Tuning ranges of regular temperaments: Difference between revisions

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There are various methods which have been suggested for defining tuning ranges appropriate to a given [[regular temperament]].

Given a rank-''r'' ''p''-limit regular temperament, we may define a tuning range by finding the [[Wikipedia: Convex hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest a set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], when this tuning is defined. This is the '''nice tuning range'''. We may define another tuning range by requiring that the ''p''-odd limit tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also [[Wikipedia: Monotonic function|monotone]] nondecreasing. This we may call the '''valid tuning range'''. A tuning which is both nice and valid is a ''strict'' tuning and this defines the '''strict tuning range'''.

Given a rank-''r'' ''p''-limit regular temperament, we may define a tuning range by finding the [[Wikipedia: Convex hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest a set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], when this tuning is defined. This is the '''nice tuning range'''. We may define another tuning range by requiring that the ''p''-odd limit tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also [[Wikipedia: Monotonic function|monotone]] increasing (i.e. nondecreasing). This we may call the '''valid tuning range'''. A tuning which is both nice and valid is a ''strict'' tuning and this defines the '''strict tuning range'''.

While nice tunings are always guaranteed to occur, valid tunings are not. For instance, from the tuning map [{{val| 1 0 5 }}, {{val| 0 1 -2 }}] for the temperament tempering out 45/32 we find that all tunings are of the form {{val| 1 0 5 }} + ''a''{{val| 0 1 -2 }} = {{val| 1 ''a'' 5-2''a'' }}. Applying this to the list of steps between the notes of the 5-limit tonality diamond, [6/5, 25/24, 16/15, 9/8], we obtain [3''a'' - 4, 7 - 5''a'', ''a'' - 1, 2''a'' - 3] from which it follows that ''a'' ≥ 4/3, ''a'' ≤ 7/5, ''a'' ≥ 1, ''a'' ≥ 3/2, the solution set of which is empty. Hence there are no valid tunings of this temperament.

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 There are various methods which have been suggested for defining tuning ranges appropriate to a given [[regular temperament]].
-Given a rank-''r'' ''p''-limit regular temperament, we may define a tuning range by finding the [[Wikipedia: Convex hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest a set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], when this tuning is defined. This is the '''nice tuning range'''. We may define another tuning range by requiring that the ''p''-odd limit tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also [[Wikipedia: Monotonic function|monotone]] nondecreasing. This we may call the '''valid tuning range'''. A tuning which is both nice and valid is a ''strict'' tuning and this defines the '''strict tuning range'''.
+Given a rank-''r'' ''p''-limit regular temperament, we may define a tuning range by finding the [[Wikipedia: Convex hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest a set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], when this tuning is defined. This is the '''nice tuning range'''. We may define another tuning range by requiring that the ''p''-odd limit tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also [[Wikipedia: Monotonic function|monotone]] increasing (i.e. nondecreasing). This we may call the '''valid tuning range'''. A tuning which is both nice and valid is a ''strict'' tuning and this defines the '''strict tuning range'''.
 While nice tunings are always guaranteed to occur, valid tunings are not. For instance, from the tuning map [{{val| 1 0 5 }}, {{val| 0 1 -2 }}] for the temperament tempering out 45/32 we find that all tunings are of the form {{val| 1 0 5 }} + ''a''{{val| 0 1 -2 }} = {{val| 1 ''a'' 5-2''a'' }}. Applying this to the list of steps between the notes of the 5-limit tonality diamond, [6/5, 25/24, 16/15, 9/8], we obtain [3''a'' - 4, 7 - 5''a'', ''a'' - 1, 2''a'' - 3] from which it follows that ''a'' ≥ 4/3, ''a'' ≤ 7/5, ''a'' ≥ 1, ''a'' ≥ 3/2, the solution set of which is empty. Hence there are no valid tunings of this temperament.