Since one feature of human hearing has been mentioned, it is convenient to have a short digression on musical *pitch* and its relation to frequency. Just as we hear loudness on a logarithmic scale, so we respond to pitch on a logarithmic scale too: when a musician talks about the relation of two different notes, they use language which describes the *ratio* of the two frequencies. At least, this is more or less true. For the moment I will ignore quibbles of detail and tell the simple but approximate story. Some of the resulting inaccuracies will be corrected in Chapter ? when we return to questions of hearing and its relation to music. Our brains are wired in a way which means that we have a special liking for frequencies in simple whole-number ratios, and there are musical terms to describe these. The simplest is the ratio 2:1: two notes with frequencies in this ratio are said to be an *octave* apart. The ratio 3:2 describes a *fifth*, 4:3 a *fourth*, 5:4 a *major third*, 6:5 a *minor third*. Notes with a frequency ratio close to one of these, but not quite accurate, are likely to sound out of tune.

Now there is an arithmetic snag. If you choose a starting note and then go up in seven octaves and back down again in 12 fifths, you arrive back almost at the starting point, but not quite. If you are a singer or a violinist this is no problem. You can vary the pitch continuously, and it doesn’t matter if these two notes are slightly different. But if you play an instrument with fixed and defined pitches (via a keyboard, for example) then you need to play the same note for both ends of this sequence. That can only be achieved by some kind of fudging of the frequencies, known as a *tempering system*. To tune a keyboard instrument, it is simply not possible to make all note combinations (or *intervals*) be perfectly in tune. Some compromise is needed, and different choices may be made depending on the style and period of music to be played. We won’t go into any detail here, we will only describe the simplest and most common system, called *equal temperament*.

The argument goes like this. We hear in ratios, so let’s try to find a way of tuning so that the ratio from each note to the one above is the same. We will keep octaves at exactly the 2:1 ratio, so the question is: how do we subdivide the octave into a number of equal steps, so that we include reasonably good approximations to the simple ratios described above: 3:2, 4:3 and so on? A bit of empirical exploring with a calculator reveals that the smallest number of subdivisions which works well enough for most purposes is 12. So that is why a piano has 12 keys to the octave (ignoring any distinction between black and white keys, which are purely for ergonomic convenience of the player). In a similar way, a guitar has 12 frets to the octave. The frequency ratio needed to achieve this can easily be worked out. We need a number which when multiplied by itself 12 times ends up with the value 2 (for the octave). So we need the “twelfth root of 2”, written $\sqrt[12]{2}$, which is equal to approximately 1.059, or about 6%. This ratio is the *equal-tempered semitone*. Any two adjacent notes on a piano have frequencies different by about 6%.

Applied to the earlier example: to go up seven octaves and come back down in 12 equal steps to reach exactly the starting point, each step needs to have a ratio corresponding to seven equal-tempered semitones, which is 1.498 rather than the “perfect fifth” ratio 1.5. Close, but slightly tweaked. For this interval, equal temperament does pretty well. Its most conspicuous failing comes with the major third: four equal-tempered semitones gives a ratio 1.2599, whereas the perfectly tuned major third should have the ratio 5/4 = 1.25. That sounds fairly close, but not close enough for an expert. To describe small frequency discrepancies like this, it is usual to further subdivide the semitone into 100 small intervals called *cents*. The most acute musical ears can discriminate about 2 cents, 1/50 of a semitone, or a frequency change of about 0.1%. But the discrepancy between 1.2599 and 1.25 is about 14 cents, very clearly audible. This has consequences for the design of the piano and other fixed-pitch stringed instruments, which we will come to in Chapter ?.

Since we will be showing a lot of graphs of things plotted along a frequency axis, it will be useful to have a translation table between musical notes and frequency in Hz. So far we have only defined ratios of frequencies, so to get an agreed pitch standard we need to adopt a convention. The usual modern convention is to tune one of the notes A to the frequency 440 Hz. This is the note the oboe will usually play before a concert starts, to allow the orchestra to tune up. Starting from this value, it is easy to work out the frequency of any other equal-tempered note. For reference, a table is given here. Musical notes are labelled with their usual names A, B, C etc. for the “white notes”, with intervening semitones, the “black notes”, called things like “A sharp” or “B flat”, written A$\sharp$ or B$\flat$. Within equal temperament those two names mean exactly the same note, with a frequency one semitone above A and one semitone below B. The table also shows the conventional scheme for numbering the octaves, so we can refer to the A in a particular octave as, for example, A$_4$.

Octave: | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

C | 32.70 | 65.41 | 130.8 | 261.7 | 523.3 | 1047 | 2093 |

C$\sharp$ | 34.65 | 69.30 | 138.6 | 277.2 | 554.4 | 1109 | 2218 |

D | 36.71 | 73.42 | 146.8 | 293.7 | 587.4 | 1175 | 2350 |

E$\flat$ | 38.89 | 77.79 | 155.6 | 311.2 | 622.3 | 1245 | 2489 |

E | 41.20 | 82.41 | 164.8 | 329.7 | 659.3 | 1319 | 2637 |

F | 43.65 | 87.31 | 174.6 | 349.2 | 698.5 | 1397 | 2794 |

F$\sharp$ | 46.25 | 92.50 | 185.0 | 370.0 | 740.0 | 1480 | 2960 |

G | 49.00 | 98.00 | 196.0 | 392.0 | 784.0 | 1568 | 3136 |

A$\flat$ | 51.91 | 103.8 | 207.6 | 415.3 | 830.6 | 1661 | 3323 |

A | 55.00 | 110.0 | 220.0 | 440.0 | 880.0 | 1760 | 3520 |

B$\flat$ | 58.27 | 116.5 | 233.1 | 466.2 | 932.3 | 1865 | 3729 |

B | 61.73 | 123.5 | 246.9 | 493.9 | 987.7 | 1976 | 3951 |