Perhaps second only to the Arban Complete Celebrated Method is the Complete Modern Method for Trumpet by Ernest S. Williams. This massive tome of exercises, etudes, duets and solos is a must have in every trumpet player's library. Williams explains:
From the Preface:
In this Method the studies begin in the register of the instrument where there is very little tension required on the part of the student to produce the sounds. Gradually, higher and lower tones are added as endurance increases and tone quality improves. By practicing the studies and solos in the order and manner indicated, the student will cultivate a proper setting and will, at the same time, exercise and gradually strengthen the muscles used in playing the trumpet or cornet. T his development requires patience and perseverance. In this Volume, ample opportunity is given the student to overcome many difficulties and to improve his musicianship, while attaining good tone quality, technic and endurance. The psychological and pedagogical aspects are given much consideration. The material as presented, is designed to promote Music Education and to develop musicianship while scientifically cultivating the technique. Thus equipped, the performer then expresses himself as a musician through the vehicle of the trumpet or cornet.
-Ernest S. Williams