Werckmeister's (mostly) helpful monochord numbers

Reference is Chapter 31 of Musikalische Temperatur. I obtained a copy of relevant pages of the facsimile (80-81 and Errata) and compared with the Hehr translation.

The first question is - What is Werckmeister talking about? Chapter 31 comes after the end of Chapter 30 where the tuning procedure for 'Werckmeister IV' (the tuning with 1/3 comma temperings) is described verbally. C-G tempered, G-D pure, D-A tempered, A-E pure etc.

Then it says 'This temperament is also represented on the monochord [with] a little discrepancy, without any laborious procedure.' So he is still talking about 'IV' and giving some monochord numbers that are nearly equivalent to the temperament but easier to apply.

Just after the numbers he talks of the practical problems that may arise when using a monochord, and says it could be used to tune a harpsichord or clavichord ('Instrument oder clavichordium'). Elsewhere in his works Werckmeister mentions how difficult it is to tune organs to monochords because of the difference in their characters of sound.

So let's look at the numbers. In the facsimile they are

C 120 C# 114 1/2 d 107 1/5 D# 101 1/5 E 95 3/5 F 90 F# 85 1/3 G 80 2/5 G# 76 2/15 A 71 7/10 Bb 67 1/5 B 64 C 60.

However Hehr's translation gives 80 1/5 for G. This is a mistake and leads to the erroneous idea that G-D is tempered, which obscures the derivation from 'IV'. Johnny R. converts the numbers to decimals and gets G 80.2 - which ought to be 80.4. (In cents, 693.3 not 697.6).

Since the pattern of pure and tempered fifths is now clearly supposed to be the same as 'IV', we can deduce a further misprint as C# 114.5 is not a pure fifth from G# and is a very flat fifth from F# 85.33. (Remember these are string lengths!). But if we enforce C#-G# to be a pure fifth we get

C# = 114.2 or 114 1/5

It is easy to allow for a misprint 1/5 -> 1/2 here. Having taken care of this we get a scale

0
85.8
195.3
295.0
393.5
498.0
590.2
693.3
787.7
891.6
1003.8
1088.3

the approximation to 'IV' using monochord numbers, which W. thinks can be acceptable for stringed keyboard instruments. There is no 'extra' Werckmeister tuning, unless you count the result of superimposed typos as an extra...

http://en.wikipedia.org/wiki/Werckmeister_temperament

Best,
~~~T~~~