tuning forks

Want to play better in tune on the violin (or viola, cello, etc)? In addition to having a good sense of relative pitch and left hand fundamentals, string players also need to have a basic understanding of both vertical and horizontal intonation. The bad news is there is no such thing as perfect intonation. The good news is that with experience, string players have the ability to quickly adjust their intonation in real time to stay the best in tune with an ensemble (or themselves).

There are have been many tuning systems throughout music history, many of which are still being used today. If it interests you to explore this topic in greater detail, I strongly recommend Ross Duffin’s book How Equal Temperament Ruined Harmony (and why you should care).

In addition to the “universally-accepted” equal temperament, here is a summary of two tuning systems that non-tempered instrumentalists (this includes vocalists) should be aware of.

If you prefer a video version of this, just scroll past the text.

Pythagorean Intonation

The pythagorean intonation system is based on the perfect 5th intervals tuned to the the ratio of 3:2, which gives it its pure quality. To get the precise ratio, take the starting note and multiply its Hz value by 1.5 to get the Hz of the upper note, a perfect fifth above. For example, a pure 5th above the open A string at 440Hz is E at 660Hz. By comparison, on a piano tuned to equal temperament (at A=440), this same E would be ever so slightly flatter; somewhere between 659 and 660 Hz. In this case, the fifths are tempered in order to achieve 12 equally-spaced semitones across all the octaves. In the pythagorean system, the notes are tuned in the circle of 5ths, sequentially. A great model is the traditional tuning of the open strings of the the violin, viola, and cello (tuned to perfect 5ths). The double bass tunes to perfect 4ths (inversion of the 5th), so it fits into this category as well.

The pythagorean system is most frequently used in order to play single-line melodic material that’s based on scales. Staggering notes by the 3:2 ratio results in a large major 3rd interval, which supports the idea of “expressive intonation” in which leading tones lead, and other neighbor tones can create resolutions with narrow half steps. This allows melodies to sound most in tune to a human ear. However, there are two problems with the pythagorean system:

  1. The large major 3rd interval helps us tune a linear melody, but if it was played as the 3rd of a chord, the chord on its own would sound out of tune. (ie – To play a major 3rd double stop in tune, the top note would have to be placed slightly lower than normal to create a “pure” sonority.)
  2. If we continue staggering notes to the 3:2 ratio beyond just 4 strings the overall intonation gets sharper. (ie – If we start with the note C and go around the circle of 5ths, by the time we return back to C, it will no longer be in tune with the starting note, but significantly sharper.)


Just Intonation

The just intonation system is based on the harmonic series (also known as overtone series). This is a sequence of tones based on a fundamental pitch, in which each frequency of the series is a multiple of the starting note, creating all pure intervals. An example of such a “fundamental pitch” can be one of our open strings. Just intonation is used primarily to tune chords, which is very helpful in an ensemble like a string quartet, or while playing double stops. While this system helps the 3rd and 6th intervals to be tuned in their purest forms vertically (harmonically), unfortunately it makes the horizontal melodic line sound out of tune to the human ear.

This is the paradox of intonation. When studying or performing solo repertoire that combines double stops with melodic material (ie – Unaccompanied Bach, Ysaye Sonatas, etc) string players frequently encounter this conflict, and sometimes need to make compromises and decide what to prioritize. This is also why tuning in string quartets can be such a daunting process. If a piano is involved in the ensemble, it then becomes necessary to also compromise certain notes to equal temperament. When different systems clash, the otherwise-objective musical element of intonation becomes somewhat subjective. For a well-trained musical ear, self-recording is a great tool to help make a decision during such moments.



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