21edo: Difference between revisions

← Older edit Newer edit →

VisualWikitext

@@ Line 5: / Line 5: @@
 | ja = 21平均律
 }}
+{{Infobox ET
+| Prime factorization = 3 * 7
+| Step size = 57.143
+| Fifth type = 12\21 = 685.714¢ = [[7-edo]]
+| Major 2nd = 3\21 = 171¢
+| Minor 2nd = 3\21 = 171¢
+| Augmented 1sn = 0\21 = 0¢
+}}
 ==Theory==
+{| class="wikitable"
-Twenty-one equal divisions of the octave provides both 7-edo 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 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other EDO &lt;26.
+! colspan="2" |
+!prime 2
+!prime 3
+!prime 5
+!prime 7
+!prime 11
+!prime 13
+!prime 17
+!prime 19
+|-
+! rowspan="2" |Error
+!absolute (¢)
+|0
+| -16.24
+|13.7
+|2.6
+|20.1
+|16.6
+|9.3
+| -11.8
+|-
+![[Relative error|relative]] (%)
+|0
+| -28
+|24
+|5
+|35
+|29
+|16
+| -21
+|-
+! colspan="2" |[[nearest edomapping]]
+|21
+|12
+|7
+|17
+|10
+|15
+|2
+|5
+|}
+-edo provides both 7-edo 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 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other EDO &lt;26.
 In diatonically-related terms, 21-EDO 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.