7/4

Frequency ratio 7/4, measuring approximately 968.8 cents, named harmonic seventh or natural seventh, represents the interval between the 4th and 7th harmonics in the harmonic series. It is also called a septimal minor seventh or subminor seventh – the word "septimal" referring to the presence of a 7 as the highest prime in the ratio, and the word "subminor" referring to the harmonic seventh's narrowness compared with a traditional minor seventh (such as 9/5 or 16/9, 12edo's 1000-cent interval, or a minor seventh found in a meantone system). It is traditionally seen as a seventh, though it may show up as a sixth in rare uses, specifically as the sum of 11/8 and a 14/11 neomajor third as per this example.

7/4 has seen use in blues music, barbershop quartet music, and some musical traditions of the world, but has mostly not been recognized as a "consonance" in Western music theory. In most Just Intonation systems, the harmonic seventh is treated as a fundamental consonance in its own right, with its own distinct quality.

Harmonic seventh chord

7:4 appears in an otonal tetrad that forms the basis of much JI music, commonly called a "harmonic seventh chord". It consists of a ptolemaic major triad (4:5:6) plus a harmonic seventh: 4:5:6:7(:8). This tetrad, a hallmark of blues and barbershop harmony, not to mention modern Just Intonation practice, represents a sequence of overtones from the fourth to the seventh. (8, being a doubling of 4, represents an octave above the root.) The intervals between adjacent members of the chord decrease in size:

5:4	approx. 386 cents	major third
6:5	approx. 316 cents	minor third
7:6	approx. 267 cents	septimal subminor third
8:7	approx. 231 cents	septimal supermajor second

This chord is similar to the "dominant seventh chord" in 12edo, the most significant difference being the mistuning of the harmonic seventh. In 12edo, the interval closest to the harmonic seventh is the minor seventh at 1000 cents. The difference of about 31 cents is striking, and especially noticable when the chords are presented next to one another. While the "dominant seventh chord" of 12edo is treated as a dissonance that needs to resolve (usually in the chord pattern V7 to I), the "harmonic seventh chord" has a much different flavor and is often treated by composers in Just Intonation as a consonance.

Another interval found in a harmonic seventh chord is the septimal tritone of 7/5, which represents the interval between the major third (5) and the harmonic seventh (7). This interval, at 583 cents, sounds distinct from 12edo's half-octave tritone of 600 cents. In just intonation, 7/5 is treated as a consonant tritone, and has a much mellower and sweeter sound than the 600-cent tritone we are used to hearing.

Since 12edo does not distinguish between a minor and subminor third or a major and supermajor second, the intervals between adjacent members of the chord do not have the pattern of decreasing step size which characterizes the harmonic seventh chord:

5:4 becomes 400 cents.
6:5 becomes 300 cents.
7:6 becomes 300 cents.
8:7 becomes 200 cents.

Meantone augmented sixth

See also: Meantone

In meantone systems – which are generated by repeatedly stacking a slightly flattened (from just) perfect fifth such that four fifths gives a near-just major third of 5/4 – there is sometimes a good approximation of the harmonic seventh in the form of an augmented sixth. Quarter-comma meantone (aurally identical, for most intents and purposes, to 31edo) is one such system. In quarter-comma meantone, the interval of C to A# approximates a harmonic seventh, and is a distinct interval from C to Bb, a meantone minor seventh falling somewhere between 16/9 and 9/5. The augmented sixth appears in tonal harmony in the augmented sixth chord. The so-called German sixth chord, in quarter-comma meantone, would approximate the harmonic seventh chord of 4:5:6:7(:8).

Note that a good approximation of the harmonic seventh is not available in every meantone system. In 19edo (aurally identical, more or less, to 1/3-comma meantone), the "augmented sixth" is an interval of 947 cents – about 22 cents flat of 7:4, and so less effective as a consonance. Systems on the flat end of reasonable meantone tunings, flatter than 19edo, have the augmented sixth closer to 12/7, while the diminished seventh is closer to 7/4. Mapping the harmonic seventh to A6 is known as septimal meantone and mapping it to d7 is known as flattone.

Approximations by EDOs

Following EDOs (up to 200) contain good approximations^[1] of the interval 7/4. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (↑) or flat (↓).

EDO	deg\edo	Absolute error (¢)	Relative error (r¢)	↕	Equally acceptable multiples ^[2]
21	17\21	2.6026	4.5547	↑
26	21\26	0.4049	0.8772	↑	42\52, 63\78, 84\104, 105\130, 126\156, 147\182
31	25\31	1.0839	2.8003	↓	50\62
36	29\36	2.1592	6.4777	↓
47	38\47	1.3868	5.4319	↑
57	46\57	0.4049	1.9231	↓	92\114, 138\171
73	59\73	1.0371	6.3091	↑
83	67\83	0.1512	1.0459	↓	134\166
88	71\88	0.6441	4.7233	↓
109	88\109	0.0186	0.1687	↓
135	109\135	0.0630	0.7086	↑
140	113\140	0.2545	2.9689	↓
145	117\145	0.5500	6.6464	↓
161	130\161	0.1182	1.5858	↑
187	151\187	0.1581	2.4630	↑
192	155\192	0.0759	1.2145	↓
197	159\197	0.2980	4.8920	↓

↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
↑ Super EDOs up to 200 within the same error tolerance

7/4

Contents

Harmonic seventh chord

Meantone augmented sixth

Approximations by EDOs

See also

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