7/4: Difference between revisions

| Name = harmonic seventh, natural seventh, septimal minor seventh, subminor seventh

Frequency ratio '''7/4''', measuring approximately 968.8 [[cent]]s, 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 (sub)minor 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 minor seventh, though it may show up as an augmented sixth in some cases.

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:

== Meantone augmented sixth ==

{{See also| Meantone }}

In [[Meantone family #Septimal meantone|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]].