guerino mazzola (fall 2015 ): introduction to music technology iiacoustic reality ii.6 (m sept 30)...
Post on 18-Jan-2018
219 Views
Preview:
DESCRIPTION
TRANSCRIPT
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
IIII Acoustic RealityAcoustic Reality
II.6 II.6 (M Sept 30) (M Sept 30) The Euler Space and TuningsThe Euler Space and Tunings
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
De harmoniae veris principiis per speculum musicum repraesentatis(1773)p.350
Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae(1739)
The Euler Space and TuningsThe Euler Space and Tunings
1707-17831707-1783
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy f = ff = f00.2.2oo.3.3qq.5.5t t
log(2)log(2)log(3)log(3)
log(5)log(5)
frequency for c below middle c (132 Hz)frequency for c below middle c (132 Hz)
o, q, t rationals, o, q, t rationals, i.e. fraction numbers p/r of integers, e.g. 3/4, -2/5 i.e. fraction numbers p/r of integers, e.g. 3/4, -2/5
pitch(f) ~pitch(f) ~ log(f) = log(flog(f) = log(f00) + o.) + o.log(2)log(2) + q. + q.log(3)log(3) +t.+t.log(5)log(5) ~ o.~ o.log(2)log(2) + q. + q.log(3)log(3) +t.+t.log(5)log(5)
o, q, t are o, q, t are uniqueunique for each f for each f prime number factorization!prime number factorization!
132 = 440.2 132 = 440.2 -1-1. 3 . 5 . 3 . 5 -1-1
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
(o/12).log(2), o = integers (o/12).log(2), o = integers f = ff = f00.2.2o/12o/12
(3/12).log(2)(3/12).log(2)
o, q, t = 1, 0, 0o, q, t = 1, 0, 0
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
frequency ratios in 12-tempered tuningfrequency ratios in 12-tempered tuning
0011223344556677889910101111
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
very old: frequency ratios in Pythagorean tuning (2-, 3-based)very old: frequency ratios in Pythagorean tuning (2-, 3-based)
0011223344556677889910101111
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
frequency ratios in just tuning (2-, 3-, 5-based)frequency ratios in just tuning (2-, 3-, 5-based)
0011223344556677889910101111
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
frequency ratios in Pythagorean tuning (2-, 3-based)frequency ratios in Pythagorean tuning (2-, 3-based)
0011223344556677889910101111
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
log(2)log(2)log(3)log(3)
log(5)log(5)
Euler spaceEuler space
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
6
00 1122
33
445577
88
991010
1111
33 4400
99
88
77
consonances <—> dissonances!consonances <—> dissonances!
d = 5 c + 2⨉d = 5 c + 2⨉
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
a?
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
5/4
g
ad
f
g
a
b
mean-tone tempered scalemean-tone tempered scale
c ➡ d ➡ e → f ➡ g ➡ a ➡ b → c’
f
g
a
b
5/4 5/4
=
c’
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
frequency ratios in mean-tone tempered scalefrequency ratios in mean-tone tempered scale
0011223344556677889910101111
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
Gioseffo Zarlino (Gioseffo Zarlino (1517 - 15901517 - 1590): major and minor): major and minor
180o
pitch classes in just tuningpitch classes in just tuning
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
pitch classes in just tuningpitch classes in just tuning
b♭
b♭
1 third (+2 octaves) – 4 fifths= third comma= syntonic comma= -21.51 Ct
12 fifths – 7 octaves= fifth comma= Pythagorean comma= 23.46 Ct
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
calculating and hearing commatacalculating and hearing commata
third comma, syntonic comma1 third (+2 octaves) – 4 fifths ~ 5/4 × (2/1)2 × (3/2)-4
= -21.51 Ct2-21.51/1200 = 0.987652
fifth comma, Pythagorean comma12 fifths – 7 octaves ~ (3/2)12 × (2/1)-7
= 23.46 Ct223.46/1200 = 1.01364
pitch(f) = 1200/log(2) × log(f) + const. [Ct], Take log-basis = 2:pitch(f) = 1200 × log2(f) + const. [Ct]
pitch(f/g) = 1200 × log2(f/g) [Ct]
f/g = 2pitch(f/g)/1200 [Hz]
440 Hz 446.003 Hz ⇒
440 Hz 434.567 Hz ⇒
Gue
rino
Maz
zola
(Fal
l 201
5G
uerin
o M
azzo
la (F
all 2
015©©
): In
trodu
ctio
n to
Mus
ic T
echn
olog
y ):
Intro
duct
ion
to M
usic
Tec
hnol
ogy
pitch classes in 12-tempered tuningpitch classes in 12-tempered tuning
6
00 1122
33
445577
88
991010
1111
cc
gg
Ÿ12
Ÿdd = 5 = 5 xx kk +2 +2
unique formulaunique formulathat exchanges that exchanges consonancesconsonances
and and dissonancesdissonances of counterpoint! of counterpoint!
top related