The ups and downs of tuning our musical instruments with equal temperament

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 (ET) 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.

Just intervals are the easiest to sing and most natural… tempering is done for the reason of convenience… nature has no regard for our convenience

Marin Marsenne - Harmonie Universelle, 1636

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.

Music theory is an ingenious manifestation of the human mind, which allows us to create awe inspiring art. However, it comes with a typical limitation of the human mind: the need to systemise and reduce to a graspable size.

Music as we know it would be impossible without a temperament that allows all notes of the musical scale to sit in one octave that is repeatable in lower or higher registers.

If you take the notes of a musical scale from C to B, you will be able to continue with a new, higher C after you reach the note B… and so on. Each note of this higher scale (higher octave) will have a frequency of vibration that is exactly double as the same note in the lower octave. All the intervals in each octave will sound the same in each key. For example, the major third C-E sounds the same as the major third F#-A.

This is only possible because the scale has been equally tempered.

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.

The gain for equal temperament is a homogenized neutral grey coloring that is completely dependable… Western music now exists under the dictatorship of this one homogenized temperament.

Owen H. Jorgensen - Tuning: Containing the Perfection of Eighteenth-Century Temperament, the Lost Art of Nineteenth-Century Temperament and the Science of Equal Temperament, 1991

ET is a compromise which is presented to us by the keyboard as an aid in mastering the tonal world, and then pretends to be that world itself

Paul Hindemith - The Craft of Musical Composition, 1942

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 ET). 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.

With meantone, this was not possible because the keyboard was tuned to one main key (usually C) and all other keys would sound inherently different. In fact, music was composed “by key”, meaning that each key would have its own unique character and combinations of notes would sound different from any other key.

Transposing a piece of music written in one key to another key would change the music completely, often in a way that would result unacceptable.

These restrictions were the reason why there had been a constant effort throughout history to find a sort or temperament that would smooth the differences between keys, therefore allowing modulation (transposing between keys).

Such tunings were known as “well temperaments”. Johann Sebastian Bach’s famous Well-Tempered Clavier, a collection of 48 pieces, was composed in one of the variants of the well temperament. Such tuning achieved the goal of making modulation of key more tolerable, although a piece of music would still sound different in a new key. In fact the 48 pieces where composed to highlight the differences between the minor and major mode of each key, with two pieces for each tonality (minor and major mode of each of 12 keys = 24 x 2 pieces for each tonality = 48). This effect is completely lost if Bach’s work is performed on an equally tempered instrument.

At the contest in Jena in 1706, Johann Georg Neidhardt advocated the system of 12 equal semitones, developed after studying the work of Andreas Werckmeister (the other famous pioneer of ET) as well as that of all the luminaries of this subject throughout the ages (such as Ptolemy, Gioseffo Zarlino, Marin Mersenne, etc.). Only 21, he ad just published his Die beste und leichteste Temperature des Monochordi (The best and easiest temperament of the monochord).

On the other side, choir master Nicholas Bach advocated the well temperaments, with its compromises that would still sound acceptable to the ears acquainted to the pure harmonies of the vocal tradition (just intonation).

Not a matter of beauty

It is important to keep in mind that the proposal of ET as a tuning system was not based on its beauty in terms of harmony, rather on the technical solutions it would provide for the manufacture of keyboards as well as the possibility to really modulate key without differences. It was commonly accepted at that time (as it is now) that ET would provide technical advantages at the expense of pure harmony. That wasn’t a question. And it was also the reason why on that particular day in Jena, it was refused in favour or Nicholas Bach’s well temperament.

When I go from my justly-intoned harmonium to a grand pianoforte, every note of the latter sounds false and disturbing… these are unpleasant symptoms for the further development of art. The mechanism of instruments and attention to their convenience, threaten to lord it over the natural requirements of the ear, and to destroy once more the principle upon which modern musical art is founded

Herman Helmholtz - On the Sensations of Tone as a Physiological Basis for the Theory of Music, 1877

In fact, ET would not really establish itself in western music for another 200 years, being always deemed too “out of tune” to be acceptable.

The invention of the tuning fork in 1711 was a great help in establishing this tuning system as it helped instrument makers and tuners to tune intervals that were perceived as unpleasant to ears used to the more natural options offered by meantone and well temperament.

In any case, the practical advantages of this tuning system allowed it to slowly but surely take more and more space in western music, a presence that has never been fully accepted by everyone. The discussion on how inadequate ET really is in terms of harmony has never really stopped and it ranges from the purely technical aspects of the tuning itself to the fact that many great compositions of the past are nowadays performed on equally temped pianos, whereas the composers created them on well tempered and even meantone tuned instruments.

…tho’ the Octave may be divided into 12 equal Semitones, ’tis impossible that such a Scale could express any true Musick… Temperaments derived not from the Nature of the System of Musick itself, but the Accident of limiting it to fixt Sounds

Alexander Malcolm - A Treatise of Musick, Speculative, Practical and Historical, 1721

Turning points

At some point in history, music divided into two streams: keyboard and fret based music which requires temperament on one side, pure harmonic music as performed by a cappella choirs and certain chamber string ensembles on the other.

The turning point that created this division may be identified in the 3rd century BC with the invention of the hydraulos organ, the first keyboard instrument, attributed to Ktesibios of Alexandria.

In their book The Story of Harmony, Rex Weyler and Bill Gannon summarise the situation like this:

Musicians playing polyphonic and harmonic music from the Renaissance to the 20th century could choose from the following:

  • play pure harmonic music, but don’t use fixed-tone instruments

  • play pure harmonic music, but add extra digitals to the fixed-tone instruments to allow for enharmonic shifts of some notes

  • temper the notes in favor of relatively pure triads in the common keys, and accept some unusable “wolf” intervals in the distant keys

  • equally temper all the notes, and accept the compromises of all harmonic intervals, except the octaves

It was not until the end of the 20th century that a new option would present itself, as instruments were able to take advantage of electronic and computer technology:

  • build keyed or fretted instruments in which the tones of all notes can change as needed

Modern music softwares can allow keyboard synthesisers to produce a variety of different tunings, thus allowing composers and musicians to experiment with the different effects that each tuning can elicit in the listener.

It could also potentially provide a solution to the ubiquitous use of the piano as a vocal training instrument.

As composers and educators we give them [singers] an accompanying instrument – the piano – which is continually at odds with their instincts. After they have mastered this incongruity we pose them in an a cappella choir or before an orchestra, where they are at the mercy of each intonational whim of concertmasters and conductors, and proceed to criticize them for their “bad intonation”. The great need for a better instrument than the piano in the training of singers and for accompaniment of songs is too self-evident to be labored.

Harry Partch - Genesis of a Music, 1949


If there are so many different tuning systems and some of those are closer to harmonic ratios than others, could it be possible that music performed in a tuning that is closer to the intervals naturally produced by the overtone series is more suitable to us as living beings evolved in the same environment as those natural harmonies?

It has been observed that we respond more positively to listening to music when we listen to music that we like, rather than to a specific kind of music. So our emotional bond with music itself determines the quality of the effects upon us. In this case, the fact that the music is performed in equal temperament seems irrelevant.

So I’d like to close this article with a provocative question (actually two): what if us deriving pleasure from music tuned to equal temperament is similar to us deriving pleasure from eating foods that taste delicious but may not contain the best ingredients for the health of our bodies?

What if our emotional attachment elicits some positive physiological responses while the inherent incongruity of the tuning has an adverse impact on our nervous systems?

Copyright 2017/24 - Simone Vitale