Currywurst44: Created page with "{{Interwiki | en = Consistent | de = konsistent | es = | ja = 一貫性 }} An edo represents the ''q''-odd-limit '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, if the difference between the closest 7/4 and the closest 5/4 is also the closest 7/5. An edo is '''distinctly consistent'''..."

2026-04-10T01:38:24Z

Created page with "{{Interwiki | en = Consistent | de = konsistent | es = | ja = 一貫性 }} An edo represents the ''q''-odd-limit '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, if the difference between the closest 7/4 and the closest 5/4 is also the closest 7/5. An edo is '''distinctly consistent'''..."

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{{Interwiki
| en = Consistent
| de = konsistent
| es =
| ja = 一貫性
}}
An [[edo]] represents the [[odd limit|''q''-odd-limit]] '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, if the difference between the closest [[7/4]] and the closest [[5/4]] is also the closest [[7/5]]. An edo is '''distinctly consistent''' (or '''uniquely consistent''') in the ''q''-odd-limit if every interval in that odd limit is consistent and mapped to a distinct edostep. For example, an edo cannot be distinctly consistent in the [[7-odd-limit]] if it maps 7/5 and [[10/7]] to the same step (in this case, the semi-octave of [[2edo]], [[tempering out]] [[50/49]]).

The page ''[[Minimal consistent edos]]'' shows the smallest edo that is consistent or distinctly consistent in a given odd limit while the page ''[[Consistency limits of small edos]]'' shows the largest odd limit that a given edo is consistent or distinctly consistent in.

== Mathematical definition ==
''S'' shall be a set of intervals and ''M'' a tuning's pitch mapping of these intervals. ''r''''1'' and ''r''''2'' shall be in ''S'' with ''r''''1'' * ''r''''2'' = ''r''''3'' also in S. A tuning is '''consistent''' to distance ''d'' when the error of all ''M''(''r''''1'' * ''r''''2'') < 1/(2''d'') and the interval mapping is linear with ''M''(''r''''1'' * ''r''''2'') = ''M''(''r''''1'') + ''M''(''r''''2'').

A tuning is '''distinctly consistent''' when all ''M''(''r''''3'') are different.

User:Currywurst44/Consistency Rewrite - Revision history