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 [[ | 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 [http://en.wikipedia.org/wiki/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-limit [[Tonality_diamond|tonality diamond]], when this tuning is defined. This is the ''nice'' tuning range. We may define another tuning range by requiring that the tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also [http://en.wikipedia.org/wiki/Monotonic_function monotone] weakly increasing. 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 [http://en.wikipedia.org/wiki/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-limit [[Tonality_diamond|tonality diamond]], when this tuning is defined. This is the ''nice'' tuning range. We may define another tuning range by requiring that the tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also [http://en.wikipedia.org/wiki/Monotonic_function monotone] weakly increasing. 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. | ||
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While nice tunings are always guaranteed to occur, valid tunings are not. For instance, from the tuning map [<1 0 5|, <0 1 -2|] for the temperament tempering out 45/32 we find that all tunings are of the form <1 0 5| + a<0 1 -2| = <1 a 5-2a|. 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 [3a-4, 7-5a, a-1, 2a-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 [<1 0 5|, <0 1 -2|] for the temperament tempering out 45/32 we find that all tunings are of the form <1 0 5| + a<0 1 -2| = <1 a 5-2a|. 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 [3a-4, 7-5a, a-1, 2a-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. | ||
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 <1 a b 2a+ab-5 12-a-3b|. 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 ≤ 4a-4} with {49/31 ≤ a ≤ 35/22, 2+a/5 ≤ b ≤ 3-3a/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 quadralateral, 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 verticies 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 41p/41, 53p/53, 72p/72 etc.; however 19p/19, 22p/22 and 31p/31 are not in the nice range. | 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 <1 a b 2a+ab-5 12-a-3b|. 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 ≤ 4a-4} with {49/31 ≤ a ≤ 35/22, 2+a/5 ≤ b ≤ 3-3a/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 quadralateral, 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 verticies 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 41p/41, 53p/53, 72p/72 etc.; however 19p/19, 22p/22 and 31p/31 are not in the nice range. | ||
Milne, Sethares and Plamodon define valid tunings in '''''Tuning Continua and Keyboard Layouts''''' in the premiere issue of ''Journal of Mathematics and Music''; they discuss nice tunings in '''''X_System''''' in the Open University’s repository. | |||
[[Category:math]] | [[Category:math]] | ||
[[Category: | [[Category:temperament]] | ||
[[Category:theory]] | [[Category:theory]] | ||
[[Category:todo:simplify]] | [[Category:todo:simplify]] | ||
[[Category:tuning]] | [[Category:tuning]] | ||