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where ${f}_{0}$ is the fundamental frequency (in Hz), $n$ denotes the harmonic number, and ${a}_{0}$ is the DC (constant) offset.
When an instrument produces overtones whose frequencies are essentially integer multiples of the fundamental, you do not perceive all of the overtones as distinct frequencies. Instead, you perceive a single tone; the harmonics fuse together into a single sound. When the overtones follow some other arrangement, you perceive multiple tones. Consider the screencast video in which explains why physical instruments tend to produce overtones at approximately integer multiples of a fundamental frequency.
Musicians broadly categorize combinations of tones as either harmonious (also called consonant ) or inharmonious (also called dissonant ). Harmonious combinations seem to "fit well" together, while inharmonious combinations can sound "rough" and exhibit beating . The screencast video in demonstrates these concepts using sinusoidal tones played by a synthesizer.
Please refer to the documents Consonance and Dissonance and Harmony for more information.
A tuning system defines a relatively small number of pitches that can be combined into a wide variety of harmonic combinations; see Tuning Systems for an excellent treatment of this subject.
The vast majority of Western music is based on the tuning system called equal temperament in which the octave interval (a 2:1 ratio in frequency) is equally subdivided into 12 subintervals called semitones .
Consider the 88-key piano keyboard below. Each adjacent pair of keys is one semitone apart (you perhaps are more familiar with the equivalent term half step ). Select some pitches and octave numbers and view the corresponding frequency. In particular, try pitches that are an octave apart (e.g., A3, A4, and A5) and note how the frequency doubles as you go towards the higher-frequency side of the keyboard. Also try some single semitone intervals like A0 and A#0, and A7 and A#7.
The frequency values themselves may seem rather mysterious. For example, "middle C" (C4) is 261.6 Hz. Why "261.6" exactly? Would "262" work just as well? Humans can actually perceive differences in the sub-Hz range, so 0.6 Hz is actually noticeable. Fortunately an elegantly simple equation exists to calculate any frequency you like. The screencast video of explains how to derive this equation that you can use in your own music synthesis algorithms. Watch the video, then try the exercises to confirm that you understand how to use the equation.
What is the frequency seven semitones above concert A (440 Hz)?
659.3 Hz (n=7)
What is the frequency six semitones below concert A (440 Hz)?
311.1 Hz (n=-6)
1 kHz is approximately how many semitones away from concert A (440 Hz)? Hint: ${\mathrm{log}}_{2}(x)=\frac{{\mathrm{log}}_{a}(x)}{{\mathrm{log}}_{a}(2)}$ . In other words, the base-2 log of a value can be calculated using another base (your calculator has log base 10 and natural log).
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