7/4: Difference between revisions

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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]].

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 ==

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 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]].
+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 ==