21edo: Difference between revisions

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21edo supports {{w|tertian harmony}} with 7edo's flat fifth, containing both 7edo's neutral chords and inflected major and minor chords. The [[5/4]] major third is mapped to 400{{c}}, identical to 12edo's, but the minor third is more extreme in 21edo due to the flatness of the fifth (closer to subminor), so that the chords might be more comparable to [[neogothic]] chords. In fact, [[6/5]] is slightly closer to the 6-step neutral third than the 5-step minor third, meaning 21edo lacks [[consistency]] to the [[5-odd-limit]].

21edo closely approximates the [[octave-reduced]] [[harmonic]]s [[7/4]] (a subminor seventh), [[15/8]] (a major seventh), [[23/16]] (a wide tritone), [[29/16]] (a supraminor seventh), [[31/16]] (a supermajor seventh), [[33/32]] (a quartertone), [[39/32]] (a neutral third), and [[43/32]] (an acute fourth). The intervals [[17/16]], [[19/16]], [[27/16]] are approximated less accurately, but are still usable. 21edo can be crudely treated as a no-11s 31-limit temperament, though the lack of consistency will give some unusual results, such as [[10/9]] being mapped wider than [[9/8]]. However, treating 21edo as a 2.15.7.33.39.23.29.31.43 subgroup temperament allows for a more accurate JI interpretation of the tuning, with a maximum error of any 43-odd-limit interval in this subgroup being 6.4{{c}}. These approximations derive from and are inherited by [[84edo]], which covers a large number of primes in high limits. 21edo also works well on the 2.27.9/5.7.11/5.13/5.17/5 subgroup, which is derived from [[63edo]], ~~which~~ is possibly a more sensible way to treat it.

21edo closely approximates the [[octave-reduced]] [[harmonic]]s [[7/4]] (a subminor seventh), [[15/8]] (a major seventh), [[23/16]] (a wide tritone), [[29/16]] (a supraminor seventh), [[31/16]] (a supermajor seventh), [[33/32]] (a quartertone), [[39/32]] (a neutral third), and [[43/32]] (an acute fourth). The intervals [[17/16]], [[19/16]], [[27/16]] are approximated less accurately, but are still usable. 21edo can be crudely treated as a no-11s 31-limit temperament, though the lack of consistency will give some unusual results, such as [[10/9]] being mapped wider than [[9/8]]. However, treating 21edo as a 2.15.7.33.39.23.29.31.43 subgroup temperament allows for a more accurate JI interpretation of the tuning, with a maximum error of any 43-odd-limit interval in this subgroup being 6.4{{c}}. These approximations derive from and are inherited by [[84edo]], which covers a large number of primes in high limits. 21edo also works well on the 2.27.9/5.7.11/5.13/5.17/5 subgroup, which is derived from [[63edo]], and is possibly a more sensible way to treat it.

In terms of interval regions, 21edo possesses four types of 2nds (subminor, minor, submajor, and supermajor), three types of 3rds (subminor, neutral, and major), a "third-fourth/naiadic" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.

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 edo supports {{w|tertian harmony}} with 7edo's flat fifth, containing both 7edo's neutral chords and inflected major and minor chords. The [[5/4]] major third is mapped to 400{{c}}, identical to 12edo's, but the minor third is more extreme in 21edo due to the flatness of the fifth (closer to subminor), so that the chords might be more comparable to [[neogothic]] chords. In fact, [[6/5]] is slightly closer to the 6-step neutral third than the 5-step minor third, meaning 21edo lacks [[consistency]] to the [[5-odd-limit]].
-edo closely approximates the [[octave-reduced]] [[harmonic]]s [[7/4]] (a subminor seventh), [[15/8]] (a major seventh), [[23/16]] (a wide tritone), [[29/16]] (a supraminor seventh), [[31/16]] (a supermajor seventh), [[33/32]] (a quartertone), [[39/32]] (a neutral third), and [[43/32]] (an acute fourth). The intervals [[17/16]], [[19/16]], [[27/16]] are approximated less accurately, but are still usable. 21edo can be crudely treated as a no-11s 31-limit temperament, though the lack of consistency will give some unusual results, such as [[10/9]] being mapped wider than [[9/8]]. However, treating 21edo as a 2.15.7.33.39.23.29.31.43 subgroup temperament allows for a more accurate JI interpretation of the tuning, with a maximum error of any 43-odd-limit interval in this subgroup being 6.4{{c}}. These approximations derive from and are inherited by [[84edo]], which covers a large number of primes in high limits. 21edo also works well on the 2.27.9/5.7.11/5.13/5.17/5 subgroup, which is derived from [[63edo]], which is possibly a more sensible way to treat it.
+edo closely approximates the [[octave-reduced]] [[harmonic]]s [[7/4]] (a subminor seventh), [[15/8]] (a major seventh), [[23/16]] (a wide tritone), [[29/16]] (a supraminor seventh), [[31/16]] (a supermajor seventh), [[33/32]] (a quartertone), [[39/32]] (a neutral third), and [[43/32]] (an acute fourth). The intervals [[17/16]], [[19/16]], [[27/16]] are approximated less accurately, but are still usable. 21edo can be crudely treated as a no-11s 31-limit temperament, though the lack of consistency will give some unusual results, such as [[10/9]] being mapped wider than [[9/8]]. However, treating 21edo as a 2.15.7.33.39.23.29.31.43 subgroup temperament allows for a more accurate JI interpretation of the tuning, with a maximum error of any 43-odd-limit interval in this subgroup being 6.4{{c}}. These approximations derive from and are inherited by [[84edo]], which covers a large number of primes in high limits. 21edo also works well on the 2.27.9/5.7.11/5.13/5.17/5 subgroup, which is derived from [[63edo]], and is possibly a more sensible way to treat it.
 In terms of interval regions, 21edo possesses four types of 2nds (subminor, minor, submajor, and supermajor), three types of 3rds (subminor, neutral, and major), a "third-fourth/naiadic" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.