Otonal heptad

The otonal heptad is the seven-note just intonation scale 8:9:10:11:12:13:14:16: the harmonics 8 through 14 and the octave. It is a common diatonic-like scale, and also sometimes called "the overtone scale" (see overtone scale). It is a rotation of harmonics 7–14: starting that scale on its second degree, the 8th harmonic. It is also harmonics 8–16 without the 15th harmonic. As a 13-limit scale built from consecutive harmonics, all of its notes share the same fundamental.

Intervals

#	Cents	Ratio	Interval name
0	0.000	1/1	unison
1	203.910	9/8	major second
2	386.314	5/4	just major third
3	551.318	11/8	undecimal superfourth
4	701.955	3/2	just perfect fifth
5	840.528	13/8	tridecimal neutral sixth
6	968.826	7/4	harmonic seventh
7	1200.000	2/1	octave

Properties

The steps of the scale are 9/8, 10/9, 11/10, 12/11, 13/12, 14/13 and 8/7. All but the last are superparticular steps between consecutive harmonics, and the last is the 14:16 step that closes the octave.

Above the root, the scale contains the harmonic seventh chord (1/1, 5/4, 3/2, 7/4), as well as 4:5:6 (the major triad) and 4:5:7:9:11:13 (an extended harmonic 13th chord). It has no perfect fourth and no major seventh.

Relation to the acoustic scale

The scale approximates the acoustic scale (also called the overtone scale or lydian dominant scale in jazz): a diatonic scale with a raised fourth and a lowered seventh, the fourth mode of the ascending melodic minor scale. The name is commonly said to come from the scale's resemblance to this segment of the harmonic series. In terms of the diatonic degrees, the otonal heptad is 1, 2, 3, 4, 5, 6, 7, with some deviations from 12edo:

5/4 (386.3 cents) is 13.7 cents flat of the 12edo major third
11/8 (551.3 cents) is 48.7 cents flat of the augmented fourth
13/8 (840.5 cents) lies between the minor and major sixth, 40.5 cents above the minor sixth
7/4 (968.8 cents) is 31.2 cents flat of the minor seventh

Relation to the Carlos harmonic scale

The otonal heptad can be seen as the diatonic version of the Carlos harmonic scale, which is a chromatic scale. The notes of the heptad are the even harmonics 16, 18, 20, 22, 24, 26 and 28 of that scale, which has five more: the harmonics 17, 19, 21, 27 and 30.

Scala file

! oheptad.scl
!
Otonal heptad 8:9:10:11:12:13:14:16
7
!
9/8
5/4
11/8
3/2
13/8
7/4
2/1
!

View • Talk • EditOvertone scales
Small modes	2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24
Larger modes	30 • 32 • 36 • 48 • 60 • 128
Families	/2: 2 • 4 • 8 • 16 • 32 • 128 /3: 3 • 6 • 9 • 12 • 15 • 18 • 21 • 24 /5: 5 • 10 • 15 • 20 • 25 • 30 • 35 • 60 • 80 /7: 7 • 14 • 21 • 28 • 35 • 56 /11: 11 • 22 • 33 /13: 13 • 26
Related	Harmonic series • Subharmonic series • Carlos harmonic scale • Ringer scale • Primodality