# Diapason

The word diapason is used in a number of musical contexts. Generally it can mean the range of a musical instrument or voice. It also has more specific uses:

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## The organ stop

The diapason is the principal stop of the pipe organ, which has pipes throughout the entire range of the instrument. Diapason pipes give the organ its characteristic sound.

Diapasons come in two varieties: open, where the end of the pipe is clear, producing a bright sound; and stopped where the end of the pipe is blocked, producing a more muffled, sweeter sound. The name diapason is also used on some electric organs for voices which imitate the pipe organ stop.

## In harmony

In harmony, diapason, also called the octave, is the ratio of 2:1 between a pair of frequencies or, equivalently, of 1:2 between a pair of wavelengths. It is the simplest ratio other than unison. It is the foundation of the system of base-2 logarithms used in music (the mind naturally perceives pitches in terms of the logarithm of their frequency, so the notations used in music denote the logarithms of the frequencies they represent).

When constructing scales, a pair of notes related by diapason are considered to be equivalent. They are the same note, but on different octaves. This relation is called octave equivalency.

Having the notation repeat allows representing frequencies as wrapping around in a circle. The circle starts at unison and ends at diapason, with unison and diapason at the same position on the circle. Any frequency [itex]f[itex] which is smaller than 1 (unison) or larger than 2 (diapason) has an octave-equivalent frequency [itex]f_0[itex] within the circle:

[itex] 1 \le f_0 = {f \over 2^n} \le 2 [itex]

where

[itex] n = \lfloor {\log}_{2} f \rfloor [itex].

Conversely, every frequency [itex]f_0[itex] within the circle has an infinite number of octave-equivalent frequencies outside it, all related by the ratio of diapason:

[itex] f(n) = {f_0 \cdot 2^n} [itex]

where

[itex] n \in \{\mbox{all integers other than 0}\} [itex].

Diapason is 10 in binary, and it is the sum of all the reciprocals of triangular numbers:

[itex] {1 \over 1} + {1 \over 3} + {1 \over 6} + {1 \over 10} + {1 \over 15} + {1 \over 21} + {1 \over 28} + ... = 2 [itex].

## Pitch standard

Diapason is also used to refer to a physical implementation of a pitch standard, for instance a tuning fork used as a standard.

## The diapason normal

Diapason Normal is the name given to the historical pitch standard where the A above middle C is tuned to 435 Hz. This standard was set by law in 1859 in France, and became popular throughout Europe. In 1939, A = 440 Hz was codified as the international standard of concert pitch for broadcast music, and has replaced the diapason normal.

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