THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

18.8 Musical Instruments 521Table 18.1. Interval Nomenclature and Frequency Ratios.Largest Integer OccurringInterval (second) Frequency Ratio in the RatioUnison 1:1 1Octave 2:1 2Fifth 3:2 3Fourth 4:3 4Major third 5:4 5Major sixth 5:3 5Minor third 6:5 6Minor sixth 8:5 8Second 9:8 9simplistic, but the Chinese philosophers in Confucius’s time also regarded smallnumbers 1, 2, 3, 4 as the source of all perfection.The Swiss mathematician Leonhard Euler adopted the psychological approachin declaring that the human mind takes pleasure in law and order, particular innatural phenomena. His theory of harmony is this: the smaller the numbers requiredto express the ratio of two frequencies, the easier it is to find this law and order,thus making it more pleasant to hear the combined sounds. Euler went so far as topropose a definitive measure of the dissonance of a chord. His idea was to expressthe frequency ratio of a specific chord by the smallest number possible and then tofind the common denominator for these frequencies. For example, the frequencyratio of the common chord CEG c ′ is 4:5:6:8. The least common denominator is120, since it is the smallest number of which 4, 5, 6, and 8 are all factors. Butthis theory falls apart when the same denominator is assigned to the chord ofthe seventh CEFGB (frequency ratios 8:10:12:15) that turned out to be far moreunpleasant to listen to.18.8 Musical InstrumentsMusical instruments fall into four categories: string, wind, percussion, and electricalinstruments. A string instrument may have its strings struck, bowed, orplucked. Wind instruments can be sub-classified as single-mechanical reed,double-mechanical reed, lip reed, air reed, and vocal-cord reed. Percussion instrumentsare classified as being either definite pitch or indefinite pitch. The adventof electronics has given rise to a whole new class of instruments, such as synthesizerswhich can effectively replicate the sounds of conventional strings, winds,and percussion instruments as well as generate unusual sounds not heard from anyother instruments. Even personal computers can function as musical instrumentsprovided they are equipped with special soundboards and speakers and they areprogrammed to simulate various types of instruments. Table 18.2 lists a numberof musical instruments and their respective classifications.