continued…

One of the basic applications of frequency ratio analysis of intervals is verification of the fact that the more consonant or “solid” sounding ratios are the lowest numerically or simplest. A brief application to simple melodies and chords shows that as ratios get more complex, dissonance and therefore tension is built. The stereotypical idea of a cadenza and resolution is the ratios falling in complexity back to the more stable sounding consonant, simpler ratios.

No matter how complicated and dissonant the works of many of the classical composers, they almost always ended the piece with a pure major or minor triad closed by the root note on the bottom and an octave of the root on top. This is probably the mathematical basis of the Picardy third ending Bach used, the major third being more consonant than the minor third.

One complication of this analysis is the possible confusion of mathematically simple ratios versus the physiological basis of hearing. It may be that even though a ratio is relatively simple it will sound dissonant to the ear because that specific ratio is harder for the mind to distinguish because of the nature of auditory physiology. Another interesting point is that musically the midway point of the octave is the flat five while mathematically the midway point is the fifth. This occurs because of the logarithmic nature of the major scale on a frequency scale. Human hearing is of course also very logarithmic in structure. Remember that the cochlea is not only curved but also a pattern of circular diminishing tubular shape (thus the snail shell analogy). The placement of the little hairs (cilia) on the tube walls is crucial because sound waves move these little objects and this movement is what triggers the nervous tissue that gets translated into electrically impulses that the brain identifies as “hearing sounds”. The overtone series is also logarithmic in nature with wider intervals at the bottom and with the intervals becoming progressively smaller as the pitches climb higher. The overtone series is another great source of information on the relationship between frequency ratios and musical ideas and structures.