We hear the size of the difference in pitch between C' (middle C) and
C'' as the same as the difference between C'' and C'''. We call this difference
"one octave" (made of 12 semitones) in both cases. Yet, the *frequency*
difference between C' and C'', as measured in Hertz, is not the same as
the *frequency* difference between C'' and C''', as the following
table and figure
1 show:

Note | No. of semitones above C' | Multiplier of C' frequency | F (Hz) | log_{10} F |

C''' | 24 | 4 | 1056 | 3.0237 |

B | 23 | 3.75 | 990 | 2.9956 |

A | 21 | 3.333 | 880 | 2.9445 |

G | 19 | 3 | 792 | 2.8987 |

F | 17 | 2.667 | 704 | 2.8476 |

E | 16 | 2.5 | 660 | 2.8195 |

D | 14 | 2.25 | 594 | 2.7738 |

C'' | 12 | 2 | 528 | 2.7226 |

B | 11 | 1.875 | 495 | 2.6946 |

A | 9 | 1.667 | 440 | 2.6435 |

G | 7 | 1.5 | 396 | 2.5977 |

F | 5 | 1.333 | 352 | 2.5465 |

E | 4 | 1.25 | 330 | 2.5185 |

D | 2 | 1.125 | 297 | 2.4728 |

C' | 0 | 1 | 264 | 2.4216 |

The frequency difference between C' and C'' is 264 Hz; betwen C'' and
C''' it is 528Hz, twice as large. One octave is not a fixed frequency *difference
*but
a frequency *ratio* of 2:1. The size of each semitone in Hz gets larger
as we go higher up the musical scale (figure
1). If we could squash the frequency scale so that higher semitones
are the same size as lower semitones we would be using a **logarithmic**
scale (figure
2). On the logarithmic frequency scale, the difference between C' and
C'' is the same as the difference between C'' and C'''.

**Amplitude and loudness**

The perception of loudness is also more nearly logarithmic than linear.
In honour of Alexander Graham Bell, the inventor of the telephone, the
term **bel** was given to a unit on the logarithmic scale of *acoustical
power (intensity) ratio*. The interval between the measured intensity
*W
*1
and the baseline (reference) intensity
*W*_{0}
in bels is log_{10} (*W*_{1}
/ *W*_{0}). 1 bel is a ratio of 10:1,
2 bel a ratio of 100:1. This unit is too large for practical purposes,
so the **decibel** (dB), one-tenth of a bel, is more commonly used.
1 dB (intensity level, or IL) = 10 log_{10}
(*W*_{1} / *W*_{0}).
In speech acoustics, the reference intensity *W*_{0}
is usually the threshold of hearing, 10^{-10}
W/m^{2}, which corresponds to a sound pressure
*P*_{0}
of 20 µPa (20 millionths of a Pascal, or 0.00002 Pa). Since intensity
is proportional to the square of sound pressure, a decibel is 20 times
the logarithm of the measured pressure divided by the reference pressure:
1 dB (sound pressure level, or SPL) = 20 log_{10}
(*P*_{1} / *P*_{0}).