Major sixth

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major sixth
Inverse minor third
Name
Other namesseptimal major sixth, supermajor sixth, major hexachord, greater hexachord, hexachordon maius
AbbreviationM6
Size
Semitones 9
Interval class 3
Just interval 5:3, 12:7 (septimal), 27:16 [1]
Cents
Equal temperament 900
Just intonation 884, 933, 906
Major sixth
Play (help*info) Major sixth on C.png
Major sixth Loudspeaker.svg Play  
Pythagorean major sixth
Play (help*info)
, 3 Pythagorean perfect fifths on C. Pythagorean major sixth on C.png
Pythagorean major sixth Loudspeaker.svg Play  , 3 Pythagorean perfect fifths on C.

In music from Western culture, a sixth is a musical interval encompassing six note letter names or staff positions (see Interval number for more details), and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the larger of the two. The major sixth spans nine semitones. Its smaller counterpart, the minor sixth, spans eight semitones. For example, the interval from C up to the nearest A is a major sixth. It is a sixth because it encompasses six note letter names (C, D, E, F, G, A) and six staff positions. It is a major sixth, not a minor sixth, because the note A lies nine semitones above C. Diminished and augmented sixths (such as C to A and C to A) span the same number of note letter names and staff positions, but consist of a different number of semitones (seven and ten, respectively).

Contents

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]

A commonly cited example of a melody featuring the major sixth as its opening is "My Bonnie Lies Over the Ocean". [3]

The major sixth is one of the consonances of common practice music, along with the unison, octave, perfect fifth, major and minor thirds, minor sixth, and (sometimes) the perfect fourth. In the common practice period, sixths were considered interesting and dynamic consonances along with their inverses the thirds. In medieval times theorists always described them as Pythagorean major sixths of 27/16 and therefore considered them dissonances unusable in a stable final sonority. We cannot know how major sixths actually were sung in the Middle Ages. In just intonation, the (5/3) major sixth is classed as a consonance of the 5-limit.

A major sixth is also used in transposing music to E-flat instruments, like the alto clarinet, alto saxophone, E-flat tuba, trumpet, natural horn, and alto horn when in E-flat, as a written C sounds like E-flat on those instruments.

Assuming close-position voicings for the following examples, the major sixth occurs in a first inversion minor triad, a second inversion major triad, and either inversion of a diminished triad. It also occurs in the second and third inversions of a dominant seventh chord.

The septimal major sixth (12/7) is approximated in 53 tone equal temperament by an interval of 41 steps or 928 cents.

Frequency proportions

Many intervals in a various tuning systems qualify to be called "major sixth," sometimes with additional qualifying words in the names. The following examples are sorted by increasing width.

In just intonation, the most common major sixth is the pitch ratio of 5:3 ( Loudspeaker.svg play  ), approximately 884 cents.

In 12-tone equal temperament, a major sixth is equal to nine semitones, exactly 900 cents, with a frequency ratio of the (9/12) root of 2 over 1.

Another major sixth is the Pythagorean major sixth with a ratio of 27:16, approximately 906 cents, [4] called "Pythagorean" because it can be constructed from three just perfect fifths (C-A = C-G-D-A = 702+702+702-1200=906). It corresponds to the interval between the 27th and the 16th harmonics. The 27:16 Pythagorean major sixth arises in the C Pythagorean major scale between F and D, [5] [ failed verification ] as well as between C and A, G and E, and D and B. Loudspeaker.svg Play  

Another major sixth is the 12:7 septimal major sixth or supermajor sixth , the inversion of the septimal minor third, of approximately 933 cents. [4] The septimal major sixth (12/7) is approximated in 53-tone equal temperament by an interval of 41 steps, giving an actual frequency ratio of the (41/53) root of 2 over 1, approximately 928 cents.

The nineteenth subharmonic is a major sixth, A 19 upside down.png = 32/19 = 902.49 cents.

See also

Sources

  1. Jan Haluska, The Mathematical Theory of Tone Systems (New York: Marcel Dekker; London: Momenta; Bratislava: Ister Science, 2004), p.xxiii. ISBN   978-0-8247-4714-5. Septimal major sixth.
  2. Bruce Benward and Marilyn Nadine Saker, Music: In Theory and Practice, Vol. I, seventh edition ([ full citation needed ] 2003): p. 52. ISBN   978-0-07-294262-0.
  3. Blake Neely, Piano For Dummies, second edition (Hoboken, NJ: Wiley Publishers, 2009), p. 201. ISBN   978-0-470-49644-2.
  4. 1 2 Alexander J. Ellis, Additions by the translator to Hermann L. F. Von Helmholtz (2007). On the Sensations of Tone, p.456. ISBN   978-1-60206-639-7.
  5. Oscar Paul, A Manual of Harmony for Use in Music-Schools and Seminaries and for Self-Instruction , trans. Theodore Baker (New York: G. Schirmer, 1885), p. 165.

Further reading

Related Research Articles

Just intonation Musical tuning based on pure intervals

In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. Any interval tuned in this way is called a just interval. Just intervals consist of members of a single harmonic series of a (lower) implied fundamental. For example, in the diagram at right, the notes G and middle C, are both members of the harmonic series of the lowest C and their frequencies will be 3 and 4 times, respectively, the fundamental frequency; thus, their interval ratio will be 4:3. If the frequency of the fundamental is 64 Hertz, the frequencies of the two notes in question would be 192 and 256.

Pythagorean tuning term in music

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈702 cents wide.

In music theory, an interval is the difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

Perfect fifth musical interval

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.

Wolf interval Dissonant musical interval

In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including most meantone temperaments.

Major second musical interval

In Western music theory, a major second is a second spanning two semitones. A second is a musical interval encompassing two adjacent staff positions. 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.

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.

Semitone musical interval

A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale. For example, C is adjacent to C; the interval between them is a semitone.

Major third musical interval

In classical music, a third is a musical interval encompassing three staff positions, and the major third is a third spanning four semitones. Along with the minor third, the major third is one of two commonly occurring thirds. It is qualified as major because it is the larger of the two: the major third spans four semitones, the minor third three. For example, the interval from C to E is a major third, as the note E lies four semitones above C, and there are three staff positions from C to E. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones.

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.

Minor third musical interval

In the music theory of Western culture, 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. 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. The minor third is a skip melodically.

Minor sixth musical interval

In Western classical music, a minor sixth is a musical interval encompassing six staff positions, and is one of two commonly occurring sixths. It is qualified as minor because it is the smaller of the two: the minor sixth spans eight semitones, the major sixth nine. For example, the interval from A to F is a minor sixth, as the note F lies eight semitones above A, and there are six staff positions from A to F. Diminished and augmented sixths span the same number of staff positions, but consist of a different number of semitones.

Minor chord chord having a root, a minor third, and a perfect fifth; e.g. A–C–E or C–E♭–G

In music theory, a minor chord is a chord having a root, a minor third, and a perfect fifth. When a chord has these three notes alone, it is called a minor triad. For example, the minor triad built on C, called a C minor triad, has pitches C–E–G:

Comma (music) small musical interval, the difference between two tunings of the same note

In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning system, and another F tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B and A are both approximated by the same interval although they are a septimal kleisma apart.

Quarter-comma meantone, or 14-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80), with respect to its just intonation used in Pythagorean tuning ; the result is . The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

53 equal temperament musical tuning system with 53 pitches equally-spaced on a logarithmic scale

In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps. Play  Each step represents a frequency ratio of 2153, or 22.6415 cents, an interval sometimes called the Holdrian comma.

In music, 22 equal temperament, called 22-TET, 22-EDO, or 22-ET, is the tempered scale derived by dividing the octave into 22 equal steps. Play  Each step represents a frequency ratio of 222, or 54.55 cents.

Septimal major third musical interval

In music, the septimal major thirdplay , also called the supermajor third and sometimes Bohlen–Pierce third is the musical interval exactly or approximately equal to a just 9:7 ratio of frequencies, or alternately 14:11. It is equal to 435 cents, sharper than a just major third (5:4) by the septimal quarter tone (36:35). In 24-TET the septimal major third is approximated by 9 quarter tones, or 450 cents. Both 24 and 19 equal temperament map the septimal major third and the septimal narrow fourth (21:16) to the same interval.

Septimal minor third musical interval

In music, the septimal minor thirdplay , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5. In 24-tone equal temperament five quarter tones approximate the septimal minor third at 250 cents. A septimal minor third is almost exactly two-ninths of an octave, and thus all divisions of the octave into multiples of nine have an almost perfect match to this interval. The septimal major sixth, 12/7, is the inverse of this interval.

Regular diatonic tuning

A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.

Septimal third tone

A septimal 1/3-tone is an interval with the ratio of 28:27, which is the difference between the perfect fourth and the supermajor third. It is about 62.96 cents wide. The septimal 1/3-tone can be viewed either as a musical interval in its own right, or as a comma; if it is tempered out in a given tuning system, the distinction between these two intervals is lost. The septimal 1/3-tone may be derived from the harmonic series as the interval between the twenty-seventh and twenty-eighth harmonics. It may be considered a diesis.

Five-limit tuning

Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.