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''.

See also [[Minimal_consistent_EDOs|this list]] of odd limits, with the smallest edo that is consistent or uniquely consistent in that odd limit. And [[Consistency_levels_of_small_EDOs|this list]] of edos, with the largest odd limit that this edo is consistent or uniquely consistent in.

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One 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. 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.

Examples on consistency vs. unique 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 uniquely consistent only up to the [[5-odd-limit]]. Another example or non-unique 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 uniquely consistent only up to the [[11-odd-limit]].

Examples on consistency vs. unique 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 uniquely consistent only up to the [[5-odd-limit]]. Another example or non-unique 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 uniquely consistent only up to the [[11-odd-limit]].

==Generalization to non-octave scales==

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* [http://www.tonalsoft.com/enc/c/consistent.aspx consistent (TonalSoft encyclopedia)]

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[[Category:Edo]]

[[Category:~~temperament~~]]

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[[Category:~~term~~]]

[[Category:Term]]

[[Category:~~todo~~:~~discuss_title~~]]

[[Category:~~todo~~:~~reduce_mathslang~~]]

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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''.
 See also [[Minimal_consistent_EDOs|this list]] of odd limits, with the smallest edo that is consistent or uniquely consistent in that odd limit. And [[Consistency_levels_of_small_EDOs|this list]] of edos, with the largest odd limit that this edo is consistent or uniquely consistent in.
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 One 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. 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.
-Examples on consistency vs. unique 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 uniquely consistent only up to the [[5-odd-limit]]. Another example or non-unique 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 uniquely consistent only up to the [[11-odd-limit]].
+Examples on consistency vs. unique 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 uniquely consistent only up to the [[5-odd-limit]]. Another example or non-unique 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 uniquely consistent only up to the [[11-odd-limit]].
 ==Generalization to non-octave scales==
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 * [http://www.tonalsoft.com/enc/c/consistent.aspx consistent (TonalSoft encyclopedia)]
-[[Category:edo]]
+[[Category:Edo]]
-[[Category:temperament]]
+[[Category:Temperament]]
-[[Category:term]]
+[[Category:Term]]
-[[Category:todo:discuss_title]]
-[[Category:todo:reduce_mathslang]]
+[[Category:Todo:discuss title]]
+[[Category:Todo:reduce mathslang]]