1. II.3 Mental Reality
II.3.2 (M Sept 25) The Euler Space

2. The Euler Space De harmoniae veris principiis per speculum musicum
repraesentatis
(1773)
p.350
Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae
(1739)

3. frequency for middle c
f = f0.2o.3q.5t
o, q, t integers, i.e. numbers ...-2,-1,0,1,2,...
pitch(f) ~ log(f) = log(f0) + 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!
log(5)
log(3)
log(2)

4. log(5)
Euler space
log(3)
log(2)

5. pitch classes in just tuning

6. Gioseffo Zarlino (1517 - 1590): major and minor
pitch classes in just tuning
180o
Gioseffo Zarlino ( ): major and minor

7. third (or syntonic) comma

8. Big Problem!!! 440 Hz ⇒ 434.567 Hz 440 Hz ⇒ 446.003 Hz
calculating and hearing commata
third comma, syntonic comma
1 third (+2 octaves) – 4 fifths ~ 5/4 × (2/1)2 × (3/2)-4
= —21.51 Ct
/ =
440 Hz ⇒ Hz
fifth comma, Pythagorean comma
12 fifths – 7 octaves ~ (3/2)12 × (2/1)-7
= Ct
223.46/1200 =
Big Problem!!!
440 Hz ⇒ Hz

9. Solution (i/12).log(2), i integer (3/12).log(2)
f = f0.2o.3q.5t
pitch(f) = log(f0) + o.log(2) + q.log(3) +t.log(5)
also admit fractional exponents o, q, t = r/s, e.g. 6/5, -2/3
Solution
(3/12).log(2)
fractions also ok for independence of directions!
(i/12).log(2), i integer

11. pitch classes in 12-tempered tuning6 1 2 3 4 5 7 8 9 10 11 c g

12. 0 <—> 2 3 <—> 5 4 <—> 10 7 <—> 1 8 <—> 6
consonances <—> dissonances! 7 8 4 3 9 6 1 2 3 4 5 7 8 9 10 11
0 <—> 2
3 <—> 5
4 <—> 10
7 <—> 1
8 <—> 6
9 <—> 11
d = 5 ⨉ c + 2

13. pitch classes in 12-tempered tuning6 1 2 3 4 5 7 8 9 10 11 c g
d = 5 x k +2
unique formula
that exchanges consonances
and dissonances of counterpoint!