From a letter to Forkel, first published in 1871 in the Allgemeine Musikalische Zeitung as the third in a series of articles by H. Bellermann.
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Oder wenn man von C nach e 80 : 81 in vier Quinten vertheilen will, kann es folgender Art geschehen:
Or if one prefers to split 80:81 into four fifths from C to e, it can happen by the following method:
C-G 216 : 323 temperirte Quinte = 2/3 - 1/324 216 : 324 reine Quinte ------------------------------------- G-d 215 1/3 : 322 temperirte Quinte = 2/3 - 1/323 215 1/3 : 323 reine Quinte ------------------------------------- A-e 214 2/3 : 321 temperirte Quinte = 2/3 - 1/322 214 2/3 : 322 reine Quinte ------------------------------------- D-A 214 : 320 temperirte Quinte = 2/3 - 1/321 214 : 321 reine Quinte -------------------------------------[The above is a convoluted way of saying that the fifths are narrow by amounts equal to the ratios 323/324, 322/323, 321/322, and 320/321 respectively, each of which is essentially 1/4 of the syntonic comma. The order of them is curious.]
Auf eine andere Art der Quinten-Excess vertheilt in drei Quinten, wie auch in fünf Quinten:
From another method the Pythagorian Comma splits into 3 fifths, in the same way into 5 fifths:
| | oder | oder | ---------------------------------| C-G | 0 | 0 | -2 | Cis-Gis | 0 | 0 | 0 | D-A | -6 | -5 1/2 | -3 1/2 | Dis-B | 0 | 0 | 0 | E-H | 0 | 0 | 0 | F-c | 0 | 0 | 0 | Fis-cis | -1 | -1 | -1 | G-d | 0 | 0 | -2 1/2 | Gis-dis | 0 | 0 | 0 | A-e | -5 | -5 1/2 | -3 | B-f | 0 | 0 | 0 | H-fis | 0 | 0 | 0 | ---------------------------------| | 12 | 12 | 12 |[The numbers in the last column are novel. Nothing in the letter prepares for them nor explains them.]
Gleichschwebende Temperatur:
C. | Cis | D | Dis | E | F | Fis | G | Gis | A | B | H | c --------------------------------------------------------------------------- | 7 | 2 | 9 | 4 | 11 | 6 | 1 | 8 | 3 | 10 | 5 | 0 ---------------------------------------------------------------------------[The numbers are the amount by which each note differs from its pitch in a series of pure fifths starting from C and proceeding sharpwise.]
Abweichung meiner beiden Temperaturen gegen die angezeigte gleichschwebende:
Divergence of both of my temperaments from the equal one shown:
| C |Cis| D |Dis| E | F |Fis| G |Gis| A | B | H | c |---|---|------|---|---|---|---|------|---|------|---|---|--- I. | 0 |-5 |+2 |-3 |-7 |-1 |-5 |+1 |-4 |-2 1/2|-2 |-6 | 0 i.e., "K II" |---|---|------|---|---|---|---|------|---|------|---|---|--- II. | 0 |-5 |-3 1/2|-3 |-7 |-1 |-5 |-1 3/4|-4 |-5 1/4|-2 |-6 | 0 i.e., "K III" ------------------------------------------------------------[This may be the first time that anyone ever defined a temperament in terms of its deviation from ET. Electronic tuner users: double these numbers. The results are your cents offsets (add a constant if not starting from C).]