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II.3 Mental Reality II.3.2 (Thu Sept 19) The Euler Space

II.3 Mental Reality II.3.2 (Thu Sept 19) The Euler Space. 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). frequency for middle c.

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II.3 Mental Reality II.3.2 (Thu Sept 19) The Euler Space

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  1. II.3 Mental Reality II.3.2 (Thu Sept 19) 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 log(5) log(3) log(2) 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!

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

  5. pitch classes in just tuning

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

  7. third (or syntonic) comma a?

  8. 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 2-21.51/1200 = 0.987652 440 Hz ⇒ 434.567 Hz fifth comma, Pythagorean comma 12 fifths – 7 octaves ~ (3/2)12× (2/1)-7 = 23.46 Ct 223.46/1200 = 1.01364 Big Problem!!! 440 Hz ⇒ 446.003 Hz

  9. (3/12).log(2) fractions also ok for independence of directions! 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 (i/12).log(2), i integer

  10. 6 0 11 1 10 2 9 3 8 4 7 5 pitch classes in 12-tempered tuning

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

  12. 6 0 11 1 10 2 c 9 3 g 8 4 7 5 pitch classes in 12-tempered tuning d = 5 xk +2 unique formula that exchanges consonances and dissonances of counterpoint!

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