This dissertation is in two unrelated parts: the first part is the Dissertation Essay (p. 1-49) and the second part is the Dissertation Composition (p. 50-178). The Dissertation Essay is an analysis of György Ligeti's Étude 1: Désordre (Disorder) (1985) for piano. In the first part of the essay, there is a short introduction to Ligeti's Piano Etudes, quotations by him, and basic information about the Désordre. In the second part the etude's melodic lines are discussed, with help of figures of the phrases of the melodic lines. The main focus of the composition is in rhythm, which is given consideration in the third and most extensive part. The rhythm is approached from a perspective of different rhythmic levels, also with the help of figures. Finally in the fourth, shifting and form are scrutinized. The last part is followed by Conclusions, Bibliography / Works Cited, and Appendix (Analysis sheets of Désordre). In the analysis sheets of Désordre (Appendix) the etude is approached from the points of view of rhythmic levels of two individual lines, phrasings of these melodic lines, shifting of bar lines (between right and left hands) and overall form. Throughout this essay it is recommended that one follows Appendix. Désordre is a refined combination of systematic and non-systematic music. Ligeti bends, changes and breaks the system along the way. The fine balancing between the order and disorder, is a core idea of the composition. The Dissertation Composition is Velinikka, Concerto for Quarter-tone Accordion and Chamber Orchestra (2008). The composition was commissioned by Gaudeamus Music Week and is dedicated to Veli Kujala. Velinikka, 25-minute work, was premiered on September 1st, 2008 by Veli Kujala, quarter-tone accordion, and Insomnio conducted by Ulrich Pöhl at Muziekgebouw aan 't IJ in Amsterdam. The cadenza of the concerto is improvised or/and composed by the soloist. Cooperation between Veli Kujala and the undersigned has led to an elaboration of a new instrument, the quarter-tone accordion. The most significant difference between the quarter-tone accordion and a standard chromatic button accordion with free-bass system is that the reed blocks inside the accordion are replaced with quarter-tone reed blocks. The quarter-tone reed blocks are able to produce a complete quarter-tone scale of nearly five-octave range. The quarter-tone accordion was invented by Veli Kujala in 2004. The quarter-tone reed blocks were designed by Veli Kujala and built by Pigini, an accordion factory in Italy, in 2006. The tuning system used is quarter-tone equal temperament, in which the scale divides an octave into 24 equal-ratio steps. Today many composers are interested in using micro-intervals. This increases demand for instruments able to produce them. The quarter-tone accordion is one of the answers for this demand.