guerino mazzola (fall 2015 © ): honors seminar ii.3mental reality ii.3.2 (w sept 30) the euler...
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Gue
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II.3II.3 Mental RealityMental Reality
II.3.2 II.3.2 (W Sept 30) (W Sept 30) The Euler SpaceThe Euler Space
Gue
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The Euler SpaceThe Euler Space
De harmoniae veris principiis per speculum musicum repraesentatis(1773)p.350
De harmoniae veris principiis per speculum musicum repraesentatis(1773)p.350
Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae(1739)
Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae(1739)
Gue
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Gue
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ar f = ff = f00.2.2oo.3.3qq.5.5t t
log(2)log(2)log(3)log(3)
log(5)log(5)
frequency for middle cfrequency for middle c
o, q, t integers, o, q, t integers, i.e. numbers ...-2,-1,0,1,2,... i.e. numbers ...-2,-1,0,1,2,...
pitch(f) ~pitch(f) ~ log(f) = log(f) = log(flog(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!
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log(2)log(2)log(3)log(3)
log(5)log(5)
Euler spaceEuler space
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pitch classes in just tuningpitch classes in just tuning
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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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a?
third (or syntonic) comma third (or syntonic) comma
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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 440 Hz 446.003 Hz ⇒
440 Hz 434.567 Hz ⇒
Big Pro
blem!!!
Big Pro
blem!!!
Gue
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Gue
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(i/12).log(2), i integer (i/12).log(2), i integer (3/12).log(2)(3/12).log(2)
fractions also ok for independence of directions!fractions also ok for independence of directions!
f = ff = f00.2.2oo.3.3qq.5.5tt
pitch(f) = log(fpitch(f) = log(f00) + o.) + o.log(2)log(2) + q. + q.log(3)log(3) +t.+t.log(5)log(5)
also admit fractional exponents o, q, t = r/s, e.g. 6/5, -2/3 also admit fractional exponents o, q, t = r/s, e.g. 6/5, -2/3
Solutio
n
Solutio
n
Gue
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Gue
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Gue
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Gue
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6
0011
22
33
44
5577
88
99
1010
1111
33 4400
99
88
77
consonances <—> dissonances!consonances <—> dissonances!
d = 5 c + 2⨉d = 5 c + 2⨉
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pitch classes in 12-tempered tuningpitch classes in 12-tempered tuning
6
0011
22
33
44
5577
88
99
1010
1111
cc
ggdd = 5 = 5 xx kk +2 +2
unique formulaunique formulathat exchanges that exchanges consonancesconsonances
and and dissonancesdissonances of counterpoint! of counterpoint!