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Major second

In music theory, a major second is a diatonic interval encompassing two semitones, or one whole step, between two pitches, representing the distance from the tonic to the supertonic in a major scale.^[1]^[2] This interval can occur as a harmonic interval, where the two notes sound simultaneously, or a melodic interval, where they are played or sung in succession.^[1] It is denoted in shorthand as "M2" and contrasts with the minor second, which spans only one semitone.^[2] Common examples of the major second include C to D, D to E, F to G, G to A, and A to B within the C major scale, each separated by two half steps on a piano keyboard.^[3]^[2] In staff notation, it appears as the skip from a line to the adjacent space or space to line in the diatonic scale, such as from the bottom line (E in treble clef) to the space above (F♯, adjusted for key).^[1] The interval's size is numerically classified as a second due to encompassing two scale degrees, with its "major" quality determined by alignment with the major scale's structure—natural to natural, sharp to sharp, or flat to flat without alteration.^[1]^[3] Historically rooted in ancient Greek music theory, the major second derives from Pythagorean tuning, where it approximates a frequency ratio of 9:8, forming the "whole tone" as a foundational building block of the diatonic scale alongside the minor second (256:243).^[4]^[5] In just intonation, this ratio yields a pure, consonant sound for stepwise motion, though equal temperament slightly adjusts it to 200 cents for uniformity across the chromatic scale.^[4] The interval's prominence persisted through Western music's evolution, appearing in medieval modes, Renaissance polyphony, and modern compositions as a staple for melodic contours and chord voicings.^[3] As a versatile element, the major second contributes to the stepwise progression in major and minor scales, facilitating smooth transitions in melodies while occasionally introducing mild dissonance in harmonic contexts, such as suspensions or appoggiaturas.^[1] Its recognition and ear-training importance is emphasized in pedagogical resources, where it is often the first interval taught after the unison due to its prevalence in familiar tunes like "Happy Birthday" (starting on the first two notes).^[2]

Definition and Properties

Interval Size and Notation

The major second is defined as the musical interval between the first (tonic) and second (supertonic) degrees of the major scale, spanning two semitones or a whole step.^[6] In twelve-tone equal temperament, this interval measures exactly 200 cents, calculated as two semitones of 100 cents each within the 1200-cent octave.^[7] By comparison, the minor second spans only one semitone, or 100 cents in equal temperament.^[6] Standard notations for the major second include the abbreviations M2 or simply "major second," as well as terms like "whole tone" or "whole step" to emphasize its size relative to half steps.^[2] These notations distinguish it from smaller intervals like the minor second, which is a half step.^[8] In staff notation, the major second appears as adjacent scale degrees, such as from C to D in the key of C major on the treble clef, where D is positioned one line above C.^[1] This simple ascending or descending placement highlights its role as a foundational diatonic interval. In Pythagorean tuning, the major second is derived from the ratio 9:8, which corresponds to approximately 203.91 cents, slightly larger than the equal-tempered version and based on stacking perfect fifths.^[7] This ratio underscores its historical significance in early tuning systems prioritizing consonant fifths.^[5]

Acoustic and Harmonic Characteristics

The major second interval, in just intonation, is defined by a frequency ratio of 9:8 between the higher and lower tones, a simple integer ratio that contributes to its relative consonance compared to more complex intervals.^[9] This ratio arises from the acoustic properties of vibrating strings or air columns, where the overtones of the tones interact such that the second harmonic of the lower tone (at 2f, where f is the fundamental frequency) aligns in proximity to partials of the higher tone, though not perfectly, leading to a sense of stability tempered by mild roughness in complex timbres.^[10] In the harmonic series, the major second corresponds to the ratio between the ninth and eighth partials (9:8), positioning it as a natural occurrence in the overtone structure of a single tone.^[11] Perceptually, the major second produces a bright, open sound quality that often conveys a sense of forward motion or tension release within melodic contexts, such as stepwise progressions in the diatonic scale.^[12] This perception stems from its position as a small interval with sufficient harmonic coherence to avoid extreme dissonance, yet it can evoke subtle instability due to acoustic interactions between overtones. In untempered tunings deviating from the pure 9:8 ratio, such as meantone systems, beat frequencies emerge from the slight mismatch in partials; for instance, a detuning of just a few cents can produce audible beats at rates of 10-20 Hz for mid-range pitches, enhancing the interval's dynamic expressiveness.^[12] Mathematically, the interval's size is quantified in cents using the formula for logarithmic frequency ratios:

\text{cents} = 1200 \times \log_2 \left( \frac{f_2}{f_1} \right)

Applied to the major second with ratio 9/8, this yields approximately 203.91 cents, distinguishing it from the equal-tempered approximation of 200 cents and underscoring its acoustic purity in just intonation.^[13]

Historical Development

Ancient Greek Origins

The major second, known in ancient Greek music theory as the epogdoon (ἐπόγδοον), was recognized as the interval with a ratio of 9:8, literally meaning "one-and-eighth" to reflect its superparticular proportion relative to the unison. This interval, approximating 204 cents, emerged from Pythagorean experiments with the monochord, where Pythagoras (c. 570–495 BCE) demonstrated that dividing a string in the ratio 9:8 produced a whole tone, the foundational step beyond the unison in scalar construction. Philolaus of Croton (c. 470–385 BCE), a key Pythagorean, explicitly described the epogdoon as the difference between the perfect fifth (3:2) and the perfect fourth (4:3), establishing its mathematical basis in harmonic science.^[14] In the tetrachord system, which formed the core of Greek scalar organization, the epogdoon was the whole tone interval used twice, along with a limma (256:243), to divide the perfect fourth (4:3) in diatonic genera. The tetrachord, spanning four notes with fixed outer pitches, allowed for conjunct or disjunct arrangements to build larger systems like the heptachord or octave, and the epogdoon's role ensured consonant progression within these structures. Aristoxenus of Tarentum (c. 375–335 BCE), in his Harmonics, analyzed the epogdoon through perceptual and spatial methods rather than pure ratios, emphasizing its auditory magnitude as greater than the semitone but less than the third tone, thus prioritizing practical intonation over strict Pythagorean arithmetic. Ptolemy (c. 100–170 CE), building on this in his own Harmonics, cataloged tetrachord divisions incorporating the epogdoon in various genera, refining its application for melodic coherence while critiquing overly rigid Pythagorean constraints.^[14] The epogdoon held profound cultural significance in Greek modes, such as the Dorian, where it contributed to the ethos of dignity, courage, and discipline, influencing emotional expression in drama, philosophy, and civic life. In tragic theater, for instance, its placement in modal frameworks evoked solemnity and moral reflection, as seen in Euripides' works, aligning music with the philosophical ideal of harmony between soul and cosmos. Pythagoras himself employed the epogdoon-based paeans on the lyre for therapeutic serenity, underscoring its role in ethical education and communal rituals.^[14]

Evolution in Western Music Theory

The theoretical treatment of the major second in Western music began with early medieval adaptations of ancient Greek concepts, as seen in Boethius' De institutione musica (c. 500–520 CE), which transmitted Pythagorean ideas of intervals, including the whole tone as a foundational diatonic step derived from the ratio 9:8.^[15] Building on this legacy, the medieval period saw the major second formalized within practical pedagogy through Guido d'Arezzo's hexachord system in the 11th century, where it constituted the "whole tone" interval in solmization syllables, such as between re and mi, enabling singers to navigate overlapping hexachords (C-D-E-F-G-A, F-G-A-B♭-C-D, and G-A-B-C-D-E) across the gamut without fixed pitches.^[16] This approach emphasized the major second's role as a consistent stepwise motion in modal chant, distinct from the semitone (mi-fa), and became a cornerstone of sight-singing education in monastic and scholastic settings.^[17] In the Renaissance, Gioseffo Zarlino advanced the major second's theoretical status in Le Istitutioni harmoniche (1558), advocating its realization as the just intonation ratio 9:8 within synthetic scales to achieve harmonic purity and sensory appeal, positioning it as a "sweet" diatonic interval essential to the senario (1:1 to 6:1) and consonant progressions in polyphony.^[18] Zarlino's framework integrated the major second into modal structures, arguing it complemented perfect consonances like the octave and fifth while supporting the emerging emphasis on vertical harmony in Venetian polychoral music.^[19] The Baroque era introduced temperamental adjustments to the major second amid expanding tonal practices, with meantone systems—prevalent in keyboard and ensemble music—rendering it as a uniform "mean tone" slightly smaller than the Pythagorean 9:8 (approximately 193.2 cents in quarter-comma meantone) to enhance consonance in common keys, particularly by purifying major thirds at the expense of remote fifths.^[20] This variation allowed the major second to function flexibly across transpositions, as in the works of composers like Frescobaldi and Sweelinck, where it facilitated smoother voice leading in affected keys without the "wolf" intervals of pure Pythagorean tuning.^[21] Concurrently, Jean-Philippe Rameau's Traité de l'harmonie (1722) incorporated the major second into functional harmony, viewing it as an essential component of major-mode scales and dominant-tonic progressions, where it often appeared in inverted seventh chords or as a passing tone reinforcing the fundamental bass and tonal resolution. Rameau's theories thus elevated the major second from a mere scalar interval to a dynamic element in chordal sequences, influencing the galant style's emphasis on clear harmonic motion.^[22]

Tuning Systems and Variations

Just Intonation

In just intonation, the major second is defined by the pure frequency ratio of 9:8, which corresponds to approximately 203.91 cents above the fundamental pitch.^[23] This ratio arises from the acoustic principles of simple integer proportions, prioritizing harmonic purity over uniform spacing. The derivation of the 9:8 ratio involves stacking two perfect fifths, each with a 3:2 ratio, and then reducing the result by one octave to bring it within the standard interval range:

\left( \frac{3}{2} \times \frac{3}{2} \right) \div 2 = \frac{9}{4} \div 2 = \frac{9}{8}.

^[4] This process aligns with Pythagorean tuning principles, a subset of just intonation limited to 3-limit ratios (powers of 2 and 3), where the major second emerges naturally from successive fifths. While the full Pythagorean scale introduces the Pythagorean comma—calculated as (3/2)^{12} / 2^7 \approx 531441/524288 (23.46 cents)—for closing the octave circle, the 9:8 second itself requires no such adjustment due to its direct simplicity from two fifths.^[24] The 9:8 ratio enhances consonance because its low prime factors (3 and 2) align closely with the harmonic series, producing beats that are minimal and pleasing to the ear.^[23] This purity makes it particularly suitable for a cappella vocal ensembles and early instruments like lutes or viols, where performers can adjust pitches dynamically to achieve these exact proportions without fixed tempering. Just intonation with the 9:8 major second was a cornerstone of Renaissance music theory and practice.^[18] In related tuning systems like meantone temperament, which aims to approximate just intonation for consonant chords such as pure major thirds (5:4), the major second is adjusted using fractions of the syntonic comma (81/80, approximately 21.51 cents).^[24] For instance, quarter-comma meantone tempers fifths slightly flat, resulting in a major second of approximately 193.2 cents.^[25]

Equal Temperament and Modern Usage

In twelve-tone equal temperament, the major second encompasses exactly two semitones, equivalent to 200 cents or one-fifth of an octave, as the octave is divided into twelve equal parts.^[26] This precise measurement arises from the formula for interval size in cents:

$1200 \times \log_2 \left( 2^{2/12} \right) = 200

where the frequency ratio for the major second is $2^{1/6}, ensuring consistent semitone steps throughout the scale.^[13] The widespread adoption of equal temperament began in the 18th century, evolving from earlier meantone systems and gaining prominence through Johann Sebastian Bach's The Well-Tempered Clavier (1722), a collection of preludes and fugues in all major and minor keys that showcased the versatility of a well-tempered tuning.^[27] This work highlighted the system's ability to modulate freely without retuning, paving the way for its dominance in Western music.^[28] In modern usage, equal temperament offers uniformity across all keys on fixed-pitch keyboard instruments like the piano and organ, enabling seamless performance in any tonal center without the harmonic biases of unequal tunings.^[29] However, this approximation renders the major second slightly detuned relative to just intonation's pure ratio of 9:8 (approximately 203.91 cents), resulting in a flattening of 3.91 cents that prioritizes modulation over interval consonance.^[23]

Musical Applications

Role in Scales and Harmony

In both major and minor scales, the major second serves as the foundational interval from the tonic to the supertonic (scale degree 2), forming the first whole step in the diatonic pattern and contributing to the establishment of tonality by providing a stable, consonant extension above the root that outlines the key's basic stepwise motion. In the major scale, this interval initiates the characteristic sequence of whole-whole-half-whole-whole-whole-half steps, creating the bright, resolved sound associated with major keys.^[30] Similarly, the natural minor scale begins with a major second in its whole-half-whole-whole-half-whole-whole pattern, reinforcing the minor key's tonality through a comparable initial ascent that sets the modal framework without the raised leading tone found in other minor variants.^[31] The supertonic's position as a major second above the tonic thus plays a pivotal role in tonal orientation, acting as a structural anchor that differentiates the scale from chromatic or atonal constructions.^[32] Harmonically, the major second integrates into chord structures, particularly as part of the major triad, where the major third from root to third degree encompasses two stacked major seconds within the scale (e.g., from tonic to supertonic, then supertonic to mediant). This stacking underscores the triad's consonant framework, with the major second providing the initial layer of the imperfect consonance that defines major harmony. In equal temperament, the major second measures approximately 200 cents, offering a consistent intervallic size that supports these harmonic builds across keys. In voice leading and counterpoint, the major second functions as an imperfect consonance or mild dissonance, often employed in passing tones during second-species counterpoint, where it must connect stepwise to adjacent notes while adhering to rules that prioritize smooth linear motion between stronger consonances like thirds or fifths.^[33] In functional harmony, the major second embodies the supertonic's role in progressions, frequently appearing in voice-leading motions within I-IV or ii-V sequences, where it resolves to more stable intervals such as the unison or third to maintain tonal balance and drive cadential flow. This resolution tendency—typically by contrary or oblique motion—enhances the progression's coherence, as the supertonic's inherent pull toward the dominant or subdominant reinforces the cycle of tension and release central to tonal music. The interval's dissonance potential emerges in contexts like modal interchange, where an augmented second (enlarged by a semitone) arises from borrowing elements such as the raised seventh in the harmonic minor scale, introducing chromatic color while remaining distinct from the narrower minor second in resolution and stability.^[34]^[35]

Examples in Compositions and Genres

In classical music, the major second serves as a foundational melodic interval for stepwise motion, providing smooth transitions in themes and chorales. For instance, the opening theme of Wolfgang Amadeus Mozart's Eine kleine Nachtmusik, K. 525 (1787), begins with a major second ascent from G to A in G major, establishing the graceful and elegant character of the serenade.^[36] Similarly, in J.S. Bach's four-part chorales, such as those in his St. Matthew Passion (BWV 244, 1727), major seconds appear frequently in soprano and alto lines as part of diatonic stepwise progressions, contributing to the flowing, hymn-like quality of the vocal writing. In jazz and blues, the major second often features in pentatonic and blues scales to create tension and resolution, particularly when blue notes bend toward or resolve via whole-step approaches. For example, in the blues standard "The Thrill Is Gone" by B.B. King (1969), the guitar melody employs major seconds within the minor pentatonic framework to heighten emotional expressiveness, as the interval bridges bent notes and chord tones for idiomatic phrasing.^[37] This usage underscores the interval's role in improvisational lines, where it contrasts with chromatic minor seconds for color. In folk and non-Western traditions, the major second plays a key structural role in melodic contours. In Hindustani classical music, the shuddha rishabha (Re) represents a major second above the tonic (Sa) in ragas like Bilawal, facilitating ascending phrases that evoke stability and ascent; for instance, it appears prominently in the arohana (ascending scale) of Raga Yaman, enhancing the raga's serene mood.^[38] Likewise, in many sub-Saharan African musical practices, call-and-response patterns incorporate major seconds in stepwise exchanges between leader and group, as seen in Yoruba talking drum ensembles or Akan folk songs, where the interval supports rhythmic dialogue and communal participation.^[39] In modern pop and rock, the major second drives catchy hooks through simple whole-step leaps. The melody of "Happy Birthday to You" (traditional, 1893) exemplifies this, with the opening "Hap-py" sung as a major second (e.g., C to D), making it instantly recognizable and easy to sing in unison.^[40] This interval's prominence in verse-chorus structures, such as the stepwise motifs in The Beatles' "Yesterday" (1965), highlights its versatility in creating memorable, accessible phrases across genres.^[41]

References

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