In music theory, a major third is a consonant interval that spans four semitones (or half steps) in the chromatic scale and encompasses three diatonic scale degrees, such as from C to E in the key of C major.[1][2] This interval is distinguished from the minor third, which spans only three semitones, by its brighter, more stable sonic quality, often evoking a sense of resolution or majortonality.[3] In just intonation, the major third corresponds to a frequencyratio of 5:4 (approximately 386 cents), providing a pure and harmonious consonance when tuned accordingly, though in equal temperament it approximates this ratio at 400 cents.[4][5]The major third forms a foundational element of major triads, which consist of a root note, a major third above it, and a perfect fifth, creating the characteristic sound of major keys prevalent in Western classical, popular, and jazz music.[6] It appears prominently in the first three notes of the major scale (e.g., do-mi) and contributes to the structural harmony of progressions like the I-IV-V chord sequence.[7] Historically, the major third's role evolved from approximations in Pythagorean tuning by ancient Greek theorists, through Renaissance just intonation practices emphasizing its acoustic purity, into the standardized equal-tempered systems of the common practice era.[8] In performance and composition, tuning the major third precisely can enhance emotional expressiveness, as deviations—such as the wider tempered version—may introduce subtle tension or color.[8]
Definition and Fundamentals
Basic Definition
In music theory, the major third is a fundamental interval that spans three diatonic degrees within a scale, such as from the tonic note C to the mediant E in the C major scale.[7] This interval encompasses four semitones in the twelve-tone equal temperament system commonly used in Western music.[1] It is one of the two primary types of thirds, distinguished from the minor third, which spans only three semitones and three diatonic degrees but with a flattened third degree (e.g., C to E♭).[2]The nomenclature "major" originates from the Latin word maior, meaning "greater," reflecting that this interval is larger in size compared to its counterpart, the minor third derived from minor, meaning "lesser."[9][10] This distinction in terminology arose in medieval and Renaissance music theory to classify intervals based on their relative widths within diatonic contexts.[11]Visually, the major third can be represented on a musical staff in treble clef, where the lower note C is placed on the ledger line below the staff, and the upper note E is placed on the bottom line of the staff.[7] On a piano keyboard diagram, it appears as the distance from the white key C to the white key E, passing over the white D and the black keys D♯/E♭, covering four half-steps.[6]As a core element of Western music theory, the major third functions as a building block for constructing scales, harmonies, and melodic lines, providing the characteristic bright and stable quality to major tonalities.[7] Its frequency ratio approximates 5:4 in just intonation systems.[12]
Interval Size and Ratio
The major third spans exactly four semitones in twelve-tone equal temperament, corresponding to a size of 400 cents.[13] In just intonation, the major third is defined by the simple frequency ratio of 5:4, equivalent to approximately 386.31 cents.[14] This just ratio derives from the Pythagorean major third (81:64, approximately 407.82 cents) by flattening it by the syntonic comma (81:80, or 21.51 cents), a adjustment commonly applied in just intonation systems to achieve purer thirds from the fifth-based Pythagorean framework.[15]The standard measure of interval size in cents uses the formula\text{cents} = 1200 \times \log_2 \left( \frac{f_2}{f_1} \right),where f_2 / f_1 is the frequency ratio; applying this to the 5:4 ratio yields $1200 \times \log_2(1.25) \approx 386.31 cents.[14] For comparison, the minor third—spanning three semitones or 300 cents in equal temperament—has a just intonation ratio of 6:5, approximately 315.64 cents.[16] In equal temperament, these thirds deviate from their just counterparts, producing audible beats between close overtones when sounded simultaneously. For the major third, beats arise primarily from the mismatch between the fifth harmonic of the lower note and the fourth harmonic of the upper note, with frequency difference \Delta f \approx f \cdot |5 - 4 \cdot 2^{4/12}|, where f is the lower note's frequency. For the minor third, beats occur between the sixth harmonic of the lower note and the fifth harmonic of the upper note, \Delta f \approx f \cdot |6 - 5 \cdot 2^{3/12}|.[17]The following table compares the sizes and typical beat frequencies (for a base frequency near 262 Hz, as in middle C) in equal temperament relative to just intonation:
Beat rates increase roughly proportionally with pitch height due to higher fundamental frequencies.[17]
Harmonic Properties
Consonance and Stability
In traditional music theory, particularly from the medieval period onward, the major third is classified as an imperfect consonance, distinguished from perfect consonances such as the unison, octave, and perfect fifth due to its frequency ratio of 5:4, which, while simple, is more complex than the 2:1 or 3:2 ratios of perfect intervals.[18][19] This classification emerged as polyphony developed around the 9th to 13th centuries, when theorists like Walter Odington began recognizing the major third as an imperfect consonance in polyphonic structures such as discant, though it was treated with more caution than perfect consonances in counterpoint rules.[18]The psychoacoustic basis for the major third's consonance lies in its low beating rates between interacting partials and its alignment within the overtone series, where the interval approximates the relationship between the fourth and fifth harmonics (ratios 4:1 and 5:1 relative to the fundamental), promoting perceptual stability through minimal interference and neural synchronization.[20] When two tones form a just major third (5:4ratio), the beating frequency between their fundamentals is relatively slow compared to dissonant intervals like the minor second, reducing auditory roughness, while partial coincidences in the harmonic series enhance fusion and harmony.[21] This alignment contributes to the interval's sense of repose, as the upper tone's lower partials closely match overtones of the lower tone, minimizing dissonance from inharmonic clashes.[20]In the consonance hierarchy of music theory, the major third ranks below perfect consonances (unison, octave, perfect fifth, and sometimes perfect fourth) but above dissonant intervals such as seconds and sevenths, and on par with other imperfect consonances like the minor third and sixths, reflecting its intermediate perceptual stability.[22] Experimental studies confirm this ordering, with listeners rating imperfect consonances like the major third as more pleasant and stable than dissonances but less so than perfect ones, due to greater harmonic simplicity in the latter.[22] This hierarchy influences compositional practices, where major thirds provide moderate resolution without the full repose of a perfect consonance.The major third contributes to stability in common chord progressions such as I-IV-V in major keys, where it forms the defining interval in the root-position triads of the tonic (I) and subdominant (IV), offering a sense of harmonic color and partial resolution that supports the overall tonal center before the dominant (V) introduces tension.[19] In the authentic cadence resolving from V to I, the major third in the tonic triad reinforces closure by aligning with the harmonic series of the root, creating a stable, bright sonority that contrasts with the leading-tone pull of the dominant.[23] This role underscores the interval's function in establishing diatonic harmony without overpowering the structural primacy of perfect consonances.[24]
Harmonic vs. Non-Harmonic Contexts
In harmonic contexts, the major third functions vertically as a stacked interval within chords, particularly as the upper partial in root-position major triads such as C-E-G, where it defines the chord's characteristic brightness and major quality.[25][26] This vertical alignment contributes to the triad's consonance and stability, evoking an uplifting and stable sonority central to tonal harmony.[25]In contrast, non-harmonic contexts treat the major third linearly, as a melodic interval in horizontal lines without simultaneous vertical stacking, often appearing as a skip in melodies or as part of contrapuntal motion.[7] Here, it serves as a connective element, such as in stepwise progressions or passing skips, emphasizing motion rather than harmonic support.[19]Theoretically, this distinction traces to medieval treatises, where the major third was classified as an imperfect consonance—less stable than perfect consonances like the fifth—and thus restricted in strict counterpoint to avoid overuse or consecutive parallels, reflecting its transitional role between consonance and potential dissonance.[18] By the 13th century, theorists like Franco of Cologne classified it as an imperfect consonance, permitting its use in discant and counterpoint, though with caution to alternate with perfect consonances in strict styles.[18]Examples illustrate this divide: in tonal harmony, the major third reinforces vertical structure, as in the major triad's role within common-practice progressions like I-IV-V-I; in modal music, such as Gregorian chant, it appears melodically as a linear skip without triad formation, prioritizing contour over harmony; and in atonal contexts, it functions purely as a non-functional interval in serial or twelve-tone compositions, detached from any stacking implications.[25][18]
Auditory and Perceptual Aspects
Sound Characteristics
The major third exhibits a bright, open, and uplifting timbre arising from the reinforcement of overtones, where the frequency of the upper note coincides with a prominent harmonic partial of the lower note, enhancing perceived clarity and resonance.[27] This alignment contributes to its overall sense of stability and appeal in auditory perception.[28]In Western musical contexts, the major third carries strong emotional connotations of joy and resolution, often evoking feelings of happiness and affirmation.[29] A familiar example appears in the opening phrase of the tune "When the Saints Go Marching In," where the interval between "Oh" and "when" delivers an immediate sense of uplift and celebration.[30]When demonstrated through isolated pure tones, a major third at the 5:4 frequency ratio produces a smooth, unwavering sound free of interference, highlighting its inherent purity.[31] In contrast, the equal-tempered major third introduces a gentle beating effect due to the slight mistuning of its partials, imparting a dynamic yet less stable quality often described as shimmering.[32]Relative to the minor third, which conveys a darker and more introspective tone, the major third's brighter character underscores its role in creating optimistic and resolved auditory experiences.[29]
Tuning System Variations
In just intonation, the major third achieves its purest form with a frequency ratio of 5:4, corresponding to approximately 386.3 cents, which produces a highly consonant interval well-suited to the natural adjustments made by vocalists and string players during performance.[16][33] This tuning system prioritizes simple integer ratios derived from the harmonic series, minimizing beats and enhancing harmonic stability in contexts like a cappella singing or unaccompanied strings.[16]Pythagorean tuning, constructed by stacking pure perfect fifths (3:2 ratio, 702 cents each), results in a major third of about 408 cents (81:64 ratio), which is noticeably wider than the just version and introduces a sharper, less consonant quality.[33] This deviation, known as the syntonic comma (approximately 21.5 cents), arises because the tuning favors fifths over thirds, often necessitating "wolf" intervals—such as a compromised fifth—in remote keys to close the circle of fifths.[33]Equal temperament compromises the major third to exactly 400 cents to ensure uniform semitones across all keys, introducing a slight dissonance compared to just intonation but enabling modulation without retuning.[33] This 14-cent flattening from the Pythagorean third (or 13.7-cent sharpening from just) creates a balanced but less pure sound, widely adopted in modern keyboard instruments for its versatility.[33]Other historical systems like meantone and Werckmeister temperaments offer variations that temper this compromise. Quarter-comma meantone purifies the major third to around 386.3 cents in common keys by flattening fifths, though it produces wider "wolf" thirds (e.g., ~427 cents) in distant keys, limiting its use to fewer tonalities.[33] Werckmeister III, a well temperament, yields major thirds ranging from 390.2 cents (e.g., C-E) to 407.8 cents (e.g., F#-A#), providing usable purity across all keys without extreme wolves, bridging toward equal temperament.[33]
In the major scale, the major third forms the interval between the first and third scale degrees, establishing the tonal foundation of the major mode. For instance, in the C major scale, the notes C and E create this major third, which distinguishes the major scale from its minor counterpart by providing a sense of brightness and resolution.[2] This interval is essential for defining the scale's overall character, as it appears in key positions such as the tonic triad.[34]Within triads, the major third serves as the interval from the root to the third in a root-position major chord, differentiating it from the minor third in minor chords. In the C major triad (C-E-G), the major third between C and E imparts a consonant, stable quality to the harmony, forming the basis of major tonality in Western music.[35] This structure is evident in the tonic, subdominant, and dominant triads of the major scale, where the major third contributes to the chord's uplifting sonority.[36]In extended harmonies, the major third plays a key role in seventh chords, particularly the major-major seventh, where it spans from the root to the third (e.g., C-E in Cmaj7: C-E-G-B), maintaining the major quality while adding tension through the major seventh above the root.[37] Chord inversions further highlight its function in voice leading; for example, in first inversion (e.g., E-G-C for C major), the major third becomes the bass note, facilitating smoother transitions between chords by allowing stepwise motion in inner voices.[38] This positioning aids in common-practice voice leading principles, where the third's placement minimizes parallel octaves and promotes contrary motion.[39]Notational conventions reflect these roles through chord symbols such as "Cmaj" or "CΔ" for major triads and seventh chords, indicating the presence of the major third in lead sheets and scores.[40] Scale degree analysis, using Roman numerals (e.g., I for the tonicmajor triad), underscores the major third's position as the ^3 in the major scale, essential for harmonic analysis.[3]
Historical Development
In ancient Greekmusic theory, particularly within the Pythagorean tuning system attributed to Pythagoras around the 6th century BCE, the major third was derived from stacking four perfect fifths (81:64 ratio), resulting in a sharp interval approximately 22 cents wider than the just major third (5:4). This mistuning caused the major third to beat 6–32 times per second when sounded harmonically, rendering it severely dissonant and thus avoided in simultaneous use, though it was melodically tolerable in scalar contexts. The major third's ratio (81:64) was viewed as imperfect compared to the pure consonances of octaves, fifths, and fourths, reflecting a mathematical emphasis on small integer ratios that prioritized perfect intervals over thirds.[41] During the medieval period, the major third transitioned to classification as an imperfect consonance in organum and early counterpoint treatises, marking a gradual acceptance beyond mere melodic roles. In the 11th century, Guido d'Arezzo's Micrologus categorized intervals into perfect (unisons, fourths, fifths, octaves) and imperfect (major and minor thirds, major sixths), allowing thirds in parallel organum but restricting their use to avoid dissonance with prevailing Pythagorean tuning.[42] By the late 12th century, English organum practices increasingly incorporated major thirds as consonant intervals, influencing continental developments in polyphony where they served as supportive harmonies in discant and motets.[43]The Renaissance brought a pivotal shift, with the major third fully embraced in polyphonic textures due to advancements in tuning that enhanced its purity. Composers like Josquin des Prez employed major thirds prominently in motets such as [Ave Maria... virgo serena](/page/Ave_Maria_ ..._Virgo_serena) (c. 1475–1480), where they form the basis of triadic harmonies tuned syntonically (5:4 ratio) to minimize beating and support imitative counterpoint.[44] This acceptance was facilitated by meantone temperaments, proposed by Bartolomé Ramos de Pareja in 1482 and refined by Gioseffo Zarlino in 1558, which tempered fifths to yield just major thirds, favoring the harmonic sweetness of thirds and sixths over the sharpness of Pythagorean intervals in increasingly complex vocal polyphony.[43]From the Baroque era onward, the major third achieved central status in tonal harmony, underpinning major triads and key definitions across all modulating contexts. Johann Sebastian Bach's The Well-Tempered Clavier (1722, 1742) exemplifies this integration, using well-tempered systems to explore major thirds in preludes and fugues through all 24 keys, enabling fluid tonal progressions where the interval defines chord quality and resolution. In the 20th century, even atonal compositions repurposed the major third devoid of tonal function, as in atonal triads featuring augmented fourths or "major" variants for coloristic effects, seen in works by composers like Arnold Schoenberg who treated it as one interval among equals in pitch-class sets.[45]