Twelve-Tone Equal Temperament

A photograph of several ranks of organ pipes encased in ornate golden architecture, with a set of 23 pipes with trumpet bells flaring out from a center pillar. Ranks of organ pipes in the Igrejas dos Carmelitas, a church in Porto, Portugal. Organ pipes are tuned by adjusting the pipes' lengths in various ways, often by moving stops, caps, and slides built into the pipes themselves.
(John Samuel | CC BY-SA-4.0)

Twelve-tone equal temperament, commonly abbreviated as 12-TET, describes a tuning system considered as a standard for many types of modern music.

Historical Context

The common modern practice of using a musical scale with twelve equal intervals represents a system which has evolved over millenia.

Pythagorean Tuning

Pythagorean tuning is named for the Greek music theorist Pythagoras but was used by Mesopotamian theorists as far back as 2000 BCE. The system is built around the ratio between the first and second harmonic of a vibrating string, or 3:2.

A diagram showing two waveforms of the same width; the top one shows one peak and one valley; the bottom one shows a peak, valley, and another peak. The distances of the first peak are measured and the ratio between their measurements, 3:2, is shown.
Figure 1: When the frequencies of the first and second harmonic of a note are compared, the resulting ratio is 3:2.

The tuning system is constructed by starting with a base pitch, using the ratio to find a second pitch, and repeating the process with each new pitch found. Doing so gives us 12 different pitches before arriving at the original pitch.

Figure 2: The twelve notes of the chromatic scale, created by applying a 3:2 ratio to successive notes starting from C2. Transposing each note downward yields the twelve divisions of the octave; however, the cycle does not return to the same exact frequency. The difference between the first and last note — in this case, about 0.45 Hz — is called the Pythagorean comma.

If this process is followed accurately, however, there is a slight difference between the original pitch and the result of the complete cycle. This distance is referred to as a Pythagorean comma.

When the resulting notes of Pythagorean tuning are placed in the same octave, the steps in the resulting scale are not equal. As a result, playing in keys built around notes other than the base note will have different, sometimes dissonant, internal proportions.

Figure 4: A keyboard tuned to a Pythagorean tuning of the second octave as shown in Figure 2.

Just Intonation

While Pythagorean tuning uses the distance between the first and second harmonics of a vibrating string, just intonation makes use of the other intervals present in the harmonic series to determine the intervals of the scale.

Figure 5: A keyboard tuned to five-limit just tuning, which uses interval ratios drawn from powers of 2, 3, 4 and 5. In this tuning, the interval from C to G is consonant but from F# to C# is more dissonant.

There are many different types of just intonation, each of which uses a different selection of ratios to find a balance that creates desired levels of consonance.

Figure 6: Suite for Microtonal Piano, a 1977 piece by American composer Ben Johnston. For this piece, Johnston employs a piano tuned in a just intonation across the entire range, where pitch classes in different octaves do not necessarily correspond to one another.

Temperament

Often, "pure" tuning systems like Pythagorean tuning and just intonation are modified by slightly changing certain intervals in the scale to achieve specific goals. This process is called tempering pitches or intervals, and the systems built around it are called temperaments.

Well Temperament

Because tuning systems like just intonation are tuned around a specific tonic note, instruments cannot effectively play in other keys without being retuned to that key. These systems can be made more flexible by slightly altering specific scale degrees, introducing a slight amount of dissonance in the base key to make other keys more consonant.

Figure 7: A keyboard tuned to one of several temperaments devised by Andreas Werckmeister, in which intervals in just intonation are altered to facilitate playing in all twelve keys.

Well temperament is any type of temperament applied to just intonation which decreases the level of dissonance in keys other than the base key. German music theorist Andreas Werckmeister specialized in designing popular systems of well temperament, including the system used by German composer Johann Sebastian Bach, in a 1722 collection of keyboard works entitled The Well-Tempered Clavier. In this collection, as well as in a second collection completed in 1742, Bach wrote preludes and fugues in every possible key to demonstrate the flexibility of the tuning system.

Figure 8: Prelude No. 3 in C Sharp Major from The Well-Tempered Clavier, Book I by Johann Sebastian Bach, performed by American pianist Carl Radford on a Steinway grand piano tuned to a well temperament devised by British scientist Thomas Young.

Equal Temperament

The tuning system most commonly used by professional musicians today is a form of well temperament in which every interval is the same size. Unlike other well temperaments, in which each key has a slightly different character due to the slight differences in intervals, this system, called equal temperament, ensures the same level of consonance in every key.

Figure 9: A keyboard tuned to equal temperament.

Pitch Reference

Even in musical communities which had consistent practices regarding tuning systems or temperaments, pitch references — accepted standard frequencies for the tuning system's fundamental pitch — would vary among locations, groups, and even instruments as late as the 18th century.

A photograph of a metal tuning fork mounted on a wooden, open-ended box.
Figure 10: A tuning fork mounted on a resonator box. Tuning forks were invented in 1711 to create consistent pitch references for tuning musical instruments.
(Brian0918 | CC BY-SA-3.0)

As scientific methods for measuring frequency became available, professional musical groups and even governments began promoting pitch standards, usually by specifying the frequency of A4. The most common standard today is A4 = 440 Hz, though groups might adopt a different standard to achieve a more brilliant sound or to accurately replicate historical performances.

Figure 11: A keyboard tuned to equal temperament which can be adjusted to a different pitch reference.

Twelve-Tone Equal Temperament: Summary

  • The common modern practice of a twelve-note scale is the result of centuries of changing standards.
    • Pythagorean tuning is built around the 3:2 ratio between the first and second harmonics of a vibrating string.
      • Following Pythagorean tuning through an entire cycle results in a slight discrepancy in frequencies, called a Pythagorean comma.
    • Just intonation describes any tuning process which uses intervals drawn from the harmonic series.
      • Just intonations often provide consonant intervals in one key but dissonant intervals in other keys.
    • Temperament is the process of deliberately adjusting specific pitches within an intonation system.
      • Well temperament describes any system where temperament is used to soften dissonances and allow performance in multiple keys.
      • Equal temperament is a system of well temperament in which pitches are equally spaced through the octave.
      • The term Twelve-tone equal temperament is often used to describe the system most widely used by musicians today, and is commonly abbreviated as 12TET.
  • Many modern musical cultures adhere to a standard pitch reference around which instruments are designed and built.
    • The most common pitch reference used today is A440, which refers to a system where A4 is tuned to 440 Hz.
    • Musicians will sometimes use other pitch references in order to reflect historical performance practices or to create brighter or darker sounds.

Exercises

Exercise 1: Description of Exercise

Exercise 2: Description of Exercise