Hearing range, the concept of the octave and diatonics
1) The limits of human hearing depend on both frequency and intensity of the sound. The sensitivity of the human ear peaks within the range of approximately 2000 to 4000 oscillations per second (20004000 Hz). Humans can detect frequencies up to 270,000 Hz, if the signal is strong enough, but what is commonly referred to as the hearing range is (roughly) between 20 and 20,000 Hz.
2) A certain range may also be examined separately. The range between 110 and 220 Hz forms an octave. The two tones at both ends of the range are recognised as the same regardless of the culture of the listener. The numerical values of the frequencies have a simple numerical ratio: the frequency of the higher sound is twice the frequency of the lower sound. The mathematical ratio and the perception of sameness are clearly related. The relation between the tones cannot, however, be explained only in terms of mathematics, because human beings tend to perceive two frequencies with a numerical ratio of 2:1 as being just slightly too close to each other, whereas the tones with a ratio of approximately 2.02:1 are perceived as forming a pure octave.
In Western cultures, the octave (which refers not only to the distance between the two tones at the either end but also to the range of tones between them) has been divided diatonically since antiquity. Each octave has seven tone positions that are not distributed evenly, but form two different types of intervals instead: whole tones (or whole steps) and semitones (or half steps). The size of the whole tone (from F to G, for example) equals the size of two semitones (such as from B to C).
3) In a medieval letter-based musical notation used since approximately the 10th century onward, the same letters were first used to denote tones one octave apart from each other. The names we now give to tones also originate from this period. In this letter-based notation, the same letters (usually the first letters of the alphabet) appeared first as single capital letters, were then repeated as single small letters and then (for example) as two small letters. A, a and aa would therefore all be one octave apart from each other.
4) On the diatonic scale, each octave contains 5 whole tones and 2 semitones. The principles of musical notation, as well as many musical instruments such as the piano, are based on the diatonic scale. The major and minor scales as well as all church modes are diatonic, or, in other words, can be understood as parts of the diatonic scale.
We are so used to the diatonic scale that we cannot hear any difference between whole tones and semitones when, for example, going up the major scale in a singing exercise. Deviating from the diatonic scale (such as being able to sing several successive whole steps) requires musical experience or training.
5) How the tone positions are defined in relation to their frequencies is not manifest in musical notation or the names we give to the tones. The tone positions, nevertheless, are best understood as narrow ranges instead of exact points within each octave. The size of the whole tones, semitones and the intervals created by adding them together may differ, though within certain limits. The notions of pure and out of tune are, therefore, highly relative. An octave can be divided into 12 semitones. If the distance between each semitone is equal, the tuning system is called equally tempered.