Musical Offerings, Fall 2018

Musical Offerings ⦁ 2018 ⦁ Volume 9 ⦁ Number 2 65 Figure 3 : A comparison of just and Pythagorean intonations. 11 The difference in the wolf fifth is noticeable. The Pythagorean wolf fifth expressed as a decimal is 1.480 or 24-cents flat as discussed before (Pythagorean comma). The just wolf fifth expressed as a decimal is 1.536 or 41-cents sharp. When any type of chromatic modulation occurs, these issues are essentially irresolvable. Because the ratios between the notes are not consistent, shifting the tonic “do” results in many odd intervals. These can sound very dissonant, and thus systems like just or Pythagorean intonation limit the possibilities of musical performance. Although its roots are much earlier, equal temperament (ET) is the system that has been widely used and adopted since the mid-eighteenth century . 12 ET first defines the perfect octave to be a 2:1 frequency ratio. Next, the musical scale is broken into twelve notes each equidistant from each other. Each half step, or semitone, corresponds to changing the frequency by a factor of 2 1 / 12 . A whole step indicates a frequency change of 2 2 / 12 or 2 1 / 6 . After twelve of these equidistant half-step changes, the resulting frequency is 2 12 / 12 or 2—simply twice the frequency of the original note. Thus, the perfect octave has been reached. ET allows for incredibly easy modulation, because chromatic tones are far less dissonant. While ET may seem like the perfect solution to the problem of the Pythagorean comma, it is not quite that simple. As 11 Liern, “On the Construction, Comparison, and Exchangeability of Tuning Systems,” 201. 12 Amanda N. Staufer, “The Unifying Strands: Formalism and Gestalt Theory in the Musical Philosophies of Aristoxenus, Descartes, and Meyer,” Musical Offerings 9 , no. 1 (2018): 34, doi : 10.15385/jmo.2018.9.1.3 .