In the learned Renaissance and Baroque music circles, it was common to study counterpoint. Theorists, composers, organists, and other musicians got together and immersed themselves in various metaphysical speculations. This all comes down from the fact that music was part of the classical education students received from universities.
This system included the education in 7 liberal arts (just like in the Ancient Greece): the Trivium (grammar, dialectics, and rhetoric) and the Quadrivium (arithmetic, geometry, astronomy, and music). To make it easier to understand, you can imagine that as part of the Trivium, students learned everything about the words and as part of the Quadrivium they learned all about the numbers.
The most fascinating fact (and the most different from today's perspective) is that the study of music was concerned with numbers and not with emotions. This all makes sense when you think that all music consists of intervals and various proportions. Just as architecture was concerned with the numbers embedded in space, music basically was concerned with sounding numbers. The study of Trivium and Quadrivium was in preparation of the philosophy and theology, of course.
So students of such old polyphonic masters as Jan Pieterszoon Sweelinck and Gioseffo Zarlino liked to study counterpoint together. For example, Sweelinck's students Matthias Weckmann and Johann Adam Reincken wrote down everything the old master (who was called the Great Orpheus of Amsterdam and Maker of German Organists) taught them about the art of counterpoint. The result is Sweelinck's Composition's Regeln (the Rules of Composition - vol. 10 of the "Werken") - an indispensable source of inspiration for anybody interested in the style of composition and improvisation of the 16th-17th centuries.
We have to remember that in those days the art of contrapuntal composition was a science in itself with many seemingly mysterious rules and techniques.
Invertible counterpoint is one of those techniques. You know that if we invert an interval of the third, we get a sixth and vice versa. This is done by exchanging two voices by the octave. This is called invertible counterpoint at the octave. It's simple to understand. Here is a list of intervals and their inversions:
1 - 2 - 3 - 4 - 5 - 6 - 7 - 8
8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
So you see that if we invert a fourth (C-F), we get a fifth (F-C) and vice versa or instead of a second (C-D), we get a seventh (D-C). That's why thirds (C-E) and sixths (E-C) are so valuable here. They are the sweetest intervals.
But Reincken, the creator of the longest chorale fantasia in the North German organ school on record ("An Wasserflussen Babylon" - 327 measures), uses even more advanced counterpoints - invertible counterpoint at the tenth and the twelfth. Here's how the intervals are inverted in these cases:
Invertible counterpoint at the tenth:
1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10
10-9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
Here you can't play two consecutive thirds or sixths because in inversion you will get forbidden parallel octaves or fifths.
Invertible counterpoint at the twelfth:
1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10-11-12
12-11-10-9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
In this counterpoint, thirds are very useful because you get tenths in inversion. But sixths are used very carefully because the inversion would create dissonant sevenths.
Remember that you, like those students of Sweelinck can experiment with invertible counterpoint by writing short melodies and supplying them with counter-melodies with desired intervals. Always play everything you write down to really discover what works and what doesn't.
Marcia funèbre, Op.59 by Johan Adam Krygell (1835-1915) who was a Danish organist and composer of the Romantic period.
God, That Madest Earth and Heaven
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