FIFTEENTH

In music, a 'fifteenth' (sometimes abbreviated '15ma') is the interval between one musical note and another with one-quarter or quadruple the frequency. It corresponds to two octaves. It is the fourth harmonic. It is referred to as a fifteenth because, in the diatonic scale, there are 15 notes between them if one counts both ends (as is customary). Two octaves (based on the Italian word for eight) do not make a sixteenth, but a fifteenth. In other contexts, the term ''two octaves'' is likely to be used.
For example, if one note has a frequency of 400 Hz, the note a fifteenth above it is at 1600 Hz, and the note a fifteenth below is at 100 Hz. The ratio of frequencies of two notes a fifteenth apart is therefore 4:1.

As the fifteenth is a multiple of octaves, the human ear tends to hear both notes as being essentially "the same", as it does the octave. Like the octave, in the Western system of music notation notes a fifteenth apart are given the same name—the name of a note an octave above A is also A. However, because of the large frequency distance between the notes, it is less likely than an octave to be judged the same pitch by non-musicians. Passages in parallel fifteenths is much less common than parallel octaves, though it does happen. In particular, sometimes an organist will use two stops a fifteenth away (notated as 2').
Like the notation ''8va'' for octave, '15ma' (''quindicesima'') means "play two octaves higher than written." It could also mean two octaves lower, but two octaves below the low notes of the bass clef is below the range of nearly all instruments. This direction can be cancelled with the word ''loco'', but often a dashed line or bracket indicates the extent of the music affected.

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The 16va and 16vb notations

The 16va and 16vb notations

The notation '16va' is an alternative notation often found in music notation to indicate "play two octaves higher than written." Most consider this usage improper. It seems to have arisen from the folk beliefs that if one octave is 8 steps then two octaves must be 16 steps, and the suffix "va" must mean something like "play higher." Thus if ''8va'' means "play one octave higher" then ''16va'' must mean "play two octaves higher."
Some musicians and music publishers who well understand that two octaves is the interval of a 15th and ''8va'' is an abbreviation for the Italian "ottava" still use the notation ''16va'', especially in music intended for amateurs or children, because they feel ''16va'' is more immediately comprehensible than ''15ma'', especially to non-Italian speakers.
Similarly the notation '16vb' is an alternative notation, considered by some to be improper but found in many printed scores, indicating "play two octaves lower than written".