Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler...
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Transcript of Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler...
![Page 1: Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfda1a28abf838cb01bd/html5/thumbnails/1.jpg)
Gue
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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
![Page 2: Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfda1a28abf838cb01bd/html5/thumbnails/2.jpg)
Gue
rino
Maz
zola
(F
all 2
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Gue
rino
Maz
zola
(F
all 2
015©©
): H
onor
s S
emin
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): H
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emin
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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)
![Page 3: Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfda1a28abf838cb01bd/html5/thumbnails/3.jpg)
Gue
rino
Maz
zola
(F
all 2
015
Gue
rino
Maz
zola
(F
all 2
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): H
onor
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emin
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): H
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emin
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!
![Page 4: Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfda1a28abf838cb01bd/html5/thumbnails/4.jpg)
Gue
rino
Maz
zola
(F
all 2
015
Gue
rino
Maz
zola
(F
all 2
015©©
): H
onor
s S
emin
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): H
onor
s S
emin
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log(2)log(2)log(3)log(3)
log(5)log(5)
Euler spaceEuler space
![Page 5: Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfda1a28abf838cb01bd/html5/thumbnails/5.jpg)
Gue
rino
Maz
zola
(F
all 2
015
Gue
rino
Maz
zola
(F
all 2
015©©
): H
onor
s S
emin
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): H
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s S
emin
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pitch classes in just tuningpitch classes in just tuning
![Page 6: Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfda1a28abf838cb01bd/html5/thumbnails/6.jpg)
Gue
rino
Maz
zola
(F
all 2
015
Gue
rino
Maz
zola
(F
all 2
015©©
): H
onor
s S
emin
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): H
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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
![Page 7: Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfda1a28abf838cb01bd/html5/thumbnails/7.jpg)
Gue
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Maz
zola
(F
all 2
015
Gue
rino
Maz
zola
(F
all 2
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a?
third (or syntonic) comma third (or syntonic) comma
![Page 8: Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfda1a28abf838cb01bd/html5/thumbnails/8.jpg)
Gue
rino
Maz
zola
(F
all 2
015
Gue
rino
Maz
zola
(F
all 2
015©©
): H
onor
s S
emin
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): H
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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!!!
![Page 9: Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfda1a28abf838cb01bd/html5/thumbnails/9.jpg)
Gue
rino
Maz
zola
(F
all 2
015
Gue
rino
Maz
zola
(F
all 2
015©©
): H
onor
s S
emin
ar
): H
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s S
emin
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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
![Page 10: Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfda1a28abf838cb01bd/html5/thumbnails/10.jpg)
Gue
rino
Maz
zola
(F
all 2
015
Gue
rino
Maz
zola
(F
all 2
015©©
): H
onor
s S
emin
ar
): H
onor
s S
emin
ar
![Page 11: Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfda1a28abf838cb01bd/html5/thumbnails/11.jpg)
Gue
rino
Maz
zola
(F
all 2
015
Gue
rino
Maz
zola
(F
all 2
015©©
): H
onor
s S
emin
ar
): H
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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⨉
![Page 12: Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space.](https://reader036.fdocuments.in/reader036/viewer/2022062409/5697bfda1a28abf838cb01bd/html5/thumbnails/12.jpg)
Gue
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Maz
zola
(F
all 2
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Gue
rino
Maz
zola
(F
all 2
015©©
): H
onor
s S
emin
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): H
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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!