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

René Descartes: compendium musicae (1618) 1596-1650 psychological foundation of music: 8 axiomatic rules e.g. music must be simple to please the soul...

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

M.C. Escher: Balcony

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

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

notation: white keys = C-major scale 1 1/2 1/4 1/8 1/16 1/32

tuning!!! MusicalInstrumentDigitalInterface MIDI: pitch symbols 0,1,2,... 127

modern: frequency ratios in 12-tempered tuning 1 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

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 5 294.135 81/64 -6 4 407.82 4/3 -1 498.045 729/512 -9 6 611.73 3/2 701.955 128/81 7 -4 792.18 27/16 3 905.865 16/9 -2 996.09 243/128 -7 1109.78

45/32 = 2-5.32.51 classical: frequency ratios in just tuning 1 2 3 4 5 6 7 8 9 10 11

MUTABOR by Rudolf Wille

? Leonhard Euler‘s gradus suavitatis function Plomp & Levelt 1965 10/ interval 1707-1783 (2e.3f.5g) = 1 + (2-1)|e| + (3-1)|f| + (5-1)|g| = 1 + |e| + 2|f| + 4|g| Euler‘s substitution theory ? counterpoint 0 1 2 3 4 5 6 7 8 9 10 11