ΜΟΥΣΙΚΑ ΔΙΑΣΤΗΜΑΤΑ Κ ΜΑΘΗΜΑΤΙΚΑ
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. . . : 45/06
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:
. : 2009-2010
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...4
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7
..16
18
.21
Zarlino..22
..23
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24
27
.28
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(-)..36
42
o ) 45
Fibonacci. 49
Fourier51
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2/3 , 3/2 (
5
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3/4 , 4/3 (
4
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2/3 : 3/4 = 8/9 ( 5 4)
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:
, .
27/32 , 3/4 : 8/9
. 243/256 ,
3/4 (8/9 *8/9=64/81), : 3/4 :
64/81= 243/256 .
,
8 5. 4
5 8 .
: 2
8 5 5
...
2 3, :
1/1=20*3
0, 8/9=2
3*3
-2, 64/81=2
6*3
-4, 3/4=2
-2*3
1, 2/3=2
1*3
-1,
16/27=24*3
-3, 128/243=2
7*3
-5, 1/2=2
-1*3
0, 256/243=28*3-5.
, , 1,
2, 3.6
6 , . 6
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12
7:
cents
2/1 1200
( ) 3/2 701,95
( ) 4/3 498,04
9/8 203,91
65536/59049 180,45
( ) 2187/2048 113,68
( ) 256/243 90,22
531441/524288 23,46
cent 1/100
1200 cents.
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1/8 9/8 81/64 4/3 3/2 27/16 243/128 2
9/8 9/8 256/243 9/8 9/8 9/8 256/243 9/8
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: 1, 2, 4, 8 1, 3, 9, 27.
, (1, 2, 3, 4, 8, 9, 27)
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() .
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. : 3/2*4/3=2
()
( ). : 3/2*2/1=3
.
. : 3/2:4/3=9/8
, ,
. :
(9/8)6 >2. , 12 5
7 8: (3/2)12=129,746,
27=128.
(=531441/524288)25
13 , , . 17
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. : 9/8*9/8*16/15>3/2
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(9/8) (16/15)
(4/3) (10/9)
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(9/8*10/9*16/15=1440/1080=(360*4)/(360*3)=4/3)
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. 2/1 , 3/2 , 4/3
5/4 .
9/8
.
1 9/8 5/4 4/3 3/2 27/16 15/8 2
9/8 10/9 16/15 9/8 9/8 10/9 16/15
, 10/9
-
23
16/15.
81/80.
Zarlino
Zarlino (1517-
1590, ),
,
.
. just
intonation.
1 9/8 5/4 4/3 3/2 5/3 15/8 2
9/8 10/9 16/15 9/8 10/9 9/8 16/15
-
24
, 2 . . .,
30 , 12 6
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: 12+12+6=30,
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9216 - 8192 - 7776 - 6912 - 6144 - 5832 - 5184 - 4608
:
9216/8132=9/8, 8192/7776=256/243, 7776/6912=9/8, 6912/6144=9/8,
6144/5832=256/243, 5832/5184=9/8 5184/4608=9/8.
(9/8)
(256/243). , ,
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, - 12,
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7. 12
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7 .
, :
12 9 7
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27
,
12+9+7=28 . , 28 , (
) 498 cents,
498/2817,7857 cents. ,
12*17.7857213 cents.
10 cents . ,
9*17,7857160 cents 7*17,7857125 cents.
, 12,9 7
, . ,
, :
1 9/8 27/22 4/3 3/2 18/11 16/9 2
, 9/8= 1,125 = 204 cents , 12/111,0909151 cents 88/81 1,08642 143
cents. ,
, ,
204-151-143, 213-160-
125. ,
Zarlino,
:
1 9/8 5/4 4/3 3/2 5/3 15/8 2
-
28
, : - = 5/4 Zarlino
= 27/22 1,2272 < 1,25 = 5/4,
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1881-85, : . ,
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512 256 Hz.
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1
8/9
22/27
3/4
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18:
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1 8/9 81/100 3/4 2/3 16/27 27/50 1/2
, 9/8, 800/729
27/25. :
18
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-
31
800/729 = (9*8*800)/(9*8*729) = 9/8*(8*10*80)/9*9*81) = 9/8*802/81
2=
9/8*(80/81)2
= 9/8 : (81/80)2
, 27/25 = 27/24 * 24/25 = 9/8 * 24/25 = 9/8 : 25/24
)2 2 2 2 2 2 2 3
1 4/9 81/200 3/8 1/3 8/27 27/100 1/4
,
.
1 1 1 1 1 1 1 2
1 9/8 100/81 4/3 3/2 27/16 50/27 2
2 2 2 2 2 2 2 3
2 9/4 200/81 8/3 3/1 27/8 100/27 4
1888
.
. , 81/80,
. , 25/24, (
24/25 ) 1/3
(25/24 = 1,0416666.71 cents).19 ,
19
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-
32
,
. : 9/8 = 1,125 =
204 cents, 800/729 = 1,09739 161 cents 27/25 = 1,08 133 cents.
(204-151-143).
:
5/4 : 81/80 (-) 5/4 : (81/80 *25/24) (-)
5/4 * 81/80 : 25/24 (-) 5/4 * 81/80 (-)
4/3 (-) 4/3 * (81/80)2 (-) 4/3 * 25/24 (-)
3/2 (-) 3/2 : (81/80)2 (-) 3/2 : 25/24 (-)
5/3 * 81/80 (-) 5/3 : 81/80 (-)
5/3 * 81/80 : 25/24 (-) 5/3 : 981/80 * 25/24) (-)
-
33
15/8 : 81/80 (-) 15/8 : (81/80*25/24) (-)
15/8 * 81/80 : 25/24 (-)20
, 9/8, 5/4, 4/3,
3/2, 5/3 15/8. , ,
.
Zarlino . ,
, ,
.
Zarlino
.
20
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-
34
21
Cents Cents Cents
9/8 = 1.125 204 12/111.091 151 88/811.086 143
9/8 = 1.125 204 800/7291.097 161 27/25=1.08 133
Zarlino 9/8 = 1.125 204 10/91.111 182 16/151.067 112
36
. , 6
, 5 4.
:
6 + 5 + 4 = 15 ,
21
, ., . 40
-
35
498 / 15 = 33,2 cents. 33,2 * 6 = 199,2 cents,
33,2 * 5 = 166 cents , 33,2 * 4 = 132,8 cents. ,
, .
, ,
72,
.
.
71 ,
:
256/243 5,29
18/17 5/82
17/16 6,18
16/15 6,59
2187/2048 6,71
15/14 7,00
14/13 7,53
13/12 8,18
12/11 8,88
11/10 9,71
-
36
65536/59049 10,59
10/9 10,71
512/459 11,12
272/243 11,47
9/8 12,00
:
12 12 5,29 (
29,29), :
, 12 9,71 7,53 ( 29,24)
12 8,88 8,88 ( 29,76)
12 -10,71 6,59 ( 29,30)
10,71 9,71 8,88 ( 29,30) ...
,
: 12 9,71 7,53
12 -10 7,5. , , 12 -12 -
5,5. 72 ,
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12 12 6, .
-
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9/8 12
11/10 ~10
14/13 ~7,5 ( 8)
. 2187/2048 ~6,5 ( 6)
. 256/243 ~5,5 ( 6)
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(
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(2048/2025).
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: = k *
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f1, f2 f2>f1. k
.
fa / fb fx / fy .,
. : (fa/fb)(fx/fy) = fafx / fbfy
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(fa*fy/fb*fx)
-
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a R*
: (fa/fb)a
*
: fa/fb37
k (
),
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:
k
12 .................
53 .................
68 ................. ,
72 .................
301 ...............Savart
665 ...............Delfi unit
1200..cents
cents. , , 1200 cents.
200 cents 100. (6
37
25 / , . 545
-
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x 200 = 1200) 5 (-, -, -, -, -)
2 (-, -).
12
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(5/4) , 5/4 * 5/4 * 5/4
= 53/43 = 125/64, 2/1, 128/64. ,
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-
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312/212 = 531441/4096 = 129.746, 27/17 = 128
524288/4096.
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, .38
Fibonacci
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.
musica
quadrivium, (,
, ) (Eli Maor 1998).
Fibonacci
.
1170 1240.
Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
38
http://panadam.wordpress.com/2008/04/29/welltempered/
-
51
89, 144, 233, 377, 610, 987, 1597, 2584, 4181, ... (
).
,
, , =1.618033989.
1/= 0.618033989, : 1/=-
1. 1/
.
Fibonacci f(1) = f(2) = 1 , f(n+1) = f(n) + f(n-
1),
. ,
.
:
.
, .
Leonardo Da Vinci Mona Lisa
.
Mozart ,
, ,
.
ela Bartok Le Corbusier
.
-
52
.
Fourier
.
Euler, o D'Alembert Langrange
. Daniel Bernoulli
,
Fourier 1822
"Theorie analytique de la chaleur".
Fourier :
1. f, g
f +g .
2. f , f(x + k*T) = f(x), =0,1,2...
,
.
, 3
4 5 2 .
-
53
()
. Furier, f
.
19
Fourier
.
von Helmholtz Fourier.
Helmholtz .
.
sinx, 1/2sin2x, 1/3sin3x. (
Science & Music)
.
-
54
3 '
. 39
()
(29 1922 - 4 2001)
20 ,
Iannis Xenakis.
,
,
, , .
,
, ,
1950 1960.
39
http://www.telemath.gr/mathematical_articles/mathematical_articles_music/maths_articles_music.php
-
55
,
:
glissanto,
, .
,
.
, ,
.40
, .
, , ,
:
.
1960,
, ( )
Musiques Formelles.
- ,
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( )
40
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-
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.41
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,
1960 ( )
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1970,
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59
, , , ,
2005
, , University studio Press,
1995
, 25 /
,
(Finale 2008),
, 2008
,
, , 1994
,
,
, .
.,
., ,
Fagotto, 1996
-
60
,
,
, ,
, Montpellier 2008
, , ,
2006
http://www.oud.gr/music_arabe_gr.htm
http://panadam.wordpress.com/2008/04/29/welltempered/
http://www.telemath.gr/mathematical_articles/mathematical_articles_mu
sic/maths_articles_music.php