Technical Library

TEMPERAMENTS II: Spiral of Keys

Entire Contents Copyright © 2012 CBH
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Francesco Antonio Vallotti (1697–1780) 24K jpeg
WIKIMEDIA COMMONS / PUBLIC DOMAIN

Pythagoras of Samos (c570 BC–c495 BC)
detail of woodcut from
Gaffurio’s Theorica musicae, 1492

Pythaogrean spiral ©2012 CBH 8K gifUnderstanding the Pythagorean comma before beginning to tune a few popular historic temperaments…

If you search hard enough on the net, you might even find some tuning directions. It is sometimes hard to make much sense of them because while they are intended to be helpful, they are often written by people who already know how to tune! Perhaps you will find the hints on these pages a little easier: There’s not too much science here.

Pythagoras discovered in the sixth century BC that if you tune a series of acoustically pure perfect fifths (C–G–D–A–E–B–F–C–G–D–A–E–B…) by the time it looks like you are getting back to the beginning you are actually out of tune: The B is not the same note as the C you began with. (This can be proven mathematically.) In fact, the B is sharper than the C by almost a quarter of a semitone, an amount we call the Pythagorean comma. What you have isn’t a Circle of Fifths, but a spiral, much like an orange peel.

In Equal Temperament (or evil temperament, if you like—the way modern pianos are tuned), this error is divided amongst all the fifths, so that each one is narrowed by an almost imperceptible amount to make the B equal a C. There, the spiral has been squashed into circle. As a result, each key is equally out of tune—you are paying your piano tuner not to tune your piano, but to actually make it equally out of tune!

When they were running the piano tuning course at the Sydney Conservatorium a few years back, they locked the six or seven applicants in rooms and told them to tune the pianos six days a week, no holidays. At the end of the year, they could tune equal temperament pretty well indeed.

The earlier temperaments are much easier to set, because fewer intervals are tempered, and those that are, are tempered to a much greater degree than all those fifths in equal temperament. This means that even ordinary musicians can hear them. You don’t need perfect pitch, just to be able to listen. And you needn’t lock yourself away for a year to learn how. Just read on…


Temperament index:


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