Consistency: Difference between revisions
Skipping three primes is too much for most in the 12edo example and its integer-limit consistency that follows immediately isn't useful before introducing the very concept of integer-limit consistency. Misc. cleanup |
Move the 46edo compression example to the right section |
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The closest approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the closest approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives {{nowrap|(3/2)(7/6) {{=}} [[7/4]]}}, the harmonic seventh, for which the closest 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 closest approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the closest approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives {{nowrap|(3/2)(7/6) {{=}} [[7/4]]}}, the harmonic seventh, for which the closest 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. | ||
As another notable example, [[46edo]] is not consistent in the [[15-odd-limit]]. The 15/13 interval is 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. | |||
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]]. | ||
An example of the difference between consistency vs distinct consistency: In 12edo the [[7-odd-limit]] intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12edo is consistent up to the [[9-odd-limit]], it is distinctly consistent only up to the [[5-odd-limit]]. Another example of non-distinct consistency is given by the intervals [[14/13]] and [[13/12]] in [[72edo]] where they are both mapped to 8 steps. Although 72edo is consistent up to the [[17-odd-limit]], it is distinctly consistent only up to the [[11-odd-limit]]. | An example of the difference between consistency vs distinct consistency: In 12edo the [[7-odd-limit]] intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12edo is consistent up to the [[9-odd-limit]], it is distinctly consistent only up to the [[5-odd-limit]]. Another example of non-distinct consistency is given by the intervals [[14/13]] and [[13/12]] in [[72edo]] where they are both mapped to 8 steps. Although 72edo is consistent up to the [[17-odd-limit]], it is distinctly consistent only up to the [[11-odd-limit]]. | ||
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The concept of integer limits means that octave inversion and octave equivalence no longer apply: for example, [[13/10]] and [[11/7]] are in the 16-integer-limit, but [[20/13]] and [[22/7]] are not. | The concept of integer limits means that octave inversion and octave equivalence no longer apply: for example, [[13/10]] and [[11/7]] are in the 16-integer-limit, but [[20/13]] and [[22/7]] are not. | ||
As a result, [[stretched and compressed tuning|octave stretch and compression]] can be employed to improve an equal tuning's consistency limits: if we compress the octave of 46edo slightly (by about a cent), we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating Mode 8 of the harmonic series. | |||
It is possible to extend the concept of odd limits to other equaves, such as the "''q''-throdd-limit" with 3/1 (tritave) equivalence, but because an [[edt]] that is consistent to a certain throdd limit will also be consistent to the corresponding integer-limit, there is little reason to complicate the analysis with additional types of infinite interval sets. This wiki measures consistency in the special case of edos with odd limits instead of integer limits for ease of explanation, but the two types of consistency are effectively equivalent for edos anyways (an edo that is consistent to the ''q''-odd-limit will be consistent to the {{nowrap|(''q'' + 1)}}-integer-limit and vice versa) unless intervals or primes are skipped or if a [[JI subgroup]] is used. | It is possible to extend the concept of odd limits to other equaves, such as the "''q''-throdd-limit" with 3/1 (tritave) equivalence, but because an [[edt]] that is consistent to a certain throdd limit will also be consistent to the corresponding integer-limit, there is little reason to complicate the analysis with additional types of infinite interval sets. This wiki measures consistency in the special case of edos with odd limits instead of integer limits for ease of explanation, but the two types of consistency are effectively equivalent for edos anyways (an edo that is consistent to the ''q''-odd-limit will be consistent to the {{nowrap|(''q'' + 1)}}-integer-limit and vice versa) unless intervals or primes are skipped or if a [[JI subgroup]] is used. | ||