How equal temperament ruined harmony (and why you should care)

In ancient times, music, under the title of harmony, belonged to the same class as the studies of arithmetic, geometry and astronomy: all disciplines involving the numbering of space and time. Numbers were believed to be the noblest thing the human mind could concern itself with; scientific principles were thought to be a reliable path to objective knowledge, from which Truth, Goodness and Beauty could be explained.

It is probably a mere anecdote (meanwhile it has been proven not to work in this way with hammers), but the story goes that Pythagoras (ca. 570 – ca. 500 b.c.), upon passing by a smithy, heard harmonies each time the hammers hit the anvil. He noticed that the difference in the pitches he heard stood in relation to the mass of the hammers. A hammer half the mass of another produced a tone one octave higher; at 1/3 of the mass, what sounded was the fifth of the original tone. An illustration from Gaffurius’ Theoretica Musica (1492) shows the different instruments Pythagoras allegedly then used to investigate the mathematical relationships between musical intervals.

Although again it cannot be proven, we assume that it was also Pythagoras who invented the monochord.  This is a one-stringed instrument with a bridge that can be shifted to study the intervals shown above. When the bridge is placed in the middle, thus dividing the string exactly in two (2:1), the octave can be heard, placed at 1/3 of the string length (3:2), one hears the fifth, at 1/64 the fourth (4:3) and at 1/5 the major third (5:4). When the root note or fundamental is struck, not only does the string vibrate in its entire length, but simultaneously also between the so-called ‘nodes’ at 1/2, 1/3, 1/4 etc. of the string length. These make the so-called ‘pure’ intervals, and their consecutive series is referred to as the ‘overtones’. The places where the waves swing outward are called ‘antinodes’.

Each root note contains a major chord in the first four overtones. By using the second and third overtones (the fifth and the fourth), Pythagoras calculated the seven tones of a scale. Starting from C we thus have: C-E-G, F-A-C and G-B-D.

However, nature seems to have saddled us with a frustrating problem, because the natural proportions are not simply beautiful or even functional. It should seem logical that pure fifths piled one upon the other – we need 12 to reach the root note again (but then 7 octaves higher) – should result in the same note, but curiously, we end up substantially higher, namely 1/4 of a semitone. This difference, between the note on which the circle began and the twelfth fifth that turns out too high, is called the Pythagorean comma.   

It is difficult to know what people would have done in daily practice on instruments that were not bound to a fixed tuning; furthermore we can ask ourselves whether a travelling minstrel was really troubled by this comma. The Greek philosopher Aristoxenos (360-300 b.c.) rated the ear more reliable than the calculations of Pythagoras. Still, from all that was written about tuning, we can conclude that people followed Pythagoras’ line of thought until well into the 16th century – after all, mathematical principles were high on the list during the Renaissance. The so-called Pythagorean tuning was viewed as the most perfect of all. In this system, all the fifths are tuned purely, but for the last or second last, in which one must compensate in order not to end up too high: this fifth became so small and false-sounding, like a wild cry, that it was called the ‘wolf fifth’.

One of the consequences of this tuning in pure fifths was the occurrence of larger and smaller semitones; the sharps were higher than the flats. The disadvantage of this tuning was that the thirds could not be used and were also not in accordance with the perfect thirds: the major third was too big, the minor third too small. In fact, thirds were not even regarded as consonants. The difference between the pure major third and the third resulting from the stacking of pure fifths is called the syntonic comma. Composers of the time naturally took into account the possibilities and impossibilities of the tuning: in music from the Renaissance for example, final chords consist only of pure intervals. Yet the wish to be able to employ fuller harmonies, thus with the third, led people to search for other tuning systems wherein the thirds would be usable.

The word ‘temperament’ (also called ‘temperature’) comes from the Latin word temperare, in which one can recognize the verb ‘to temper’. Temperare means ‘to mix well’, ‘to mix/bring to the right proportion’, ‘to moderate’, ‘soften’ or ‘regulate’. To obtain good thirds one thus needed to ‘temper’ the fifths. In the circle of fifths, it takes four fifths to reach the major third (for example c-g, g-d, d-a, a-e). By tuning all fifths 1/4 of a syntonic comma smaller (although some sources mention the Pythagorean comma here, but the difference is almost negligible), eight pure thirds can be created. All keys to a maximum of four sharps or flats are usable in this tuning; however other problems arise from having so many small fifths. This tuning is called 1/4 comma ‘meantone’, because the major second amounts to precisely half of a pure major third.

In 1/4’ meantone there are also bigger and smaller semitones; here, in fact, the flats are much higher and the sharps much lower. This is because when tuning for keys with flats, one moves down along the circle of fifths and for keys with sharps, one moves up along the circle. Thus the diatonic semitones are bigger (for example d-e♭) while the chromatic semitones are very small (for example d to d♯). Flats and sharps are therefore not interchangeable; on instruments with fixed tuning one had to choose which notes would be tuned as flats and which as sharps. In the past, makers even went so far as to build harpsichords with split keys to enable the use of g♯ and a♭, d♯ as well as e♭.

As time passed, people wanted to be able to modulate more freely, and to make this possible, instrument tuning had to be more and more ‘tempered’. One example of a milder variant of 1/4’ meantone is 1/6’ meantone, wherein all fifths are tuned 1/6 smaller; there are several other temperaments based on this principle. The above-mentioned meantone tuning systems are examples of regular tunings, because all fifths are tuned equally small. What also occurred, were temperaments in which some fifths were tuned smaller and some pure: the so-called irregular temperaments. The tuning systems of Werckmeister and Neidhart are well-known examples of this. Different variations on tempered tuning are described already from the end of the 15th century. Some sources from this time indicate that singers and players of instruments with non-fixed tuning would adapt their intonation automatically to the pure intervals. As always, there are other sources that contradict this phenomenon or the desirability thereof.

Naturally, it seems that the most obvious solution to the problem is the tuning system most commonly used in the Western classical world nowadays, namely equal temperament. Here, the Pythagorean comma is divided equally over all 12 semitones in the octave, so that all intervals are 1/12  of this comma too small. Written descriptions that seem to describe equal temperament already exist from the time the meantone temperaments were described as well (and even earlier), however, it is not always easy to understand if what they describe is really an equal distribution of the comma. In any case, judging from many sources, it appears that the equal division of the comma, through which there was no longer a difference in pitch between flats and sharps, thus making all the intervals sound the same (if all  semitones are identical, all intervals will be identical), was not considered to be varied enough by everybody.  

We must also not forget that the transition from modal to tonal music was still in full swing. In church modes, the intervals within a scale are much more varied than in tonal music, where everything revolves more or less around major and minor. Equal temperament would make tonal music even less varied. Bach’s Das Wohtemperierte Klavier was not written with equal temperament in mind. The term Wohltemperierung was first mentioned in Andreas Werckmeister’s Orgelprobe (1691) and defined as a softening of meantone tuning in such a way that harmonies of the most bearable, most pleasing (literally: “most pure”) kind possible can be produced in all keys.

Let us return to the onion soup: music from the beginning of the 17th century is also referred to as rhetorical music, because everything notated in a composition implies a certain emotional response. This sounds logical enough, but actually, very specific tools were applied to obtain specified affects; one could speak of an aesthetic blueprint. One of these tools concerns the key(s) of a piece – in fact, many theorists ascribe specific characters to the different keys.

Despite the fact that there were always controversies around this, many people held the opinion that keys owed their characteristics chiefly to the smaller and larger semitones, which, in the different keys, fall on different places in the scale. This gives each harmony and interval its own colour. Composers consciously used specific harmonies and intervals, including those sounding less pleasant in a certain key, to strengthen a particular affect (emotional response). Even if we cannot always know in which temperament a composer imagined his composition, we must assume that we miss some part of the affects when we perform or listen to music in a temperament different from the one the composer had in mind. Consider, for example, a composition with text, where, on a word like morire (to die), the composer uses a harmony that could, in a temperament other than equal, sound even more unpleasant. Another example would be chromaticism: this is usually associated with pain and sounds extra agonising with semitones of different sizes.

While in theory there is no variation within equal temperament, the prevailing tendency nowadays is to play sharps higher and flats lower. This is generally done to strengthen the function of the leading note. In equal temperament, the differences in intonation are no longer inherent to the tuning, but rather a question of choice, or interpretation.  

Now listen to these two recordings of Sweelinck’s Fantasia Cromatica and Rossi’s Toccata Septima. Try to sense how the temperament is affecting you, and also try to sense if the second link is affecting you as much, and listen to the first link again:

On YouTube one can find numerous other videos demonstrating how the different temperaments work. The internet also contains a great supply of articles with more extensive explanations.

Antoinette Lohmann, April 2021

Translation: Manu Huyssen, Antoinette Lohmann