24edo: Difference between revisions

Line 18:

=Theory=

{~~| class="wikitable"~~

~~|+ Prime intervals~~ in ~~24 EDO~~

|-

~~! colspan="2" | Prime interval~~

~~! 2~~

~~! 3~~

~~! 5~~

~~! 7~~

~~! 11~~

~~! 13~~

~~! 17~~

~~! 19~~

|-

~~! rowspan="2" | Error~~

~~! absolute ([[Cent|¢]])~~

~~| 0.0~~

~~| -2.0~~

~~| +13.7~~

~~| -18.8~~

~~| -1.3~~

~~| +9.5~~

~~| -5.0~~

~~| +2.5~~

|-

~~! [[Relative error|relative]] (%)~~

~~| 0~~

~~| -4~~

~~| +27~~

~~| -38~~

~~| -3~~

~~| +19~~

~~| -10~~

~~| +5~~

|-

~~! colspan="2" | [[nearest edomapping]]~~

| 24

~~| 14~~

~~| 8~~

~~| 19~~

~~| 11~~

~~| 17~~

~~| 2~~

~~| 6~~

|}

The [[Harmonic_Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals ([[7/4|7:4]], [[7/5|7:5]] and [[7/6|7:6]]) are almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like [[36edo|36-tET]], [[72edo|72-tET]], [[84edo|84-tET]] or [[156edo|156-tET]]. However, it should be noted that 24edo, like [[22edo]], ''does'' temper out [[Quartisma|117440512/117406179]], linking the otherwise sub-par 7-limit harmonies with those of the 11-limit, and speaking of 11-limit representation in 24edo, the 11th harmonic, and most intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in this EDO. The 24-tone interval of 550 cents is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half the perfect fifth.

@@ Line 18: / Line 18: @@
 =Theory=
-{| class="wikitable"
-|+ Prime intervals in 24 EDO
+{{Primes in edo|24}}
-|-
-! colspan="2" | Prime interval
-! 2
-! 3
-! 5
-! 7
-! 11
-! 13
-! 17
-! 19
-|-
-! rowspan="2" | Error
-! absolute ([[Cent|¢]])
-| 0.0
-| -2.0
-| +13.7
-| -18.8
-| -1.3
-| +9.5
-| -5.0
-| +2.5
-|-
-! [[Relative error|relative]] (%)
-| 0
-| -4
-| +27
-| -38
-| -3
-| +19
-| -10
-| +5
-|-
-! colspan="2" | [[nearest edomapping]]
-| 24
-| 14
-| 8
-| 19
-| 11
-| 17
-| 2
-| 6
-|}
 The [[Harmonic_Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals ([[7/4|7:4]], [[7/5|7:5]] and [[7/6|7:6]]) are almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like [[36edo|36-tET]], [[72edo|72-tET]], [[84edo|84-tET]] or [[156edo|156-tET]].  However, it should be noted that 24edo, like [[22edo]], ''does'' temper out [[Quartisma|117440512/117406179]], linking the otherwise sub-par 7-limit harmonies with those of the 11-limit, and speaking of 11-limit representation in 24edo, the 11th harmonic, and most intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in this EDO. The 24-tone interval of 550 cents is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half the perfect fifth.