24edo: Difference between revisions

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=Theory=

The 24edo system divides the octave into 24 equal parts of exactly 50 cents each. It is also known as '''quarter-tone tuning''', since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes, and also because of its use in theory and occasionally in practice in [[Arabic,_Turkish,_Persian|Arabic]] music. It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments - [[~~DIY_Quartertone_Composition_with_12_equal_tools|see this page~~]].

The 24edo system divides the octave into 24 equal parts of exactly 50 cents each. It is also known as '''quarter-tone tuning''', since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes, and also because of its use in theory and occasionally in practice in [[Arabic,_Turkish,_Persian|Arabic]] music. It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments as illustrated in ''[[DIY Quartertone Composition with 12 equal tools]]''.

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.

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 =Theory=
-The 24edo system divides the octave into 24 equal parts of exactly 50 cents each. It is also known as '''quarter-tone tuning''', since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes, and also because of its use in theory and occasionally in practice in [[Arabic,_Turkish,_Persian|Arabic]] music. It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments - [[DIY_Quartertone_Composition_with_12_equal_tools|see this page]].
+The 24edo system divides the octave into 24 equal parts of exactly 50 cents each. It is also known as '''quarter-tone tuning''', since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes, and also because of its use in theory and occasionally in practice in [[Arabic,_Turkish,_Persian|Arabic]] music. It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments as illustrated in ''[[DIY Quartertone Composition with 12 equal tools]]''.
 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.