21edo: Difference between revisions

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== Theory ==

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{{~~cent~~}} is only off in 21edo by 2.60{{~~cent~~}} from just (968.83{{~~cent~~}}), which is better than any other [[edo]] below 26.

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.

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.

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{{~~cent~~}} or less), as well as a very reasonable approximation of the 27th harmonic (around 8{{~~cent~~}} 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.

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.

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

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=== Rank-3 scales ===

The rank-3 scale [[diasem]] (323132313 or 313231323 in 21edo) is the 21edo tempering of [[Zarlino]] diatonic with 1\21 comma steps added, resulting in two "major seconds" (171{{~~cent~~}} and 228{{~~cent~~}}), two "minor thirds" (286{{~~cent~~}} and 343{{~~cent~~}}) and two "fourths" (457{{~~cent~~}} and 514{{~~cent~~}}). 21edo is the smallest edo to support a non-degenerate diasem (with L:M:S ratio 3:2:1).

The rank-3 scale [[diasem]] (323132313 or 313231323 in 21edo) is the 21edo tempering of [[Zarlino]] diatonic with 1\21 comma steps added, resulting in two "major seconds" (171{{c}} and 228{{c}}), two "minor thirds" (286{{c}} and 343{{c}}) and two "fourths" (457{{c}} and 514{{c}}). 21edo is the smallest edo to support a non-degenerate diasem (with L:M:S ratio 3:2:1).

=== Tetrachordal scales ===

@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|21}}
+{{ED intro}}
 == Theory ==
 {{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{{cent}} is only off in 21edo by 2.60{{cent}} from just (968.83{{cent}}), which is better than any other [[edo]] below 26.
+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.
 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.
-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{{cent}} or less), as well as a very reasonable approximation of the 27th harmonic (around 8{{cent}} 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.
+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.
 Thanks to its sevenths, 21edo is an ideal tuning for its size for [[metallic harmony]].
@@ Line 443: / Line 443: @@
 === Rank-3 scales ===
-The rank-3 scale [[diasem]] (323132313 or 313231323 in 21edo) is the 21edo tempering of [[Zarlino]] diatonic with 1\21 comma steps added, resulting in two "major seconds" (171{{cent}} and 228{{cent}}), two "minor thirds" (286{{cent}} and 343{{cent}}) and two "fourths" (457{{cent}} and 514{{cent}}). 21edo is the smallest edo to support a non-degenerate diasem (with L:M:S ratio 3:2:1).
+The rank-3 scale [[diasem]] (323132313 or 313231323 in 21edo) is the 21edo tempering of [[Zarlino]] diatonic with 1\21 comma steps added, resulting in two "major seconds" (171{{c}} and 228{{c}}), two "minor thirds" (286{{c}} and 343{{c}}) and two "fourths" (457{{c}} and 514{{c}}). 21edo is the smallest edo to support a non-degenerate diasem (with L:M:S ratio 3:2:1).
 === Tetrachordal scales ===