# Basic Facts Week 5

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 1 octave +a fifth 15th or 2 octaves 17th or 2 octaves +a third 19th or 2 octaves +a fifth - three octaves Frequency 440 880 1320 1760 2200 2640 3080 3520

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 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 21/12 = 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.