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As we all know, the standard music keyboard consists of 12 basic notes: C, C♯, D, D♯, E, F, F♯, G, G♯, A, A♯, and B.

My question is, can there be any intermediate notes? For example, the note after C is C♯. They have a gap of one semitone. Can there be other notes that exist in between C and C♯? If so, will they be musical or noisy?

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    Dividing an octave into 12 equal parts is only one way to tune a musical instrument. Some the intervals which don't fit into that system sound more "musical" than those which do, if you have an open mind about such things. Historically, dividing an octave into 19 equal parts goes back a few hundred years, and a few keyboard instruments using that system were built. In the computer era the problems of making something which a human can physically play don't exist, and finer subdivisions have been used. 31, 43, and 51 notes per octave have all been use. Google for "just intonation." – user19146 Oct 29 '17 at 7:22
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    ... of course there has never been any physical limitation on the pitches that humans can sing, and some folk music traditions (in western cultures as well as eastern) are certainly not limited to 12 notes per octave. Unfretted stringed instruments can also play any pitch, and some wind istruments can "bend" the pitches of notes even though the instrument is nominally designed for a 12-note-per-octave scale. – user19146 Oct 29 '17 at 7:25
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Absolutely there are pitches in between. Technically, there are an infinite number of pitches in between each two successive semitones. In practice, quarter tones have been used more than most other intermediate pitches. These are pitches that are "halfway" between two successive semi-tones, or 50 cents away their lower and upper neighbors.

The use of pitches that are between the common 12 tones is often called microtonality, although sometimes that term is applied to using alternate tunings of the same 12 tones.

Microtonality is very common in musical traditions other than the Western European tradition. Quarter tones (more or less) are used very frequently in Indian music, for example. And many, if not most, non-European traditions do not use 12-tone equal temperament. The Balinese scales like the pelog scale use fewer than 12 pitches but they are not tuned in such a way that they line up with the 12 tones of the European tradition.

Even in the European tradition, 20th century "classical" music has many pieces that use microtonality and/or atonality. Threnody For The Victims of Hiroshima has many sections that don't specify definite pitches at all, and features many slides between different pitch areas. Some might disagree about whether such compositions fit their concept of what is "music", but music that works outside of the 12 tone equal tempered framework is more common and popular than many people realize.

  • And then there's music that's pitch-shifted away from A440 (the Undertale OST is particularly infamous for that; for instance, ASGORE plays back flat for D minor). – Dekkadeci Oct 29 '17 at 15:48
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Australian/UK composer Percy Grainger made a microtone machine...

http://120years.net/the-free-music-machinepercy-grainger-burnett-crossusaaustralia1948-2/

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    Also see book How Equal Temperament Ruined Harmony by Ross Duffin for a scientific treatment. – AJP Oct 29 '17 at 13:06
  • Not necessary to go so far afield. Every violin is a "microtone machine". – Stinkfoot Nov 4 '17 at 8:16

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