Minor Third in Just Intonation vs. Equal Temperament: Frequency Ratios Explained

In music theory, a minor third is a fundamental interval that spans three semitones, but its exact frequency ratio depends on the tuning system used. In just intonation, the minor third is tuned to a pure harmonic ratio of 6:5, which produces a consonant, beat-less interval. In equal temperament, the interval is approximated to 300 cents (2^3/12:1), which sacrifices purity for the ability to modulate freely across all keys. This article explains the mathematical basis, historical development, and practical implications of these two approaches, offering clarity on a subtle yet consequential difference.

Overview

The minor third is an interval that sounds melancholic and is found in minor triads, scales, and countless melodic phrases. Its definition varies between tuning systems: just intonation uses simple whole-number ratios derived from the harmonic series, while equal temperament divides the octave into 12 equal parts. The discrepancy between the two—about 15.64 cents—is small enough to pass as the same interval in casual listening, but large enough to affect harmony’s stability and color. Understanding the distinction is crucial for performers, composers, and audio engineers who need to make informed tuning choices.

How It’s Notated

In standard staff notation, a minor third is notated as a third scale degree that is a whole tone plus a semitone above the root. For example, from C to E♭ (C–E flat) is a minor third. The notation itself does not indicate the tuning system; it merely specifies the interval’s place within the diatonic structure. However, in just intonation contexts, composers may add a small horizontal arrow or a plus/minus sign above the note to indicate a deviation from equal temperament. In microtonal notation, a downward or upward arrow can signify a quarter-tone or other microtonal adjustment, but the conventional minor third symbol remains unchanged.

How It’s Performed

Performers have varying degrees of freedom depending on their instrument. Singers, string players, and trombone players can adjust pitch in real time. In a cappella ensembles or string quartets, musicians often intuitively shift toward just intonation intervals to minimize beats and enhance consonance. A violinist playing the minor third above a sustained root will tend to lower the pitch slightly to match the 6:5 ratio, especially if harmonic purity is prioritized. Keyboard players, in contrast, must rely on the fixed tuning of their instrument; on a piano in equal temperament, they cannot alter the minor third’s pitch. Guitarists and other fretted instruments also face fixed tuning, though they can use alternate tunings or capos to approximate just intervals.

How It Works

The physics of pitch is based on frequency ratios. In just intonation, the minor third is defined as the ratio 6:5—that is, the upper note vibrates 6 times for every 5 vibrations of the lower note. This ratio yields a frequency factor of 6/5 = 1.2, which corresponds to 315.64 cents (where 1 cent is one hundredth of a semitone). In equal temperament, each semitone is exactly 100 cents, so a minor third of three semitones is 300 cents. The equal-tempered ratio is 2^3/12 = 2^0.25 ≈ 1.189207, which is slightly smaller than 6/5 (1.2). The difference between 315.64 and 300 cents is 15.64 cents, a subtle but audible discrepancy that causes beating when both systems are used together. The just minor third is perfectly consonant and has no amplitude modulation; the equal-tempered minor third, due to its mistuning, produces a gentle beating effect that becomes noticeable in sustained chords.

Origin / Etymology

The term “minor” comes from the Latin minor, meaning “smaller,” contrasting with “major” (larger). In music theory, intervals are named by ordinal numbers counting scale steps: a third spans three diatonic scale degrees. The distinction between minor and major refers to the size of the interval: a minor third contains three semitones, while a major third contains four. The concept of “just intonation” traces back to ancient Greek music theory, particularly the work of Pythagoras, who discovered simple ratios like 2:1, 3:2, and 4:3. However, the specific 6:5 ratio for the minor third was later formalized by Renaissance theorists such as Gioseffo Zarlino, who advocated for Ptolemaic tuning based on the first six integers. Equal temperament, by contrast, emerged gradually from the need to modulate without retuning; its mathematical basis was described by theorists like Simon Stevin in the 16th century and Zhu Zaiyu in China around the same time, though widespread adoption occurred in the 18th century with the rise of keyboard instruments.

Historical Context

The history of tuning systems is a tale of compromise. Ancient Greek music used Pythagorean tuning, which generates all notes from perfect fifths (3:2). In that system, a minor third is the result of three descending fifths (for example, E–C in Pythagorean) and yields a ratio of 32:27 ≈ 1.185185, which is even narrower than equal temperament. This interval sounded harsh to many ears, leading to the search for a more consonant minor third. The Just Intonation movement of the Renaissance, championed by Zarlino, introduced the pure 6:5 ratio, making the minor third consonant and beat-free. However, just intonation limited modulations because the comma (small tuning discrepancy) accumulates and prevents a closed cycle of fifths. As composers in the Baroque era sought to use all keys freely, they turned to various meantone temperaments and eventually to equal temperament, which distributes the comma evenly across all semitones. Bach’s The Well-Tempered Clavier (1722) is often cited as a showcase for a tuning system—though not necessarily equal temperament in modern sense—that allowed all 24 major and minor keys to be used without retuning. By the 19th century, equal temperament became the dominant standard for fixed-pitch instruments, while vocal and string ensembles continue to gravitate toward just intervals.

Defining Characteristics

The sonic difference between a just and an equal-tempered minor third is subtle but perceptible to trained ears. The just minor third, with its 6:5 ratio, is perfectly consonant and free of beats. It blends seamlessly with a major or minor chord, creating a serene and stable harmonic foundation. In contrast, the equal-tempered minor third is about 15.64 cents sharper than the just version. This mistuning introduces a slow beating (approximately 30 times per second for a 1000 Hz root) that adds a subtle shimmer or roughness, which many musicians associate with a more “instrumental” tone. The equal-tempered version is also slightly major sounding because the third is stretched relative to the pure ratio, which can affect the emotional character of a phrase. In practice, choirs and string quartets often employ just intonation when playing in a stable key, but they may drift toward equal temperament when modulating or when playing with fixed-pitch instruments.

Where You’ll Encounter It

The minor third appears ubiquitously in music: it forms the first two intervals of a natural minor scale (e.g., A–C), the bottom third of a minor triad, and the characteristic “blues” third. In just intonation contexts, the 6:5 minor third is used in a cappella choral music, early music performance, and contemporary microtonal compositions. String quartets and barbershop quartets often spontaneously tune to pure ratios when sustained chords are used. In equal temperament, the minor third is heard in virtually all Western music on pianos, guitars, synthesizers, and woodwinds. It is also the default interval in popular music, from rock to electronic dance, where the tempered intonation is considered “instrumental” and familiar. The difference becomes evident when a violin plays a minor third in a passage with a drone note: the pure ratio yields a smoother tone, whereas the equal-tempered version adds a nervous energy.

Key Figures

Several figures shaped our understanding of these tuning systems. Pythagoras (c. 570–495 BCE) established the mathematical basis for consonance using ratios, but his minor third (32:27) was far from pure. Gioseffo Zarlino (1517–1590) codified the 6:5 ratio in his Istitutioni harmoniche, advocating for just intonation. Simon Stevin (1548–1620) mathematically formulated equal temperament in the West, while Zhu Zaiyu of China independently derived a similar system. Johann Sebastian Bach (1685–1750) composed pieces that demonstrated the practicality of tempered tuning, although the specific tuning he used remains debated. Later, Hermann von Helmholtz (1821–1894) wrote extensively on the physics of just and equal temperament, explaining the beating phenomena and the psychological aspects of tuning. These figures collectively illuminated the delicate balance between mathematical purity and musical flexibility.

Common Misconceptions

Many musicians assume that equal temperament is the “correct” tuning because it is universal on keyboards, but it is actually a compromise. Another misconception is that just intonation is always “in tune”; while it is ideal for a single key, it becomes impossible to maintain when modulating because the ratios clash due to the syntonic comma. Some believe that the equal-tempered minor third is inherently worse, but it enables music in all keys, which is a significant artistic benefit. Another confusion is the conflation of “minor third” with “minor chord”; the chord also includes a perfect fifth, and the minor third alone does not define the chord’s quality. Finally, many think that electronic tuners displaying “in tune” use just intonation, but they usually default to equal temperament, which can lead to beatings with acoustic instruments.

Legacy & Influence

The tension between just and equal temperament continues to influence modern music. In the 20th century, composers like Harry Partch and Lou Harrison embraced just intonation, building custom instruments to explore pure ratios. Meanwhile, the rise of digital audio workstations and microtuning functions has made it easier for musicians to experiment with alternative tunings. The minor third’s dual identity remains a testament to the philosophical divide between abstract mathematics and practical utility. Understanding this interval’s ratios empowers performers to make deliberate choices about intonation, enriching their musical expression. As recording technology advances, the ability to switch between just and equal temperaments in a single piece may become more common, offering new harmonic colors to composers.

In summary, the minor third in just intonation (6:5) and equal temperament (2^3/12:1) represent two solutions to the problem of tuning. Just intonation provides sonic purity at the cost of flexibility; equal temperament provides flexibility at the cost of purity. Both systems are essential to different musical contexts, and a nuanced understanding of their ratios allows musicians to appreciate the depth and richness of this fundamental interval.