Pitch Organization of Quarter-Tone String Trio Using Frequency Ratios of 11/9 and 8/7

Primary author: Michael Williams
Faculty sponsor: Scott Blasco

Primary college/unit: Arts and Sciences
Campus: Pullman


Using just intonation, also called pure intonation, is the tuning of musical intervals as small whole number ratios (such as 3:2 or 4:3) of frequencies. The pitches of this piece are based on two different intervals, or frequency ratios, of 11/9 and 8/7. I create two separate series, one consisting of added ratios 11/9 (11/9, 11/9*11/9, 11/9*11/9*11/9, etc.) and another of added ratios of 8/7. The first organization of pitches is made from a series of 11/9 intervals divided into series of fifths, notated at the nearest quarter tone. An interval of a near-perfect fifth is created by adding two 11/9 intervals together. By adding two 8/7 ratios, the series of frequencies sounds somewhat uneven, and kind of disorienting. In order to combat this unevenness in the 8/7 series, I divide it into three different series based on the series of fifths compiled from the series of 11/9 series. The first series of 8/7 series frequencies matches closely with the beginning of the 11/9 frequency series. The second and third series are tempered by some 45 cents to make the 8/7 series more even between the frequencies. By combining the 11/9 series with the 8/7 series altogether, a microtonal quarter-tone scale is created. This scale is used for my string trio.