Guess what? Except for octaves, none of the intervals and chords played with those pitches are precisely in tune. Musicians normally don't notice that their music is minutely out of tune, because they have become accustomed to the 12 pitches during the past 200 years. To play intervals and chords that are completely in tune, the precise pitches of many notes must be shifted slightly from their normal frequencies. Microtuning is the term used to describe those tiny frequency adjustments. Trained singers, wind-instrument players, and fretless stringed-instrument players constantly perform those shifts to produce intervals that are as in tune as possible. On the other hand, keyboards, fretted strings, and mallet-percussion instruments can play only fixed frequencies and therefore are never perfectly in tune. Why did Western music settle on a set of notes that is always out of tune? How can electronic musicians overcome the tyranny of such a limited palette of pitches? To answer those questions, you must understand the nature of musical intervals and what it means to be in tune. TUNING INTERVALS A note is defined by its pitch, which corresponds directly to its fundamental frequency. Intervals consist of two notes sounding at the same time or sequentially, and chords consist of several simultaneous intervals. The relationship between those notes is often expressed as the ratio of their frequencies. In the interval of an octave, for example, the frequency of the higher note is exactly twice the frequency of the lower note; the ratio of the two frequencies is 2:1. Intervals with ratios of two whole numbers are called pure intervals. The common pure intervals include the octave (2:1), the perfect fifth (3:2), the perfect fourth (4:3), the major third (5:4), and the major second (9:8). There are many other intervals, but some can be one ratio or another, depending on the tuning system. For example, the ratio of a minor second is 16:15 in one tuning system and 17:16 in another system. Other tuning systems, including that used in Western music, use intervals that cannot be expressed as ratios of two whole numbers. Such intervals are called impure, and their ratios are called irrational. Those intervals are impossible to represent with whole-number ratios, so a different interval-measuring system was developed. The octave was divided into 1,200 equal intervals called cents, which let you measure pure and impure intervals in the same way. For example, the pure major third is approximately 386 cents, whereas the impure major third used in Western music is exactly 400 cents. As a result, modern major thirds are sharp with respect to the pure variety. All music students encounter the circle of fifths in their studies (see Fig. 1). The graphic includes all 12 notes of the standard Western tuning system in a sequence of perfect fifths. In that tuning system, the circle closes on itself, because B# is just a different name for C. Those two notes are called enharmonic equivalents. But if you use pure perfect fifths in the exercise, the final B# is 23.46 cents higher than the starting C (discounting octaves). Under those conditions, the circle of fifths becomes a spiral of fifths.
Figure 1: The circle of fifths becomes a spiral if you use pure perfect fifths. The final B# is 23.46 cents higher than the starting C (discounting octaves). That 23.46-cent discrepancy is called the Pythagorean comma, named after the ancient Greek scholar Pythagoras, who did a lot of fundamental research of musical intervals. Because most tuning systems are octave based (that is, they include a set of intervals that repeats in each octave), the Pythagorean comma must be placed in the scale to preserve the pure octave. Exactly how that is done is the art of creating a tuning system. TUNING SYSTEMS Constructing a tuning with nothing but pure intervals, you must specify each interval individually. Such a system is generally called just intonation (see Fig. 2). Each interval with the root note sounds perfectly in tune. However, like most scales other than the common Western tuning, the notes in just intonation are not equally spaced. As a result, you can play only in the key defined by the root note and a few closely related keys. For example, in just intonation with a root of C, the major third from C to E is 386 cents, but the major third from B to D# is 428 cents (42 cents sharp with respect to a pure major third). So in the key of C, everything sounds fine, but modulating to the key of B sounds terrible.
Figure 2: In just intonation, each interval with the root of the scale is pure. The scale above the line is the familiar 12-tone equal temperament. One of the first tunings to allow modulating into other keys is called meantone temperament (see Fig. 3). Temperament refers to the fact that some or all intervals are tempered, or adjusted, from their pure forms to allow performances in different keys. In meantone temperament, some perfect fifths are shortened slightly to accommodate the comma. However, they are not shortened by the same amount, so some keys sound distinctly better than others.
Figure 3: Meantone temperament was one of the first attempts to create a 12-tone tuning that would allow modulating into other keys. It was not entirely successful. By the beginning of the 18th century, Western music was becoming more complicated and modulating into increasingly distant keys. Many musicians and theorists devised temperaments to allow modulation into any key. Among the most successful was Andreas Werckmeister (see Fig. 4), whose temperaments were used by J. S. Bach and others. The notes were still not equally spaced in the scale, so each key had a distinct character. In fact, Bach wrote The Well-Tempered Clavier to demonstrate the character of each key in a temperament.
Figure 4: Andreas Werckmeister created many temperaments, including this one, which is now called Werckmeister III. During the same period in history, other musicians experimented with equal temperament, in which the 12 notes were equally spaced within the octave. That "equality" is achieved by shortening each perfect fifth in the spiral of fifths by about 2 cents, making each one exactly 700 cents. The interval between consecutive notes in the chromatic scale is exactly 100 cents, which collapses the spiral into the circle of fifths. With that compromise, you can play in any key with equal ease. Each key sounds identical, with no change in character from one to another. Unfortunately, they also sound equally out of tune. Compared with their pure forms, perfect fifths are 2 cents flat, major thirds are 14 cents sharp, and minor thirds are 16 cents flat. The other intervals are similarly out of tune compared with their pure forms. Other scales with equal steps come closer to producing pure intervals. Some musicians divide the octave into 19, 31, or 53 equal steps, and those scales include many almost-pure intervals. Wendy Carlos has taken a slightly different approach, assembling a series of equal steps that doesn't repeat in each octave. Her alpha scale (see Fig. 5) includes steps of 78 cents each. The tuning produces nearly pure thirds, fourths, fifths, and minor sevenths, though there is no pure octave.
Figure 5: Wendy Carlos’s alpha tuning uses equal steps of 78 cents. The tuning produces perfect fifths and fourths, major and minor thirds, and minor sevenths that are very close to pure in any key. As Western musicians converged on 12-tone equal temperament, the rest of the world was using many different tunings, some of which survive to this day. The musics of Indonesia, India, Asia, and the Middle East sound exotic and foreign because they are based on intervals different from those in Western music. For example, Indonesian music primarily uses one of two scales: Pelog or Slendro (see Fig. 6).
Figure 6: Indonesian music uses two main scales: Pelog (a) and Slendro (b). The tunings probably arise from the harmonics of the gong and the struck-bar instruments used to play them. TUNING SYNTHS One primary reason to adopt 12-tone equal temperament is the historical tendency toward music that is intended to be played on a fixed-pitch keyboard and that modulates into diverse keys. With early tunings that are highly key dependent, you must retune the keyboard instrument each time you play in a different key. That is not something you'd want to do with a harpsichord or an acoustic piano in the middle of a piece of music. Equal temperament eliminates that requirement, so it found favor among Western musicians. Retuning digital synthesizers is easy. All it takes is the appropriate software to recalibrate the oscillators to produce any set of frequencies you desire. The Yamaha DX7II was the first widely available synth to offer that capability. Since then many electronic-keyboard manufacturers have included the ability to use tunings other than equal temperament. Most of those instruments - which include models from E-mu, Korg, and Kurzweil - can retune only the 12 notes in an octave, and those tunings are repeated in all octaves. For key-dependent tunings, you can usually specify the desired root note. In a few instruments, you can retune each note in the entire MIDI range independently. That capability lets you construct larger tunings, such as 53-tone equal temperament or the Indian 22-note scale from which ragas are derived. Synthesizers with alternate tunings usually can't share their tuning data with dissimilar instruments or retune on the fly, so Robert Rich and Carter Scholz developed the MIDI Tuning Standard (MTS), which was added to the official MIDI specification. The standard includes two major parts: bulk dumps and single-note retuning. It outlines the messages by which an instrument can be retuned during a performance. The specified resolution is 0.0061 cent, which is fine for most researchers and musicians. USING TUNINGS Alternate tunings can be used in many ways, particularly with synths. Early and ethnic music can be played with more authenticity, and you can achieve better consonance in all forms of music, particularly if you don't modulate into widely divergent keys. Even if you do modulate, you often can change tunings at the same time. For example, you might create two synth patches with the same sound and different tunings, such as just intonation in the keys of C and B, and select the patch that is tuned to the key you are playing in. Another important application of microtuning is education. If you're a music teacher, you can impart a greater sense of historical perspective to your students by playing music with appropriate tunings from different periods and locations. For example, play a sequence with equal temperament followed by the same sequence in just intonation. The difference is startling. You also can explore the world of sound and acoustics with greater ease and precision. Using alternate tunings has never been easier, thanks to modern music technology. Hopefully, manufacturers will continue to offer that capability in their instruments and include support for MTS, which brings microtuning into the MIDI fold and provides musicians with even greater resources for composition and experimentation. After all, if electronic musicians don't push the musical envelope, who will?
You can read more about microtonality in Scott Wilkinson's book Tuning In: Microtonality in Electronic Music, published by Hal Leonard Books.
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