6.1 Tuning systems  (Page 9/9)

All intervals work in this same way. So, in order for twelve semitones (half steps) to equal one octave, the size of a half step has to be a number that gives the answer "2" (the size of an octave) when you multiply it twelve times: in other words, the twelfth root of two. And in order for a hundred cents to equal one semitone, the size of a cent must be the number that, when you multiply it 100 times, ends up being the same size as a semitone; in other words, the hundredth root of the twelfth root of two. This is one reason why most musicians prefer to talk in terms of cents and intervals instead of frequencies.

Beats and wide tuning

One well-known result of tempered tunings is the aural phenomenon known as beats . As mentioned above , in a pure interval the sound waves have frequencies that are related to each other by very simple ratios. Physically speaking, this means that the two smooth waves line up together so well that the combined wave - the wave you hear when the two are played at the same time - is also a smooth and very steady wave. Tunings that are slightly off from the pure interval, however, will result in a combined wave that has an extra bumpiness in it. Because the two waves are each very even, the bump itself is very even and regular, and can be heard as a "beat" - a very regular change in the intensity of the sound. The beats are so regular, in fact, that they can be timed; for equal temperament they are on the order of a beat per second in the mid range of a piano. A piano tuner works by listening to and timing these beats, rather than by being able to "hear" equal temperament intervals precisely.

It should also be noted that some music traditions around the world do not use the type of precision tunings described above, not because they can't, but because of an aesthetic preference for wide tuning . In these traditions, the sound of many people playing precisely the same pitch is considered a thin, uninteresting sound; the sound of many people playing near the same pitch is heard as full, lively, and more interesting.

Some music traditions even use an extremely precise version of wide tuning. The gamelan orchestras of southeast Asia, for example, have an aesthetic preference for the "lively and full" sounds that come from instruments playing near, not on, the same pitch. In some types of gamelans, pairs of instruments are tuned very precisely so that each pair produces beats, and the rate of the beats is the same throughout the entire range of that gamelan. Long-standing traditions allow gamelan craftsmen to reliably produce such impressive feats of tuning.

Further study

As of this writing:

• The Just Intonation Network has much information about Just Intonation, including some audio examples.
• Kyle Gann's An Introduction to Historical Tunings is a good source about both the historical background and more technical information about various tunings. It also includes some audio examples.
• The Huygens-Fokker Foundation has a very large on-line bibliography of tuning and temperament.
• Musemath has several animations illustrating equal temperament and the math necessary to understand it.
• Alfredo Capurso, a researcher in Italy, has developed the Circular Harmonic System (c.ha.s), a tempered tuning system that solves the wolf fifth problem by adjusting the size of the octave as well as the fifth. It also provides an algorithm for generating microtonal scales. You can read about it at the Circular Harmonic System website or download a paper on the subject. You can also listen to piano performances using this tuning by searching for "CHAS tuning" at YouTube.