Common-tone relationships constructed among scales tuned in simple ratios of the harmonic series and expressed as values in cents of twelve-tone equal temperament
Peter Lucas Hulen · 2006
Abstract: – Intervals between observable harmonics in the frequency spectra of identifiably pitched sounds occur in a serial pattern of simple ratios. A new musical scale has been invented with intervals in these ratios, and a series of scales created to share common tones, allowing for musical modulation. Twelve-tone equal temperament (12Tet) is the assumed standard for musical instruments of recent Euro-American origin. It limits division of the octave—the interval between frequencies having a 2:1 ratio—to twelve equal, smaller intervals. For tuning, each smaller interval (semitone or half-step) is divided into 100 cents, resulting in division of the octave into 1,200 fine increments. Simple ratios of the harmonic series, the basis for the new scale, do not conform to the standard semitones of 12Tet. Digital sound synthesis provides great flexibility, but writing programs specifying exact frequencies is tedious, time-consuming, and detracts from composing music with the results. Commercial synthesizers correlate fine-tuning with the intervals of 12Tet. Octave frequencies being a ratio of two, 12Tet dividing octaves into twelve equal intervals, the value of any one thus being the twelfth root of two, a logarithmic equation must be used to calculate values in cents for simple-ratio intervals. Commercial synthesis equipment can be adjusted accordingly, and new music composed with the new scale.