Per Nørgård - Musikalske stemningssystemer


				Tuning systems in music By Jørgen Mortensen

				When one considers how radically innovative Per Nørgård's thinking is with regard to many concepts in music - harmonics, melody, rhythm, for example - it is not surprising that he also has a special approach to the tuning of instruments. Musicians have been occupied for centuries with the question of how to tune keyboard instruments. There is always a limit, always something or other that does not quite fit, so that people are always trying to find a better compromise. A real Gordian knot! Nørgård, too, was confronted with this Gordian knot, but the way he cut it was very surprising!

Equal temperament				The way in which a piano is normally tuned today is called equal temperament. According to this method, all fifths are a little too small, though this applies to all fifths, and all major thirds are far too wide, though again this applies to all major thirds. In fact, all intervals are false, apart from the octave. In this way, it does not matter what key one plays in - there is no difference between the keys in the sense that some are more false than others. On the other hand, no intervals apart from the octave are 'just', and this is really quite a violent attack on our powers of hearing, which we only discover when we hear a completely true major triad.
Pythagorean tuning				One can decide that all fifths are to be just. Up to a certain point this works out well: we tune the fifths up from C (C, G, D, E, B, F sharp) and the fifths down from C (F, B flat, E flat, A flat, D flat). But the fifth F sharp - C sharp has to be read as F sharp - D flat (a diminished sixth - the so-called 'rogue fifth'). It will be terribly false, but this is unavoidable; whichever way you look at it, 12 true fifths are larger than 7 octaves. This kind of tuning based on true fifths is called Pythagorean tuning, but its importance is purely theoretical. The major thirds become far too wide - even wider than those created by tuning in equal temperament - and the 'rogue fifth' is truly frightening! The following sample score shows Pythagorean tuning with its divergence from equal temperament tuning noted in cents for each note. One cent is one hundredth of a semitone (in equal temperament). A true fifth is two cents larger than an equal temperament fifth, so that 12 fifths together will be 24 cents too high - The Pythagorean Comma.


True tuning				In true tuning, the first task is to make sure that the most important triads are true. From C one tunes a true C major triad, C-E-G; from G one tunes a pure G major triad, G B D; from C one tunes F a pure fifth below, and A a pure major third above, so that the F major triad also becomes pure. At this point one has in fact a C major scale, and all the triads of the scale are pure - apart from the D minor triad! To achieve a pure D minor triad one has to abandon the G major triad by re-tuning the note D. Not even a diatonic scale in which all triads are pure is possible on a keyboard instrument! Time and time again we come up against this barrier… The score sample below shows all 12 notes in true tuning. Note that a true major third is 14 cents less than the corresponding major third in equal temperament - in other words, it is far too wide: