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Abstract
The article studies the problem of generalizing the concept of ‘diatonic scale’ for a given ambient chromatic of N tones: ‘Which subset A⊂ℤ N shall be considered as a generalized diatonic scale?’ Each generic type of well-formed scale has exactly two specific manifestations in chromatic universes, which are large ME*-scales, i.e. which are maximally even non-degenerate well-formed scales, whose cardinality exceeds half of the chromatic cardinality. A qualitative distinction between these two large ME*-scales of the same type can be comfortably made on the basis of the shuffled Stern–Brocot tree, which is introduced in Section 2. The shuffled Stern–Brocot tree represents the same abstract binary tree as the traditional Stern–Brocot tree, but has a different planar arrangement. Candidates and final choices for generalized diatonic scales are studied in Section 3.
Candidates are those large ME*-scales A⊂ℤ N which are tightly generated by a prime residue class m mod N. According to this property there is no non-degenerate m-generated well-formed scale W properly included between the translation T 1(A c ) of the complement of A and A itself. This property is equivalent to the fact that the associated ratio m/N is a chromatic number, i.e. that its penultimate predecessor on the Stern–Brocot tree is a convergent for m/N, which again is equivalent to the fact that the ratio m/N corresponds to a right branching on the shuffled Stern–Brocot tree. Tight generatedness of a large ME*-scale is also equivalent to the fact that the small-scale step of chromatic size 1 is at the same time also the rarer step, while the large step of size 2 is at the same time also the more frequent step. A generalized diatonic scale A minimizes the cardinality difference between |A| and |A c | among all tightly generated large ME*-scales in ℤ N . For N divisible by 4, this definition reproduces exactly the family of hyperdiatonic scales, as studied by Agmon (Journal of Music Theory, 33(1), 1–25, 1989) and Clough and Douthett (Journal of Music Theory 35, 93–173, 1991). For N odd we exclude the trivial case of 2-generated scales and obtain a rich inventory of generalized diatonic scales, such as the 7-generated 11-tone scale in ℤ19. An interesting point for N odd is also the generalization of the tritone as a minimal limited transposition subset.
We show that Ivan Wyschnegradsky has already done pioneering work on this subject (1916). His 11-generated 13-tone scale in the quarter-tone chromatic ℤ24 is a hyperdiatonic scale in the sense of Agmon (1989) and Clough and Douthett (1991) as above, and his argumentation in favour of this scale anticipates central points of the discussion in this article.
Acknowledgements
I would like to thank Thomas Noll, Mark Gould and the anonymous reviewers for valuable remarks and comments. Thomas Noll pointed out the relationship between tightly generated scales and chromatic numbers. I would like to thank Thomas warmly for his thorough revision of this paper.
Notes
1. The generic level of description in Citation4 Citation3 is algebraically linked to what I call cyclic tone systems as a special cases of tuning groups (see Citation12 Citation13). The generic interval system controls the counting of steps and—in the case of generated scales—of generator intervals within the generalized circle of “fifths”. Well-formed scale theory—as mentioned above—is essentially built upon a linear automorphism of the generic interval system which exchanges the circle of steps with the generalized circle of fifths. In contrast to the external definition of the generic interval system (like a metalanguage for counting in a scale, so to say), cyclic tone systems are defined as factor groups of the free group, generated by intervals such as octave and fifth. This is an intrinsic definition where the cyclic tone system is constructed within the specific level. The group isomorphism between the generic interval system and the cyclic tuning group provides a music-theoretically interesting connection between these levels.
2. The general theory of well-formed scales does not require the choice of an equally tempered 12-chromatic universe from the outset. The diatonic scale is generated by a fifth (e.g. with frequency ratio ω=3/2 and spanning the linear pitch interval within the octave with frequency ratio 2/1 and linear pitch interval log
2(2)=1. But any generator “close enough” to
, such as g=7/12 yields the same structural results. The mathematical condition for “close enough” is that the ratio 4/7 needs to be a semi-convergent of the number g.
3. Seventh chords in the generic diatonic ℤ7 are the white keys of such a trivial generalized diatonic scale, while the complementary triad forms the black keys. The same observation drives Noll's definition of a pseudo-diatonic system Citation16.
4. The microtonal composer Mark Gould (see Citation23) confirmed in e-mail conversations that he favours this scale.
5. Recall that we restrict our considerations to the left half of the entire tree below the node 1/2.
6. The harmonic minor scale also contains two tritones.
7. According to his son Dimitri, Ivan Wyschnegradsky—who lived in Paris—wanted the transliteration of his name written without accents. I follow his preference (see also Citation26).
8. Clough and Douthett Citation2, pp. 168–69 dedicate particular investigations to the study of interval circles in the quarter-tone diatonic set M 24, 13, i.e. in ‘Wyschnegradsky's’ hyperdiatonic scale.
9. This is even true for k=1 with the word .