Vibration of matter produces sound energy. Only when the vibrational rate falls between 20 vibrations and 20,000 vibrations per second (20 Hz to 20,000 Hz) are we able to hear this sound energy. Such energy vibrates our eardrums, from which the energy is transferred through several anatomical mechanisms to the cilia--small hairs of varying length--in our inner ears. Only certain cilia, those corresponding to the sound frequencies reaching our ears, vibrate sympathetically, sending signals to our brain, which then interprets those vibrations as sound.
The wave patterns of the periodic functions sine and cosine lend themselves perfectly as a model for describing the cyclical nature of vibrational energy, including sound. Since the sine function y = sin t begins at the origin (when t = 0), sine is the more convenient of the two for this purpose. Here is a graph of a single note:
As an example, note A above middle C is the note on which most tunings of instruments is based: 440 Hz. In radians, we could describe this note A with
In degrees, this would be
Musical tones are restricted to a limited set of frequencies we call notes. How we judge combinations of notes is partly subjective, but most people would agree on some simple basics. For example, notes C and G blend nicely together and produce a stable harmony. Change note G to F# and now the combination of C and F# is unstable, producing tension. Adjacent note combinations such as C and C# simply clash and are unpleasant to the ear. Can these effects be seen mathematically? That is the purpose for graphing note combinations here.
For purposes of this project musical notes are assigned relative frequencies with the lowest note, C, having a frequency of 1 cycle per second, or 1Hz. (Hz is a "Hertz.") While these are not true frequencies, the math is easier and the comparative results the same as if true frequencies were used. (Actual middle C has a frequency close to 262 Hz.) Subsequent notes are tuned by the well-tempered system of tuning in use since 1720. The ratio of frequencies of consecutive chromatic notes is the twelfth root of 2, approximately 1.05946. This means that multiplying the frequency of a note by 1.05946 will yield the frequency of the next note above it on the chromatic (all-inclusive) scale.
For a chart of relative frequencies over two octaves (with lowest C being 1 Hz) click here: Frequencies. For a more comprehensive discussion of tuning notes, see notes C and G.
Here are graphs of two note combinations. Click on any picture for a larger graph and more commentary.