A neutral sixth is a musical interval wider than a

minor sixth In Western classical music, a minor sixth is a musical interval encompassing six staff positions (see Interval number for more details), and is one of two commonly occurring sixths (the other one being the major sixth). It is qualified as ''mi ...

but narrower than a

major sixth 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 ...

. Three distinct intervals may be termed neutral sixths: * The ''undecimal neutral sixth'' has a ratio of 18:11 between the

frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

of the two tones, or about 852.59 cents. * A ''tridecimal neutral sixth'' has a ratio of 13:8 between the

of the two tones, or about 840.53 cents. This is the smallest neutral sixth, and occurs infrequently in music, as little music utilizes the 13th

harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...

. * An ''

equal-tempered An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, w ...

neutral sixth'' is 850 cents, a hair narrower than the 18:11 ratio. It is an equal-tempered

quarter tone A quarter tone is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide (aurally, or logarithmically) as a semitone, which itself is half a whole tone. Quarter tones divide the octave by 50 cents each, a ...

exactly halfway between the equal-tempered minor and major sixths, and half of an equal-tempered perfect eleventh (octave plus fourth). These intervals are all within about 12 cents of each other and are difficult for most people to distinguish. Neutral sixths are roughly a

sharp from

12 equal temperament Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That result ...

(12-ET) minor sixths and a quarter tone flat from 12-ET major sixths. In just intonation, as well as in tunings such as 31-ET, 41-ET, or

72-ET In music, 72 equal temperament, called twelfth-tone, 72-TET, 72- EDO, or 72-ET, is the tempered scale derived by dividing the octave into twelfth-tones, or in other words 72 equal steps (equal frequency ratios). Each step represents a frequency ...

, which more closely approximate just intonation, the intervals are closer together. A neutral sixth can be formed by subtracting a

neutral second In music theory, a neutral interval is an interval that is neither a major nor minor, but instead in between. For example, in equal temperament, a major third is 400 cents, a minor third is 300 cents, and a neutral third is 350 cents. A neutral ...

from a

minor seventh In music theory, a minor seventh is one of two musical intervals that span seven staff positions. It is ''minor'' because it is the smaller of the two sevenths, spanning ten semitones. The major seventh spans eleven. For example, the interval f ...

. Based on its positioning in the harmonic series, the undecimal neutral sixth implies a

root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...

one minor seventh above the higher of the two notes.

Thirteenth harmonic

The pitch ratio 13:8 (840.53 cents) is the ratio of the thirteenth

and is notated in Ben Johnston's system as A¹³. In 24-ET is approximated by A. This note is often corrected to a just or

Pythagorean Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Ne ...

ratio on the

natural horn The natural horn is a musical instrument that is the predecessor to the modern-day (French) horn (differentiated by its lack of valves). Throughout the seventeenth and eighteenth century the natural horn evolved as a separation from the trump ...

, but the pure thirteenth harmonic was used in pieces including

Britten Edward Benjamin Britten, Baron Britten (22 November 1913 – 4 December 1976, aged 63) was an English composer, conductor, and pianist. He was a central figure of 20th-century British music, with a range of works including opera, other ...

's ''

Serenade for tenor, horn and strings The ''Serenade for Tenor, Horn and Strings'', Op. 31, is a song cycle written in 1943 by Benjamin Britten for tenor, solo horn and a string orchestra. Composed during the Second World War at the request of the horn player Dennis Brain, it is a ...

''.Fauvel, John; Flood, Raymond; and Wilson, Robin J. (2006). ''Music And Mathematics'', p.21-22. .

References

{{Intervals Neutral intervals Sixths (music) Quarter tones

Thirteenth harmonic

See also

References