Is our music out of tune?

The question of tuning

The question of tuning and temperament is a fascinating one in that it is partially scientific (because it involves objective facts about the nature of sound) and partially aesthetic (because it involves subjective taste as well as cultural aspects that change overtime).

Therefore, the whole subject, although highly scientific, can’t be only approached through the sole rationality of science.

Today, in the western world, music relies predominantly on a tuning system known as equal temperament. With the exception of those choral groups and chambers ensembles that use just intonation (based on harmonic tuning) and certain progressive composers who use a system called extended just intonation, most of the music we listen to in western society is composed, recorded and performed in equal temperament.

In ancient times and up until the 18th century, the situation was very different. Musicians in Europe used a variety of tuning systems. Equal temperament, although theorised, was not really achievable because of the difficulty of tuning such an “artificial” system by ear.

At the time of the great modern western composers, therefore, keyboard instruments were tuned either in mean-tone or well temperament. These systems, not being equally tempered, conferred each key with a unique quality because the “sizes” of the intervals in each key were different, therefore would sound different. Equal temperament (EQ) evens out all those differences to the point that each key has the same character as all other keys. This is what makes it possible, for example, to transpose a song in various keys depending on the specific vocal range of a singer.

If, for example, a man wants to sing a song originally written by or for a female voice (or vice versa), he can transpose the song to a more comfortable key and the harmony will sound exactly the same.

But what is really the “problem” with temperaments?

What is usually known as just intonation is based on the musical intervals as they occur naturally in the overtone series. It is the tendency of the singing voice to naturally tune to these “pure” intervals as, in a way, they are embedded in the fabric of sound. They represent, so to speak, a law of nature in the physics of sound.

When attempting to explain the overtone series, things can get abstract and a bit complex very quickly, so I will attempt to provide an explanation that is as simple as possible.

Let’s start by saying that whenever you hear a single sound, say one note on a piano or sung by a voice, you are not only hearing one sound but rather a combination of various sounds. Even though in music you would call that one note (or one tone), in reality it is a combination of tones.

So let’s say that the note is a C on a piano keyboard. The note we identify as C is only the fundamental or lower tone of an ascending series of tones with increasing frequencies. These additional frequencies are called overtones.

The overtone series is a physical phenomenon and the “distances” between the frequencies of each overtone are fixed and can’t be changed.

The overtones follow a specific sequence. When we compare this to music, the sequence is such that if we take C as fundamental (as we said before), the first overtone is said to be the same C (in music, this interval is called unison). The second overtone is again a C but one octave higher (double the frequency). The third overtone is a G above the second C (the musical interval of a perfect fifth). The fourth is again a C, one octave higher than the previous one. The fifth overtone will be an E (major third). The overtone series goes on and it is virtually infinite, with octaves repeating in between other intervals that become less and less consonant as the frequency increases. Interval, in music, is the name given to the distance between two notes. Intervals are as many as the possible combinations of different notes.

It is important to keep in mind that the frequencies of the the notes mentioned above, differ from the frequencies of the same notes on a piano exactly because pianos are usually tuned to equal temperament)

The “problem” started with the invention of fixed tuning instruments, mainly keyboards. In fact, the voice as well as the instruments of the strings section (violin, violas, etc), some brass instruments such as the trumpet and trombones, ancient instruments such as the horn, are all capable of adjusting the pitch by ear, as they play.

Fixed tuning instruments such as keyboards and guitars can’t do that. Once they are tuned, each key produces a pitch that can’t be changed, unless re-tuned (yes, one can “bend” notes on the guitar, but it would be impossible to do that for all notes in a piece of music).

So these instruments have to use a “tempered” system, where some of the intervals have to be altered in order to fit into an octave.

What is a pure interval?

In just intonation, no series of a single pure harmonic interval will match up with another series of a single pure harmonic interval. If you take a series of 12 consecutive pure fifths, the last note will not be an exact octave of the first. If you take three pure thirds (C-E, E-G#, G#-C), the last note will not be an exact octave of the first… and so on for all pure intervals.

An interval can be called “pure” when the frequency of vibration of each note in the interval matches the simple acoustic ratios found in the harmonic (or overtone) series:

Fundamental (note C), harmonic ratio 1:1,

Octave (note C’), harmonic ratio 2:1,

Fifth (note G), harmonic ratio 3:2,

Fourth (note F), harmonic ratio 4:3,

Major third (note E), harmonic ratio 5:4

Minor third (note Eb), harmonic ratio 6:5

These very simple ratios, found by careful observation of vibrating strings or the length of resonating pipes, when combined together in musical intervals, create the least amount of “beats” between the notes. The effect is what we experience as harmonic tuning.

March of the keyboard instruments

Seeing as the driving force of Europe’s 18th century music was the keyboard, it is easy to understand how the need for an elegant solution to the problem of temperament was greatly needed. In fact, keyboards up to that time came in all sorts of forms and temperaments. There were keyboards with seven, twelve, seventeen and more keys per octave, arranged in various modes and scales in attempts to combine harmonic intervals with tempered ones.

Looking back at those keyboards today, it is easy to understand why there was a strive for simplifying the manufacture of such instruments.

One of the many variants of keyboards was the one completed by Nicholas Faber in February 1361 for the organ of the town of Halberstadt in Germany. The upper of the three keyboards on this organ featured a diatonic major scale in the key of C with five “accidentals” slightly raised behind the main keys.

This was only one of the many designs of keyboards ever invented, but it ended up being the prototype of the one we still use today.

The history of musical tuning is long and complex. We tend to take for granted the tones we hear when we place our hands on a piano keyboard. In a way, for many of us westerners, that’s all we know and all we will mostly hear throughout our lives.

However, the journey to this widely used system (equal temperament) was a long one and, for many, was surely not the end. In fact, many composers and scholars (past and contemporary) disagree with the losses brought about by this system.

Well-temperament vs Equal temperament: two systems compared

It is important to note that every tuning system has its advantages and its shortcomings. Equal temperament has solved some practical problems, but it has done so at the expense of pure harmony.

At the beginning of the 18th century (1706) the city of Jena in Germany was the stage of an important event, a tuning contest where two systems where tested and compared.

At that time, keyboard tuning was transitioning from meantone to well temperament (in a way a precursor of EQ). One of the main reasons behind the need to find a better temperament was the possibility to “modulate” keys, that is the possibility to play the same chords in different keys and have them sound the same in terms of character an relationship between the notes.