Historical Tuning of Keyboard Instruments

Whereas guitars are tuned by setting the dominant frequency of each string to a fixed reference, which can be easily automated, this is not the case for a piano.

This is because each piano string is governed by a non-linear differential equation describing its motion. The overtones therefore are not simple multiples of the fundamental frequency but instead appear displaced from where they would naively be expected. Although there is a strong systematic variation to these displacements, there is also a random component, rooted in difficult-to-characterize variations in the manufacture of each string.

Thus, each string unto itself has a unique Fourier spectra, and not only do the fundamentals need to be matched in frequency so as to minimize dissonance, but the higher modes as well.

Thus, ach piano has its own tuning, unique its constituent parts. (And, there may be multiple tunings for a single piano.)

Further, we do not have an adequate mathematical characterization of the sensation of 'dissonance' and thus the job of a professional piano tuner is to minimize this sensation when a pair of strings is played.

Though there are electronic systems for selecting the proper piano string fundamental, they can differ by as much as ten cents (10% of the distance between one note and the next) from a professional tuning.

Therefore, we can put a man on the moon, but we do not know how to tune a piano!

Many modern digital tuners offer the ability to tune by various historical temperaments, making the use of historical tuning both easy and fast. However, it is important IMO to understand how the different tuning methods came about, the math behind them and the harmonic implications they entail.

Trying to explain historical tuning using the circle of fifth is actually incorrect, even when it pertains to keyboard instruments. The concept of a cycle of fifths is relatively recent, is not compatible with harmonic theory before the 18th century, and is useless for explaining meantone for example.

It's also important to take into account that the different tuning methods go hand in hand with specific kinds/styles/periods of music. Therefore it's important to present them in a historical context. The website touches on this briefly, but a more comprehensive discussion can be found elsewhere, for example on wikipedia.

In western music one can discern the following periods which correspond with basically universal (disclaimer: the following is grossly generalized):

1. Ancient Greece - the discovery of the mathematical rules of harmony (both musical and celestial) is attributed to Pythagoras, based on the superparticular ratios based on the numbers 1, 2, 3, 4: 2:1 - diapason (octave) 3:2 - diapente (fifth) 4:3 - diatessaron (fourth)

This mathematical framework was superimposed on the Greek system of tetrachords (4-note scales), in their various genera, which always covered a diatessaron (fourth), the most common genus being the diatonic (composed of 2 whole tones and 1 semitone). The so called Pythagorean system divided the diapason into two tetrachords, divided by a whole tone. It is interesting to note that the modern major scale is still basically the same construct as that of the ancient greeks: two diatonic tetrachords with a whole tone in the middle.

2. Middle Ages (up until early 15th century) - the musical and mathematical thought of the middle ages came of course from the classical era, and in the music of the middle ages the prevalent tuning method for all intents and purposes was based on Pythagorean ideas. We now call this method the Pythagorean tuning.

The basic idea is that all ratios are derived from the fifth (3:2). In the key of C, D would be tuned at a ratio of 9:8 (fifth minus fourth), E would be tuned at (9:8)^2 (the product of two whole tones), etc. The advantage of this system, compared to the Greek system, is the ability to wander into other keys. With the advent of Guido d'Arezzo hexachord system with its moveable hexachords, knowing how to tune Bb or F#, or go even further to Ab or G# became a matter of tuning a sequence of fifth.

It is important to note that this does not imply a closed cycle of fifth. In fact, tuning a sequence of 12 fifths (and then canceling the 7 octaves we climbed), starting from C and ending on B#, would give us a note significantly higher than the one we started with. This difference is called the pythagorean comma (about 1/8 tone).

The implications of this system are manifold: - no enharmonic identity - C is not B#, Eb is not D# - # is (relatively) high, b is low - Pythagorean thirds are dissonant: 9:8 * 9:8 = 81:64. Just thirds (based on the natural harmonic series) are 5:4, or 80:64. The resulting difference, 81:80, is called the syntonic comma (about 1/9 tone). In medieval music, cadences include only the stable, consonant intervals of fifths and octaves (with resulting fourths). Thirds and sixths are considered mid-way between consonant and dissonant, and are considered unstable.

3. Renaissance and early baroque - the age of meantone (15th century to late 17th century) - while various experiments have been made during the renaissance with just intonation and with equal tuning, they never took hold. Just intonation means that we lose the ability to move between keys. Equal temperament means that we have lousy thirds (almost as lousy as pythagorean thirds). The musical taste of the renaissance demanded on the one hand consonant, sonorous thirds, and on the other hand the ability to move between keys.

To that end the meantone tuning was developed, with a very simple idea: if four 3:2 fifths give us a dissonant third that is too high, why don't we temper each of the four fifths so we'll get a pure third? The amount of tempering is actually very small, only a quarter of a syntonic comma, or about 5 cents in present-day terms. Contrary to just intonation, where the interval C-D is 9:8 and the interval C-E (where E is a pure third) is 10:9, in meantone tuning the D is right in the middle between C and E, therefore the name of this system.

The tempering of the fifths is continued in the sequence of fifths in both directions: C-G-D-etc, and C-F-Bb-etc. The implications are: - no enharmonic identity - like in Pythagorean tuning. - # is low, b is high (the opposite of Pythagorean tuning). - fifths are slightly beating, but the pure thirds more than make up for it - 1/4 comma meantone has an indescribable sweetness to it. - Since all fifths are equal, all keys sound the same, and you can easily move between keys. This also means that instruments with different pitches (as was the situation during the late 17th century and even later), could play together without problems on intonation, at least theoretically. In that regard meantone is a kind of equal temperament. - When tuning keyboard instruments in meantone, the tuner has to select whether to tune each black key as sharp or flat. Also, one fifth would be unusable (it would be way too wide). Meantone temperament on a keyboard with 12 notes per octave is actually quite limiting.

It is important however to note that meantone is not a cyclical temperament, you can theoretically continue the sequence of fifth in both directions (sharp and flat) and you'll never close the circle.

A different way of thinking about meantone temperament is the division of the octave into X equal intervals. It turns out that 1/4 comma meantone fits almost exactly a division of the octave into 31 equal parts, where a whole tone is 5 parts, the diatonic semitone (e.g. C-Db) is 3 parts, the chromatic semitone (e.g. C-C#) is 2 parts.

This idea of dividing the octave into more than 12 equal parts led in the 16th century to much experimentation, mostly in Italy, in building keyboard instruments with split sharps, allowing the player to play both sharps and flats and removing the 12-notes per octave limit.

Renaissance theoreticians experimented with tempering fifths by varying amounts such as 1/5 comma, 1/6 comma, 2/7 comma, and leading to a division of the octave to 19 parts, 43 parts and 55 equal parts.

The 1/6 comma meantone (and corresponding division of the octave into 55 equal parts), became popular towards the end of the 17th century, and gained new adherents towards the end of the 18th century, with both Mozarts (father and son) being some of its proponents.

4. Baroque - towards the end of the 17th century meantone temperament became increasingly seen as limiting and unsuitable to contemporary musical taste. As players and composers became more attached to the different affects of the different keys, and wanted to be released from the limits of meantone on keyboards, musicologists started looking for a way to temper the fifths such that all fifths would be usable.

This led to the invention of the closed temperament, with Werckmeister being the first to offer a recipe for the tuning of keyboards, where 4 of the fifths are each tempered by a 1/4 of a pythagorean comma. The rest of the fifths are pure. This creates a system where all keys are usable, but some would be more dissonant than others.

Werckmeister was soon followed by others such as Neidhardt, Valotti, Kirnberger, etc, each with his own recipe for tempering some of the fifths by varying amounts. The principle is always the same: the total amount of tempering is equal to a Pythagorean comma.

Some instructions, such as the various instructions for the French "tempérament ordinaire" are much more vague, and just start with a pure third in one of the main keys (C, F or G), tempering its constituent fifths, and becoming more and more dissonant as one goes farther away in the cycle of fifths.

It is important to understand that all these temperaments were intended for keyboard instruments, and that players of other instruments, as well as singers, were expected to follow certain conventions based on meantone ideas, but that's another topic of discussion.

5. Equal temperament - equal temperament, although a very old idea (it goes back at least to the 16th century), only gained popularity with the rise of the piano, being the first mass-produced musical instrument. Of course it also has to do with the fact that composers of the romantic era became increasingly adventurous in their harmonies and fond of employing enharmonic modulations, thus erasing any notion of difference between sharps and flats, from Chopin and Liszt to Wagner, Mahler and beyond.

Although the historical tuning methods came back into use together with the revival of early music and historical instruments, it is a sad fact that today equal temperament is so universal, so omnipresent that we have become completely desensitized to the fact that basically all the intervals we hear today, except for the octave, are dissonant to some degree.

If anybody has more questions I'd be happy to help. I've quite a bit of experience in this field.

I am not musical. This is a resource someone else suggested to me years ago to add to a small homeschooling site I run. It is listed on my site as a math resource. I thought math+music might appeal to some niche of the geek crowd, even though I am tone deaf and can't really appreciate it.