Constrained tuning: Difference between revisions
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'''Constrained tunings''' are tuning [[optimization]] techniques using the constraint of some purely tuned intervals (i.e. [[eigenmonzo]]s). '''CTE tuning''' ('''constrained Tenney-Euclidean tuning''') is the most typical instance. It has a more sophisticated variant, '''CTWE tuning''' ('''constrained Tenney-Weil | '''Constrained tunings''' are tuning [[optimization]] techniques using the constraint of some purely tuned intervals (i.e. [[eigenmonzo]]s). '''CTE tuning''' ('''constrained Tenney-Euclidean tuning''') is the most typical instance. It has a more sophisticated variant, '''CTWE tuning''' ('''constrained Tenney-Weil-Euclidean tuning'''). These two tunings will be the focus of this article. Otherwise normed tunings can be defined and computed analogously. | ||
All constrained tunings are standard temperament optimization problems. Specifically, as [[TE tuning]] can be viewed as a [[Wikipedia: Least squares|least squares problem]], CTE tuning can be viewed as an equality-constrained least squares problem. | All constrained tunings are standard temperament optimization problems. Specifically, as [[TE tuning]] can be viewed as a [[Wikipedia: Least squares|least squares problem]], CTE tuning can be viewed as an equality-constrained least squares problem. | ||
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== CTE tuning vs CTWE tuning == | == CTE tuning vs CTWE tuning == | ||
Consider the fact that TE tuning does not treat divisive ratios as more important than multiplicative ratios – 5/3 and 15/1 are taken as equally important, for example. To address that, a skew on the space may be introduced, resulting in [[TWE tuning]]. Constraining the equave to pure on top of TWE gives CTWE | Consider the fact that TE tuning does not treat divisive ratios as more important than multiplicative ratios – 5/3 and 15/1 are taken as equally important, for example. To address that, a skew on the space may be introduced, resulting in [[TWE tuning]]. Constraining the equave to pure on top of TWE gives CTWE tuning. | ||
[[POTE tuning]] works as a quick approximation to CTWE. As POTE destretches the equave, it keeps the angle in the tuning space unchanged, and thus sacrifices multiplicative ratios for divisive ratios. On the contrary, CTE sticks to the original design book of TE as its result remains TE optimal. | [[POTE tuning]] works as a quick approximation to CTWE. As POTE destretches the equave, it keeps the angle in the tuning space unchanged, and thus sacrifices multiplicative ratios for divisive ratios. On the contrary, CTE sticks to the original design book of TE as its result remains TE optimal. | ||
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== Systematic name == | == Systematic name == | ||
In D&D's guide to RTT, the [[Dave Keenan & Douglas Blumeyer's guide to RTT: alternative complexities#Naming|systematic name]] for the CTE tuning scheme is | In D&D's guide to RTT, the [[Dave Keenan & Douglas Blumeyer's guide to RTT: alternative complexities #Naming|systematic name]] for the CTE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: all-interval tuning schemes #Held-octave minimax-.28E.29S|held-octave minimax-ES]]'', and the systematic name for the CTWE tuning scheme is ''[[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: tuning fundamentals #Held-intervals|held-octave]] [[Dave_Keenan %26 Douglas Blumeyer%27s guide to RTT: alternative complexities #Tunings used in 7|minimax-E-lils-S]]''. | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:Tuning]] | [[Category:Tuning]] | ||
[[Category:Tuning technique]] | [[Category:Tuning technique]] | ||