7/4: Difference between revisions
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{{Wikipedia| Harmonic seventh }} | |||
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Frequency ratio '''7/4''', measuring approximately 968.8 [[cent | 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 [[overtone 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). While many microtonalists see 7/4 as being strictly a type of seventh, [[User:Aura|Aura]] is one of perhaps only a handful of microtonalists at most to [[User:Aura/Aura's Ideas on Functional Harmony #Interstep Functions|offer a counterargument]]. | ||
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. | 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 | == 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 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: | 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 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: | ||
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* [[8/7|8:7]] becomes 200 cents. | * [[8/7|8:7]] becomes 200 cents. | ||
== Meantone | == Meantone augmented sixth == | ||
In [[Meantone family #Septimal meantone|meantone systems]] – which are generated by repeatedly stacking a slightly flatted (from just) [[perfect fifth]] such that four fifths gives a near-just [[major third]] – 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," and is treated as a rare and special dissonance. The so-called "German | In [[Meantone family #Septimal meantone|meantone systems]] – which are generated by repeatedly stacking a slightly flatted (from just) [[perfect fifth]] such that four fifths gives a near-just [[major third]] – 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," and is treated as a rare and special dissonance. The so-called "German sixth", in quarter-comma meantone, would approximate the harmonic seventh chord of 4:5:6:7(:8). In Just Intonation, this augmented sixth is likely to be [[225/128]] – the Neapolitan augmented sixth. | ||
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 | 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. | ||
{{See also| Meantone }} | |||
== Approximations by EDOs == | == Approximations by EDOs == | ||
Following [[EDO]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 7/4. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (↑) or flat (↓). | Following [[EDO]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 7/4. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (↑) or flat (↓). | ||
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* [[8/7]] – its [[octave complement]] | * [[8/7]] – its [[octave complement]] | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
[[Category:7-limit]] | [[Category:7-limit]] | ||
[[Category:Just interval]] | [[Category:Just interval]] | ||
[[Category:Overtone]] | [[Category:Overtone]] | ||
[[Category:Theory]] | [[Category:Theory]] | ||
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[[Category:Subminor seventh]] | [[Category:Subminor seventh]] | ||
[[Category:Over-2]] | [[Category:Over-2]] | ||
[[Category:Pages with internal sound examples]] | |||
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[[de:Naturseptime]] | [[de:Naturseptime]] | ||