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Minor third
Minor third
Minor third
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Minor third
Inversemajor sixth
Name
Other namessesquitone
Abbreviationm3
Size
Semitones3
Interval class3
Just interval6:5, 19:16, 32:27[1]
Cents
12-Tone equal temperament300
Just intonation316, 298, 294
Minor third
equal tempered
just (6:5)
19th harmonic (19:16), E19♭
Comparison, in cents, of intervals at or near a minor third
Jazz and rock bassist Joseph Patrick Moore introducing a cycle of minor thirds

In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two commonly occurring thirds. It is called minor because it is the smaller of the two: the major third spans an additional semitone. For example, the interval from A to C is a minor third, as the note C lies three semitones above A. Coincidentally, there are three staff positions from A to C. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones (two and five). The minor third is a skip melodically.

Notable examples of ascending minor thirds include the opening two notes of "Greensleeves" and of "Light My Fire".

The minor third may be derived from the harmonic series as the interval between the fifth and sixth harmonics, or from the 19th harmonic.

The minor third is commonly used to express sadness in music, and research shows that this mirrors its use in speech, as a tone similar to a minor third is produced during sad speech.[2] It is also a quartal (based on an ascendance of one or more perfect fourths) tertian interval, as opposed to the major third's quintality. The minor third is also obtainable in reference to a fundamental note from the undertone series, while the major third is obtainable as such from the overtone series. (See Otonality and Utonality.)

The minor scale is so named because of the presence of this interval between its tonic and mediant (1st and 3rd) scale degrees. Minor chords too take their name from the presence of this interval built on the chord's root (provided that the interval of a perfect fifth from the root is also present or implied).

A minor third, in just intonation, corresponds to a pitch ratio of 6:5 or 315.64 cents. In an equal tempered tuning, a minor third is equal to three semitones, a ratio of 21/4:1 (about 1.189), or 300 cents, 15.64 cents narrower than the 6:5 ratio. In other meantone tunings it is wider, and in 19 equal temperament it is very nearly the 6:5 ratio of just intonation; in more complex schismatic temperaments, such as 53 equal temperament, the "minor third" is often significantly flat (being close to Pythagorean tuning (playⓘ)), although the "augmented second" produced by such scales is often within ten cents of a pure 6:5 ratio. If a minor third is tuned in accordance with the fundamental of the overtone series, the result is a ratio of 19:16 or 297.51 cents (the nineteenth harmonic).[3] The 12-TET minor third (300 cents) more closely approximates the nineteenth harmonic with only 2.49 cents error.[4] M. Ergo mistakenly claimed that the nineteenth harmonic was the highest ever written, for the bass-trumpet in Richard Wagner's Der Ring des Nibelungen (1848 to 1874), when Robert Schumann's Op. 86 Konzertstück for 4 Horns and Orchestra (1849) features the twentieth harmonic (four octaves and a major third above the fundamental) in the first horn part three times.[5]

Other pitch ratios are given related names, the septimal minor third with ratio 7:6 and the tridecimal minor third with ratio 13:11 in particular.

The minor third is classed as an imperfect consonance and is considered one of the most consonant intervals after the unison, octave, perfect fifth, and perfect fourth.

The sopranino saxophone and E♭ clarinet sound in the concert pitch ( C ) a minor third higher than the written pitch; therefore, to get the sounding pitch one must transpose the written pitch up a minor third. Instruments in A – most commonly the A clarinet, sound a minor third lower than the written pitch.

Pythagorean minor third

[edit]
Semiditone as two octaves minus three justly tuned fifths
Semiditone (32:27) on C

In music theory, a semiditone (or Pythagorean minor third)[6] is the interval 32:27 (approximately 294.13 cents). It is the minor third in Pythagorean tuning. The 32:27 Pythagorean minor third arises in the 5-limit justly tuned major scale between the 2nd and 4th degrees (in the C major scale, between D and F).[7] Playⓘ

It can be thought of as two octaves minus three justly tuned fifths. It is narrower than a justly tuned minor third by a syntonic comma. Its inversion is a Pythagorean major sixth.[citation needed]

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Minor third

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from Grokipedia
A minor third is a musical interval that spans three semitones (or half steps) and encompasses three letter names in the diatonic scale, such as from C to E♭ or D to F. It represents the smaller of the two possible thirds, differing from the by one , which spans four semitones. In terms of notation, it is typically written on adjacent lines or spaces in staff notation, comprising one whole step followed by one half step. In acoustic terms, the minor third corresponds to a frequency ratio of 6:5 in just intonation, derived by subtracting a major third (5:4) from a perfect fifth (3:2). This interval is classified as consonant, producing a stable and harmonious sound, though it carries a darker, more somber emotional quality compared to the brighter major third. It serves as a foundational element in Western music theory, particularly in constructing minor triads—chords built from a root, minor third, and perfect fifth—which underpin minor keys and evoke melancholy or tension in compositions. The minor third appears prominently in natural phenomena and cross-cultural music, such as the pentatonic scale and even human speech patterns, where it approximates intervals of emotional expression. In equal temperament, its size is 300 cents, approximately 15.64 cents narrower than the just intonation version (approximately 315.64 cents), influencing its role in modern harmonic progressions and melodic lines.

Definition and Properties

Interval Basics

The minor third is a fundamental musical interval in Western music theory, defined as the distance spanning three semitones, or half steps, and encompassing three letter names in the diatonic scale, between two pitches. For instance, the interval from C to E♭ (spanning C, D, E♭) exemplifies a minor third, where E♭ is the minor third above C. This interval forms a key building block in scales, chords, and harmonies, distinguishing it from larger or smaller intervals in the diatonic system. In equal temperament, the standard tuning system for most Western music, the minor third measures exactly cents, with each equaling 100 cents. This precise ensures consistent intonation across instruments like , where the interval is fixed by the equal division of the into 12 semitones. By comparison, the —a closely related interval—spans four semitones and 400 cents in equal temperament, making the minor third relatively narrower and creating a distinct sonic profile. The minor third is typically notated as m3 in interval shorthand, though it may appear as ♭3 in chord symbols to indicate the flattened third degree relative to the root. Its frequency ratio, approximately 6:5 in just intonation, contributes to this character but is explored in greater detail elsewhere.

Frequency Ratios

The minor third in just intonation is defined by the frequency ratio of 6:5, where the frequency of the upper note is 6/5 times that of the lower note. This ratio arises from simple integer harmonics in the overtone series, specifically as the interval between the fifth harmonic (fundamental times 5) and the sixth harmonic (fundamental times 6) of a single tone, yielding a consonant interval based on low partials. To quantify this interval in cents—a logarithmic unit where one octave equals 1200 cents—the size is calculated using the formula: cents=1200×log⁡2(fupperflower)\text{cents} = 1200 \times \log_2 \left( \frac{f_\text{upper}}{f_\text{lower}} \right) For the 6:5 ratio, this yields exactly 315.64 cents. In other tuning systems, the minor third deviates from this just value. The Pythagorean minor third uses the ratio 32:27, approximately 294 cents, which is narrower than the equal-tempered approximation. Equal temperament approximates the minor third as 23/12≈1.18922^{3/12} \approx 1.1892, or precisely 300 cents, providing a compromise for fixed-pitch instruments across all keys.

Tuning Systems

Pythagorean Tuning

In Pythagorean tuning, the scale is constructed by stacking pure perfect fifths with a frequency ratio of 3:2, reducing by octaves (2:1) as necessary to fit within a single octave, which results in the minor third emerging as a derived interval rather than a primary one. For instance, starting from a reference note C at frequency 1, the second scale degree D is reached by two successive fifths reduced by one octave, yielding a ratio of 9:8; the fourth degree F is obtained via a pure fourth (equivalent to an octave minus a fifth), at 4:3. The minor third from D to F thus has the ratio (4/3) ÷ (9/8) = 32/27. This construction prioritizes the purity of fifths and octaves, treating the minor third as a secondary consequence of the tuning system. The 32:27 ratio corresponds to approximately 294 cents, calculated as 1200 × log₂(32/27) ≈ 294.13 cents, which is narrower than the just minor third of 6:5 (about 316 cents). This deviation arises from the inherent properties of the Pythagorean system, where completing the circle of 12 fifths exceeds seven octaves by the Pythagorean comma, a small interval of (3/2)¹² / 2⁷ ≈ 23.46 cents (or 531441:524288), leading to inconsistencies in third sizes across keys. The comma represents the tuning's "closure error," compressing some intervals like the minor third relative to harmonic ideals based on simpler integer ratios. Historically, the Pythagorean minor third was viewed as impure and dissonant by some ancient and early medieval theorists due to its complex ratio and deviation from the smoother 6:5 just intonation, which better aligns with the harmonic series. Figures like Boethius (c. 480–524 CE), drawing on Greek traditions, classified such thirds as dissonances because their ratios (32:27 for the minor third) were less simple than those of fifths and octaves, limiting their use in early polyphony to avoid perceived roughness. Later theorists, such as Ptolemy (c. 100–170 CE), critiqued Pythagorean thirds for this reason, advocating adjustments to achieve greater consonance in melodic and harmonic contexts.

Just Intonation

In just intonation, the minor third is defined by the simple frequency ratio of 6:56:5, derived from the interval between the fifth and sixth harmonics in the harmonic series of a fundamental tone. This ratio yields an interval of approximately 315.64 cents, providing a pure consonance that minimizes acoustic beats due to the small integer values involved, making it particularly suitable for unaccompanied vocal ensembles or string groups where performers can adjust pitches dynamically for harmonic clarity. Unlike the Pythagorean minor third of 32:27 (about 294 cents), which derives from stacked perfect fifths and introduces noticeable dissonance, the 6:5 ratio prioritizes harmonic purity over melodic consistency across keys. In practice, just intonation allows minor thirds to be tuned adaptively—sharper or flatter depending on the key—to maintain this purity, as seen in systems like meantone temperaments that approximate just intervals while accommodating multiple tonalities on fixed-pitch instruments. For example, in a just intonation tuning centered on C, the note E♭ forms a minor third above C at the 6:5 ratio, creating a resonant foundation for minor triads within that key. This approach enhances the interval's acoustic stability in acoustic settings, where performers rely on natural overtones for precise alignment.

Equal Temperament

In 12-tone equal temperament, the standard tuning system for most modern Western music, the minor third spans three semitones, yielding a frequency ratio of 23/12≈1.18922^{3/12} \approx 1.1892 and exactly 300 cents. This divides the octave evenly into 12 logarithmically equal steps, providing a consistent interval size regardless of the starting pitch. On instruments like the piano, the minor third from C to E♭ thus encompasses three equal semitones, serving as a fundamental building block for scales, chords, and melodies. Compared to the just intonation minor third of 6:5 (approximately 315.64 cents), the equal-tempered version is about 15.64 cents narrower, rendering it slightly flat and introducing subtle beating in sustained chords where harmonic purity is expected. This deviation compromises some acoustic consonance but remains imperceptible in fast passages or ensemble settings. Unlike the pure intervals of just intonation, this approximation prioritizes uniformity over exact ratios. Equal temperament was widely adopted for keyboard instruments in the late 18th century, enabling fixed-pitch tuning that functions equally well across all keys without retuning. A key advantage is its support for modulation and transposition in complex compositions, allowing the minor third to maintain consistent intonation in remote keys and enharmonic equivalents. This versatility revolutionized keyboard music, making it the dominant system for fixed-pitch instruments today.

Historical Development

Ancient and Medieval Periods

In ancient Greek music theory, the minor third, known as the semiditone, was a key interval in the chromatic genus, where Aristoxenus described the tetrachord as comprising two semitones followed by a semiditone, emphasizing perceptual judgment over numerical ratios in his Harmonic Elements. Later, Ptolemy refined this in his Harmonics (c. 150 CE) by approximating the minor third in syntonic tuning with the just ratio of 6:5, integrating it more harmoniously into diatonic scales while critiquing purely Pythagorean approaches. Pythagorean theorists, prioritizing perfect consonances like the (2:1) and fifth (3:2), largely avoided thirds altogether, viewing the Pythagorean (32:27, approximately 294 cents) as imperfect and dissonant compared to the (81:64), which they tolerated but did not emphasize in melodic practice. This preference reflected a broader where only superparticular ratios were deemed fully , relegating the to secondary status in early theoretical texts. During the medieval period, Boethius transmitted and adapted Greek ideas in his De institutione musica (c. 510 CE), classifying the minor third as a composite interval of one whole tone plus one , positioning it among the imperfect concords suitable for melodic progression but not full harmony. By the , Guido d'Arezzo incorporated the minor third into his solmization system, particularly in the molle or soft starting on F (F-G-A-B♭-C-D), where intervals like A to C formed a minor third, facilitating sight-singing in modal while avoiding the tritone. In plainchant practices from the 9th to 12th centuries, the minor third played a subtle cultural role, enhancing the melancholic or contemplative ethos of modes like the Hypodorian and Hypophrygian, where it appeared in cadential figures to evoke in liturgical texts. This usage aligned with the period's modal framework, derived from earlier Greek influences via , underscoring the interval's expressive potential in unaccompanied vocal traditions.

Renaissance to Baroque

During the Renaissance, the minor third emerged as a foundational element in polyphonic music, marking a shift from the parallelism of earlier medieval styles—where voices moved in parallel fourths and fifths—to a greater emphasis on vertical harmonic intervals. Composers like Josquin des Prez integrated minor thirds into motets, creating richer textures through the interplay of thirds and sixths, which served as "colored dissonances" resolving to more stable consonances. In Josquin's four-voice motet Ave Maria (c. 1475–1500), for instance, a phrase employs a chain of sixths resolving to fifths, with minor thirds contributing to dynamic tension within the prevailing Pythagorean tuning, where they appear narrow and contribute to the era's evolving harmonic sensibility. This harmonic development coincided with the adoption of mean-tone tuning around the early , which prioritized purer thirds for keyboard and fretted instruments. Documented by theorist Pietro Aaron in 1523, quarter-comma mean-tone rendered the minor third narrower at a of 81:68 (approximately 310 cents), slightly flatter than the ideal of 6:5 (316 cents), to achieve sweeter major thirds at (386 cents) and enhance chordal consonance on organs and lutes. This system, prevalent in and early , facilitated the vertical stacking of thirds in , producing a more resonant and agreeable sound compared to the wider Pythagorean thirds. Theoretical advancements further solidified the minor third's role, as seen in Gioseffo Zarlino's Le Istitutioni harmoniche (1558), where he advocated for just minor thirds (6:5) as essential consonances derived from natural proportions, arguing they formed the basis of perfect harmony in both vocal and instrumental music. Zarlino proposed tempering these intervals minimally—by one-seventh of a syntonic comma—for practical use on fixed-pitch instruments, ensuring the minor third's (semiditone) consonance without excessive alteration, thus influencing compositional practices toward greater harmonic sweetness. In the Baroque era, the minor third became central to the establishment of minor keys and affective expression, particularly in the works of Johann Sebastian Bach (1685–1750). Bach's Passions, such as the St. Matthew Passion (1727), employ minor keys like E minor to evoke profound sorrow and crucifixion imagery, with minor thirds underpinning the harmonic foundation for emotional depth in choruses and arias, aligning with the doctrine of affections. By the early 18th century, the transition to well-temperament systems around 1680–1700 allowed minor thirds to be intonated more flexibly across all keys, departing from mean-tone's limitations and enabling freer modulation while preserving relative purity in common tonalities. This shift paved the way for the later standardization of equal temperament, broadening the minor third's versatility in tonal music.

Musical Applications

In Scales and Modes

The natural minor scale, also known as the , is characterized by its third scale degree, which forms a minor third above the tonic, creating the fundamental minor tonality that contrasts with the major scale's major third. This interval establishes the scale's melancholic quality and serves as the basis for minor keys in Western music. For example, the A natural minor scale consists of the pitches A-B-C-D-E-F-G, where the note C lies a minor third above A. The harmonic minor scale builds on the natural minor by raising the seventh scale degree by a semitone, primarily to facilitate a stronger dominant chord, yet it retains the minor third from the tonic to preserve the overall minor modality. This adjustment affects melodic and harmonic motion without altering the defining third-degree interval. In A harmonic minor, the scale is A-B-C-D-E-F-G♯, with C remaining a minor third from A. In modal contexts, the minor third contributes distinct timbres to several modes, particularly the Dorian and Phrygian, both of which are classified as minor modes due to this interval. The Dorian mode incorporates a minor third followed by a major sixth, lending a brighter yet introspective flavor; for instance, D Dorian unfolds as D-E-F-G-A-B-C, emphasizing the F as a minor third from D. Similarly, the Phrygian mode features a minor third after its signature minor second, evoking an exotic or tense character, as in E Phrygian: E-F-G-A-B-C-D, where G forms the minor third from E. These modes highlight the minor third's role in modal color without relying on key signatures. The minor pentatonic scale, widely used in blues, rock, and folk traditions, underscores the minor third as one of its core intervals, deriving from the natural minor scale by omitting the second and sixth degrees. This results in a concise five-note structure that prioritizes the root-minor third-fifth framework for expressive melodies. The A minor pentatonic scale, for example, comprises A-C-D-E-G, with C providing the essential minor third from A. Parallels to the minor third exist in non-Western systems, such as Indian classical music, where the komal ga (flattened third swara) approximates a minor third interval from the tonic sa, infusing ragas with pathos or introspection. This komal ga appears prominently in thaats like Kafi and ragas such as Bhimpalasi, mirroring the minor third's scalar function in evoking emotional depth.

In Harmony and Chords

The minor triad, a foundational element of Western harmony, is constructed by stacking a minor third above the root followed by a major third above that interval, resulting in a perfect fifth from root to highest note. For example, in C minor, the triad comprises the notes C (root), E♭ (minor third), and G (perfect fifth). This structure distinguishes it from the major triad, which uses a major third over the root, and imparts a characteristic somber or melancholic quality to the harmony. In harmonic progressions, the minor triad plays a central role in defining the tonal center and emotional trajectory of minor keys, particularly as the tonic (i) and subdominant (iv) chords. A common progression is i–iv–V–i, as in C minor (Cm–Fm–G–Cm), where the V chord is often major due to the raised leading tone in the harmonic minor scale; this sequence builds tension toward resolution and evokes pathos, especially in cadential contexts. Inversions of the minor triad, such as the first inversion (e.g., E♭–G–C for Cm/E♭), facilitate smoother voice leading by placing the third in the bass, allowing for fluid transitions in polyphonic textures. Extended harmonies incorporating the minor third include the minor seventh chord, formed by adding a minor seventh above the root to the minor triad (e.g., Cm7: C–E♭–G–B♭), which enriches the chord's color and function, often serving as the tonic or subdominant in jazz and classical progressions.

Acoustics and Perception

Consonance Characteristics

The minor third holds a prominent position in the hierarchy of consonant intervals, ranked third in overall consonance after the octave and perfect fifth, yet slightly below the major third due to the increased complexity of its frequency ratio of 6:5 compared to the major third's simpler 5:4 ratio. This ranking, established in foundational acoustic theory, reflects how simpler integer ratios correlate with greater perceptual stability and reduced roughness in interval perception. In terms of acoustic properties, the minor third in just intonation produces near-zero beat frequencies between aligned partials, as the interval's pure 6:5 ratio allows harmonics to coincide precisely without interference, fostering a smooth sonic texture. In equal temperament, however, the approximated ratio introduces a subtle detuning, resulting in low beat rates—approximately 4.75 Hz when the lower note is at A=440 Hz—that manifest as a gentle pulsation rather than harsh dissonance. The beat frequency for this detuning is given by the equation ∣fupper−(6/5)flower∣|f_\text{upper} - (6/5) f_\text{lower}|, which approaches zero in just intonation but yields the observed minor discrepancy in tempered systems. The minor third's consonance is further bolstered by its alignment in the harmonic series, spanning the interval between the fifth and sixth partials, where the frequencies reinforce each other closely at these low overtones. Spectral analysis of complex tones confirms this, showing strong partial overlap and minimal interference in the lower harmonic regions, which contributes to the interval's relative stability compared to more complex ratios. This reinforcement at early partials distinguishes the minor third's acoustic profile, emphasizing its role as an imperfect but effective consonance in harmonic contexts.

Cultural and Psychological Aspects

In Western music, the minor third interval is frequently associated with emotions of sadness and tension, contributing to the affective quality of minor keys and modes. This connotation arises from its role in minor triads and melodies, where it evokes a sense of melancholy or unease compared to the brighter major third. For instance, the traditional English folk tune "Greensleeves," set in a minor key, prominently features descending minor thirds in its melody, enhancing its haunting and sorrowful character. Psychological research has long examined the perceptual qualities of the minor third, identifying it as more dissonant than the major third when presented in isolation. Hermann von Helmholtz, in his seminal 1863 treatise On the Sensations of Tone, proposed that dissonance in intervals like the minor third stems from the roughness caused by closely spaced overtones, ranking it lower in consonance than the major third due to increased beating sensations. This view, based on physiological acoustics, has influenced subsequent studies on interval perception, confirming the minor third's tendency to produce a sense of instability or discomfort in listeners. Culturally, the minor third and similar intervals carry varied emotional significances beyond Western traditions. In Arabic maqam music, Maqam Saba incorporates a minor third (e.g., from the tonic to the third scale degree) within its jins structure, often evoking themes of longing, grief, and spiritual depth, as heard in traditional performances expressing romantic yearning or sorrow. Similarly, in African American blues music, "blue notes" involve microtonal bends toward the minor third (the flattened third degree in a major context), infusing the genre with a poignant, expressive tension that blends major and minor tonalities to convey hardship and emotional intensity. In modern media, the minor third is employed in film scores to heighten and unease, particularly through repetitive ostinati that build tension. Composers like utilize pulsing minor third patterns in action-thriller soundtracks, such as those in series, to create a relentless, ominous atmosphere that underscores dramatic conflict. This technique leverages the interval's inherent dissonance to amplify psychological dread in horror and genres. From an evolutionary perspective, minor intervals may elicit innate responses linked to distress signaling, as suggested by parallels between human vocalizations and musical structures. Research indicates that infant cries feature a higher proportion of minor seconds—a closely related dissonant interval—compared to neutral babbling, mirroring the prevalence of such intervals in sad Western music and potentially reflecting an adaptive mechanism for conveying emotional urgency across species. This shared acoustic signature supports theories that minor-like intervals in music tap into primal affective cues for sadness or alarm.

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