As shown by the University of New South Wales and the Peabody Conservatory of Music, every pitch has a distinct frequency. This means that every note has a specific wavelength. Mathematicians are able to graph these waves, creating visual, numeric representations of sound.
A music metronome marking indicates how many beats occur per minute. For instance, if the metronome marking indicates sixty beats per minute, then every beat has the duration of one second. This means that all rests and notes in music involve mathematical divisions and multiplications of time duration. Specific note types indicate these divisions and multiplications. For example, the duration of a whole note lasts four times as long as a quarter note.
Musicians count beats as they perform as this ensures that they will give the correct duration of time to each rest and pitch. Counting beats also identifies precisely where the person is in the music. For example, if a conductor tells an orchestra to start "at the middle of the bar" this can be ambiguous, since some music uses multiple time signatures and complex, overlapping rhythms. If the conductor can explain with numbers where he is in the music based on counting procedures (e.g., "on beat four"), then he has eliminated this ambiguity.
The University of New South Wales shows that musical acoustics involves how sound waves bounce off surfaces in the environment. Engineers thus examine the precise angles at which sound waves will hit architectural surfaces in order to get the clearest and most efficient sound production.
In order to achieve a uniform pitch and establish playing standards, instrument and reed manufacturers have to make sure that instruments and reeds have the same basic dimensions. This means that they have to make specific calculations to keep instruments and reeds the same shape and size. For example, oboe players want the length of their reeds to be about 72 mm, since reed length affects pitch.
Since the time of Pythagoras, music developed as a relationship between ratios of frequencies. For example, doubling the frequency of any pitch will give a pitch exactly one octave higher than the original pitch. Human ears hear these ratios as harmony. Over thousands of years, changes in the treatment of what ratios make up a scale have changed what sounds "in tune." If it weren't for these changes, musicians wouldn't be able to transpose well or have scales that start on different pitches sound the same. Musicians also wouldn't have the modern 12-tone row, where mathematical addition and subtraction is the basis of every scale degree, since treatment of pitch ratios is what determines how many notes are in a scale. Math thus is the foundation for much of music theory and harmony.