What are the differences between tone, note, and pitch?

Answer

A pitch is a particular frequency of sound, for example 440 Hz. Wikipedia goes into a lot of detail about how pitch is subjective, and frequency is objective; the frequency that you think you hear (the pitch) might not be the real frequency at all, due to overtones (see below) and other factors. You can read that article for more details, though for our purposes the definition as "a particular frequency" is sufficient.
A note is a named pitch. Arbitrarily named, of course, by us humans. For example, Western music generally refers to the 440 Hz pitch as A, specifically A4. A note can refer to an occurrence of such a pitch as well. Playing A4 twice can either be talked about as "playing one note twice" or "playing two notes", depending on the context and how specific you want to be. Notes that are even multiples of other notes share the same name; for example, 880 Hz (double 440 Hz) is also called A, specifically A5.
As Kos points out in the comments, a note can also carry temporal information. For example, given the same tempo, a whole note is held twice as long as a half note, which is in turn held twice as long as a quarter note, etc.
When a sound consists of multiple pitches is when things get slightly messy. If a sound consists of multiple pitches, it could either be multiple notes played at once or a single note with overtones (see my answer here for more information on overtones and harmonics). You cannot determine the difference purely from the sound alone, you have to look at the actual physical action. Striking two strings on guitar will produce two notes; striking one will only produce one note, even if that single note consists of a fundamental frequency plus overtones.
Timbre has to do with the other qualities of the sound. Only a sound produced electronically can have only one pitch; all other sounds consist of multiple pitches. The mix of frequencies in a sound results in the timbre. For example, playing A4 on a guitar will actually result in a sound composed of the following frequencies: 440 Hz, 880 Hz, 1320 Hz, 1760 Hz, etc. The particular strength, or amplitude, of the frequencies results in the timbre. One sound might have very little 880 Hz present in it while another has a lot, for example, and we can pick up on that difference. It's how we can tell that an A4 played on a piano and an A4 played on a guitar sound different.
"Tone" is sometimes informally used as a synonym for "timbre", but more often "tone" is synonymous with "note". It can basically replace "note" any time you would say "note"; e.g. "playing two tones". Some would argue that it more commonly refers to a single pitch — e.g. a note is a tone plus overtones, and a tone does not consist of overtones — but unfortunately the usage is inconsistent. "Tone" can also be synonymous with "step": A half-tone is a half-step, and a whole-tone is a whole-step. For example, from C to C# is a half step or half tone, whereas from C to D would be a whole step.
I don't like to use the term "tone" because of the possible confusion about its meaning. It's better to use timbre, pitch, note, or step when appropriate and leave tone out altogether.

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