Consistency: Difference between revisions
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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) 2edo). | 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) 2edo). | ||
Note that we are | Note that we are looking at the closest direct approximation for each interval, and trying to find a val to line them up. If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent. | ||
While the term "consistency" is most frequently used to refer to some odd-limit, sometimes one may only care about 'some' of the intervals in some odd-limit; this situation often arises when working in JI [[subgroup]]s. We can also "skip" certain intervals when evaluating consistency. For instance, [[12edo]] is consistent in the no-11's, no-13's [[19-odd-limit]], meaning for the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, and 19, where we deliberately skip 11 and 13. | While the term "consistency" is most frequently used to refer to some odd-limit, sometimes one may only care about 'some' of the intervals in some odd-limit; this situation often arises when working in JI [[subgroup]]s. We can also "skip" certain intervals when evaluating consistency. For instance, [[12edo]] is consistent in the no-11's, no-13's [[19-odd-limit]], meaning for the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, and 19, where we deliberately skip 11 and 13. | ||
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== Mathematical definition == | == Mathematical definition == | ||
Formally, if ''N''-edo is an [[equal division of the octave]], and if for any interval ''r'', ''N''-edo (''r'') is the closest ''N''-edo approximation to ''r'', then ''N'' is '''consistent''' with respect to a set of intervals S if for any two intervals '' | Formally, if ''N''-edo is an [[equal division of the octave]], and if for any interval ''r'', ''N''-edo (''r'') is the closest ''N''-edo approximation to ''r'', then ''N'' is '''consistent''' with respect to a set of intervals S if for any two intervals ''r''<sub>''i''</sub> and ''r''<sub>''j''</sub> in S where ''r''<sub>''i''</sub>''r''<sub>''j''</sub> is also in S, ''N''-edo (''r''<sub>''i''</sub>''r''<sub>''j''</sub>) = ''N''-edo (''r''<sub>''i''</sub>) + ''N''-edo (''r''<sub>''j''</sub>). | ||
; Alternative formulation using val | |||
If for any interval ''r'', ''N''-edo (''r'') is the closest ''N''-edo approximation to ''r'', and if V (''r'') is ''r'' mapped by a val V, then ''N'' is consistent with respect to a set of intervals S if there exists a val V such that ''N''-edo (''r'') = V (''r'') for any ''r'' in S. | |||
{{Databox|Proof for equivalence| | |||
Let us denote the monzo of any ratio ''r'' by m. Due to the linearity of the interval space, for any intervals ''r''<sub>''i''</sub>, ''r''<sub>''j''</sub>, and ''r''<sub>''i''</sub>''r''<sub>''j''</sub> in S, their monzos are m<sub>''i''</sub>, m<sub>''j''</sub>, and m<sub>''i''</sub> + m<sub>''j''</sub>, respectively. | |||
The ratio ''r'' mapped by the val V is the tempered step number V (''r'') = V·m. There is the following identity: | |||
<math>V\cdot(\vec {m_i} + \vec {m_j}) = V\cdot\vec {m_i} + V\cdot\vec {m_j}</math> | |||
Hence, | |||
<math>V (r_i r_j) = V (r_i) + V (r_j)</math> | |||
If ''N''-edo satisfies | |||
<math>\operatorname {\mathit {N}-edo} (r_i r_j) = \operatorname {\mathit {N}-edo} (r_i) + \operatorname {\mathit {N}-edo} (r_j)</math> | |||
then ''N''-edo is a member of the function space formed by all vals. Therefore, there exists a val V such that ''N''-edo (''r'') = V (''r'') for any ''r'' in S. | |||
}} | |||
Normally, S is considered to be some set of [[odd limit|''q''-odd-limit intervals]], consisting of everything of the form 2<sup>''n''</sup> ''u''/''v'', where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''N'' is then said to be ''q-odd-limit consistent''. | Normally, S is considered to be some set of [[odd limit|''q''-odd-limit intervals]], consisting of everything of the form 2<sup>''n''</sup> ''u''/''v'', where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''N'' is then said to be ''q-odd-limit consistent''. | ||
If each interval in the ''q''-odd-limit is mapped to a unique value by ''N'', then it said to be ''uniquely q-odd-limit consistent''. | If each interval in the ''q''-odd-limit is mapped to a unique value by ''N'', then it is said to be ''uniquely q-odd-limit consistent''. | ||
== Examples == | == Examples == | ||
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Alternatively, we can use "modulo-''n'' limit" if the [[equave]] is ''n''/1. Thus the tritave analogue of odd limit would only allow integers not divisible by 3 below a given number, assuming tritave equivalence and tritave invertibility. | Alternatively, we can use "modulo-''n'' limit" if the [[equave]] is ''n''/1. Thus the tritave analogue of odd limit would only allow integers not divisible by 3 below a given number, assuming tritave equivalence and tritave invertibility. | ||
== | == External links == | ||
* [http://www.tonalsoft.com/enc/c/consistent.aspx | * [http://www.tonalsoft.com/enc/c/consistent.aspx TonalSoft encyclopedia | Consistency / consistent] | ||
* [https://docs.google.com/spreadsheets/d/1yt239Aeh26RwktiI9Nkli87A0nmBcRri1MXrFb4hE-g/edit?usp=sharing Consistency and relative error of edo] | * [https://docs.google.com/spreadsheets/d/1yt239Aeh26RwktiI9Nkli87A0nmBcRri1MXrFb4hE-g/edit?usp=sharing Consistency and relative error of edo] | ||