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8/13/2019 Hang Tulajd

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< >

Lp = 10 log10 pp02

p

p0


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G 4 4

4

4

4

G

2

2

2

2

2


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n

n

n

n

(

)

*

= 80


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G=

g1, g2G g3 G, g1g2= g3

g1, g2, g3 G g1(g2g3) = (g1g2)g3

e G,

g G

eg= g

g G g1 G, gg1 =e


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g g

g1, g2 G

g1g2 = g2g1

G H H G

H G G H

G K

g G g G

G G=< g >

G

G {g0 = e; g; g2; g3; . . .}

gk =g l


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A B (a; b)

a A b B

A B := {(a, b) :a A, b B}

A B R A B

(a, b) R aRb

A AA

a A a a

a, b A a b b a

a,b,c A a b b c a c.

A x, y

x y

R1 A B R2C D R1 R2A D

(a, d) A D

c B C aR1c cR2d

R A {R(x)|x A} R(x)

x A A/R A R


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H G a G aH G

H a aH aH

H G a G a

aH=H a H

H G

N G N

(aN)(bN) := (ab)N

G N G/N

a b m a

b m m|a b

a b (mod m)


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m a a

m

m


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T :={0, 1, 2, . . . , 79, 80} h(t) := 100 5 t

25 (t T)

K T K

|K| = n

n= 2 (t1; t2)

g : T2 R+

g(t1; t2) :=h(t2) :h(t1).


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g

g(t1; t

1) =440 2

t

112

440 2t112

= 2t

1t112

12t1 t1

g t1 t1


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(T; h) A T h

t 0

T

A := {9, 8, . . . , 2}

C


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h

h(t) = 440 2t12

h(t) := 262 2t12

h h

262 2t+912 = 440 2

t12

h

A = {9, 8, . . . , 2} A T

A:= 0, 1, 2, . . . 11

A h(A) =h(A)

A h h


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A

0, 1 . . . 11

T

T Z

mod 12

mod 12

A

A

a1, a2 A

a1 a2:= a1+a2 (mod 12).

A

mod m

a1, a2 Z (a1+a2) Z k Z

a:= a1+a2 km 0 a m 1

(a1+a2) +a3 a1+ (a2+a3).

a A

a+ 0 a (mod m)


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a A a1

a+a1 0 (modm)

a

a1 a m a (mod m)

0 m a m 1 m a A

mod m m = 12

Z+m Z+

12

Z+12

Z+12 C12 g Z+

12 g

g := 1 < g >=Z+12

mod 12

C , C i s z, . . . , H

C,Cisz,. . .

Z


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12Z 12Z Z 12Z

12Z

12Z

Z/12Z mod 12

C := 0 + 12Z Cisz:= 1 + 12Z

. . . H:= 11+ 12Z

C12

n C12

1 + 1 + + 1 n

Cm m

g Cm m (m; g) = 1

k Z kg 0 (mod 12)

g

l = (m; g) l = 1

< g >=Cm


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1, 5, 7, 11

(mod 12)


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7

7 + 5 0 (mod 12)

7 5(mod 12)

5

C

F = G

7 6 42

5 6 30

42 30 = 12 0 (mod 12)


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GS

G4S


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C 1D

1E

12F

1G

1A

1H

12C


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G

k(T) (T; h)

k(T)

K, K

k(T)

KK t1K t

2 K

t1 K

t2 K z1, z2 Z

g(t1, t

1) = 2z1

g(t2, t

2) = 2z2

KK K K K K

G

G

(T; h)

k(T) 1 K 2 K


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KK

T

K, K T K={9; 2; 7} K ={2;3;7}

t1:= 9 t

1 := 3

g(t1; t

1) =440 2

t

112

440 2t112

= 2t

112

t112 = 2

3(9)12 = 21

z1= 1 Z

g(t; t) =h(t)

h(t)= 20

0 Z

g(tK; tL) = h(tL)

h(tK)= 2tLtK = 2z

g(tL; tK) =h(tK)

h(tL) = 2tKtL = 2z.

z Z zZ

g(tK; tL) = 2i

g(tL; tM) = 2j

g(tK; tM) = 2i+j.


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i, jZ i+j Z.

k(T)/

k(T)/

G

G4

r (T; h)

r : TT (r R+)

r := {(t1; t2)|t1, t2 T g(t1, t2) =r}

T T

D(r) = T

k(T)


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K, K k(T) KK

r R+ r(K) =K

k(T)

r:= 1 r(K) =K Kk(T) KK.

K, L k(T)

KL

r R

+

r(K) =L

1/r R

+

1/r(L) =K LK.

K,L,M k(T) KL LM r, qR+

r(K) =L q(L) =M rq(K) =M KM

k(T) k(T)/


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G2 2 2 2 2

G 2 22 4

G 44

k(T) t1 K

t2 L zZ g(t1, t2) = 2z KL

r R+ t2L t3 K g(t2, t3) =r LK

K( )K t1 K z Z r R+ t3 K

g(t1, t3) = 2zr

K(T)


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A


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y(x, t) =A sin(t kx)

A k x t

y t

x

T


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f

y

x

y= 0

1 = 2

2= 3

c=


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340m/s c= f

g(t1; t2)

g(t1; t2) Q+


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2 : 1, 3 : 2, 4 : 3

oktav kvint= kvart

(2 : 1) : (3 : 2) = (4 : 3)

C : 528Hz C: 264Hz F : 352Hz G: 396Hz

528 : 396 = 352 : 264

C

C

C

C(1)

32 G(3/2) G(3/2)

32 D(9/4)

C

D(9/8)


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H(243/128)


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22/3

5/22

2 3/5

23/5

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3 5/23

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24/3 5

26/3 5

n= p11 p22 p

mm


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d(n) =1(p1 1) +2(p2 1) + +m(pm 1) + 1, d(1) := 1

n= p d(n) = 1(p 1)+ 1 =p

m = n d(m) = d(n)

d(a: b) :=d(lkkt(a; b))


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C , E, G

d(4 : 5 : 6) = 9

d(8 : 9) = 8

Q+

f(q) =f(2zq) q Q+z Z

f(q) =f(1q

) q Q+

f(u) f(uv

) u, v lnko(u; v) = 1

f(u)< f(u+ 2) u


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C

G

2

2

6

k: Q+ Z

k(q) = 12

(max{u; v} 1) q := 2zuv

u, v u v

(t1, t2)

k(t1; t2) :=

k(g(t1; t2)) g(t1; t2) Q+

1

1

2

1

3

2

4

3

5

3

5

4

6

5

8

5

9

8

15

8

16

9

16

15

64

45

k(q)


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K := (t1; t2; . . . ; tn)

k(K) :=max{k(ti; tj)|1 i j n}

KK KK K( )K

k(K) =k(K)


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32

121

2

7=

531441

524288= 1.0136

x y a cos(t+)

a


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x(t) =a cos(1t) y(t) =b cos(2t)

12

= pq p q p

q

pq =

4

3


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12 : 9 = 8 : 6


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