Septimal minor third

Infobox Interval| main_interval_name = Septimal Minor Third
inverse = Septimal major sixth
complement = Septimal major sixth
other_names = - | abbreviation = m3
semitones = ~3 | interval_class = ~3 | just_interval = 7:6
cents_equal_temperament = 300| cents_just_intonation = 267In music, the septimal minor third audio|Septimal minor third on C.mid|play, also called the "subminor third"(by eg Helmholtz) 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.

The septimal minor third 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 audio|Septimal minor triad on C.mid|play; 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 1/4 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.

In Equal-Temperament

The widely used 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.

Listening

Because of its position in the harmonic series, the sixth overtone being a perfect fifth above the root, the septimal minor third implies a root note 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.

References

*Hermann Helmholz and Alexander Ellis (trans), "On the Sensation of Tone", Dover Publications.