Quartertones are the “notes between the notes,” found by dividing tempered half-steps (semitones) into equal quarter-steps. For performers, they are an entry point into a wider compositional technique called Microtonality. There are many different microtonal systems, each dividing the tonal spectrum according to a specified logic.
The quartertones covered in this section are but one way of thinking about microtonality. The tuning system that we generally use in Western classical music is called 12-tone equal temperament, which basically means that the octave is divided into twelve intervals that are perceived as equidistant from each other, all half-steps being the same. Using a similar logic we can divide each semitone (or half-step) further in half, resulting in quartertones. But it does not stop there! Some composers have divided semitones by 3 or 4, resulting in 1⁄6th-tones and 1⁄8th-tones. These intervals are very difficult to hear accurately, much less play well. They do not appear frequently in practice.
As you may know, the frequencies of the pitches we hear are measured in Hertz, which is actually a logarithmic system rather than a linear system. (e.g. the octave above the pitch A-440 is 880Hz, the octave below is 220Hz). Tuning systems such as Just Intonation are based on simple frequency ratios that originate from the tonic note of the key. While this system lends itself to beautiful consonant 3rds, 4ths, 5ths, and 6ths in the home key, it doesn’t work so well for instruments with fixed tuning (keyboard instruments, harps, lutes, etc.) because adventures into other keys can produce very dissonant intervals. The equal tempered system has been around for many hundreds of years (actually first written about by Greek philosopher Aristoxenus in the 4th century B.C.) and was gradually adopted as the tuning system for keyboard instruments in the Baroque era as the best solution for being able to play in all keys.
Contemporary composers have used many other logical systems for dividing the octave. For example, American composer Harry Partch (1901-1974), developed a system based on extending the “simple” ratios out to the number 11, which yielded a 43-note scale. (The image below is a visual representation of the ratios used to create his scale.) Obviously playing 43 notes within an octave accurately on a conventional string instrument would be near impossible, so Partch created his own instruments and modified others to play his music.
One particular microtonal system is important for us string players to comprehend — the system built on the overtone (harmonic) series. If you’ve read the Harmonics Overview you may have noticed that certain notes in the harmonic series are slightly “out of tune” when compared to equal temperament tuning. The 5th and 10th overtones are 13 “cents” flat (with 100 cents equal to one half-step), and the 7th overtone is 31 cents flat (a 1⁄6th-tone). The 11th overtone is almost exactly a quartertone (50 cents) sharp — or flat, depending on how you look at it — from equal-temperament.
Many composers in the genre of music known as spectralism will write harmonics with microtonal notation in order to be very specific about the pitch that they want. So you might see something similar to this for the 7th and 11th harmonics (shown here on each string):
Sometimes the composer will write those microtonally altered pitches in other octaves, fingered not as harmonics but as regular stopped notes, indicating that it should be in tune with the pitch that occurs in the overtone series.
Notation of quartertones has not been completely standardized, so many symbols have been used in the past. Most composers use some kind of alteration on the sharp, flat, and natural signs:
Again, since notation has not been standardized you may encounter any of the above symbols, not to mention many unique solutions conceived by composers. In 1974 a recommendation was issued by the International Conference on New Musical Notation in an attempt to simplify the symbology and create a norm for composers and engravers. They settled on the following:
I find this to be the clearest and most compelling notation because it does away with the 3⁄4-tone notations, which are more difficult to read on the page and require an additional mental step. Therefore, these are the symbols that will be used in the remaining examples and exercises.
To learn how to hear and practice quartertones, read on: Orienting Your Ears