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Major second
Major second
Major second
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Step: major second (major tone) Playⓘ.
major second
Inverseminor seventh
Name
Other nameswhole tone, whole step
AbbreviationM2
Size
Semitones2
Interval class2
Just interval9:8[1] or 10:9[1]
Cents
12-Tone equal temperament200[1]
Just intonation204[1] or 182[1]
Minor tone (10:9) Playⓘ.

In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (Playⓘ). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more details). For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff positions. Diminished, minor and augmented seconds are notated on adjacent staff positions as well, but consist of a different number of semitones (zero, one, and three).

The intervals from the tonic (keynote) in an upward direction to the second, to the third, to the sixth, and to the seventh scale degrees of a major scale are called major.[2]

The major second is the interval that occurs between the first and second degrees of a major scale, the tonic and the supertonic. On a musical keyboard, a major second is the interval between two keys separated by one key, counting white and black keys alike. On a guitar string, it is the interval separated by two frets. In moveable-do solfège, it is the interval between do and re. It is considered a melodic step, as opposed to larger intervals called skips.

Intervals composed of two semitones, such as the major second and the diminished third, are also called tones, whole tones, or whole steps.[3][4][5][6][7][8] In just intonation, major seconds can occur in at least two different frequency ratios:[9] 9:8 (about 203.9 cents) and 10:9 (about 182.4 cents). The largest (9:8) ones are called major tones or greater tones, the smallest (10:9) are called minor tones or lesser tones. Their size differs by exactly one syntonic comma (81:80, or about 21.5 cents). Some equal temperaments, such as 15-ET and 22-ET, also distinguish between a greater and a lesser tone.

The major second was historically considered one of the most dissonant intervals of the diatonic scale, although much 20th-century music saw it reimagined as a consonance.[citation needed] It is common in many different musical systems, including Arabic music, Turkish music and music of the Balkans, among others. It occurs in both diatonic and pentatonic scales.

Listen to a major second in equal temperamentⓘ. Here, middle C is followed by D, which is a tone 200 cents sharper than C, and then by both tones together.

Major and minor tones

[edit]
Origin of large and small seconds and thirds in harmonic series.[10]
Lesser tone on D. Playⓘ

In tuning systems using just intonation, such as 5-limit tuning, in which major seconds occur in two different sizes, the wider of them is called a major tone or greater tone, and the narrower minor tone or, lesser tone. The difference in size between a major tone and a minor tone is equal to one syntonic comma (about 21.51 cents).

The major tone is the 9:8 interval[11] playⓘ, and it is an approximation thereof in other tuning systems, while the minor tone is the 10:9 ratio[11] playⓘ. The major tone may be derived from the harmonic series as the interval between the eighth and ninth harmonics. The minor tone may be derived from the harmonic series as the interval between the ninth and tenth harmonics. The 10:9 minor tone arises in the C major scale between D and E and between G and A, and is "a sharper dissonance" than 9:8.[12][13] The 9:8 major tone arises in the C major scale between C and D, F and G, and A and B.[12] This 9:8 interval was named epogdoon (meaning 'one eighth in addition') by the Pythagoreans.

Notice that in these tuning systems, a third kind of whole tone, even wider than the major tone, exists. This interval of two semitones, with ratio 256:225, is simply called the diminished third (for further details, see Five-limit tuning § Size of intervals).

Comparison, in cents, of intervals at or near a major second

Some equal temperaments also produce major seconds of two different sizes, called greater and lesser tones (or major and minor tones). For instance, this is true for 15-ET, 22-ET, 34-ET, 41-ET, 53-ET, and 72-ET. Conversely, in twelve-tone equal temperament, Pythagorean tuning, and meantone temperament (including 19-ET and 31-ET) all major seconds have the same size, so there cannot be a distinction between a greater and a lesser tone.

In any system where there is only one size of major second, the terms greater and lesser tone (or major and minor tone) are rarely used with a different meaning. Namely, they are used to indicate the two distinct kinds of whole tone, more commonly and more appropriately called major second (M2) and diminished third (d3). Similarly, major semitones and minor semitones are more often and more appropriately referred to as minor seconds (m2) and augmented unisons (A1), or diatonic and chromatic semitones.

Unlike most uses of the terms major and minor, these intervals span the same number of semitones. They both span 2 semitones, while, for example, a major third (4 semitones) and minor third (3 semitones) differ by one semitone. Thus, to avoid ambiguity, it is preferable to call them greater tone and lesser tone (see also greater and lesser diesis).

Two major tones equal a ditone.

Epogdoon

[edit]
Diagram showing relations between epogdoon, diatessaron, diapente, and diapason
Translation
Detail of Raphael's School of Athens showing Pythagoras with epogdoon diagram

In Pythagorean music theory, the epogdoon (Ancient Greek: ἐπόγδοον) is the interval with the ratio 9 to 8. The word is composed of the prefix epi- meaning "on top of" and ogdoon meaning "one eighth"; so it means "one eighth in addition". For example, the natural numbers are 8 and 9 in this relation (8+(×8)=9).

According to Plutarch, the Pythagoreans hated the number 17 because it separates the 16 from its Epogdoon 18.[14]

"[Epogdoos] is the 9:8 ratio that corresponds to the tone, [hêmiolios] is the 3:2 ratio that is associated with the musical fifth, and [epitritos] is the 4:3 ratio associated with the musical fourth. It is common to translate epogdoos as 'tone' [major second]."[15]

Further reading

[edit]
  • Barker, Andrew (2007). The Science of Harmonics in Classical Greece. Cambridge University Press. ISBN 9780521879514.
  • Plutarch (2005). Moralia. Translated by Frank Cole Babbitt. Kessinger Publishing. ISBN 9781417905003.

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Major second

View on Grokipedia
from Grokipedia
In music theory, a major second is a diatonic interval encompassing two s, or one whole step, between two pitches, representing the distance from the tonic to the in a major scale. 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. It is denoted in shorthand as "" and contrasts with the minor second, which spans only one . 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. 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). 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. Historically rooted in , the major second derives from , where it approximates a frequency ratio of 9:8, forming the "whole tone" as a foundational building block of the alongside the minor second (256:243). In , this ratio yields a pure, sound for stepwise motion, though slightly adjusts it to 200 cents for uniformity across the . The interval's prominence persisted through Western music's evolution, appearing in medieval modes, , and modern compositions as a staple for melodic contours and chord voicings. 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 contexts, such as suspensions or appoggiaturas. Its recognition and ear-training importance is emphasized in pedagogical resources, where it is often the first interval taught after the due to its prevalence in familiar tunes like "Happy Birthday" (starting on the first two notes).

Definition and Properties

Interval Size and Notation

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

Acoustic and Harmonic Characteristics

The major second interval, in , is defined by a frequency of 9:8 between the higher and lower tones, a simple that contributes to its relative consonance compared to more complex intervals. This arises from the acoustic properties of vibrating strings or air columns, where the overtones of the tones interact such that the second of the lower tone (at 2f, where f is the ) 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. In the series, the major second corresponds to the between the ninth and eighth partials (9:8), positioning it as a natural occurrence in the structure of a single tone. 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. 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. Mathematically, the interval's size is quantified in cents using the formula for logarithmic frequency ratios: cents=1200×log⁡2(f2f1)\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.

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 . This interval, approximating 204 cents, emerged from Pythagorean experiments with the monochord, where (c. 570–495 BCE) demonstrated that dividing a in the ratio 9:8 produced a whole tone, the foundational step beyond the in scalar construction. of Croton (c. 470–385 BCE), a key Pythagorean, explicitly described the epogdoon as the difference between the (3:2) and the (4:3), establishing its mathematical basis in harmonic science. In the 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 (4:3) in diatonic genera. The , spanning four notes with fixed outer pitches, allowed for conjunct or disjunct arrangements to build larger systems like the heptachord or , and the epogdoon's role ensured consonant progression within these structures. 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 but less than the third tone, thus prioritizing practical intonation over strict Pythagorean arithmetic. (c. 100–170 CE), building on this in his own Harmonics, cataloged divisions incorporating the epogdoon in various genera, refining its application for melodic coherence while critiquing overly rigid Pythagorean constraints. The epogdoon held profound cultural significance in Greek modes, such as the Dorian, where it contributed to the of dignity, courage, and discipline, influencing emotional expression in , , and civic life. In tragic theater, for instance, its placement in modal frameworks evoked solemnity and moral reflection, as seen in ' works, aligning music with the philosophical ideal of between and . himself employed the epogdoon-based paeans on the for therapeutic serenity, underscoring its role in ethical education and communal rituals.

Evolution in Western Music Theory

The theoretical treatment of the major second in Western music began with early medieval adaptations of concepts, as seen in ' 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. Building on this legacy, the medieval period saw the major second formalized within practical through Guido d'Arezzo's system in the , where it constituted the "whole tone" interval in 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. This approach emphasized the major second's role as a consistent stepwise motion in modal , distinct from the (mi-fa), and became a of sight-singing education in monastic and scholastic settings. In the , advanced the major second's theoretical status in Le Istitutioni harmoniche (1558), advocating its realization as the ratio 9:8 within synthetic scales to achieve harmonic purity and sensory appeal, positioning it as a "" diatonic interval essential to the senario (1:1 to 6:1) and consonant progressions in . Zarlino's framework integrated the major second into modal structures, arguing it complemented perfect consonances like the and fifth while supporting the emerging emphasis on vertical in Venetian polychoral music. 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. 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. 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.

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. 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: (32×32)÷2=94÷2=98.\left( \frac{3}{2} \times \frac{3}{2} \right) \div 2 = \frac{9}{4} \div 2 = \frac{9}{8}. This process aligns with principles, a subset of 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 —calculated as (3/2)12/27≈531441/524288(3/2)^{12} / 2^7 \approx 531441/524288 (23.46 cents)—for closing the circle, the 9:8 second itself requires no such adjustment due to its direct simplicity from two fifths. The 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. This purity makes it particularly suitable for vocal ensembles and early instruments like lutes or viols, where performers can adjust pitches dynamically to achieve these exact proportions without fixed tempering. with the major second was a cornerstone of theory and practice. In related tuning systems like , which aims to approximate for consonant chords such as pure major thirds (5:4), the major second is adjusted using fractions of the (81/80, approximately 21.51 cents). For instance, quarter-comma meantone tempers fifths slightly flat, resulting in a major second of approximately 193.2 cents.

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. This precise measurement arises from the formula for interval size in cents: 1200×log⁡2(22/12)=2001200 \times \log_2 \left( 2^{2/12} \right) = 200 where the frequency ratio for the major second is 21/62^{1/6}, ensuring consistent semitone steps throughout the scale. 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. This work highlighted the system's ability to modulate freely without retuning, paving the way for its dominance in Western music. In modern usage, offers uniformity across all keys on fixed-pitch keyboard instruments like and organ, enabling seamless performance in any tonal center without the harmonic biases of unequal tunings. However, this approximation renders the major second slightly detuned relative to just intonation's pure of 9:8 (approximately 203.91 cents), resulting in a flattening of 3.91 cents that prioritizes modulation over interval consonance.

Musical Applications

Role in Scales and Harmony

In both scales, the serves as the foundational interval from the tonic to the (scale degree 2), forming the first whole step in the diatonic pattern and contributing to the establishment of by providing a stable, consonant extension above the root that outlines the key's basic stepwise motion. In the , this interval initiates the characteristic sequence of whole-whole-half-whole-whole-whole-half steps, creating the bright, resolved sound associated with major keys. Similarly, the natural scale begins with a major second in its whole-half-whole-whole-half-whole-whole pattern, reinforcing the minor key's through a comparable initial ascent that sets the modal framework without the raised leading tone found in other minor variants. The '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. Harmonically, the major second integrates into chord structures, particularly as part of the major triad, where the from root to third degree encompasses two stacked major seconds within the scale (e.g., from tonic to , then to ). This stacking underscores the triad's framework, with the major second providing the initial layer of the imperfect consonance that defines major harmony. In , the major second measures approximately 200 cents, offering a consistent intervallic size that supports these harmonic builds across keys. In and , the major second functions as an imperfect consonance or mild dissonance, often employed in passing tones during second-species , where it must connect stepwise to adjacent notes while adhering to rules that prioritize smooth between stronger consonances like thirds or fifths. 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 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 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 ) arises from borrowing elements such as the raised seventh in the , introducing chromatic color while remaining distinct from the narrower minor second in resolution and stability.

Examples in Compositions and Genres

In , 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 's , K. 525 (1787), begins with a major second ascent from G to A in , establishing the graceful and elegant character of the . Similarly, in J.S. Bach's four-part chorales, such as those in his (BWV 244, 1727), major seconds appear frequently in and lines as part of diatonic stepwise progressions, contributing to the flowing, hymn-like quality of the vocal writing. In and , 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 "" by (1969), the guitar melody employs major seconds within the pentatonic framework to heighten emotional expressiveness, as the interval bridges bent notes and chord tones for idiomatic phrasing. This usage underscores the interval's role in improvisational lines, where it contrasts with chromatic seconds for color. In folk and non-Western traditions, the major second plays a key structural role in melodic contours. In , 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 (ascending scale) of Yaman, enhancing the raga's serene mood. 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 ensembles or Akan folk songs, where the interval supports rhythmic and communal participation. In modern pop and rock, the major second drives catchy hooks through simple whole-step leaps. The melody of "" (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 . This interval's prominence in verse-chorus structures, such as the stepwise motifs in ' "Yesterday" (1965), highlights its versatility in creating memorable, accessible phrases across genres.

References

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  10. https://courses.physics.illinois.edu/phys406/sp2017/Lecture_Notes/P406POM_Lecture_Notes/P406POM_Lect8.pdf
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  20. https://www.kylegann.com/histune.html
  21. https://casfaculty.case.edu/ross-duffin/baroque-ensemble-tuning/
  22. https://agertushistoryofmusic.com/2022/01/04/rameau-the-revolutionary/
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  24. https://www.mtosmt.org/issues/mto.98.4.4/mto.98.4.4.scholtz.html
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  35. https://www.thejazzpianosite.com/jazz-piano-lessons/jazz-chords/borrowed-chords/
  36. https://imslp.org/wiki/Serenade_in_G_major,_K.525_(Mozart,_Wolfgang_Amadeus)
  37. https://www.jazzguitar.be/blog/guitar-intervals/
  38. https://musicianself.com/big-small-and-one-more-indian-music-notation-sa-ri-ga-ma-western-interval-naming-the-two-different-systems/
  39. https://www.britannica.com/art/African-music/Musical-structure
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  41. https://www.musicca.com/interval-song-chart
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