James Huntingford

The Mathematics of Equal Temperament

So what is our principle issue with ET? It is that, save for the octave, none of the intervals are pure. But why is that? Here is a mathematical demonstration of the problem of ET with respect to pure intervals. Before we start, it is important to clarify the difference between absolute values and relationships with respect to the scale of pitch, hertz (Hz). The hertz scale is an exponential scale. This means that intervals are not defined by differences in their absolute values (e.g. 20Hz apart), but by their relationship (5:4, 3x etc).

Let x be the value of a note, C. The value of the next C in the scale is 2x, as per the ratio of 2:1 for a perfect octave. The relationship of twelve equally spaced semitones (ET semitones) within this octave can be expressed using the twelfth root of 2, which is 1.05946309436. This is the number that, if multiplied by itself 12 times will get us to 2 (the octave). Therefore the frequency of C# is 1.05946309436x. Multiplying this number by itself y number of times takes one up in pitch y number of semitones (this process is performed in the third column).

CUnisonxx (1:1)0
C#/DbMinor 2nd1.05946309436x11.73 (-)
DMajor 2nd1.122462x3.91 (-)
D#/EbMinor 3rd1.189207x1.2x (6:5)15.64 (-)
EMajor 3rd1.259921x1.25x (5:4)13.69 (+)
F4th1.334840x1.33x (4:3)1.96 (+)
F#/GbAugmented 4th1.414214x17.49 (+)
G5th1.498307x1.5x (3:2)1.96 (-)
G#/AbMinor 6th1.587401x1.6x (4:3 + 6:5)13.69 (-)
AMajor 6th1.681793x1.66x (4:3 + 5:4)15.64 (+)
A#/BbMinor 7th1.781797x3.91 (+)
BMajor 7th1.887749x11.73 (+)
C8th (octave)2x2x (2:1)0

In the playing of conventional triadic harmonies (major and minor chords), the intervals that result are thirds (C to E), sixths (E to C) fifths (C to G) and fourths (G to C). Therefore, in the playing of tonal music, our concern should be principally with the good tuning of such intervals. Other intervals such as 2nds and 7ths are inherently tensional; the tuning of such intervals is not our immediate concern.

The table above is designed to demonstrate the tuning variation between Equal Temperament (column 3) and Just (pure) tuning (column 4). The discrepancy is best described in cents (column 5). A distance of 100 cents is the equivalent of an equal tempered semitone. Therefore, two notes an octave apart would be 1200 cents apart. One can observe in the corner of the table that, as the tuning discrepancy of the octave is 0, there is no variation from pure tuning. This means that the octave is perfectly in tune. However one immediately proceeds to notice that every other interval is out of tune, either flat (-) or sharp (+).

As shown in the table, fourths and fifths are 1.96 cents out of tune – this is a negligible error. Our concern is with the other ‘building block’ intervals, the 3rds and the 6ths, which are 13.69 and 15.64 cents out of tune. For a comparison which may be helpful to orchestral players, 4 cents is roughly equivalent to 1 Hz at a pitch of 440Hz. Therefore the dissonance of the thirds and sixths in ET is, in a sense, equivalent to two musicians playing together at 440 and about 443.5Hz respectively. The sound is not unbearable, but it is certainly enough to squeeze out a considerable amount of beauty. And it is certainly enough for hundreds of years of musical history to avoid the implementation of ET, in order to continue to enjoy consonant triadic harmony. Return to the Tuning and Temperaments homepage to learn more about historical tuning systems.