In music, the septimal whole tone, septimal major second, or supermajor second is the

musical interval In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or h ...

exactly or approximately equal to an 8/7 ratio of frequencies.Andrew Horner, Lydia Ayres (2002). ''Cooking with Csound: Woodwind and Brass Recipes'', p.131. . "Super-Major Second". It is about 231 cents wide in

just intonation In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and ...

.Leta E. Miller, Fredric Lieberman (2006). ''Lou Harrison'', p.72. . 24 equal temperament does not match this interval particularly well, its nearest representation being at 250 cents, approximately 19 cents sharp. The septimal whole tone may be derived from the harmonic series as the interval between the seventh and eighth harmonics and the term ''septimal'' refers to the fact that it utilizes 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 ...

. It can also be thought of as the octave inversion of the 7/4 interval, the

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

Origin of seconds and thirds in harmonic series

No close approximation to this interval exists in the standard

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 resulting ...

used in most modern western music. The very simple

5 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 ...

is the smallest system to match this interval well. 26 equal temperament matches this interval almost perfectly with an error of only 0.4 cents, but at the cost of the significant flatness of its major thirds and fifths.

31 equal temperament 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 (equa ...

, which has much more accurate fifths and major thirds, approximates 8/7 with a slightly higher error of 1.1 cents.

References

{{Intervals Seconds (music) 7-limit tuning and intervals 0008:0007