#Octave music definition series

The first three basic intervals from this definition are consonant and very important in the foundation of our musical definitions: the octave, perfect fifth and major third, 2:1, 3:2 and 5:4 respectively. Therefore we can frame the precise definition for our interval ratio as follows: a basic interval is an interval whose ratio is p divided by two the n-th power, where p is prime and n is chosen so that 2 to the n-th power is the greatest power of two less than p. This equation shows how a prime ratio can be transposed into a single octave. So if a fifth in the second octave is 3:1 and the second octave is 2:1 then 3:2 would be a fifth in the first octave. Therefore we only need to bring it down by one octave to have the ratio be between 1 and 2. For example, the 3:1 ratio is a perfect fifth in the second octave.

Therefore, we can divide the frequency ratio by the number of octaves necessary to bring it into the first octave range. We want to express the fifth and third as frequencies within the first octave – therefore the ratio has to be greater than 1:1 (our starting tone) and less than 2:1 (the octave).

#Octave music definition series

Following the overtone series we get these basic ratios: 1:1 is our starting tone, 2:1 is an octave above that, 3:1 is a fifth above the octave, 4:1 is the second octave and 5:1 is a third in the second octave. A ratio of 2:1 is an octave, so it makes sense that all the other intervals are defined to be smaller than an octave. The standard convention is that interval ratios are greater than 1 and less than 2. In the last lesson we talked about the frequency ratios of common intervals. Guess what – all intervals can be described as different combinations of the octave, perfect fifth and major third – the first three overtones. This week we’re going to define conventions for interval sizes and then derive three variables where we can determine the composition of any frequency ratio. Previously in the “math and music” lesson we derived equations for expressing intervals as functions of relative frequencies.