Septimal minor third

In music, the septimal minor third play (help·info), also called the subminor third (by eg Helmholtz^[1]) 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 ( Play (help·info)).

The septimal minor third may be derived from the harmonic series, as the interval between the seventh and sixth harmonics, and as such is in inharmonic ratios with all notes in the regular 12TET scale.^[2] 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 play (help·info); however in fact that chord is a part of the overtone series, or in other words is an otonal chord, with ratios of 6:7:9, whereas the ordinary minor third is a utonal chord. While the septimal minor third is classed as a 7-limit interval, and the septimal minor triad as a 9-limit chord, the fact that the chord is otonal makes it sound perhaps about as consonant as the just minor triad.

In the meantone era the interval made its appearance as the alternative minor third in remote keys, under the name augmented second. Tunings of the meantone fifth in the neighborhood of quarter-comma meantone 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.

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.^[3]

The position of this note also appears on the scale of the Moodswinger. Yuri Landman 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)^[4].

In Equal-Temperament and Non Western scales

The commonly used in Western music 12-TET does not provide a good match for this interval, and even quarter tones (24-TET) do not match it well. 19-TET, 22-TET, 31-TET, 41-TET, and 72-TET each offer successively better matches (measured in cents difference) to this interval. 34-TET has a poor match to this interval.

In many Non-Western scales as well as microtonal just intoned scales such as the 43 tone scale of Harry Partch the tone appears quite often causing a typical non-Western sound in music. Together with its harmonic partners the just 7:5 tritone and the harmonic seventh the notes form all together with the fundamental the half-diminished seventh chord. If the fundamental is not included the three tones create a 6:5:4 frequency combination, which is a regular just intoned major chord.

Listening

	Septimal Minor Third (just intonation, 7:6 ratio), sawtooth wave The file plays A440, followed by 528 Hz, followed by both tones together.
Problems listening to this file? See media help.

Because of its position in the harmonic series, the sixth harmonic (frequency ratio 6:1) being a perfect fifth and two octaves above the root, the septimal minor third implies a difference tone a perfect fifth 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 wave. For comparison, the root pitch is played after the interval has been played.

	Septimal Minor Third (just intonation, 7:6 ratio), sine wave The file plays A880, followed by 1026.67 Hz, followed by both tones together, followed by the implied root frequency of 586.67 Hz, a fifth below the A.
Problems listening to this file? See media help.

References