II.3 Mental Reality II.3.2 (M Sept 25) 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 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)

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

pitch classes in just tuning

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

third (or syntonic) comma

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 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

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

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

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

pitch classes in 12-tempered tuning 6 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!