Consistency: Difference between revisions

Line 5:

| ja = 一貫性

}}

An [[EDO]] represents the ''q''-[[odd-limit]] '''consistently''' if the best approximations of the odd harmonics of the ''q''-odd-limit in that EDO also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5.

An [[EDO]] (or other [[equal-step tuning]]) represents the ''q''-[[odd-limit]] '''consistently''' if the best approximations of the odd harmonics of the ''q''-odd-limit in that EDO also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5. An [[equal-step tuning]] is '''distinctly/uniquely consistent''' in the ''q''-[[odd-limit]] if every interval in that odd limit is mapped to a distinct/unique step, so for example, an equal-step tuning cannot be distinctly consistent (AKA uniquely consistent) in the [[7-odd-limit]] if it maps [[7/5]] and [[10/7]] to the same step. (This would correspond to tempering [[50/49]], and in the case of EDOs, would mean the EDO must be a multiple of (AKA superset of) 2 EDO).

Note that we aren't using the 'patent' val for the EDO when making these approximations, but rather looking at the best approximation for each interval directly, rather than just the primes. If everything lines up, then the EDO is consistent within that odd-limit, otherwise it is inconsistent.

@@ Line 5: / Line 5: @@
 | ja = 一貫性
 }}
-An [[EDO]] represents the ''q''-[[odd-limit]] '''consistently''' if the best approximations of the odd harmonics of the ''q''-odd-limit in that EDO also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5.
+An [[EDO]] (or other [[equal-step tuning]]) represents the ''q''-[[odd-limit]] '''consistently''' if the best approximations of the odd harmonics of the ''q''-odd-limit in that EDO also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5. An [[equal-step tuning]] is '''distinctly/uniquely consistent''' in the ''q''-[[odd-limit]] if every interval in that odd limit is mapped to a distinct/unique step, so for example, an equal-step tuning cannot be distinctly consistent (AKA uniquely consistent) in the [[7-odd-limit]] if it maps [[7/5]] and [[10/7]] to the same step. (This would correspond to tempering [[50/49]], and in the case of EDOs, would mean the EDO must be a multiple of (AKA superset of) 2 EDO).
 Note that we aren't using the 'patent' val for the EDO when making these approximations, but rather looking at the best approximation for each interval directly, rather than just the primes. If everything lines up, then the EDO is consistent within that odd-limit, otherwise it is inconsistent.