Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

Chew (2000) (pp. 61 - 97) spends considerable effort in order to tune these variables with respect to music-theoretical constraints. A first such constraint deals with the total order of musical selected interval types⁸

⁸: P4 = perfect fourth, P5 = perfect fifth, M3 = major third, m6 = minor sixth, ..., d5 = diminished fifth, A4 = augmented fourth.

, namely

{P 4,P5} < {M 3,m6} < {m3, M 6}< {M 2,m7} < {m2, M 7}< {d5,A4}

If the ratio $h2 a = r2$ is chosen such that $2 2 15 < a < 7$ , the Euclidean distances of corresponding pairs of tone representations exemplify the same total order, namely

V~ ------- V~ -------- V~ ------- V~ --------- V~ --------- 2r2 + h2 < h < 2r2 +9h2 < 2 r2 + h2 < 2r2 + 25h2 < 4r2 + 36h2

Another set of constraints is concerned with the order of the Euclidean distances of the representations of triad tones and ’foreign’ tones from those of the triads themselves:

d(P (0),C (0)) < d(P(1),C (0)) < d(P (4),C (0)) Maj < d(P(l),C Maj(0)) A l /= 0,1,M4a.j Maj d(P (0),C (0)) < d(P(1),C (0)) < d(P (-3),C (0)) min < d(P(l),C min(0)) A l /= 0,1,-m3i.n min

These conditions are satisfied, if

$-2 < a <-3 15 15$
$3 < 4w1 + 3w2 < 15a + 52$
$1- 1 3 < 4w1 + 3w2 < 2a + 2$
$1(u2 + 1) < u1 3$
$( 18a + 34) .u1 + u2 < 81a + 14$

Chew studies the border case $2- a = 15$ in some detail, i.e. where fifths and major thirds are represented with the same Euclidean distance. In that case the above conditions restrict to

3 < 4w1 + 3w2 < 4, u2 + 1 < 3u1, 27u1 + 16u2 < 19.

A third type of constraints deals with the distances between tones and keys as well es between prominent intervals and keys. In particular on wants the following conditions to be satisfied for all $k /= 0$ with respect to both modes $m (- {M aj,min}$ :

pitch -key: dd(P((P0(),0)T,TMma(j(k0))))-< 1 leading tone -key: d(P(0),TMaj(0))+d(P-(5),TMaj(0))< 1 d(P(0),Tm(k))+d(P(5),Tm(k) d(P(0),TMaj(0))+d(P-(5),TMaj(0)) perfect fourth -key: d(P(0),Tm(k))+d(P(5),Tm(k) < 1