1. Guerino Mazzola (Fall 2015 © ): Honors Seminar II.3Mental Reality II.3.2 (W Sept 30) The Euler Space

2. Guerino Mazzola (Fall 2015 © ): Honors Seminar The Euler Space De harmoniae veris principiis per speculum musicum repraesentatis (1773) p.350 De harmoniae veris principiis per speculum musicum repraesentatis (1773) p.350 Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae (1739) Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae (1739)

3. Guerino Mazzola (Fall 2015 © ): Honors Seminar f = f 0.2 o.3 q.5 t log(2) log(3) log(5) frequency for middle c o, q, t integers, i.e. numbers...-2,-1,0,1,2,... pitch(f) ~ log(f) = log(f 0 ) + o.log(2) + q.log(3) +t.log(5) ~ 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. Guerino Mazzola (Fall 2015 © ): Honors Seminar log(2) log(3) log(5) Euler space

5. Guerino Mazzola (Fall 2015 © ): Honors Seminar pitch classes in just tuning

6. Guerino Mazzola (Fall 2015 © ): Honors Seminar Gioseffo Zarlino ( 1517 - 1590 ): major and minor 180 o pitch classes in just tuning

7. Guerino Mazzola (Fall 2015 © ): Honors Seminar a? third (or syntonic) comma

8. Guerino Mazzola (Fall 2015 © ): Honors Seminar 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 fifth comma, Pythagorean comma 12 fifths – 7 octaves ~ (3/2) 12 × (2/1) -7 = 23.46 Ct 2 23.46/1200 = 1.01364 440 Hz ⇒ 446.003 Hz 440 Hz ⇒ 434.567 Hz Big Problem!!!

9. Guerino Mazzola (Fall 2015 © ): Honors Seminar (i/12).log(2), i integer (3/12).log(2) fractions also ok for independence of directions! f = f 0.2 o.3 q.5 t pitch(f) = log(f 0 ) + 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

10. Guerino Mazzola (Fall 2015 © ): Honors Seminar

11. 601 2 3 4 57 8 9 10 11 3 4 0 9 8 7 consonances dissonances! d = 5 c + 2

12. Guerino Mazzola (Fall 2015 © ): Honors Seminar pitch classes in 12-tempered tuning 601 2 3 4 57 8 9 10 11 c g d = 5 x k +2 unique formula that exchanges consonances and dissonances of counterpoint!