Musical Offerings, Fall 2018

62 True ⦁ Intonation and Modulation determines how consonant or dissonant they sound together. As noted by Pythagoras, the most consonant sounds are generated when the frequencies of the notes can be expressed as a simple ratio (i.e. 2:1, 3:2, 4:3, etc.) . 2 Two notes sound pleasant together if their frequencies can be written as integer ratios of each other. What does this mean? Suppose the note A4 (440 Hz) is played. The frequency of the most consonant note would be either 220 Hz (2:1 below) or 880 Hz (2:1 above). Musically, this 2:1 ratio corresponds to an octave; thus, A3 has a frequency of 220 Hz, A2 has a frequency of 110 Hz, and so on. The next simplest ratio comes from the next lowest pair of positive integers—3:2. When the two frequencies 440 Hz and 660 Hz (3 ÷ 2 × 440 = 660) are sounded together, the result is consonant. Musically, this ratio corresponds to a perfect fifth. Other simple ratios can be used to produce the perfect fourth (4:3), major third (5:4), minor third (6:5), and major sixth (5:3). Figure 1 : Chart of Different Frequency Ratios . 3 Interval Frequency Ratio Decimal Cents Perfect Octave 2:1 2.00 1200 Perfect Fifth 3:2 1.50 702 Perfect Fourth 4:3 1.33… 498 Major Third 5:4 1.25 386 Minor Third 6:5 1.20 316 Major Sixth 5:3 1.66… 884 Figure 1 provides a succinct summary of the basic consonant frequency relationships. The final column expresses the frequency ratios in a slightly different way, using cents . 4 Cent value = 3986 × log (frequency ratio) Note that here, log (frequency ratio) represents the base 10 logarithm of the frequency ratio. Thus, an octave has 1200 cents because 3986 × log (2) = 3986 × 0.301 = 1200. Similar calculations can be done for the other ratios. While this process may seem complex, it can be done rather quickly and easily using a calculator. To summarize, cent values are just a more precise way of indicating how far apart two notes are. Cent values 2 Fauvel, Flood, and Wilson, Music and Mathematics , 13. 3 Thomas Donahue, A Guide to Musical Temperament (Lanham, Maryland: The Scarecrow Press, 2005), 5. 4 Ibid.