Frequency and Intervals   Hyperlinks clicked on this page open a new window, which may refer to demonstration features, such as buttons to play music, which are contained on another screen. Please refer to the menu sequence at the foot of that new window, if you wish to go to that other screen.

This screen explains the acoustical and mathematical background to musical intervals. It also provides a practical demonstration of the scale of C major on a piano keyboard, with all the frequencies, and the names for the intervals in this scale, arranged in a chart below the keyboard.  This frequency chart assumes the piano is tuned to "Concert Pitch", where A is set to 440 cycles per second.

Clicking the mouse over any white or black key will play that note. You can also type the note names C, , E, F, G, A, B on the computer keyboard to hear the corresponding notes played. Just to the left of the picture of the piano keyboard, there is a button for playing the C scale and another for the C triad. You will hear the notes and see the piano keys depressed as the scale or triad is played.

The natural scale is formed by notes which sound pleasant when played in close succession. The degrees of the scale can be numbered or named in several ways, described here. This modern major scale has evolved over time from early Greek modes. The frequencies of the notes in the natural scale bear simple ratios to the tonic or root note, based on multiples of 2, 3, 5. This is related to the harmonics or overtones of a fundamental tone.

The note which sounds the most similar to the root note, fundamental tone, or tonic, is the octave, whose frequency is exactly twice that of the tonic. This is the 2nd harmonic of the fundamental or root.

The fifth, or dominant, is the next most similar. It sounds a little fuller and richer than the octave, when played together with the tonic. Its frequency bears a ratio of 3/2 to the tonic. If a string is divided into 3 parts, and plucked, the resultant note will have a frequency 3 times that of the un-divided string, and this will be one octave plus one fifth above the root note. This is the 3rd harmonic. So we divide this frequency by 2 to bring it down an octave, resulting in the Dominant.

The third of the scale, or mediant, sounds very sweet when played with the tonic and the dominant. This forms the "Major Triad" which is the most fundamental chord of all. The mediant's frequency has a ratio of 5/4 to the tonic. This is equivalent to dividing a string into 5 parts, then bring the note down 2 octaves. The frequencies of the 3 notes of the major triad are in the ratio 4:5:6 to each other.

The equi-tempered scale divides the octave (a doubling of frequency) into 12 geometrically equal parts for each semi-tone. So each semi-tone has a frequency which is higher than its predecessor by a factor of the twelfth root of two, that is 2^(1/12)


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