Major sixth
In music theory, the major sixth is a consonant interval that spans nine semitones (half steps) between two pitches, equivalent to the distance from the tonic to the submediant in a major scale.[1] For example, the notes C and A form a major sixth, as A is the sixth scale degree above C in C major.[2] This interval measures approximately 900 cents in twelve-tone equal temperament, though in just intonation it corresponds to a simpler frequency ratio of 5:3 (about 884 cents), producing a pure, stable sound often described as warm and harmonious.[3][4] The major sixth belongs to the class of major intervals, which apply to seconds, thirds, sixths, and sevenths, and is one half step larger than the minor sixth (eight semitones).[5] It can appear melodically, as in the opening phrase of "Jingle Bells" ("Dash-ing through the snow"), or harmonically in structures like the added-sixth chord (e.g., C-E-G-A), which extends a major triad by including the sixth scale degree for a lush, jazz-influenced color.[6] In compound form, it extends to the major thirteenth (octave plus major sixth), common in extended harmonies.[2] The major sixth has been valued for its consonance in Western music theory; in just intonation, the 5:3 ratio produces a pure, stable sound, and it remains a foundational element in Western tonal music, appearing in scales, melodies, and chords across classical, jazz, and popular genres.[4] Its inversion is the minor third, and it often resolves smoothly in voice leading due to its stable yet expressive quality.[7]Definition and Properties
Classification and Nomenclature
The major sixth is a musical interval encompassing six consecutive letter names, such as C to A, and is classified as a sixth because it spans the distance from the tonic to the sixth scale degree in the diatonic scale, following the unison, second, third, fourth, fifth, and preceding the seventh.[8][2] This interval is part of the category of imperfect consonances that admit major and minor qualities, distinguishing it from perfect intervals like the fourth and fifth, which do not vary in size within the diatonic scale.[8] The major sixth is differentiated from the minor sixth by its larger size; both span the same six letter names, but the minor sixth is narrower by one half step.[2][8] Nomenclature for the major sixth includes variations such as the septimal major sixth, a term used in just intonation for the interval derived from the seventh harmonic, and the supermajor sixth in microtonal theory for an augmented version beyond the standard major size.[9] It is also known historically as the major hexachord or greater hexachord (hexachordon maius) in medieval solmization systems, referring to the interval spanning the six-note diatonic segment.[10] In the context of pitch-class set theory, the major sixth is assigned to interval class 3, reflecting the minimal distance between its pitches when measured circularly around the chromatic scale, accounting for octave equivalence and inversion.[11]Inversion and Semitone Structure
The major sixth spans exactly 9 semitones in the chromatic scale, measured from the root note to the sixth degree above it, such as from C to A.[12] This measurement positions it as a step below the minor seventh, which spans 10 semitones, and above the minor sixth at 8 semitones.[12] Inversion of the major sixth occurs by moving the lower note up an octave or the upper note down an octave, resulting in a minor third that spans 3 semitones, such as inverting C to A yields A to C (an octave higher).[1] This complementary relationship adheres to the principle that the sum of an interval and its inversion equals the octave (12 semitones), confirming the major sixth's inversion as the minor third rather than a major third.[1] The major sixth belongs to interval class 3 in pitch-class set theory, determined by the minimum distance between its pitches modulo the octave—either 9 semitones directly or the shorter 3 semitones around the octave (12 - 9 = 3).[13] This classification groups it with other intervals of equivalent minimal span, including the minor third and augmented second, emphasizing structural equivalence over directional or qualitative differences.[13] Enharmonically, the major sixth is equivalent to the diminished seventh, as both occupy the same 9-semitone space (e.g., C to A equals C to B-double-flat), producing identical pitches in equal temperament. However, in functional theory, they are distinguished by context and spelling: the major sixth functions as a consonant interval within diatonic harmony, while the diminished seventh often serves as a dissonant leading-tone chord requiring resolution, despite their acoustic identity.[14] This distinction underscores the role of notation in guiding harmonic progression and voice leading.[14]Acoustic Properties
Frequency Ratios in Just Intonation
In just intonation, the major sixth is defined by the simple frequency ratio of 5:3, corresponding to a proportion of approximately 1.666.... This ratio yields an acoustically pure interval that aligns with the natural resonances of sounding bodies, producing minimal beating when the two tones are played simultaneously.[15] The 5:3 ratio for the major sixth can be derived as the compound of a just major third (5:4) and a perfect fourth (4:3), since \frac{5}{4} \times \frac{4}{3} = \frac{5}{3}. This construction reflects the additive nature of intervals in just intonation systems, where smaller consonant intervals build larger ones through multiplication of their ratios. Alternatively, the ratio emerges directly from the harmonic series as the interval between the third partial (frequency multiple of 3) and the fifth partial (multiple of 5), illustrating its foundation in the overtone structure of a single tone.[15] (Helmholtz, 1885, p. 194) This 5:3 major sixth has roots in ancient Greek harmonic theory, where the 5:4 major third was incorporated into scales like Ptolemy's intense diatonic genus, allowing for the derivation of compound intervals such as the sixth through theoretical synthesis. In Renaissance tuning practices, theorists like Gioseffo Zarlino explicitly advocated the 5:3 ratio as part of just intonation for polyphonic music, emphasizing its sensory purity over Pythagorean approximations.[16][15]Measurements in Equal Temperament and Other Systems
In various tuning systems, the major sixth interval is quantified using cents, a unit dividing the octave into 1200 equal parts, with the conversion from a frequency ratio r given by the formula $1200 \log_2 r. This allows precise comparison of interval sizes across temperaments, relative to the just intonation baseline of approximately 884 cents for the 5:3 ratio (as established in the prior section on frequency ratios). In 12-tone equal temperament (12-TET), the standard tuning for most Western instruments, the major sixth consists of exactly nine semitones, measuring 900 cents with a frequency ratio of $2^{9/12} \approx 1.6818. This approximation renders it about 16 cents sharper than the pure just major sixth.[17][18] Pythagorean tuning constructs intervals by successive applications of the 3:2 perfect fifth ratio, resulting in a major sixth of 27:16 (\approx 1.6875) after three such stackings adjusted by an octave downward, equating to roughly 906 cents. Compared to the just 5:3, this variant is 22 cents sharp.[18][19] Septimal tuning, or 7-limit intonation, extends to include the seventh harmonic, approximating the major sixth as 12:7 (\approx 1.7143) at about 933 cents. This interval deviates by 49 cents sharper from the just major sixth and appears in select microtonal music systems.[18][19] The following table summarizes these measurements and deviations from the just intonation reference:| Tuning System | Ratio | Cents | Deviation from Just (884 cents) |
|---|---|---|---|
| 12-TET | $2^{9/12} | 900 | +16 |
| Pythagorean | 27:16 | 906 | +22 |
| Septimal (7-limit) | 12:7 | 933 | +49 |