Septimal minor third

# Septimal minor third

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In music, the septimal minor third , also called the subminor third
(e.g., by Helmholtz), is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cent
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

s, it is 267 cents, a quartertone of size 36/35
Septimal quarter tone
A septimal quarter-tone is an interval with the ratio of 36:35 , which is the difference between the septimal minor third and the Just minor third , or about 48.77 cents wide. The name derives from the interval being the 7-limit approximation of a quarter tone...

flatter than a just minor third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...

of 6/5. In 24-tone equal temperament
Quarter tone
A quarter tone , is a pitch halfway between the usual notes of a chromatic scale, an interval about half as wide as a semitone, which is half a whole tone....

five quarter tones approximate the septimal minor third at 250 cents .

The septimal minor third may be derived from the harmonic series
Harmonic series (music)
Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...

, as the interval between the seventh
Harmonic seventh
The harmonic seventh interval , also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio . This is somewhat narrower than and is "sweeter in quality" than an "ordinary" minor seventh, which has a just-intonation ratio of 9:5 , or an equal-temperament ratio of...

and sixth harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

s, and as such is in inharmonic ratios with all notes in the regular 12TET scale. It has a darker but generally pleasing character when compared to the 6/5 third. A triad formed by using it in place of the minor third is called a septimal minor or subminor triad .

In the meantone
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...

era the interval made its appearance as the alternative minor third in remote keys, under the name augmented second
Augmented second
In classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning...

. Tunings of the meantone fifth in the neighborhood of quarter-comma meantone
Quarter-comma meantone
Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...

will give three septimal minor thirds among the twelve minor thirds of the tuning; since the wolf fifth appears with an ordinary minor third, this entails there are three septimal minor triads, eight ordinary minor triads and one triad containing the wolf fifth arising from an ordinary minor third followed by a septimal major third
Septimal major third
In music, the septimal major third , 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 by the septimal...

.

Composer Ben Johnston uses a small "7" as an accidental to indicate a note is lowered 49 cents, or an upside down "ㄥ" to indicate a note is raised 49 cents.

The position of this note also appears on the scale of the Moodswinger
Moodswinger
The Moodswinger is a twelve string electric zither with an additional third bridge designed by Yuri Landman. The rod which functions as the third bridge divides the strings into two sections to cause an overtone multiphonic sound...

. Yuri Landman
Yuri Landman
Yuri Landman is a Dutch experimental luthier who has made several experimental electric string instruments for a list of artists including Lee Ranaldo of Sonic Youth, Liars, Jad Fair of Half Japanese and Liam Finn...

indicated the harmonic positions of his instrument in a color dotted series. The septimal minor third position is cyan blue as well as the other knotted positions of the seventh harmonic (5/7
Tritone
In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...

, 4/7
Harmonic seventh
The harmonic seventh interval , also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio . This is somewhat narrower than and is "sweeter in quality" than an "ordinary" minor seventh, which has a just-intonation ratio of 9:5 , or an equal-temperament ratio of...

, 3/7, 2/7 and 1/7 of the string length of the open string).

## In equal temperament and non-Western scales

Twelve-tone equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

(12-TET), as commonly used in Western music, does not provide a good approximation for this interval, and quarter tones (24-TET) do not match it well either. 19-TET
19 equal temperament
In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...

, 22-TET
22 equal temperament
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 . Each step represents a frequency ratio of 21/22, or 54.55 cents ....

, 31-TET
31 equal temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...

, 41-TET
41 equal temperament
In music, 41 equal temperament, often abbreviated 41-tET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally-sized steps . Each step represents a frequency ratio of 21/41, or 29.27 cents , an interval close in size to the septimal comma. 41-ET can be seen as a...

, and 72-TET
72 equal temperament
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...

each offer successively better matches (measured in cents difference) to this interval.

Several non-Western and just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

tunings, such as the 43-tone scale developed by Harry Partch
Harry Partch
Harry Partch was an American composer and instrument creator. He was one of the first twentieth-century composers to work extensively and systematically with microtonal scales, writing much of his music for custom-made instruments that he built himself, tuned in 11-limit just intonation.-Early...

, do feature the (exact) septimal minor third.

## Listening

Because of its position in the harmonic series
Harmonic series (music)
Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...

, the sixth harmonic (frequency ratio 6:1) being a perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

and two octaves above the root
Root (chord)
In music theory, the root of a chord is the note or pitch upon which a triadic chord is built. For example, the root of the major triad C-E-G is C....

, the septimal minor third implies a difference tone a perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

below the lower note in the interval. Depending on the timbre of the pitches, humans sometimes perceive this root pitch even if it is not played. The phenomenon of hearing this root pitch is evident in the following sound file, which uses a pure sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

wave. For comparison, the root pitch is played after the interval has been played.