1. II Acoustic Reality
II.5 (M Sept 25) The Euler Space and Tunings

2. The Euler Space and Tunings
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 c below middle c (132 Hz)
f = f0.2o.3q.5t
o, q, t rationals, i.e. fraction numbers p/r of integers, e.g. 3/4, -2/5
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)
132 =
log(2)

4. f = f0.2o/12 (o/12).log(2), o = integers (3/12).log(2)
o, q, t = 1, 0, 0

5. frequency ratios in 12-tempered tuning1 2 3 4 5 6 7 8 9 10 11 1
21/12
1/12
100
22/12
2/12
200
23/12
3/12
300
24/12
4/12
400
25/12
5/12
500
26/12 = √2
6/12
600
27/12
7/12
700
28/12
8/12
800
29/12
9/12
900
210/12
10/12
1000
211/12
11/12
1100

6. very old: frequency ratios in Pythagorean tuning (2-, 3-based)1 2 3 4 5 6 7 8 9 10 11 1
256/243 8 -5
90.225
9/8 -3 2
203.91
32/27 5
81/64 -6 4
407.82
4/3 -1
729/512 -9 6
611.73
3/2
128/81 7 -4
792.18
27/16 3
16/9 -2
996.09
243/128 -7

7. frequency ratios in just tuning (2-, 3-, 5-based)1 2 3 4 5 6 7 8 9 10 11

8. frequency ratios in Pythagorean tuning (2-, 3-based)1 2 3 4 5 6 7 8 9 10 11 1
256/243 8 -5
90.225
9/8 -3 2
203.91
32/27 5
81/64 -6 4
407.82
4/3 -1
729/512 -9 6
611.73
3/2
128/81 7 -4
792.18
27/16 3
16/9 -2
996.09
243/128 -7

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

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

11. a?

12. mean-tone tempered scalef g a b c’ f g a b a d g =
5/4
5/4
5/4
c ➡ d ➡ e → f ➡ g ➡ a ➡ b → c’

13. frequency ratios in mean-tone tempered scale1 2 3 4 5 6 7 8 9 10 11 1
21/12
1/12
100
√5/2 -1
1/2
23/12
3/12
5/4 -2
5/4 x 8/55/4
-1/4
26/12 = √2
6/12
51/4
1/4
28/12
8/12
53/4/2
3/4
210/12
10/12
55/4/4

14. b♭ b♭ pitch classes in just tuning 12 fifths – 7 octaves = fifth comma
= Pythagorean comma
= Ct
1 third (+2 octaves) – 4 fifths
= third comma
= syntonic comma
= Ct b♭

15. calculating and hearing commata
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]
fifth comma, Pythagorean comma
12 fifths – 7 octaves ~ (3/2)12 × (2/1)-7
= Ct
223.46/1200 =
440 Hz ⇒ Hz
third comma, syntonic comma
1 third (+2 octaves) – 4 fifths ~ 5/4 × (2/1)2 × (3/2)-4
= Ct
/ =
440 Hz ⇒ Hz

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

18. 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

19. Ÿ12 Ÿ pitch classes in 12-tempered tuning d = 5 x k +2 unique formula6 1 2 3 4 5 7 8 9 10 11 c g
Ÿ12 Ÿ
d = 5 x k +2
unique formula
that exchanges consonances
and dissonances of counterpoint!