24edo: Difference between revisions

Line 16:

The [[5-limit]] approximations in 24edo are the same as those in 12edo, so 24edo offers nothing new as far as approximating the 5-limit is concerned.

The 7th harmonic and its intervals ([[7/4]], [[7/5]], [[7/6]], and [[9/7]]) are almost as inaccurate in 24edo as in 12edo. To achieve a satisfactory level of approximation to intervals of 7 while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]], [[156edo|156et]], or [[192edo|192et]]. However, 24edo excels at the 11th harmonic and most intervals involving 11 ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]). 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~~. Additionally, like [[22edo]], 24edo tempers out the [[quartisma]], linking the otherwise sub-par [[7-limit]] harmonies with those of the [[11-limit]]~~.

The 7th harmonic and its intervals ([[7/4]], [[7/5]], [[7/6]], and [[9/7]]) are almost as inaccurate in 24edo as in 12edo. To achieve a satisfactory level of approximation to intervals of 7 while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]], [[156edo|156et]], or [[192edo|192et]]. However, 24edo excels at the 11th harmonic and most intervals involving 11 ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]). 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.

The tunings supplied by [[72edo]] cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11, 17, and 19. Expanding this, one will find that 24edo is consistent in the no-7s 19-odd-limit, though the 2.3.11.17.19 [[subgroup]] is where it is the most accurate.

@@ Line 16: / Line 16: @@
 The [[5-limit]] approximations in 24edo are the same as those in 12edo, so 24edo offers nothing new as far as approximating the 5-limit is concerned.
-The 7th harmonic and its intervals ([[7/4]], [[7/5]], [[7/6]], and [[9/7]]) are almost as inaccurate in 24edo as in 12edo. To achieve a satisfactory level of approximation to intervals of 7 while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]], [[156edo|156et]], or [[192edo|192et]]. However, 24edo excels at the 11th harmonic and most intervals involving 11 ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]). 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. Additionally, like [[22edo]], 24edo tempers out the [[quartisma]], linking the otherwise sub-par [[7-limit]] harmonies with those of the [[11-limit]].
+The 7th harmonic and its intervals ([[7/4]], [[7/5]], [[7/6]], and [[9/7]]) are almost as inaccurate in 24edo as in 12edo. To achieve a satisfactory level of approximation to intervals of 7 while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]], [[156edo|156et]], or [[192edo|192et]]. However, 24edo excels at the 11th harmonic and most intervals involving 11 ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]). 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.
 The tunings supplied by [[72edo]] cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N subgroups|3*24&nbsp;subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11, 17, and 19. Expanding this, one will find that 24edo is consistent in the no-7s 19-odd-limit, though the 2.3.11.17.19 [[subgroup]] is where it is the most accurate.