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In music we often talk about cents a cent is based on this value of (1/12) a – it is one one-hundreth of the size:Īll of our logarithms are therefore defined with the base of the frequency ratio of a single cent. Therefore b is a unit of measure! In equal tempered tuning each note is an equal division of the octave – 12 chromatic tones in an octave mean that each note is (1/12) of the interval value of the octave. The same applies for (3/2) and (5/4) – the sum of v intervals with the ratio of (b/1) would be (3/2), and the sum of t intervals with the ratio of (b/1) would be (5/4). Therefore, the sum of a intervals having the ratio of (b/1) is the interval whose ratio is 2 – an octave. Let’s think of (b/1) as the frequency ratio of an interval. Let’s define a base b such that the following is true: In our definitions of a, v and t we did not specify a base for the logarithms. We’re going to define that value based on Western equal tempered tuning. This is great in terms of raw theory – but how does it come into play with how we tune our instruments or understand music? It has to do with the value of measurement applied to the intervals and logarithms. This may not seem like much at first – but we’ve formed a precise language for describing interval relationships and calculating the composition of any interval from our fundamental values.
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This means if we start at some pitch – any pitch – go up two octaves (2*a) then up on major third (t) and down three perfect fifths (3*v) we’ll end up at a pitch whose ratio is (40/27) multiplied by our original pitches value. To give an example we can now express intervals as functions of a, v, and t and determine the ratios quickly – such as the sum of a perfect fifth (v) and a major third (t): The sum of these will be the size of the unknown interval we’re trying to determine. Here’s how it will work: we’ll use three new variables, m, n and q to multiple against our originals (a, v and t). Now that we have those three variables defined, we can prove a method to determine the composition of any other interval as a combination of those three. Therefore we can define the following variables, a, v and t: Remember that the size on an interval is the logarithm of its ratio (see the “Math and Music – Intervals” lesson). These variables will make it easier for use to define complex equations later on. We’re going to set some variables based on these intervals.
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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.
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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).
#MUSIC MATH EQUATIONS 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.