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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
32/23
3 5/23
24/32
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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