Origin of seconds and thirds in harmonic series

In music, the septimal minor third, also called the subminor third (e.g., by

Ellis Ellis is a surname of Welsh and English origin. Retrieved 21 January 2014 An independent French origin of the surname is said to derive from the phrase fleur-de-lis. Surname A * Abe Ellis (Stargate), a fictional character in the TV series ' ...

), is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of

cent Cent may refer to: Currency * Cent (currency), a one-hundredth subdivision of several units of currency * Penny (Canadian coin), a Canadian coin removed from circulation in 2013 * 1 cent (Dutch coin), a Dutch coin minted between 1941 and 1944 * ...

s, it is 267 cents, a quartertone of size 36/35 flatter than a just

minor third In music theory, 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 (see: interval number). The minor third is one of two com ...

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 (

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 (equal frequency ratios). Each step represents a frequency ...

being the most notable) have an almost perfect match to this interval. The septimal major sixth, 12/7, is the inverse of this interval. The septimal minor third may be derived in the harmonic series from the

seventh harmonic The harmonic seventh interval, also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio (about 969 cents). This is somewhat narrower than and is, "particularly sweet", "sweeter in quality" than an "ordinar ...

, and as such is in inharmonic ratios with all notes in the regular 12TET scale, with the exception of the fundamental and the octave. 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 is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. Me ...

era the interval made its appearance as the alternative minor third in remote keys, under the name

augmented second In classical music from Western culture, an augmented second is an interval that, in equal temperament, is sonically equivalent to a minor third, spanning three semitones, and is created by widening a major second by a chromatic semitone.Benwar ...

. Tunings of the meantone fifth in the neighborhood of

quarter-comma meantone Quarter-comma meantone, or -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 ...

will give three septimal minor thirds among the twelve minor thirds of the tuning; since the

wolf fifth In music theory, the wolf fifth (sometimes also called Procrustean fifth, or imperfect fifth) Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction', p.165. Theodore Baker, trans. G. Schirmer. ...

appears with an ordinary minor third, this entails there are three septimal minor triads, eight ordinary minor triads and one triad containing the

arising from an ordinary minor third followed by a

septimal major third In music, the septimal major third , also called the supermajor third (by Hermann von Helmholtz among others Hermann L. F. von Helmholtz (2007). ''Sensations of Tone'', p. 187. .) and sometimes '' Bohlen–Pierce third'' is the musical interval ...

. Composer Ben Johnston uses a small "7" as an accidental to indicate a note is lowered 49 cents, or an upside down seven ("ㄥ") to indicate a note is raised 49 cents. The position of this note also appears on the scale of the

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. One of the copi ...

Yuri Landman Yuri Landman (born 1 February 1973) is a Dutch inventor of musical instruments and musician who has made several experimental electric string instruments for a number of artists including Lee Ranaldo of Sonic Youth, Liars, Jad Fair of Half Japan ...

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, 4/7, 3/7, 2/7 and 1/7 of the string length of the open string)."Moodswinger"
''oddmusic.com''

In equal temperament and non-Western scales

Twelve-tone

equal temperament 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, wh ...

(12-TET), as commonly used in Western music, does not provide a good approximation for this interval, and

quarter tones 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 ...

(

24-TET 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 ...

) do not match it well either.

19-TET In music, 19 Tone Equal Temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), or 19 ET, is the tempered scale derived by dividing the octave into 19 equal steps (equal frequency ratios). Each step represent ...

22-TET 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 (equal frequency ratios). Each step represents a frequency ratio of , or 54.55 cents (). When composing wi ...

31-TET In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET (31 tone ET) or 31- EDO (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps (equ ...

, 41-TET, and 72-TET each offer successively better matches (measured in cents difference) to this interval. Several non-Western and

just intonation In music, just intonation or pure intonation is the tuning of musical intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to ...

tunings, such as the 43-tone scale developed by

Harry Partch Harry Partch (June 24, 1901 – September 3, 1974) was an American composer, music theorist, and creator of unique musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century com ...

, do feature the (exact) septimal minor third.

Listening

Because of its position in the harmonic series, the sixth harmonic (frequency ratio 6:1) being a

perfect fifth In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval fro ...

and two octaves above the

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, the septimal minor third implies a

difference tone A combination tone (also called resultant or subjective tone)Combination Tone
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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 In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

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

References

{{Intervals 7-limit tuning and intervals Minor intervals 0007:0006 Thirds (music)