History of Music Theory, Part Three

Last night I finished Book I of Riemann's "History of Music Theory", which took me through Chapter VII and to the beginning of Book II. Book II starts with mensural notation theory and goes through systematic countrapuntal theory. I had gotten to the point of wanting to make some more general observations, and to draw some conclusions from those observations, but I wanted to review a little more about Pythagorean Tuning versus the Just Intonation that is derived from the natural harmonic overtone series first. This lead me on a vast transcendental, transcontinental surfing session that gave me many insights into the nature of music and it's evolution as influenced by tuning and temperament that I had not percieved before. It wore my little brain out so completely that I had to take a nap, which explains the relative lateness of today's post.

But first, from The Department of Interesting Names: I particularly enjoy saying Guilemus Monachus, Giraldus Cambrensis, Petrus Picardus, and - last but not least - Magister Philippotus Andreas. I like things that are fun to say. Honorable mentions for Johannes Tinctoris and Franchinus Gafurius, and pity for poor Anonymous IV.

After my previous readings about the tenth and eleventh centuries, I pointed out that Western music theory started out as an investigation into the nature of sound as revealed by the natural harmonic overtone series, and that this information was borrowed from the Classical Greek culture. The early Western theorists also adopted the Pythagorean tuning system, which was a seven note diatonic scale built via a series of seven justly intonated perfect fifths starting on F: F, C, G, D, A, E, and B. This may have been partly because the perfectly just fifth expressed by a ratio as 3:2 has theological significance in the Christian religion. Or not. To avoid tritone relationships, the stack o' fifths was added to at each end to get the B-flat and F-sharp respectively.

In any event, by the 12th and 13th centuries, musical practice had gotten ahead of theory because of the conservative nature of the theorists connected with the Church and their reluctance to abandon the Pythagorean tuning system. If we compare diatonic scales built on the natural harmonic overtone series - so-called Just Intonation - with the "same" diatonic scale that results from the stack of Pythagorean fifths, we can easily see where the problem is.


Pythagorean:

C= 1:1
> 204 cent Wholetone
D= 9:8 (+204 cents)
> 204 cent Wholetone
E= 81:64 (+408 cents)
> 90 cent Semitone
F= 4:3 (+498 cents)
> 204 cent Wholetone
G= 3:2 (+702 cents)
> 204 cent Wholetone
A= 27:16 (+906 cents)
> 204 cent Wholetone
B= 243:128 (+1,110 cents)
> 90 cent Semitone
C= 2:1 (+1,200 cents)


Just: (Natural Harmonic Series)

C= 1:1
> 204 cent Wholetone
D= 9:8 (+204 cents)
> 182 cent Minor Wholetone
E= 5:4 (+386 cents)
> 112 cent Semitone
F= 4:3 (+498 cents)
> 204 cent Major Wholetone
G= 3:2 (+702 cents)
> 182 cent Minor Wholetone
A= 5:3 (+884 cents)
> 204 cent Major Wholetone
B= 15:8 (+1,088 cents)
> 112 cent Semitone
C= 2:1 (+1,200 cents)


At first blush the Pythagorean system would seem to have a lot in it's favor: There is only one size of Wholetone, there is only one size of Semitone, and the Perfect Fourths and Perfect Fifths agree with Just Intonation, as does the size of all of the Wholetones (The 9:8 Wholetone, that is). But when you look at the thirds and sixths, you see the problem right away: The major Third from C to E is 22 cents sharp (Nearly 1/4 of an Equally tempered Semitone!), and the Major Sixth from C to A is the same 22 cents sharp compared to the justly intonated intervals. Obviously, thirds and sixths would sound awful in this tuning on the monochord. But, since the medium of the time was primarily a vocal one, and singers tend to make their thirds and sixths conform to the natural justly intonated versions, composers and singers started using them anyway. So, a method of tempering one or the other of these tunings was needed so that theory could rationally explain practice. It would take a while.