History of Music Theory, Part Eight
Returning to the natural harmonic overtone series, if we simply number the harmonics sequentially starting with the numeral one for the fundamental, the series actually calculates all the necessary ratios for us! Above I have included the series in its sine and cosine forms, which will be necessary for following discussions. First, here is a list of the perfect and imperfect consonances.
1) Between C1 and C2 the ratio is 2:1, or the perfect consonance of an octave.
2) Between C2 and G3 the ratio is 3:2, or the perfect consonance of a fifth.
3) Between G3 and C4 the ratio is 4:3, or the perfect consonance of a fourth.
4) Between C4 and E5 the ratio is 5:4, or the imperfect consonance of a major third.
5) Between E5 and G6 the ratio is 6:5, or the imperfect consonance of a minor third.
6) Between G3 and E5 the ratio is 5:3, or the imperfect consonance of a major sixth.
7) Between E5 and C8 the ratio is 8:5, or the imperfect consonance of a minor sixth.
Note that the first five ratios are superparticular in nature (The numerals in the ratio are adjacent numbers differing by only 1), but that the sixths are not superparticular ratios. The numerals in the ratio that describes the major sixth differ by 2, and the numerals in the ratio that descibes the minor sixth differ by 3. Theorists down to and including Zarlino struggled with classifying consonances. Zarlino proposed that all superparticular ratios be considered perfect consonances, which makes only the sixths imperfect, but includes the thirds as perfect. This is obviously not right.
Just as the idea of octave inversion helped me to break through the "rules" of counterpoint - rules that previous theorists had deduced from analyzing a particular composer or school of composers - and get to the underlying laws that govern musical motion in counterpoint, octave inversion serves the same purpose here.
1) Perfect consonances maintain a superparticular ratio when inverted at the octave (Octave, fifth, and fourth).
2) Imperfect consonances are superparticulars in only one of the two possible octave inversions (Thirds and sixths).
This properly describes the difference between perfect and imperfect consonances, and furthermore displays for all to see why imperfect consonances are imperfect: They are not superparticular ratios in both positions of an octave inversion, but rather are superparticulars in only in one of the two positions.
In the previous post I mentioned that theorists wrestled with the syntonic comma, which is the difference of 81:80 between two 9:8 wholetones and a 5:4 major third. We can see where the problem arises in the natural harmonic overtone series above. The area in question is from C8 to E10.
1) Between C8 and E10 the ratio is 10:8.
2) (10:8)/2= 5:4, or the imperfect consonance of a major third.
3) There are two different sized wholetones dividing the major third in the natural series.
4) The lower and larger wholetone between C8 and D9 is 9:8, and is called a major wholetone.
5) The upper and smaller wholetone between D9 and E10 is 10:9, and is called a minor wholetone.
So, there are no commas in nature, there are rather two different sized wholetones that divide the major third. In point of fact, there are four different wholetones in the natural series: 8:7, 9:8. 10:9, and 11:10, just as there are two different minor thirds at 6:5 and 7:6. The point is, trying to apply a just tuning scheme to a chromatic idiom where you want all twenty-four major and minor keys available is frought with headaches to the point of being practicably impossible. For that reason, a means of tempering either the Pythagorean or the Just tuning system was needed. The struggle wouldn't be pretty.
Remember, all art is a reflection of nature. It is almost time to address immitation, canon, and fugue, which describe in music the phenomenon of reflected sound known as echo. I leave you with some reflected light to enjoy which mirrors the sine/cosine harmonic series above.