guerino mazzola (fall 2015 ): introduction to music technology iiacoustic reality ii.6 (m sept 30)...
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Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology f = f 0.2 o.3 q.5 t log(2) log(3) log(5) frequency for c below middle c (132 Hz) o, q, t rationals, i.e. fraction numbers p/r of integers, e.g. 3/4, -2/5 pitch(f) ~ log(f) = log(f 0 ) + o.log(2) + q.log(3) +t.log(5) ~ o.log(2) + q.log(3) +t.log(5) ~ o.log(2) + q.log(3) +t.log(5) o, q, t are unique for each f prime number factorization! 132 =TRANSCRIPT
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IIII Acoustic RealityAcoustic Reality
II.6 II.6 (M Sept 30) (M Sept 30) The Euler Space and TuningsThe Euler Space and Tunings
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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
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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
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(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
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frequency ratios in 12-tempered tuningfrequency ratios in 12-tempered tuning
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very old: frequency ratios in Pythagorean tuning (2-, 3-based)very old: frequency ratios in Pythagorean tuning (2-, 3-based)
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frequency ratios in just tuning (2-, 3-, 5-based)frequency ratios in just tuning (2-, 3-, 5-based)
0011223344556677889910101111
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frequency ratios in Pythagorean tuning (2-, 3-based)frequency ratios in Pythagorean tuning (2-, 3-based)
0011223344556677889910101111
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log(2)log(2)log(3)log(3)
log(5)log(5)
Euler spaceEuler space
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6
00 1122
33
445577
88
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33 4400
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88
77
consonances <—> dissonances!consonances <—> dissonances!
d = 5 c + 2⨉d = 5 c + 2⨉
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a?
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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’
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frequency ratios in mean-tone tempered scalefrequency ratios in mean-tone tempered scale
0011223344556677889910101111
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Gioseffo Zarlino (Gioseffo Zarlino (1517 - 15901517 - 1590): major and minor): major and minor
180o
pitch classes in just tuningpitch classes in just tuning
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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
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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 ⇒
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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!