The problem with, which this thesis is concerned is that of analyzing and comparing the fugal writing and contrapuntal style of J. S. Bach in the fugue expositions of The Well-Tempered Clavier and that of Paul Hindemith in the fugue expositions of the Ludus Tonalis. This comparison is made on the basis of a comprehensive analysis of the fugal expositions both collections of fugues mentioned ( The Well-Tempered by Bach and the Ludus Tonalis by Hindemith). Chapter I includes a discussion of the careers and compositional techniques of Bach and Hindemith. An emphasis is placed on a comparison of Bach's fugal writing with that of his immediate predecessors (composers of the seventeenth and early eighteenth centuries who were writing in the fugal style) and on a comparison Of Hindemith's theory of tonality, as expressed in The Craft of Musical Composition, with that of the traditional harmonic concept of Bach's day. Chapter II deals with the evolution of the fugal concept. In this chapter, imitative forms of composition which gradually evolved toward the fugue are traced from their very early beginnings through the sixteenth, seventeenth, eighteenth, nineteenth, and twentieth centuries. Emphasis is placed on the fugal form that Bach used and on Hindemith's neo-Baroque approach to fugal writing in the twentieth century. In Chapters III and IV, analyses are made of the expositions in the forty-eight fugues of Bach's Well-Tempered Clavier and Hindemith's Ludus Tonalis .There is a discussion of the number of voices, order of entries", order of statements of the theme, key relationships, and redundant entries. Also discussed In these chapters are the beginning and ending notes of the Subject, a change in tonality of the subject range and length of the subjects. There is, further, a discussion of the real and tonal answers and the reasons for the use of a tonal answer, recurring countersubject, invertible counterpoint, interludes, length of the exposition, and the cadences at the end of the fugue expositions. in Chapter V the fugal writing of Bach, as demonstrated in the fugue expositions of The Well-Tempered Clavier, and that of Hindemith, as demonstrated in the fugue expositions of the Ludus Tonalis, are compared. This comparison of these two styles of fugal writing shows the two composers' techniques and procedures to be very much alike except in Hindemith's expanded concept of tonality. Although Bach's set of fugues has set a standard for this type of writing, Hindemith has shown that this old form is still capable of being used with originality when adapted to twentieth-century practices.