Suppose we wanted to create a scale that permitted only justly tuned intervals. All octaves would have a frequency ratio of 2:1, all fifths would have a frequency ratio of 3:2, all major thirds would have a frequency ratio of 5:4, and so on. We might also require that all major seconds have a frequency ratio of 9:8.
It can be shown that mathematically that such an "ideal" tuning system is impossible. Consider, for example, the goal of having both perfect fifths and perfect octaves. Let's begin with the tone A-440 Hz, and tune a series of 12 fifths:
Notice that we began with A-440 Hz and ended with A-446 Hz. This difference is known as the Pythagorean comma and has been known since ancient times. The comma amounts to 23 hundreds of a semitone (or 23 cents). The Pythagorean comma arises from the fact that the powers of 2 and the powers of 3 never intersect. That is, 2 X 2 X 2 ... can never converge with the series 3 X 3 X 3 ...
A4-E5 440 X 3/2 = 660 Hz E5-B5 660 X 3/2 = 990 Hz Transposed down one octave = 495 Hz B4-F#5 495 X 3/2 = 742.5 Hz F#5-C#6 742.5 X 3/2 = 1113.75 Hz Transposed down one octave = 556.875 C#5-G#5 556.875 X 3/2 = 835.3125 Hz Transposed down one octave = 417.66 G#4-D#5 417.66 X 3/2 = 626.4844 Hz D#5-A#5 626.4844 X 3/2 = 939.7266 Hz Transposed down one octave = 469.8633 A#4-F5 469.8633 X 3/2 = 704.7949 Hz F5-C5 704.7949 X 3/2 = 1057.1824 Hz Transposed down one octave = 528.5962 C4-G4 528.5962 X 3/2 = 792.8943 Hz G4-D5 792.8943 X 3/2 = 1189.34123 Hz Transposed down one octave = 594.6707 D4-A4 594.6707 X 3/2 = 892.0061 Hz Transposed down one octave = 446.0030
Of course it is possible to continuing tuning more fifths that just 12. The "circle" of fifths approaches the starting point after 12, 41, and 53 tunings of fifths. As the number of tones increases, the size of the comma gets smaller, but it never goes away entirely. No number of tunings will return you to the precise starting point. It is therefore impossible to create a scale that provides a just octave interval from every note as well as a just fifth from every note.
A similar case can be made for just major thirds (5:4 ratio). No scale can be created where every pitch provides a just octave interval and a just major third. Nor can a scale be constructed that provides a just octave and just major third from every pitch.
There are four general ways of dealing with the Pythagorean comma: (1) ignore it, (2) minimize it, (3) adapt to it, or (4) hide it. The first approach is to simply abandon the goal of creating music using just intervals. Indeed, in many musical cultures, such as in the musics of Bali and Java, the common tuning systems bear little similarity to just intervals, so it is possible that there is no underlying preference for just intervals.
There are two ways of minimizing the Pythagorean comma. One is to create a tuning system with so many notes per octave that the mistuning is small. This approach is evident in the music of Harry Partch, who created his own musical instruments capable of playing 43 notes per octave.
A second way of minimizing the Pythagorean comma is to limit the music-making to a small number of notes. It is possible to tune a pentatonic scale so that most intervals are fairly close to their just values. Even better, one might limit music-making to a couple of drone tones tuned a fifth apart. Before the classical period, modulation to different keys was uncommon. Consequently, it was possible to use tuning systems optimized for particular keys (like C major).
A third approach is to continuously adapt the tuning as the music unfolds. This can be done with a computer, where the tuning of each successive note in a sequence is adjusted so that just intervals are always used. Unfortunately, this adaptive approach causes the "tonic" pitch to vary over the course of a melody: the "doh" you end with will not necessarily match the "doh" you begin with. For many melodies, most listeners find such adaptive tuning to be unpleasant. Another problem with this approach is that it only works for single-note melodies. Once you add harmony, it is impossible to ensure that both the harmonic and melodic intervals are just. A final disadvantage is that adaptive tuning is really only practical using a computer. It would be a significant challenge for human performers to adopt adaptive tuning.
The history of Western music has relied almost exclusively on the fourth approach: mask or hide the Pythagorean comma in some way. There are several ways to mask the effect of the non-just tunings. A simple approach is to add vibrato so mistunings are difficult to hear. Research has also shown that listeners have greater difficulty hearing mistunings of tones that are short in duration. So composing music using durations that are predominantly less than half a second will mask the effects of mistuning.
Normally, people consider hiding the Pythagorean comma by using a "compromise" tuning system. The most popular compromise tunings spread the comma over a number of notes. These compromise systems are referred to as "temperaments". Lots of temperaments have been advocated over the centuries. The following table characterizes just five. The numerical values in the table indicate how much a given interval deviates from the just interval in cents (hundredths of semitones).
Name of Tuning System Fifth Major Third Minor Third Major Second Minor Second Pythagorean Tuning 0.0 +21.5 -21.5 0.0 -21.5 Equal Temperament -2.0 +13.7 -15.6 -3.9 -11.7 Silbermann Tuning -3.9 +5.9 -9.8 -7.8 -2.0 Meantone Tuning -5.4 0.0 -5.4 -10.8 +5.4 Salinas Tuning -7.2 -7.2 0.0 -14.3 +14.3
Is there any way to judge whether one temperament is better than another? Can listeners hear the difference? If they can hear the difference, does the difference matter?
There are four main reasons why modern scholars have lost interest in the question of what is the best tuning system. First, in the 1930s, Carl Seashore measured the pitch accuracy of real performers and showed that singers and violinists are remarkably inaccurate. For non-fixed-pitch instruments, the pitch accuracy is on the order of 25 cents. Yet Western listeners (and musicians) are not noticeable disturbed by the pitch intonation of professional performers. Secondly, on average, professional piano tuners fail to tune notes more accurately than about 8 cents. This means that even if performers could perform very accurately, they would find it difficult to find suitable instruments. Thirdly, listeners seemingly adapt to whatever system they have been exposed to. Most Western listeners find just intonation "weird" sounding rather than "better". Moreover, professional musicians appear to prefer equally tempered intervals to their just counterparts. See the results of Vos (1986). Finally, the perception of pitch has been shown to be categorical in nature. In vision, many shades of red will be perceived as "red". Similarly, listeners tend to mentally "re-code" mis-tuned pitches so they are experienced as falling in the correct category. Mis-tuning must be remarkably large (>50 cents) before they draw much attention. This insensitivity is especially marked for short duration sounds -- which tend to dominant music-making.