21edo: Difference between revisions

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21edo supports tertian harmony with both 7edo's neutral chords and inflected major and minor chords. The major third is identical to 12edo's, but is a more extreme third in 21edo due to the flatness of the fifth (which makes the minor third close to subminor), so that the chords might be more comparable to neogothic chords.

In terms of just intonation, outside the 5-limit (where 21edo contains a flat fifth and the familiar but controversial 400c major third), 21edo also closely approximates the harmonics [[7/4]] (a subminor seventh), [[17/16]] (a semitone), [[19/16]] (a minor third), [[23/16]] (a tritone), and [[29/16]] (a minor seventh), with ~~its~~ 7, 23, and 29 especially accurate (~~especially its~~ 7~~, which is~~ more accurate than any other edo below 26). The intervals [[16/15]] and [[27/16]], if directly approximated, are also very accurate. 21edo can be liberally treated as a no-11s 29-limit temperament, but treating 21edo as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate JI interpretation of the tuning, since almost every interval in 21edo can be described as a ratio within the 29-odd-limit. 21edo also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.

In terms of just intonation, outside the 5-limit (where 21edo contains a flat fifth and the familiar but controversial 400c major third), 21edo also closely approximates the harmonics [[7/4]] (a subminor seventh), [[17/16]] (a semitone), [[19/16]] (a minor third), [[23/16]] (a tritone), and [[29/16]] (a minor seventh), with harmonics 7, 23, and 29 being especially accurate (and harmonic 7 being more accurate than in any other edo below 26). The intervals [[16/15]] and [[27/16]], if directly approximated, are also very accurate. 21edo can be liberally treated as a no-11s 29-limit temperament, but treating 21edo as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate JI interpretation of the tuning, since almost every interval in 21edo can be described as a ratio within the 29-odd-limit. 21edo also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.

In terms of interval regions, 21edo possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (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 tertian harmony with both 7edo's neutral chords and inflected major and minor chords. The major third is identical to 12edo's, but is a more extreme third in 21edo due to the flatness of the fifth (which makes the minor third close to subminor), so that the chords might be more comparable to neogothic chords.
-In terms of just intonation, outside the 5-limit (where 21edo contains a flat fifth and the familiar but controversial 400c major third), 21edo also closely approximates the harmonics [[7/4]] (a subminor seventh), [[17/16]] (a semitone), [[19/16]] (a minor third), [[23/16]] (a tritone), and [[29/16]] (a minor seventh), with its 7, 23, and 29 especially accurate (especially its 7, which is more accurate than any other edo below 26). The intervals [[16/15]] and [[27/16]], if directly approximated, are also very accurate. 21edo can be liberally treated as a no-11s 29-limit temperament, but treating 21edo as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate JI interpretation of the tuning, since almost every interval in 21edo can be described as a ratio within the 29-odd-limit. 21edo also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.
+In terms of just intonation, outside the 5-limit (where 21edo contains a flat fifth and the familiar but controversial 400c major third), 21edo also closely approximates the harmonics [[7/4]] (a subminor seventh), [[17/16]] (a semitone), [[19/16]] (a minor third), [[23/16]] (a tritone), and [[29/16]] (a minor seventh), with harmonics 7, 23, and 29 being especially accurate (and harmonic 7 being more accurate than in any other edo below 26). The intervals [[16/15]] and [[27/16]], if directly approximated, are also very accurate. 21edo can be liberally treated as a no-11s 29-limit temperament, but treating 21edo as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate JI interpretation of the tuning, since almost every interval in 21edo can be described as a ratio within the 29-odd-limit. 21edo also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.
 In terms of interval regions, 21edo possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (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.