Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology IIAcoustic Reality II.6 (M Sept 30) The Euler Space and Tunings

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology 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) The Euler Space and Tunings

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology f = f 0.2 o.3 q.5 t log(2) log(3) log(5) frequency for c below middle c (132 Hz) o, q, t rationals, i.e. fraction numbers p/r of integers, e.g. 3/4, -2/5 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! 132 =

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology (o/12).log(2), o = integers f = f 0.2 o/12 (3/12).log(2) o, q, t = 1, 0, 0

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology frequency ratios in 12-tempered tuning

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology very old: frequency ratios in Pythagorean tuning (2-, 3-based)

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology frequency ratios in just tuning (2-, 3-, 5-based)

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology frequency ratios in Pythagorean tuning (2-, 3-based)

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology log(2) log(3)log(5) Euler space

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology

consonances dissonances! d = 5 c + 2

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology a?

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology 5/4 g a d f g a b mean-tone tempered scale c ➡ d ➡ e → f ➡ g ➡ a ➡ b → c’ f g a b 5/4 = c’

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology frequency ratios in mean-tone tempered scale

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology Gioseffo Zarlino ( ): major and minor 180 o pitch classes in just tuning

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology pitch classes in just tuning b♭b♭ b♭b♭ 1 third (+2 octaves) – 4 fifths = third comma = syntonic comma = Ct 12 fifths – 7 octaves = fifth comma = Pythagorean comma = Ct

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology calculating and hearing commata third comma, syntonic comma 1 third (+2 octaves) – 4 fifths ~ 5/4 × (2/1) 2 × (3/2) -4 = Ct /1200 = fifth comma, Pythagorean comma 12 fifths – 7 octaves ~ (3/2) 12 × (2/1) -7 = Ct /1200 = pitch(f) = 1200/log(2) × log(f) + const. [Ct], Take log-basis = 2: pitch(f) = 1200 × log 2 (f) + const. [Ct] pitch(f/g) = 1200 × log 2 (f/g) [Ct] f/g = 2 pitch(f/g)/1200 [Hz] pitch(f) = 1200/log(2) × log(f) + const. [Ct], Take log-basis = 2: pitch(f) = 1200 × log 2 (f) + const. [Ct] pitch(f/g) = 1200 × log 2 (f/g) [Ct] f/g = 2 pitch(f/g)/1200 [Hz] 440 Hz ⇒ Hz 440 Hz ⇒ Hz

Guerino Mazzola (Fall 2015 © ): Introduction to Music Technology pitch classes in 12-tempered tuning c g Ÿ 12 Ÿ d = 5 x k +2 unique formula that exchanges consonances and dissonances of counterpoint!