Wednesday, June 01, 2005

History of Music Theory, Epilog

The last two chapters of Riemann's "History of Music Theory" address the revolution in acoustic theory that occured at the beginning of the 16th century, but in a dissapointinly cursory manner, and a lame chapter on musica ficta that I found to be less than useless finishes off Book II. No detail is given of the revolutionary nature of Pietro Aaron's 1/4 Comma Mean Tone Tuning System - the first attempt at a real temperament - and no comparison is offered with the presiding Pythagorean tuning or the natural harmonic overtone series' Just ratios. For that reason, I had to do some further research.

Pietro Aaron's 1/4 Comma Mean Tone Tuning System of 1523 was a revolution theoretically, but as usual theory was behind practice. Organ builders had been tempering their tunings for some time - probably for around a century - and tuning the organ had long before become a "black art" of sorts. This is evidenced by some passing remarks found in earlier treatises that said organ builders flatted their fifths slightly, and that the interval "tolerated" that practice quite well. In any event, Aaron was the first to enumerate this practice of temperament. The chromatic scale he proposed looked like the following (The size of semitones in cents are listed in between notes, and the comparison with the Natural Just Ratios and Pythagorean intervals are to the side):


(76 cents)

C-sharp (+76c= -36c vs. P @ +90c= -22c)

(117 cents)

D (+193c= -11c vs.P @ +204c= 0c)

(117 cents)

E-flat (+310c= -6c vs. P @ +294c= -22c)

(76 cents)

E (+386c= 0c vs. P @ +408c= +22c)

(117.5 cents)

F (+505.5C= +7.5c vs. P @ +498c= 0c)

(82.5 cents)

F-sharp (+586c= -4c vs. P @ +588= -2c)

(110.5 cents)

G (+696.5c= - 5.5c vs. P @ +702c= 0c)

(84.5 cents)

A-flat (+781c= -33c vs. P @ +792c= -22c)

(108.5 cents)

A (+889.5c= +5.5c vs. P @ +906c= +22c)

(116.5 cents)

B-flat (+1,006c= +10c vs. P @ +996= 0c)

(77 cents)

B (+1,083c= -5c vs. P @ +1,110c= +22c)

(117 cents)

C (+1,200c= 0c vs. P @ +1,200c= 0c)

The first thing that strikes one abut Aaron's system is the sheer complexity of it. There are no less than eight different sized semitones out of twelve! But, the most important thing is the obvious primacy of the third here: The major third is naturally pure, and the minor third is only flat by -6 cents, versus the Pythagorean thirds, which are +22 cents and -22 cents respectively. The perfect fourth and perfect fifth have lost their primacy, as is evidenced by their respective +7.5 cent and -5.5 cent temperings. Note how the flattened fifth corresponds with the descriptions of what was common practice for organ builders and tuners for a long time as evidenced by passing mentions in earlier theoretical treatises. Keep in mind that the technology for acurately measuring the frequency of a note was still centuries away, so this had to be done by ear, so it was still much more of an art than a science.

Another thing to note is the gross flatness of the minor sixth: It is actually eleven cents flatter than the Pythagorean minor sixth at -33 cents versus -22 cents! This is doubtless a holdover of the notion that the minor sixth was a dissonant interval.

Though the 1/4 Comma Mean Tone Temperament would seem to us to obviously be an unsatisfactory arrangement, it caught on in a big way and was still the standard method of temperament at the beginning of J.S. Bach's career almost two-hundred years later.

It is interesting to me as a guitarist that the equal temperament system, or something so close to it as to be virtually indistinguishable from it, was by this time under the very noses of these theorists. As soon as the first musician tied frets onto an al Ud to make a Lute, what else could he have done? Not only that, but even at least one ancient Greek writer, Aristonexus, had proposed equal temperament - and all of those writings were widely circulated and well known by this time - but it is still never mentioned in the Western tradition (Not that I have found or can remember, anyway).

Since Riemann leaves us at Zarlino, I will pick up that thread where he left off with Zarlino's "The Art of Counterpoint", which is Part III of his Le Istitutioni Harmoniche of 1558.


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