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== Theory ==
== Theory ==
The [[5-limit]] approximations in 24edo are the same as those in 12edo, therefore 24edo offers nothing new as far as approximating the 5-limit is concerned.   
== Theory ==
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-based intervals ([[7/4]], [[7/5]] and [[7/6]]) are almost as bad in 24edo as in 12edo. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]] or [[156edo|156et]]. However, it should be noted that 24edo, like [[22edo]], ''does'' temper 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]] and [[7/6]]) are almost as inaccurate in 24edo as in 12edo. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]] or [[156edo|156et]]. 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]].  
 
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.  


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 and 17. Another approach would be to treat 24edo as a 2.3.11.17.19 [[subgroup]] temperament, on which it is quite accurate.  
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 and 17. Another approach would be to treat 24edo as a 2.3.11.17.19 [[subgroup]] temperament, on which it is quite accurate.  
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=== Commas ===
=== Commas ===
24edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 24 38 56 67 83 89 }}.  
This is a partial list of the [[commas]] that 24edo [[tempers out]] with its patent [[val]], {{val| 24 38 56 67 83 89 }}.  


{| class="commatable wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
Line 482: Line 481:
|62.57
|62.57
|Quadgu
|Quadgu
|Major diesis, diminished domma
|Greater diesis, diminished domma
|-
|-
| 5
| 5
Line 489: Line 488:
|41.06
|41.06
|Trigu
|Trigu
|Diesis, augmented comma
|Lesser diesis, augmented comma
|-
|-
|5
|5
Line 524: Line 523:
|0.02
|0.02
|Sepbisa-quadbigu
|Sepbisa-quadbigu
|[[Atom]]
|[[Kirnberger's atom]]
|-
|7
|[[1323/1280]]
|{{monzo| -8 3 -1 2 }}
|57.20
|Lazozogu
| Septimal two-seventh tone
|-
|-
|7
|7
Line 538: Line 544:
|14.19
|14.19
|Zozoyo
|Zozoyo
|Sensamagic
|Sensamagic comma
|-
|-
|7
|7
Line 545: Line 551:
|7.32
|7.32
| Labirugu
| Labirugu
| Cataharry
| Cataharry comma
|-
|-
|7
|7
Line 552: Line 558:
|5.36
|5.36
|Sarurutrigu
|Sarurutrigu
| Porwell
| Porwell comma
|-
|11
|[[56/55]]
|{{monzo| 3 0 -1 1 -1 }}
| 31.19
|Luzogu
| Undecimal tritonic comma
|-
|-
|11
|11
Line 595: Line 608:
|Bilorugu
|Bilorugu
|Kalisma, Gauss' comma
|Kalisma, Gauss' comma
|-
|13
|[[66/65]]
|{{monzo| 1 1 -1 0 1 -1 }}
|26.43
| Thulogu
|Winmeanma
|-
|-
|13
|13
Line 602: Line 622:
| Thozogu
| Thozogu
|Superleap
|Superleap
|-
|13
|[[512/507]]
|{{monzo| 9 -1 0 0 0 -2 }}
|16.99
| Thuthu
|Tridecimal neutral thirds comma
|-
|13
|[[105/104]]
|{{monzo| -3 1 1 1 0 -1 }}
|16.57
| Thuzoyo
|Animist comma
|-
|13
|[[144/143]]
|{{monzo| 4 2 0 0 -1 -1 }}
|12.06
| Thulu
|Grossma
|-
|-
|13
|13
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