II Acoustic Reality II.5 (M Sept 25) The Euler Space and Tunings

The Euler Space and Tunings 1707-1783 De harmoniae veris principiis per speculum musicum repraesentatis (1773) p.350 Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae (1739)

frequency for c below middle c (132 Hz) f = f0.2o.3q.5t o, q, t rationals, i.e. fraction numbers p/r of integers, e.g. 3/4, -2/5 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) 132 = 440.2 -1. 3 . 5 -1 log(2)

f = f0.2o/12 (o/12).log(2), o = integers (3/12).log(2) o, q, t = 1, 0, 0

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

frequency ratios in just tuning (2-, 3-, 5-based) 1 2 3 4 5 6 7 8 9 10 11

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

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

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

a?

mean-tone tempered scale f g a b c’ f g a b a d g = 5/4 5/4 5/4 c ➡ d ➡ e → f ➡ g ➡ a ➡ b → c’

frequency ratios in mean-tone tempered scale 1 2 3 4 5 6 7 8 9 10 11 1 21/12 1/12 100 √5/2 -1 1/2 193.157 23/12 3/12 5/4 -2 386.314 5/4 x 8/55/4 -1/4 503.422 26/12 = √2 6/12 51/4 1/4 696.578 28/12 8/12 53/4/2 3/4 889.735 210/12 10/12 55/4/4 1082.89

b♭ b♭ pitch classes in just tuning 12 fifths – 7 octaves = fifth comma = Pythagorean comma = 23.46 Ct 1 third (+2 octaves) – 4 fifths = third comma = syntonic comma = -21.51 Ct b♭

calculating and hearing commata pitch(f) = 1200/log(2) × log(f) + const. [Ct], Take log-basis = 2: pitch(f) = 1200 × log2(f) + const. [Ct] pitch(f/g) = 1200 × log2(f/g) [Ct] f/g = 2pitch(f/g)/1200 [Hz] fifth comma, Pythagorean comma 12 fifths – 7 octaves ~ (3/2)12 × (2/1)-7 = 23.46 Ct 223.46/1200 = 1.01364 440 Hz ⇒ 446.003 Hz 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

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

Ÿ12 Ÿ pitch classes in 12-tempered tuning d = 5 x k +2 unique formula 6 1 2 3 4 5 7 8 9 10 11 c g Ÿ12 Ÿ d = 5 x k +2 unique formula that exchanges consonances and dissonances of counterpoint!