Tosi and commas

Continue of discussion from: A Capella Cantatas [General Topics]

Bradley Lehman wrote (April 24, 2004):
To clarify: Tosi here evidently did not recognize that there is a difference between the Pythagorean comma (the one that comes from tuning by pure fifths) and the syntonic comma (the one that comes from tuning by major thirds).

So, when he complains that some people assert there are only seven of them in a tone and suggests that they're wrong: he's thinking of the syntonic comma (of which there are indeed approximately 9 in an average tone), while those he's criticizing are referring to the Pythagorean comma (of which there are indeed approximately 7 in an average tone). Tosi's other remarks about D# and Eb further make it clear that he's dealing with the syntonic comma (81/80), not the Pythagorean one ((3/2)^12)/(2^7).

The issue is clouded further by the existence of at least three different sizes of tone: a 9:8 ratio, a 10:9, and a "meantone" sqrt(5):2.

So, Tosi's command of music theory was not quite solid, at least on the mathematical side. Nevertheless, Tosi clearly knew the most important thing a practical musician needs to know here, to sing either a D# or Eb in tune with colleagues: that the two pitches are quite far apart, and that the musician must be flexible enough to provide either one to a musical context as appropriate.

Further practical info about all this is available at my page: http://www-personal.umich.edu/~bpl/temper.html

Thomas Braatz wrote (April 24, 2004):
>>So, Tosi's command of music theory was not quite solid, at least on the mathematical side. Nevertheless, Tosi clearly knew the most important thing a practical musician needs to know here, to sing either a D# or Eb in tune with colleagues: that the two pitches are quite far apart, and that the musician must be flexible enough to provide either one to a musical context as appropriate.<<
And this is exactly what Agricola points to as being passé, reflecting a different time, place and performance tradition than the one to which he and his reader belonged. For singers, conforming to colleagues who played the same high note (as the one being sung) on a fixed instrument such as a church organ would still occasionally be a problem 'hanging over' from a previous age (musically intelligent organists would avoid this, of course;) but, by and large, all other instrumentalists could 'bend' the pitches as required by the new temperament standards. The singers, according to Agricola’s later commentary on Tosi, need no longer be as concerned with these specific, earlier differences but are now confronted with even more complicated pitches that conform more to what is gradually becoming the equal temperament standard, a solution to numerous previous problems that some musicians like Bach discovered were to be the ‘wave of the future,’ but a standard which, for other, more conservative musicians, took further decades of experimentation before it would be adopted completely (by the middle of the 19th century.)

Bradley Lehman wrote (April 24, 2004):
[To Thomas Braatz] Again, you're citing (and inventing) hagiography that is not proof. Meanwhile, the proof of Johann Sebastian Bach's preference for a specific UNEQUAL temperament is right here on my desk.