containing "the charm of impossibilities". The composer Tōru Takemitsu made frequent use of Messiaen's modes, particularly the third mode. In other temperaments There are no modes of limited transposition in any prime equal division of the octave, such as 19 equal temperament or 31 equal temperament. Composite divisions, such as 15 equal temperament or 22 equal temperament, have them. The 12-note chromatic scale can itself be considered such a mode when viewed as a subset of a larger system that contains it, such as quarter tones or 72 equal temperament. For example, eight equal temperament, the lowest non-prime equal temperament not completely included in 12-tet (due to a scale step in 24-tet), would have modes of limited transposition. The first would be 0, 2, 4, 6 (steps: 2222), which has only two transpositions and one mode. Another would be 0, 1, 4, 5 (steps: 1313 and 3131), which has 4 transpositions and 2 modes (the other is 0, 3, 4, 7). References Further reading Anaf, Jef. 1988. "Olivier Messiaen 80 jaar: Nog steeds actief mit zijn modale toonstelsel". Adem: Driemaandelijks tijdschrift voor muziek cultuur 24, no. 4 (October–December): 184-91. Douthett, Jack, and Peter Steinbach. 1998. "Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition". Journal of Music Theory 42, no. 2 (Fall): 241–63. Frischknecht, Hans Eugen. 2008. "Potential und Grenzen einer musikalischen Sprache: Olivier Messiaens modes à transpositions limitées unter der Lupe". Dissonanz/Dissonance, no. 104 (December): 32–34. Giesl, Peter. 2002. "Zur melodischen Verwendung des Zweiten Modus in Messiaens Subtilité des Corps Glorieux". In Musik, Wissenschaft und ihre Vermittlung: Bericht über die internationale musikwissenschaftliche Tagung der Hochschule für Musik und Theater Hannover, edited by Arnfried Edler and Sabine Meine, 259–64. Publikationen der Hochschule für Musik und Theater Hannover 12. Augsburg: Wißner. . Neidhöfer, Christoph. 2005. "A Theory of Harmony and Voice Leading for the Music of Olivier Messiaen". Music Theory Spectrum 27, no. 1 (Spring): 1–34. Schuster-Craig, John. 1990. "An Eighth Mode of Limited Transposition". The Music Review 51, no. 4 (November) : 296–306. Street, Donald. 1976. "The Modes of Limited Transposition". The Musical Times 117, no. 1604 (October): 819–23. Yamaguchi, Masaya. 2006. The Complete Thesaurus of Musical Scales, revised edition. New York: Masaya Music Services. . Yamaguchi, Masaya. 2006. Symmetrical Scales for Jazz Improvisation, revised edition. New York: Masaya Music Services. . Yamaguchi, Masaya. 2012. Lexicon of Geometric Patterns for Jazz Improvisation. New York: Masaya Music