21edo: Difference between revisions

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21edo ~~provides both~~ [[7edo]] as a ~~subset and the familiar 400-[[cent]] major third, while also giving some higher~~-~~[[limit]] [[JI]] possibilities~~. The ~~system can be treated~~ as ~~three intertwining 7edo or~~ "~~equiheptatonic~~" ~~scales~~, or as ~~seven [[3edo]] ''augmented'' triads~~. ~~The [[7/4]] at 971.43{{c}} is only off~~ in 21edo ~~by 2~~.~~60{{c}} from just (968.83{{c}})~~, ~~which~~ is ~~better than any other~~ [[~~edo~~]] ~~below 26~~.

21edo contains three [[7edo]] "equiheptatonic" scales, and can be interpreted as 7edo but with the capability to inflect up or down by a quarter-tone. The 7edo subset functions as a basic "diatonic" scale, though maximum-variety-3 options might also be preferable (such as omnidiatonic). In other words, all intervals have "minor", "neutral", and "major" variations, which makes building scales in 21edo rather interesting. If 21edo is analyzed purely diatonically, no chromatic alterations can exist because the chromatic semitone is equal to 0 cents (a fact characteristic of [[whitewood]] temperaments). So, another kind of accidental (such as ups and downs) is usually used instead.

~~In diatonically-related terms,~~ 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~~.

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.

~~Of harmonics 3~~, 5, 7, 11, and 13, ~~the only harmonic 21edo approximates~~ with ~~anything approaching a near-just flavor is the 7th harmonic. On the other hand~~, ~~21edo provides exceptionally accurate tunings of the 15th, 23rd~~, and ~~29th harmonics~~ (~~within 3{{c}} or less~~), ~~as well~~ as a ~~very reasonable approximation of the 27th harmonic (around 8{{c}} sharp). As such~~, 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 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 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.

Thanks to its sevenths, 21edo is an ideal tuning for its size for [[metallic harmony]].

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 {{Harmonics in equal|steps=21|columns=14}}
-edo provides both [[7edo]] as a subset and the familiar 400-[[cent]] major third, while also giving some higher-[[limit]] [[JI]] possibilities. The system can be treated as three intertwining 7edo or "equiheptatonic" scales, or as seven [[3edo]] ''augmented'' triads. The [[7/4]] at 971.43{{c}} is only off in 21edo by 2.60{{c}} from just (968.83{{c}}), which is better than any other [[edo]] below 26.
+edo contains three [[7edo]] "equiheptatonic" scales, and can be interpreted as 7edo but with the capability to inflect up or down by a quarter-tone. The 7edo subset functions as a basic "diatonic" scale, though maximum-variety-3 options might also be preferable (such as omnidiatonic). In other words, all intervals have "minor", "neutral", and "major" variations, which makes building scales in 21edo rather interesting. If 21edo is analyzed purely diatonically, no chromatic alterations can exist because the chromatic semitone is equal to 0 cents (a fact characteristic of [[whitewood]] temperaments). So, another kind of accidental (such as ups and downs) is usually used instead.
-In diatonically-related terms, 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.
+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.
-Of harmonics 3, 5, 7, 11, and 13, the only harmonic 21edo approximates with anything approaching a near-just flavor is the 7th harmonic. On the other hand, 21edo provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3{{c}} or less), as well as a very reasonable approximation of the 27th harmonic (around 8{{c}} sharp). As such, 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 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 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.
 Thanks to its sevenths, 21edo is an ideal tuning for its size for [[metallic harmony]].