1. II.3 Mental Reality
II.3.1 (F Sept 22) Beyond physics and psychology

2. René Descartes: compendium musicae (1618)
psychological foundation
of music:
8 axiomatic rules
e.g.
music must be simple
to please the soul...

3. Ludwig van Beethoven op. 57 „Appassionata“
Vladimir Horowitz
Glenn Gould

4. M.C. Escher:
Balcony

5. Tempo (e.g. M.M. ♩ = 120)
T(E) = (de/dE)-1 [♩ /min]
slope E e
e(E)

6. John Cage: ASLSP (http://www.aslsp.org/de)

7. notation: white keys = C-major scale
/ / /8 1/16 1/32

8. tuning!!!
MusicalInstrumentDigitalInterface MIDI: pitch symbols 0,1,2,

9. modern: 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

10. 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

11. 45/32 =
classical: frequency ratios in just tuning 1 2 3 4 5 6 7 8 9 10 11

12. MUTABOR by Rudolf Wille

13. ? Leonhard Euler‘s gradus suavitatis function
Plomp & Levelt
1965
10/
interval
(2e.3f.5g) = 1 + (2-1)|e| + (3-1)|f| + (5-1)|g|
= 1 + |e| + 2|f| + 4|g|
Euler‘s substitution theory ?
counterpoint