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Mean-Tone Temperament |
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The mean tone temperament was developed as an
adjustment on the pure intervals. Let's take a look at the pure
intervals for the diatonic scale. The diatonic scale is synonymous
with a major scale.
These are the notes and ratios for a major scale. All major scales consist of five whole steps and two half steps. By examining the whole steps that exist in this pure tone form, one can see an inconsistency in the size of the interval. (Remember, to find an interval, we take the frequency of the larger tone and divide by the frequency of the smaller tone.) The first whole step is from the root to the major 2nd. The interval is clearly 9/8. The second whole step interval is from the major 2nd to the major 3rd. Its interval is:
We have already found the discrepancy. The three remaining whole steps have intervals of 9/8, 10/9, and 9/8. These can be verified in the same fashion. The mean tone temperament is a way of resolving this conflict. This system takes the two different whole steps, and makes them into one "mean tone". Unlike the Pythagorean Scale, this system prefers the major third to the perfect fifth. Because a major third is the same as two whole steps, the mean tone tempers with the pure tones and makes the major second exactly between the root and the major third. The fifth is flattened a little as a consequence. This system would have worked fine, but it led to some ambiguity. Suppose we are in the key of C major. The major third is the note E. A major third above E is a G#. With equal whole steps, then G# is the same as two consecutive major thirds, giving an interval of :
with respect to the root. However, an Ab is a major third below C. Finding this interval, we get:
This Ab is (of course) not in the octave of 1:1 to 2:1 as C as the root, so the octave of Ab can be found:
giving an interval of 8/5 with respect to C. But since Ab and G# are the same note, their intervals should be the same. Needless to say,
This detail made the mean tone temperament extremely difficult to handle. The idea was good, but implementing it was difficult. Some keyboard designers decided to make keyboards with split black keys (making Ab and G# two separate notes), or else they were forced to choose one tuning style over the other. If this wasn't difficult enough, imagine trying to retrain your fingers to be accustom to the split black keys. The mean tone temperament did allow for some modulation, but the quality quickly fades after one leaves the key of C, F, and G (the I, IV, and V of the root).
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