Two sounds of different pitches form an **interval**. Ears, at
least those of people accustomed
to western musical styles, perceive two intervals to be the same if
the **frequency ratios** are the
same. That is, a western ear will hear sounds of 200Hz and 300Hz as
forming the same musical
interval as sounds of 240Hz and 360Hz because 300/200 = 360/240.

Western music is largely based on the intervals of the **harmonic
series**. This is the sequence of
pitches that you obtain by multiplying a single frequency by
successive integers. Thus we have
a harmonic series based on the standard pitch of 440Hz.

Ratio |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |

Harmonic |
1st |
2nd |
3rd |
4th |
5th |
6th |
7th |
8th |

Interval |
Unison |
Octave |
12th or |
15th or |
17th or |
19th or |
- |
three octaves |

Frequency |
440 |
880 |
1320 |
1760 |
2200 |
2640 |
3080 |
3520 |

Note: I expect that you all know the ratios for an octave (2:1) and a fifth (3:2). I would not memorize any others but DO memorize the intervals in the table above. Then you can figure out the thirds and the fourth any time you need them.

In antiquity, the western musical scale was constructed based on some
of these ratios.

The most
fundamental interval is the **octave**, corresponding to doubling
or halving the frequency. Two pitches
that are one octave apart sound so similar that we give the notes the
same name.

The next most important ratio is the **fifth**, corresponding to a
ration of 3:2, found between the second
and third members of the harmonic series.

Closely related is the **fourth**, a ratio of 3:4, found between the third and fourth members.

The **Pythagorean** Just scale is constructed using the **circle of fifths** so that all intervals of a fifth (and thus also a fourth) are in tune but many of the thirds are badly mistuned. The circle does not close because (3/2)^{12} is not a power of two.

The modern **equally tempered scale** is constructed from
approximate ratios that allow the division of the octave into 12 exactly equal intervals. We call the interval between
adjacent pitches 1 semitone and it corresponds to a ratio of 2^{1/12} = 1.05946. We arrange these semitones
in a regular pattern called a scale and give them alphabetic names according to an historical scheme. We start from the
convention that 440Hz will be called pitch A and then work our way through the alphabet in the following
scheme

Intervals are usually specified in terms of the number of white notes between two notes. Thus C and G form an interval of a fifth, D and F form an interval of a (minor) third.

The Pythagorean circle of fifths fails to close by an interval of ratio (3/2)^{12}/219, calls the **Pythagorean comma**. It is about 1/6th of a semitone.

For discussing fine points of tuning the semitone is divided into 100 equal parts called **cents**. This is a very tiny interval with ratio 1.0005777895.

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