II. Musical Ratios
Musical ratios are simply another way to denote pitch. In Western music we typically denote pitches by using the letters A, B, C, D, E, F, G. If we need to refer to a pitch in between one of these note names, we can use a sharp (#) or flat (b) symbol based on the context. There are, however, other ways in which to denote pitch. The most objective, and thus the one used most often by acousticians, is frequency. Sound is transmitted in waves, and frequency is a measurement of the number of complete waves (or cycles) that occur in one second. The number of cycles per second determines the pitch of atone. For example, 440 cycles per second (or 440hz) is recognized in the West as the pitch ‘A.’
It is common practice to build ratios upwards by putting the smaller number on the bottom, and Partch adheres to this. Yet, he points out that ratios can also be built downwards. If we descend from the source tone (rather than ascend from the sources tone, as in the example above) we obtain a result that is more like a simple division of the string length. In fact, the string length is one of the key factors in the pitches that it generates. If we decide that the 1/1 is the note ‘G,’ then the 3/2 above ‘G’ is ‘D.’ The 2/3 below ‘G’ is also a ‘D,’ but an octave below the previous ‘D.’ The fact the these two ratios represent the “same” pitch , while at the same time reveal a different relationship to 1/1, is a crucial principle in Partch’s theory of “Monophony.”
Another significant aspect of musical ratios is that they represent both a pitch, and an interval. In other words, the 3/2 above 1/1 tells us not only which note we have encountered, but also its relationship to (or the distance from) the source tone (1/1). It was explicitly this dual function of ratios that Partch believed to be more meaningful than letter note names. While a ‘D’ can indeed be recognized as a perfect fifth, it is only so in relation to ‘G.’ A 3/2, on the other hand, is always a3/2, regardless of its context.
The system of tuning pure (or “just”) intervals predates the equal temperament system by thousands of years. Through out Western musical history, many theorists and composers—including Partch—have preferred just intonation because it is more closely aligned with the natural “growth” of a tone. When a tone is sounded our hearing is immediately drawn to the fundamental tone, which can be described as the pitch produced by the entire string on the monochord. At successively higher frequencies, however, smaller segments (1/2, 1/3, 1/4,etc.) of the monochord’s string are also producing their own pitches at the same time. This succession of frequencies is generally referred to as the overtone series (also called the harmonic series, or upper partials). The chart bellow shows that, if a string is vibrating at 110 cycles per second, each 1/2 will also vibrate at 220 cycles per second. In a inter-related play of numbers, successive segments of the string (1/3, 1/4, 1/5 etc.) can be expressed as a successive integers multiplied by the fundamental (110 x 3 = 330, 110 x 4 =440, 110 x 5 = 550), and with each new integer, a new overtone is produced.
In just intonation the intervals above are tuned as they naturally occur. Equal temperament tunings, in contrast, must squeeze intervals slightly in order to make them synthetically equal. Although the octave and perfect fifth are nearly as pure as they are in just, on average all the other intervals in equal temperament are slight smaller than nature intended. What Partch also found intriguing about tuning in just intonation is that, if one continues with successive integer as in the method above, intervals—and thus pitches—begin to appear that do not exist in equal temperament. A 7/6, for instance, produces a tone slightly smaller than a minor third, and an 8/7produces a tone slightly larger than a major second. One can theoretically continue in this way to produce an infinite number of “new” pitches. Partch decided to stop with the number eleven, but he continued to create new microtonal resources by adding ratios together. His music is decidedly focused on forty-three tones to an octave, but its important to understand that this was his decision as a composer, not the theoretical limit of his system.