Tuning ranges of regular temperaments: Difference between revisions

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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.

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 we get inequalities ''a'' ≥ 4/3, ''a'' ≤ 7/5, ''a'' ≥ 1, ''a'' ≥ 3/2; these inequalities have no solution. Hence there are no valid tunings of this temperament.

For a more typical example, consider marvel temperament. Using the Hermite normal form tuning map again, we find that all marvel tunings are of the form {{val| 1 ''a'' ''b'' 2''a''+''ab''-5 12-''a''-3''b'' }}. Applying this to the steps of the 11-limit tonality diamond, we obtain eight inequalities, the solution set of which is the union of {30/19 ≤ ''a'' ≤ 49/31, 2 + ''a''/5 ≤ ''b'' ≤ 4''a'' - 4} with {49/31 ≤ ''a'' ≤ 35/22, 2 + ''a''/5 ≤ ''b'' ≤ 3 - 3''a''/7}, which is the triangular region bounded by the tunings for 19, 22, and 31. This is the valid range. The nice tuning range is a quadrilateral, with vertices, given in terms of frequency ratios rather than log base 2 or cents, [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)], [2, 3, 5, 225/32, 4096/375]]. The three vertices with all rational number values for the approximate 3 and 5 are not in the valid range, so that only the [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)] tuning is valid and hence strict. Other examples of strict tunings are 41''p''/41, 53''p''/53, 72''p''/72 etc.; however 19''p''/19, 22''p''/22 and 31''p''/31 are not in the nice range.

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 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.
+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 we get inequalities ''a'' ≥ 4/3, ''a'' ≤ 7/5, ''a'' ≥ 1, ''a'' ≥ 3/2; these inequalities have no solution. Hence there are no valid tunings of this temperament.
 For a more typical example, consider marvel temperament. Using the Hermite normal form tuning map again, we find that all marvel tunings are of the form {{val| 1 ''a'' ''b'' 2''a''+''ab''-5 12-''a''-3''b'' }}. Applying this to the steps of the 11-limit tonality diamond, we obtain eight inequalities, the solution set of which is the union of {30/19 ≤ ''a'' ≤ 49/31, 2 + ''a''/5 ≤ ''b'' ≤ 4''a'' - 4} with {49/31 ≤ ''a'' ≤ 35/22, 2 + ''a''/5 ≤ ''b'' ≤ 3 - 3''a''/7}, which is the triangular region bounded by the tunings for 19, 22, and 31. This is the valid range. The nice tuning range is a quadrilateral, with vertices, given in terms of frequency ratios rather than log base 2 or cents, [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)], [2, 3, 5, 225/32, 4096/375]]. The three vertices with all rational number values for the approximate 3 and 5 are not in the valid range, so that only the  [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)] tuning is valid and hence strict. Other examples of strict tunings are 41''p''/41, 53''p''/53, 72''p''/72 etc.; however 19''p''/19, 22''p''/22 and 31''p''/31 are not in the nice range.