7/4: Difference between revisions

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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~~.

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.

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 {{See also| Meantone }}
-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.
+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.