Tuning by Ratios


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Introduction

The approach of generating sound by means of a digital computer offers to musicians precise and flexible control over the four basic parameters of sound: amplitude, duration, location and perhaps the most important, frequency. Digital sound synthesis permits the tuning of frequencies to a degree of accuracy far greater than can be distinguished by the human ear. By utilizing digital audio software such as Csound, it is thus possible to explore in detail the central issue of intonation, namely, "What are the exact frequencies to use in a given passage of music?" Through various illustrations of chords, as well as a detailed tonal analyses of short pieces by J. S. Bach, this paper demonstrates that integer ratios of frequencies (such as 2:3, 5:4) are at the very heart of traditional Western tonality. Included in this paper are Csound orchestra and score files, along with computer programs in generic C that may be used in conjunction with them.

 

A Close-up Look at Frequency

Sounds produced by musical instruments are usually quite complex. The sound of a trumpet, for instance, is not a pure sound, but rather a collection of sine waves whose frequencies change in amplitude continuously over time. If the frequency of the lowest component sine wave, the fundamental, is designated as 1, the values of the other sine waves, the harmonics, will be {2, 3, 4, . . .}. For example, a note whose fundamental is 110 Hz (concert A), will have harmonics at frequencies of 220 Hz, 330 Hz, 440 Hz, etc.

With a burst of well-aimed air, a trumpeter plays a single note, setting into motion a subtle and complex cascade of events. During the rise and fall of the note's amplitude known as attack and decay, the individual harmonics will change dynamically in amplitude according to numerous factors. These factors include the frequency of the fundamental; the amount of tubing currently used; the ratio of cylindrical to exponential tubing; the acoustics of the concert hall; and the real-time input of the performer through the use of muscles in the face, lips, tongue, throat, chest, abdomen and hands. Needless to say, such a complex sound is impossible to analyze in complete detail, especially when other criteria such as combination tones are considered. It is testament to the complexity of sound that a single note from a single instrument cannot be completely understood.

An idealized steady state sine wave, on the other hand, is quite simple. There is only one frequency, the fundamental. The amplitude is constant. A digital computer driving a high-quality speaker can closely approximate this wave, which sounds like a tuning fork that is annoyingly steady and pure.

For the sake of simplicity, imagine an idealized guitar string that can produce only one steady state sine wave. (Real strings always have harmonics.) This string is stretched between two posts and set into motion with a pluck. If you viewed one cycle of this sine wave in slow motion, you would see that the string moves up, then down, then back to its center position, completing one cycle. The end of one cycle marks the beginning of the next. Figure 1 represents four distinct snapshots of this string.


Figure 1
One Cycle of an Idealized Oscillating Guitar String


The number of times the string completes this cycle in a given period of time is known as its frequency. Frequency is traditionally expressed in Hertz (Hz), also known as cycles per second (cps). A guitar string that moves up and down 440 cycles per second has a frequency of 440 Hz, the same frequency as an "A" tuning fork.

Now imagine adding between the same two posts a second guitar string identical to the first one, but strung at precisely four times the tension. (Vincenzo Galilei, father of Galileo, discovered that the ratio of an interval is proportional to string lengths, but varies as the square of the tension applied to the strings. Thus, to double the frequency of a string, either halve its length or quadruple its tension.) Both strings are plucked in an identical fashion at exactly the same moment. In a slow motion movie of this event, you would see that for every oscillation the first string makes, the second string makes exactly two. The two strings therefore align precisely at their original positions for every oscillation of the first string, and every other oscillation of the second. This frequency relationship of 2/1 is known as an octave.

Any such periodic relationship between two frequencies can be described as X/Y, where X and Y are integers. For example, an octave bears the relationship of 2/1; a perfect fifth is equivalent to 3/2; a major third can be tuned as 5/4.


Robert Asmussen
1997


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