Quarter Comma Meantone Temperament was the most commonly used method of tuning. Unlike Pythagorean tuning where the thirds are sacrificed to keep the fifths pure, the fifths are sacrificed to keep the thirds pure. Thus when we begin to construct our scale the thirds are the easiest to start with.
The thirds maintain the same ratio as their harmonic, as they are pure and this is: 5/4. Thus if we take n to be the frequency of the tonic, the frequency of the mediant will be: 5n/4. As our thirds are pure the leading note will be a third above the dominant and the submediant a third above the subdominant.
The fifths are tempered to keep the thirds pure. We shall calculate the amount of temperament as follows. The interval from C to e is two octaves and a major third. If we assume our tonic to be C, and that is tuned to the frequency n then we would expect the frequency of e to be 2.2.(5n/4): which is two octaves and a major third, and is equal to 5n. If we judge the same distance to be four perfect fifths, which have the ratio 3/2 then the frequency of e is calculated: ((3/2)^4).n which equals 5.0625n. Actually these two values can be rewritten as 80/16 and 81/16 respectively. The ratio 80:81 is one comma, and because there is one comma difference in four fifths, we shall tune each fifth a quarter comma flat, hence ‘Quarter comma meantone.’
Remembering that thirds are true we must find the ratio of our tempered fifths by equating the value of four fifths to the value of two octaves and a third, which we want to keep correct. The ratio of our tempered fifths we will define as x and can be calculated thus:
n.x^4 divide through by n
x^4 = 5
x = 5 ^(1/4)
So our new ratio for fifths is x = 5 ^(1/4) which allows us to calculate the dominant’s frequency as n.5 ^(1/4). The subdominant is tuned one fifth lower than the tonic and is equal to (2n)/(5 ^(1/4)).
The last interval we need to tune is the second. Unlike Pythagorean tuning where there is a distinction between the major and the minor second the seconds are equal in quarter comma tuning, and therefore they must also be tempered. The frequency of the major second is represented as s. As we know that two fifths up and an octave down is a major second we can calculate the value as follows:
s= (n.(5 ^(1/4)^2)
And this will prove that the thirds are perfect because two seconds make a third:
((n.(5^(1/2))/2).((n.(5^(1/2))/2) = (5n)/4
Chromatic notes are tuned in thirds, being the most useful for playing in the most common keys. Therefore F sharp is tuned a third above D, C sharp a third above A, G sharp a third above E, B flat a third below D and E flat a third below G.
Having applied all this information the following frequencies have been generated, and placed next to those of just intonation for comparison.