The musical interval between two adjacent notes in a partition of the octave into twelve (not necessarily equal) intervals is called a semitone; other names for it are a half step and a halftone. In the equal temperament, when the twelve intervals are equal, two notes are a semitone apart if the ratio between their frequencies is the 12th root of 2, or about 1.05946.
A tone is the musical intervals between two notes that have exactly one other note between them, in a partition of the octave into twelve intervals. In the equal temperament, a tone is exactly twice a semitone, and two notes are therefore a tone apart if the ratio between their frequencies is the 6th root of 2 (which is the 12th root of 2, squared), or about 1.1225.
Similarly, an octave is actually an interval of six tones, and a perfect fifth is an interval of three and a half tones – the frequencey 659.26, which is a perfect fifth above 440 in the equal temperament, is the seventh frequency after 440 in the frequency list we saw in the article, so it is seven semitones, or three and a half tones, above 440.
So an octave equals six tones, and corresponds to a frequency ratio of 1:2; then why is its name derived from the Latin word octo, meaning eight? Similarly, what does the number five have to do with the perfect fifth? If you want to know the answer, click here.
I deliberately did not use the terms “tone” and “semitone” in the explanation of the equal temperament, as I think that at first reading they confuse more than they help; somewhat paradoxically, this is true especially for those who do know some musical theory. It’s important to understand that a semitone in the equal temperament is generally different from a semitone in other temperaments. Moreover, even within a given temperament, it sometimes happens that two tones are not the same! (You can learn more about this here.) Most musicians who believe they understand well what a semitone and a tone are, are actually not aware of the fact that these terms are ill-defined, in a sense, and this is why I chose not to use them in the main explanation.
To be able to discuss in better precision notes and temperaments, musicologists
invented the term cent. A cent is a hundredth of a semitone – it is the
interval we would get if we partitioned the octave into 1,200 equal musical intervals,
and therefore correspond to a ratio of the 1,200th root of two between the frequencies
of two notes.
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