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S e r g iu s L e s k i S in at u s Pu t r a
                                0 8 14 14 0 8 8


             Tr i An g g a J iw a n d a n a
                            0 8 14 14 0 6 4
Pendahuluan
M a te m a ti k a d a n            m u sik su d ah
"b e r s a u d a r a " s e ja k z a m a n Y u n a n i
K u n o.        Pythagoras      (5 8 0 -5 0 0      SM )
s e o ra n g                 fi ls u f             d an
m a te m a ti k a w a n          te rk e n a l p a d a
zam an         Yu n a n i K u n o b e rs a m a
p a ra        m u ri d n y a          m e n e m u k an
bah w a        h a rm o n i       d a la m     m u sik
b e rk o re s p o n d e n s i                d e n g an
p e r b a n d i n g a n d u a b u a h b i la n g a n
b u la t . B i la k i t a m e m p u n y a i d u a
u ta s k a w a t y a n g d i re g a n g k a n
d e n g a n k e te g a n g a n y a n g s a m a ,
D i     z a m a n s e k a ra n g k i ta ti d a k
  t e r la lu        k e s u li t a n        u n tu k
  m e n g e r ja k a n      s e s u a tu    s e c a ra
  m a n u a l.      Cu ku p         m e n g e ta h u i
  b e b e ra p a    p e n g e ta h u a n      d asar
  y a n g s e la n ju t n y a d a p a t k i t a
  k e m b a n g k a n s e n d i ri .
 D i s i n i k i ta a k a n m e n g g u n a k a n
  p e ra n        m a te m a ti k a       se bag ai
  d a s a r p e n g e t a h u a n k i t a d a la m
  m e m b u a t s e b u a h la g u .
K ita misalkan :
N ot p e n u h d e n g an a,
N ot ½ d e n g an b,
N ot ¼ d e n g an c,
N o t 1 /8 d e n g a n d ,
N o t 1 /1 6 d e n g a n e
u n t u k m a s a la h i n i k i t a
  m e n g g u n a k a n b i r a m a 4 /4
D e n g an        m e m i s a lk a n   4 /4 = 1 ,
  m ak a       k i ta    d ap at     m e li h a t
  bah w a
a = 1 (a r t i n y a a d a 1 n o t p e n u h
  d a la m 1 b a r )
b + b = 1 (a r t i n y a a d a 2 n o t ½
  d a la m 1 b a r )
c + c + c + c = 1 ( a rti n y a a d a 4 n o t
  ¼ d a la m 1 b a r )
d + d + d + d + d + d + d + d = 1 (a r t i n y a
  a d a 8 n o t 1 /8 d a la m 1 b a r )
e+ e+ e+ e+ e+ e+ e+ e+ e+ e+ e+ e+ e+
a ta u
a= b+ b =    2b
b = c+ c =   2c
c = d+d =    2d
d = e+ e =   2e
Ja d i , k e t i k a k i t a a k a n
 m e m b u a t s e b u a h la g u d a la m
 1 b a r k i ta d a p a t m e n e n tu k a n
 b e ra p a b a n y a k n o t y a n g h a ru s
 k i t a m a s u k a n /d i b u a t
g a mb a r . 1 bar
P e r h a t i k a n g a m b a r -g a m b a r
 b e ri k u t :

 g a mb a r . a =1




 Ja d i , h a n y a a d a 1 n o t y a n g
 m u n g k i n k i t a b u a t d a la m 1
 b a r.
 Ga b r a = 2
   ma.       b




 Ja d i , h a n y a a d a 2 n o t y a n g
 m u n g k i n k i t a b u a t d a la m 1 b a r .

 Ga b r a = 2 = 4
   ma.       b    c




 Ja d i , h a n y a a d a 4 n o t y a n g
 m u n g k i n k i t a b u a t d a la m 1 b a r .
Ga b r a = 8
  ma.        d



 Ja d i , h a n y a a d a 8 n o t y a n g
 m u n g k i n k i t a b u a t d a la m 1
 b a r.

Ga b r a = 1e
  ma.       6




 Ja d i , h a n y a a d a 1 6 n o t y a n g
Ja d i k i t a s u d a h m e n g e t a h u i
  bah w a
a = b+ = 2
         b     b
b = c+ = 2
        c     c
c = d+ = 2
        d     d
d = ee = 2
       +      e
D a ri s i n i d a p a t k i ta tu ru n k a n
a = b+   b
= b + ( c + c)
= b + c + ( d + d)
= b+c+d +(e+e           )
I n i la h a d a la h s a la h s a t u c o n t o h
  p e n e ra p a n .
Ga b r c o n t o h p e n e r a p a n
  ma.




 S e la n ju t n y a k i t a a k a n
 m e n g g u n a k a n s o ft w a r e
 k o m p u te r u n tu k m e m u d a h k a n
 p e k e r ja a n k i t a .
D i s i n i k i ta a k a n m e n g g u n a k a n
           a p li k a s i g u i t a r p r o 5
B e b e ra p a       p e t u n ju k         p e n ti n g
  p e n g g u n a a n g u i ta r p ro
2. g a m b a r g i t a r , b e r fu n g s i u n t u k
  m e m u d ah k an m e n e m u k an n ad a
  yan g       in gin     d i b u a t te rm a s u k
  c h o r d y a n g a k a n d i m u a t d a la m
  b a r.
3. T e m p o ,       b e r fu n g s i          u n tu k
  m e n g a tu r s e b e ra p a c e p a t i ra m a
  la g u y a n g a k a n k i t a b u a t .
4. a d d       a       tra c k ,         b e r fu n g s i
  m e n am bah           i n s tru m e n          la i n ,
  m i s a ln y a p e r c u s s i o n / d r u m
S e t e la h       an d a       m e m bu at
g a m b a ra n          m u sik        yan g
d i i n g i n k a n , g a m b a ra n      in i
d ap at         m e n ja d i    p e d om an
a n d a b i la a n d a i n g i n s e r i u s
b e rm u s i k        a ta u       se bag ai
p en gh ibu r          k e ti k a      an d a
b e la ja r m a t e m a t i k a .
B e r i k u t a d a la h c o n t o h
p e m b u a ta n n o ta s i d a n s u a ra   K li k u n t u k
 D a ri b e n tu k m a te m a ti s a = 2 =
                                        b
  4 = 8 = 1e
   c     d    6
 An d a            d ap at           b e la ja r
  m e n g a ra n s e m e n                 d an
  m e m bu at           m u sik        se su ai
  d e n g a n s e le r a a n d a .



       Selamat mencoba dan bereksperimen !

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Pemodelan matematika sederhana dalam membuat sebuah lagu

  • 1. S e r g iu s L e s k i S in at u s Pu t r a 0 8 14 14 0 8 8 Tr i An g g a J iw a n d a n a 0 8 14 14 0 6 4
  • 2. Pendahuluan M a te m a ti k a d a n m u sik su d ah "b e r s a u d a r a " s e ja k z a m a n Y u n a n i K u n o. Pythagoras (5 8 0 -5 0 0 SM ) s e o ra n g fi ls u f d an m a te m a ti k a w a n te rk e n a l p a d a zam an Yu n a n i K u n o b e rs a m a p a ra m u ri d n y a m e n e m u k an bah w a h a rm o n i d a la m m u sik b e rk o re s p o n d e n s i d e n g an p e r b a n d i n g a n d u a b u a h b i la n g a n b u la t . B i la k i t a m e m p u n y a i d u a u ta s k a w a t y a n g d i re g a n g k a n d e n g a n k e te g a n g a n y a n g s a m a ,
  • 3. D i z a m a n s e k a ra n g k i ta ti d a k t e r la lu k e s u li t a n u n tu k m e n g e r ja k a n s e s u a tu s e c a ra m a n u a l. Cu ku p m e n g e ta h u i b e b e ra p a p e n g e ta h u a n d asar y a n g s e la n ju t n y a d a p a t k i t a k e m b a n g k a n s e n d i ri .  D i s i n i k i ta a k a n m e n g g u n a k a n p e ra n m a te m a ti k a se bag ai d a s a r p e n g e t a h u a n k i t a d a la m m e m b u a t s e b u a h la g u .
  • 4.
  • 5.
  • 6. K ita misalkan : N ot p e n u h d e n g an a, N ot ½ d e n g an b, N ot ¼ d e n g an c, N o t 1 /8 d e n g a n d , N o t 1 /1 6 d e n g a n e u n t u k m a s a la h i n i k i t a m e n g g u n a k a n b i r a m a 4 /4
  • 7. D e n g an m e m i s a lk a n 4 /4 = 1 , m ak a k i ta d ap at m e li h a t bah w a a = 1 (a r t i n y a a d a 1 n o t p e n u h d a la m 1 b a r ) b + b = 1 (a r t i n y a a d a 2 n o t ½ d a la m 1 b a r ) c + c + c + c = 1 ( a rti n y a a d a 4 n o t ¼ d a la m 1 b a r ) d + d + d + d + d + d + d + d = 1 (a r t i n y a a d a 8 n o t 1 /8 d a la m 1 b a r ) e+ e+ e+ e+ e+ e+ e+ e+ e+ e+ e+ e+ e+
  • 8. a ta u a= b+ b = 2b b = c+ c = 2c c = d+d = 2d d = e+ e = 2e
  • 9. Ja d i , k e t i k a k i t a a k a n m e m b u a t s e b u a h la g u d a la m 1 b a r k i ta d a p a t m e n e n tu k a n b e ra p a b a n y a k n o t y a n g h a ru s k i t a m a s u k a n /d i b u a t g a mb a r . 1 bar
  • 10. P e r h a t i k a n g a m b a r -g a m b a r b e ri k u t :  g a mb a r . a =1 Ja d i , h a n y a a d a 1 n o t y a n g m u n g k i n k i t a b u a t d a la m 1 b a r.
  • 11.  Ga b r a = 2 ma. b Ja d i , h a n y a a d a 2 n o t y a n g m u n g k i n k i t a b u a t d a la m 1 b a r .  Ga b r a = 2 = 4 ma. b c Ja d i , h a n y a a d a 4 n o t y a n g m u n g k i n k i t a b u a t d a la m 1 b a r .
  • 12. Ga b r a = 8 ma. d Ja d i , h a n y a a d a 8 n o t y a n g m u n g k i n k i t a b u a t d a la m 1 b a r. Ga b r a = 1e ma. 6 Ja d i , h a n y a a d a 1 6 n o t y a n g
  • 13. Ja d i k i t a s u d a h m e n g e t a h u i bah w a a = b+ = 2 b b b = c+ = 2 c c c = d+ = 2 d d d = ee = 2 + e D a ri s i n i d a p a t k i ta tu ru n k a n a = b+ b = b + ( c + c) = b + c + ( d + d) = b+c+d +(e+e ) I n i la h a d a la h s a la h s a t u c o n t o h p e n e ra p a n .
  • 14. Ga b r c o n t o h p e n e r a p a n ma. S e la n ju t n y a k i t a a k a n m e n g g u n a k a n s o ft w a r e k o m p u te r u n tu k m e m u d a h k a n p e k e r ja a n k i t a .
  • 15. D i s i n i k i ta a k a n m e n g g u n a k a n a p li k a s i g u i t a r p r o 5
  • 16. B e b e ra p a p e t u n ju k p e n ti n g p e n g g u n a a n g u i ta r p ro 2. g a m b a r g i t a r , b e r fu n g s i u n t u k m e m u d ah k an m e n e m u k an n ad a yan g in gin d i b u a t te rm a s u k c h o r d y a n g a k a n d i m u a t d a la m b a r. 3. T e m p o , b e r fu n g s i u n tu k m e n g a tu r s e b e ra p a c e p a t i ra m a la g u y a n g a k a n k i t a b u a t . 4. a d d a tra c k , b e r fu n g s i m e n am bah i n s tru m e n la i n , m i s a ln y a p e r c u s s i o n / d r u m
  • 17. S e t e la h an d a m e m bu at g a m b a ra n m u sik yan g d i i n g i n k a n , g a m b a ra n in i d ap at m e n ja d i p e d om an a n d a b i la a n d a i n g i n s e r i u s b e rm u s i k a ta u se bag ai p en gh ibu r k e ti k a an d a b e la ja r m a t e m a t i k a .
  • 18. B e r i k u t a d a la h c o n t o h p e m b u a ta n n o ta s i d a n s u a ra K li k u n t u k
  • 19.  D a ri b e n tu k m a te m a ti s a = 2 = b 4 = 8 = 1e c d 6  An d a d ap at b e la ja r m e n g a ra n s e m e n d an m e m bu at m u sik se su ai d e n g a n s e le r a a n d a . Selamat mencoba dan bereksperimen !