source file: m1607.txt Date: Fri, 18 Dec 1998 17:30:43 -0500 Subject: Assymmetrical 14-tone modes in 26-tET From: "Paul H. Erlich" I decided to look at the modes of the assymmetrical scale 0 2 4 6 8 10 12 14 15 17 19 21 23 25 in 26-tET -- because all octave species of this scale is constructed of two identical heptachords, each spanning a ~4/3, separated by either a ~4/3 or a ~3/2; and because the scale contains 10 consonant 7-limit tetrads constructed by the generic scale pattern 1, 5, 9, 12 (5 are otonal and 5 are utonal) with a maximum tuning error of 17 cents. In my paper, I introduce the notion of a charateristic dissonance. This is a dissonant interval which is the same generic size (same number of scale steps) as a consonant interval. Allowing the 7-limit to define consonance and allowing errors up to 17 cents, the 14-out-of-26 scale has three characteristic dissonances (plus their octave inversions and extensions). Two are "sevenths" of 554 cents instead of the usual 508 cents, and one is an "eighth" of 554 cents instead of the usual 600 cents. None of the modes of this scale satisfy all the properties for a strongly tonal mode according to my paper. But a few come close. The mode 0 2 4 6 8 10 11 13 15 17 19 21 23 25 or in cents, 0 92 185 277 369 462 508 600 692 785 877 969 1062 1154 has all characteristic dissonances disjoint from the tonic tetrad (0 8 15 21), which is major. The only other mode with this property is the minor equivalent: 0 2 4 5 7 9 11 13 15 16 18 20 22 24 or in cents 0 92 185 231 323 415 508 600 692 738 831 923 1015 1108. The following modes have one characteristic dissonance which shares a note with the tonic tetrad, but it approximates the 1 identity and the 11 identity when played along with the tetrad. Therefore the interval does not disturb the stability of the tonic too much, and the mode can be considered tonal: major: 0 2 4 6 8 10 12 13 15 17 19 21 23 24 or in cents 0 92 185 277 369 462 554 600 692 785 877 969 1062 1108 and minor: 0 2 3 5 7 9 11 13 15 17 18 20 22 24 or in cents 0 92 138 231 323 415 508 600 692 785 831 923 1015 1108.