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Introduction to Linear Algebra with Mathematica
Glossary
Motivated examples
5.6. Motivated Examples from Music
I. How to
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
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.
- Davis, L., What is the Circle of Fifths? Classic FM.
- Jessop, S., The historical connection of Fourier Analysis to music, The Mathematics Enthusiast, 2017, Vol. 14, No. 1, Article 7.
- Write, D., Mathematics and music, 2009.
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