Euler: From Sound to Light 159
ABbecause B – F moves strongly to the resolution E – C ( ♪ sound example 10.3). The 4:5:6:7
ratios of a dominant seventh chord involve the previously forbidden sevenths, which
Euler call “ strangers ” ( é trangers ) but invokes “ a musical license ” that allows these “ for-
eigners ” to be welcome. Thirty years after his youthful Tentamen , Euler was extending
and continuing its argument to include contemporary harmonic practice. In 1773, Euler
returned yet again to the foundations of his harmonic theory and the role of interrelated
keys in harmonic practice in “ On the True Principles of Harmony Represented in the
Mirror of Music. ”^17
Euler also put music at the center of his popular account of contemporary science for
an intelligent reader, his Letters to a German Princess (1768 – 1771). Written at the behest
of Catherine the Great (herself originally a German princess) after his return from the
Berlin Academy to the Petersburg Academy, these letters may be the first work of popu-
larization by a great scientist, but they are also an important document in the history and
philosophy of science. Euler discusses music in far more mathematical detail than any
other subject in his Letters , subjecting the princess to quite a bit of the argument of his
Tentamen. After he emphasizes the expressive, dramatic side of music and its attendant
sentiments, Euler reviews his arguments on the nature of light based on its analogy withFigure 10.3
(a) A progression cited by Euler outlining the key of C especially through the penultimate dominant seventh
chord (marked^7
3). Note the parallel octaves between the lower voices, from the second to the third (D – E) and
fourth to the fifth (F – G) chord ( ♪ sound example 10.1). (b) A dominant seventh chord built on the note C (identi-
fied by the C clef on the bottom line of the staff): C, E, G, B ♭ , as dominant seventh in the key of F ( ♪ sound
example 10.2). From Euler, “ On the True Character of Modern Music ” (1764).