This talk explores interrelationships between mathematics and music using several aural examples and demonstrations. We will discuss equal temperament, relate the chromatic scale it to modular arithmetic, and show why the musical staff is like a logarithmic scale for pitch. We will discuss how overtones are related to the integers and show how harmony derives from the overtone series. We will identify the mathematical relationships between pitches in consonant intervals and chords, and discuss the historical obstacles (going back to Pythagoras) to tuning a musical scale which gives mathematically precise harmony in all keys. The relationship between musical tones and periodic functions will be discussed, showing how a tone's timbre is determined by it's harmonics, and how this relates to trigonometry. Musical examples ranging from Tibetan throat singing to American jazz will be played to demonstrate such things as overtones, chords, timbres, and tuning.