = 3 2 .

This is the scale used by the Greeks and early medieval composers as the basis of the Ecclesiastic Mode. The scale is suitable for melodic writing but not satisfactory for harmonic writing (for modulation) (Figure 5).

We remark that the sequence { [ ( 3 2 ) n ] } is very much relevant to number

theory, esp. in connection with the Waring problem, cf. e.g. [7] , where the Gaussian symbol will be defined after (5.1).

The Verdi pitch is the scale which fixes the freq. of A to be 432. In [6] the following theorem has been found.

Theorem 6.1 The reason for the A = 432 is that in the Pythagorean scale,

C = 432 × ( 27 16 ) − 1 = 256 = 2 8 . (6.1)

This makes some of the calculations simpler but this was not used much. The reason is that the instruments are tuned not from C but from A and in just

intonation it gives a fractional frequency to A: A = 256 × 5 3 = 426 3 2 ≈ 427 . This

is approximately the standard pitch used in the time of Beethoven (Figure 7).

The Pythagorean major 3rd m = 81 64 is slightly bigger than the major 3rd 5 4 of

just intonation. The Pythagorean major 3rd is said to let the melody sound beautifully, but it diminishes harmony because of the beats contained. In { d , m } the number of beats caused by the 4 times and 5 times of m is

d ( 4 × 81 64 − 5 × 1 ) = d 16 = 0.0625 d ,

which is 16.5 times/s for d = 264 Hz, say. Hence in the Pythagorean major triad

d { 1 , m d , s d } = d { 1 , q 4 4 , q } = d { 1 , 81 64 , 3 2 } ,

there occur beats between the root and the major 3rd. To eliminate this beat, we decrease the major third by multiplying by the syntonic comma

1 Δ = 80 81

to make it m = 5 4 . This yields the just intonation, in which the major 3rd consists of

{ 1 , m d , s d } = { 1 , q 4 Δ 4 , q } = { 1 , 5 4 , 3 2 } = 1 4 { 4 , 5 , 6 } .

Figure 8 and Table 6.

On the basis of Table 2, Coxeter [3] states a speculation that the agreeable harmonics

3,4,5,6,8,10,12 (7.1)

correspond to the number of sides of regular polygons that can be constructed by a ruler and a compass. A natural inverse question is where there are corresponding harmonics to 15 and 17 since 15-gon was constructed by Euclid while 17-gon was by Gauss in 1797. The interval 15 from a low C to high B which thrilled the audience appears as the appoggiatura in the end of St. Matthew Passion (Figure 9). The interval 17 were used by the remaining two of the 3 B’s, i.e. by Beethoven and Brahms. Since regular polygons inscribed in a circle divides it into equal parts, cyclotomy, Coxeter refers to the law of cyclotomic numbers rather than the law of small numbers.

Part of the contents of this section is an elaboration of material in [4] . In this section we use the notation { d , r , m , f , s , l , t ,2 d } , which are the initials in tonic sol-fa notation, doh, ray, me fah, soh, lah, te, doh. We also think of this as an increasing sequence of frequencies denoted by the corresponding symbols. We may conveniently write

{ d , r , m , f , s , l , t , 2 d } = d { 1 , r d , m d , f d , s d , l d , t d , 2 } (8.1)

to express the ratios of each note against the basic d. We consider the modulation from C to G. In order to express the correspondence, we expand (8.1) to obtain

F = { d , r , m , f , f # , s , l , t , 2 d , 2 r , 2 m , 2 f , 2 f # , 2 s } = d { 1 , r d , m d , f d , f # d s d , l d , t d , 2 , 2 r d , 2 m d , 2 f d , 2 f # d , 2 s d } . (8.2)

To modulate C major into G major is just to replace the basic note d in (8.2) by s by which we obtain the sequence, thereby adding the prime to indicate the correspondence

{ f ′ , s ′ , l ′ , t ′ , d ′ , r ′ , m ′ , f ′ , s ′ , l ′ , t ′ , d ′ } = s { f ′ s , s ′ s , l ′ s , t ′ s d ′ s , r ′ s , m ′ s ,2, f ′ s , s ′ s , l ′ s , t ′ s ,2 d ′ s } , (8.3)

where we used the relation d ′ = s . In general by comparing the frequencies, we obtain

d ′ = s , r ′ = l , m ′ = t , f ′ = 2 d , s ′ = 2 r , l ′ = 2 m , t ′ = 2 f # . (8.4)

Also comparing the ratios in (8.2) and (8.3), we find that the ratios with prime are equal to those without:

x ′ s = x ′ d ′ = x d (8.5)

for all x ∈ F , e.g. s ′ s = s ′ d ′ = s d , the far-right side member we denote by q.

q = s d .

Then (8.5) amounts to

x ′ = q x .

With d and q as parameters, we may express all other notes in terms of them. First s = q d . Using (8.4) and (8.5), we have 2 r = s ′ = q s , whence r = 1 2 q 2 d . Similarly, from 2 d = f ′ = q f we have f = 2 q q d . Since l = r ′ = q r we deduce that l = q 1 2 q 2 d = q 3 2 d . From 2 m = l ′ = q l we deduce that m = q 1 2 q 3 d = q 4 4 d . From t = m ′ = q m , it follows that t = q 5 4 d . Finally, from 2 2 q d # = 2 f # = t ′ = q t = q 6 4 d , it follows that # = q 7 16 .

Theorem 8.1 The major chord in Table 3 is expressed as

F = { d , r , m , f , s , l , t ,2 d } = d { 1, q 2 2 , q 4 4 , 2 q , q , q 3 2 , q 5 4 ,2 }

while the minor chord as

F = { l , t , d , r , m , f , s ,2 l } = l { 1, q 2 2 , 4 q 3 , 2 q , q , 8 q 4 , 4 q 2 ,2 } .

Corollary 8.1 The Pythagorean scale is the case with q = 3 2 .

Remark 8.1 Although Theorem 8.1 explains the reason why f is chosen as 2 q ,

it is a typical hindsight since Pythagoras had no concept of modulation and he must have followed the reasoning given in Remark 5.1, 1) based on the law of small numbers.

Corollary 8.2 There are three major triads, { 1, q 4 4 , q } = { d , m , s } = CEG , { f , l , d } = FAC , { s , t , r } = GBD .

Proof. There are three major thirds q 4 4 = { m d , l f , t s } and so the piling up of minor thirds are possible only for those starting with m , l , t . □

Definition 4 The major 2nd is usually called tome and denoted by w and the minor 2nd is called semi-tome denoted s ˜ :

w = q 2 2 ,   s ˜ = 8 q 5 .

1 octave consists of 5 tones and 2 semi-tones:

2 d d = r d m r f m s f l s t l 2 d t = w w s ˜ w w w s ˜ = w 5 s ˜ 2 (8.6)

in conformity with the statement in §4. Table 7.

The law of cyclotomic numbers work well within the same key, but produces a contradiction if we modulate into other keys. In Table 6, the whole tone from Eç

to F or from G to A is 4 3 5 6 = 10 9 while for the tone from F to G is 3 2 3 4 = 9 8 and they are different by 9 8 9 10 = 81 80 , which the Greeks called a comma. To overcome

this difficulty, the well-tempered scale was invented shortly before the time of Bach (Figure 10). Bach composed “Well-tempered Clavier’’. a masterpiece

collection of 48 preludes and fugues. In this system every semitone is exactly the 12th root of 2 as given in (4.1).

Theorem 9.1 The infinite group of tempered intervals { ( 2 1 / 12 ) k | k ∈ ℤ } modulo octaves is isomorphic to the cyclic group ℤ / 12 ℤ , i.e.

{ ( 2 1 / 12 ) k | k ∈ ℤ } / 〈 2 〉 ≅ ℤ / 12 ℤ .

The proof follows from the isomorphism theorem on noting that the kernel of the homomorphism

f : ℤ → { ( 2 1 / 12 ) k | k ∈ ℤ }

is 12 ℤ .

Example 1 As defined in §1, the major third consists of 4 semi-tones, and so in equal tempered system it is ( 2 1 / 12 ) 4 . The interval from A ( f = 2 7 12 ) to C # ( f = 2 11 12 ) being 2 4 12 , it is the major third corresponding to 5/4. It is a representative of the equivalence class consisting of the intervals

( 2 1 / 12 ) 4 , ( 2 1 / 12 ) 16 , ( 2 1 / 12 ) 8 , ( 2 1 / 12 ) 20 , ( 2 1 / 12 ) 28 , ⋯

Recall Corollary 8.2. We consider Kellner’s tuning, cf. e.g. [11] . In the major

triad { 1, q 4 4 } = { d , m , s } , { f , l , d } , { s , t , r } , we make q a little smaller than natural

5th so that the number of beats caused by three times the root and twice the 5th note be equal to that of the beat caused by the four times the the 3rd and the 5th times of the root, i.e. writing such a q by q_B, we have

3 × 1 − 2 × q B = 4 × q B 4 4 − 5 × 1

whence we are led to the Quartic Equation

q B 4 + 2 q B − 8 = 0 (10.1)

called Bach’s equation. Numerically

q B = 1.495953506. (10.2)

t is well-known that equations with rational (or algebraic) coefficients of degree ≤4 are soluble algebraically, i.e. by ordinary arithmetic operations plus extraction of roots. For our curiosity we shall find the (real) roots of a quintic equation

X 4 + a X 2 + b X + c = 0 (10.3)

with a = 0 , b = 2 , c = − 8 , i.e. (15). This can be solved using its resolvent equation of degree 3:

X 3 − 2 a X 2 + ( a 2 − 4 c ) X + b 2 = 0

which is in the present case

X 3 − 32 X + 4 = 0 (10.4)

with discriminant

D = − 4 ( − 32 ) 3 − 27 × 4 2 = 16 ( 8192 − 27 ) = 16 × 8165

Since D ∉ ℚ , it follows that the Galois group of (10.3), X 4 + 2 X − 8 = 0 , is the 4th symmetric group S 4 . There exists an explicit formulas for solutions due to Cardano but it is too cumbersome to state. Here we adopt the method of Ohtuska [4] , which is indeed Cardano’s method. To solve (10.4) one puts

( u 3 + v 3 = − 4 3 u v = − 32 (10.5)

Then (10.4) becomes

0 = X 3 − 3 u v X + u 3 + v 3 = ( X + u + v ) ( X 2 + u 2 + v 2 + u X + v X + u v ) ,

whence X = − u − v . On the other hand, (10.5) is easily soluble: u 3 = − 2 ± 4 + ( 32 3 ) 3 and we choose − u = 2 + 4 + ( 32 3 ) 3 1 / 3 = 3.329053233 . At the same time, − v = 2 + 4 − ( 32 3 ) 3 1 / 3 = − 3.204114179 , so that X = − u − v = 0.124939054 .

Now recall how the resolvent Equation (10.4) was derived. With y a new variable and a a parameter, we add 2 a X 2 y + ( a y ) 2 to both sides of X 4 = − b X − c with b = 2 , c = − 8 to obtain

( X 2 + a y ) 2 = 2 a y ( x − 2 − 2 b 4 a y X + ( b 4 a y ) 2 ) = 2 a y ( 1 4 a y ) 2 , (10.6)

provided that

a 2 y 2 − b 2 8 a y − c = 0

or

y 3 − c a 2 − b 2 8 a 3 = 0 , y 3 + 8 a 2 − 1 2 a 3 = 0. (10.7)

For this to coincide with the resolvent Equation (10.4) we choose a = 1 2 and write − X for y. Hence (10.6) amounts to

X 2 + 1 2 y = − y ( X − 1 y ) , (10.8)

whence

X = 1 2 ( − y + ( 4 y − y ) 1 / 2 ) , (10.9)

which gives the value (10.2) for Bach’s fifth.

On fluctuations, there is a bunch of literature. We mention only a few [13] [14] [15] [16] [17] . Let X ( t ) be a signal and consider its discrete Fourier transform (DFT)

X ^ ( f ) = ∑ t = 1 M   e − 2 π i f t X ( t ) (12.1)

for the frequency f. Let the power spectrum be defined by

S X ( f ) = | X ^ ( f ) | 2 . (12.2)

For a more general treatment of finite power signals, we refer to [1] .

It was found (e.g. [15] ) that many natural phenomena possess the power spectrum depending on the frequency 1 / f β . This kind of phenomena is called 1/f-noise or fluctuation.

In [17] , the distribution of primes is considered from the point of view of 1/f-fluctuation. However, in [18] , there is criticism against the interpretation of power spectrum, saying that it applies to non-living.

We are to study music as something that lies at this threshold of living and non-living.

The authors declare no conflicts of interest regarding the publication of this paper.

Li, H.L., Chakraborty, K. and Kanemitsu, S. (2018) Music as Mathematics of Senses. Advances in Pure Mathematics, 8, 845-862. https://doi.org/10.4236/apm.2018.812052