Tuning ranges of regular temperaments
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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 [[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]], 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.
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.
In **Tuning Continua and Keyboard Layouts** in the premiere issue of //Journal of Mathematics and Music//, Milne, Sethares and Plamodon define valid tunings; they discuss nice tunings in **X_System** in the Open University’s repository.Original HTML content:
<html><head><title>Tuning Ranges of Regular Temperaments</title></head><body>There are various methods which have been suggested for defining tuning ranges appropriate to a given <a class="wiki_link" href="/regular%20temperament">regular temperament</a>.<br />
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Given a rank r p-limit regular temperament, we may define a tuning range by finding the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_hull" rel="nofollow">convex hull</a> in <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">tuning space</a> of the tunings with one <a class="wiki_link" href="/Eigenmonzo%20subgroup">eigenmonzo</a> 2 (pure octaves tunings) and the rest a set of r-1 members of the p-limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a>, when this tuning is defined. This is the <em>nice</em> 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monotonic_function" rel="nofollow">monotone</a> weakly increasing. This we may call the <em>valid</em> tuning range. A tuning which is both nice and valid is a <em>strict</em> tuning and this defines the strict tuning range. <br />
<br />
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.<br />
<br />
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.<br />
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In <strong>Tuning Continua and Keyboard Layouts</strong> in the premiere issue of <em>Journal of Mathematics and Music</em>, Milne, Sethares and Plamodon define valid tunings; they discuss nice tunings in <strong>X_System</strong> in the Open University’s repository.</body></html>