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Orthogonal Expansions
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
One can easily recognize notes of the same pitch class on a classical 88-key piano that has seven octaves accompanied by a minor third. A 12-key pattern with seven white keys and five black keys repeats up and down the piano. The lowest note marked A0 corresponds to a frequency of 27.5Hz. The highest note is C8 with a frequency of about 4186.01Hz. The interval between two successive piano keys is called a semitone. Therefore, an octave includes 12 semitones. A sharp ♯ raises a note by a semitone or half-step, and a flat ♭ lowers it by the same amount. In modern tuning a half-step has a frequency ratio of 212=21/12, approximately 1.059. The accidentals are written after the note name: so, for example, F♯ represents F-sharp, which is the same as G♭, G-flat. In European terminology, they are called Fa-diesis or Sol-bemolle.
Orthogonal Expansions
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
One can easily recognize notes of the same pitch class on a classical 88‐key piano that has seven octaves accompanied by a minor third. A 12‐key pattern with seven white keys and five black keys repeats up and down the piano. The lowest note marked A0 corresponds to a frequency of 27.5 Hz. The highest note is C8 with a frequency of about 4186. 01 Hz. The interval between two successive piano keys is called a semitone. Therefore, an octave includes 12 semitones. A sharp \ raises a note by a semitone or half‐step, and a flat b lowers it by the same amount. In modern tuning a half‐step has a frequency ratio of 212=21/12 $ \sqrt[{12}]{2} = 2^{1/12} $ , approximately 1.059. The accidentals are written after the note name: so, for example, F# represents F‐sharp, which is the same as Gb, G‐flat. In European terminology, they are called Fa‐diesis or Sol‐bemolle.
The ear and hearing
Published in Michael Talbot-Smith, Sound Assistance, 1999
The diagram shows a section of a typical keyboard. The letters of the notes repeat a1 to a, a to A and so on. Each of these sections covers eight white notes and is called an octave. But if the black notes are counted as well there is a total of twelve. Here is not the place to go into musical scales which can be a very complicated subject and what follows is somewhat simplified-a trained musician might raise his or her eyebrows slightly! We’ll just say that the twelve notes are, to the Western ear, equally spaced. They are called semitones. Going from C to C# is one semitone and appears to be the same increase in pitch as going from F to F# which is another semitone. (The # is called a sharp sign and means the note which is a semitone higher. C# is a semitone higher than C, and so on. Although we haven’t used it here there is another symbol b which means that the note is flattened-l5owered by a semitone.)
Combining evolutionary computation with the variable neighbourhood search in creating an artificial music composer
Published in Connection Science, 2019
Pitch, as the basic concept in music, represents twelve equally distributed semitones and is referred to as note as well, with the ratio between wave frequency of two consecutive note being equal to . With setting A4 as 440 Hz, all devices assign the same frequencies to the same notes. This implies that when a note proceeds to one octave higher, by forwarding 12 semitones, its frequency is doubled.