Consistency: Difference between revisions
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If N-edo is an [[EDO|equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is ''consistent'' with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[Odd_limit|q odd limit intervals]], consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''. | If N-edo is an [[EDO|equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is ''consistent'' with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[Odd_limit|q odd limit intervals]], consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''. | ||
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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|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. | ||
An example for a system that ''is'' consistent in the 7 odd-limit is [[ | 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: [[ | 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. | ||
==Generalization to non-octave scales== | ==Generalization to non-octave scales== | ||
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It is possible to generalize the concept of consistency to non-edo equal temperaments. Because octaves are no longer equivalent, instead of an odd limit we must use an integer limit, and the term 2^n in the above equation is no longer present. Instead, the set S consists of all intervals u/v where u <= q >= v. | It is possible to generalize the concept of consistency to non-edo equal temperaments. Because octaves are no longer equivalent, instead of an odd limit we must use an integer limit, and the term 2^n in the above equation is no longer present. Instead, the set S consists of all intervals u/v where u <= q >= v. | ||
This also means that the concept of octave inversion no longer applies: in this example, 13 | This also means that the concept of octave inversion no longer applies: in this example, [[13/9]] is in S, but [[18/13]] is not. | ||
==Links== | ==Links== | ||
[http://www.tonalsoft.com/enc/c/consistent.aspx consistent (TonalSoft encyclopedia)] | * [http://www.tonalsoft.com/enc/c/consistent.aspx consistent (TonalSoft encyclopedia)] | ||
[[Category:edo]] | |||
[[Category:temperament]] | [[Category:temperament]] | ||
[[Category:term]] | [[Category:term]] | ||
[[Category:todo:discuss_title]] | [[Category:todo:discuss_title]] | ||
[[Category:todo:reduce_mathslang]] | [[Category:todo:reduce_mathslang]] | ||