Consistency: Difference between revisions

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==Examples==

An example for a system that is ''not'' consistent in a particular odd limit is [[~~25edo|~~25edo]]:

An example for a system that is ''not'' consistent in a particular odd limit is [[25edo]]:

The best approximation for the interval of [[~~7/6|~~7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the [[3/2|perfect fifth 3/2]] is 15 steps. Adding the two just intervals gives 3/2 * 7/6 = [[~~7/4|~~7/4]], the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7 odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.

The best approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the [[3/2|perfect fifth 3/2]] is 15 steps. Adding the two just intervals gives 3/2 * 7/6 = [[7/4]], the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7 odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.

An example for a system that ''is'' consistent in the [[7-odd-limit]] is [[12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. [[12edo}} is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].

An example for a system that ''is'' consistent in the [[7-odd-limit]] is [[12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. [[12edo]] is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].

One notable example: [[46edo]] is not consistent in the 15 odd limit. The 15:13 interval is very closer slightly closer to 9 degrees of 46edo than to 10 degrees, but the ''functional'' [[15/13]] (the difference between 46edo's versions of [[15/8]] and [[13/8]]) is 10 degrees. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series.

@@ Line 11: / Line 11: @@
 ==Examples==
-An example for a system that is ''not'' consistent in a particular odd limit is [[25edo|25edo]]:
+An example for a system that is ''not'' consistent in a particular odd limit is [[25edo]]:
-The best approximation for the interval of [[7/6|7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the [[3/2|perfect fifth 3/2]] is 15 steps. Adding the two just intervals gives 3/2 * 7/6 = [[7/4|7/4]], the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7 odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.
+The best approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best approximation for the [[3/2|perfect fifth 3/2]] is 15 steps. Adding the two just intervals gives 3/2 * 7/6 = [[7/4]], the harmonic seventh, for which the best approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7 odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.
-An example for a system that ''is'' consistent in the [[7-odd-limit]] is [[12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. [[12edo}} is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].
+An example for a system that ''is'' consistent in the [[7-odd-limit]] is [[12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. [[12edo]] is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].
 One notable example: [[46edo]] is not consistent in the 15 odd limit. The 15:13 interval is very closer slightly closer to 9 degrees of 46edo than to 10 degrees, but the ''functional'' [[15/13]] (the difference between 46edo's versions of [[15/8]] and [[13/8]]) is 10 degrees. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series.