%0 Journal Article %1 3340148520080101 %A Jedrzejewski, Franck %D 2008 %J Journal of Mathematics & Music %K DIATONICISM TONALITY TRANSPOSITION_(Music) MUSICA_ficta MUSICAL_meter&rhythm diatonicism diatonicized_chromaticism generalized_diatonic_scales N-Tone_equal_temperament Stern-Brocot_tree well-formed_scales Wyschnegradsky %N 1 %P 21 - 36 %R 10.1080/17459730801995863 %T Generalized diatonic scales. %U http://search.ebscohost.com/login.aspx?direct=true&db=a9h&AN=33401485&site=ehost-live %V 2 %X The article studies the problem of generalizing the concept of 'diatonic scale' for a given ambient chromatic of N tones: 'Which subset A⊂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⊂N which are tightly generated by a prime residue class m mod N. According to this pro

@article{3340148520080101, abstract = {The article studies the problem of generalizing the concept of 'diatonic scale' for a given ambient chromatic of N tones: 'Which subset A⊂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⊂N which are tightly generated by a prime residue class m mod N. According to this pro}, added-at = {2012-09-21T10:16:09.000+0200}, author = {Jedrzejewski, Franck}, bdsk-file-1 = {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}, bdsk-url-1 = {http://search.ebscohost.com/login.aspx?direct=true&db=a9h&AN=33401485&site=ehost-live}, biburl = {https://www.bibsonomy.org/bibtex/2bcb7c027e3e2f11b36253df820aa0e24/keinstein}, date-added = {2010-03-17 13:53:00 +0100}, date-modified = {2010-03-17 13:53:26 +0100}, doi = {10.1080/17459730801995863}, file = {:Jedrzejewski/Generalized diatonic scales..pdf:PDF}, groups = {public}, interhash = {57fa60bde85ffb527d2846042c40dbaf}, intrahash = {bcb7c027e3e2f11b36253df820aa0e24}, issn = {17459737}, journal = {Journal of Mathematics \& Music}, keywords = {DIATONICISM TONALITY TRANSPOSITION_(Music) MUSICA_ficta MUSICAL_meter&rhythm diatonicism diatonicized_chromaticism generalized_diatonic_scales N-Tone_equal_temperament Stern-Brocot_tree well-formed_scales Wyschnegradsky}, number = 1, pages = {21 - 36}, timestamp = {2013-01-22T21:34:57.000+0100}, title = {Generalized diatonic scales.}, url = {http://search.ebscohost.com/login.aspx?direct=true&db=a9h&AN=33401485&site=ehost-live}, username = {keinstein}, volume = 2, year = 2008 }