Constrained tuning: Difference between revisions

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== Special constraint ==

The special eigenmonzo ''X'''''j''', where '''j''' is the all-ones monzo, has the effect of removing the weighted–skewed tuning bias. This eigenmonzo is actually proportional to the monzo of the extra dimension introduced by the skew. In other words, it forces the extra dimension to be pure, and therefore, the skew will have no effect with this constrained tuning.

The special eigenmonzo ''X''⋅'''j''', where '''j''' is the all-ones monzo, has the effect of removing the weighted–skewed tuning bias. This eigenmonzo is actually proportional to the monzo of the extra dimension introduced by the skew. In other words, it forces the extra dimension to be pure, and therefore, the skew will have no effect with this constrained tuning.

It can be regarded as a distinct optimum. In the case of Tenney weighting, it is the '''TOCTE tuning''' ('''Tenney ones constrained Tenney–Euclidean tuning''').

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$$

As a result, the [[~~Relative~~ interval error #Linearity|relative error space]] is also linear with respect to ''V''.

As a result, the [[relative interval error #Linearity|relative error space]] is also linear with respect to ''V''.

For example, the relative errors of ~~12ettoc5 (~~12et in 5-limit TOC) is

For example, the relative errors of 12et in 5-limit TOC is

$$ \mathcal{E}_\text {r}(12) = \val{-1.55\% & -4.42\% & +10.08\% } $$

That of ~~19ettoc5~~ is

That of 19et in this tuning is

$$ \mathcal{E}_\text {r}(19) = \val{+4.08\% & -4.97\% & -2.19\% } $$

As 31 = 12 + 19, the relative errors of ~~31ettoc5~~ is

As 31 = 12 + 19, the relative errors of 31et in this tuning is

$$

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== Systematic name ==

In [[D&D's guide|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]]''.

In [[D&D's guide|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]]''.

== Open problems ==

@@ Line 418: / Line 418: @@
 == Special constraint ==
-The special eigenmonzo ''X'''''j''', where '''j''' is the all-ones monzo, has the effect of removing the weighted–skewed tuning bias. This eigenmonzo is actually proportional to the monzo of the extra dimension introduced by the skew. In other words, it forces the extra dimension to be pure, and therefore, the skew will have no effect with this constrained tuning.
+The special eigenmonzo ''X''⋅'''j''', where '''j''' is the all-ones monzo, has the effect of removing the weighted–skewed tuning bias. This eigenmonzo is actually proportional to the monzo of the extra dimension introduced by the skew. In other words, it forces the extra dimension to be pure, and therefore, the skew will have no effect with this constrained tuning.
 It can be regarded as a distinct optimum. In the case of Tenney weighting, it is the '''TOCTE tuning''' ('''Tenney ones constrained Tenney–Euclidean tuning''').
@@ Line 444: / Line 444: @@
 $$
-As a result, the [[Relative interval error #Linearity|relative error space]] is also linear with respect to ''V''.
+As a result, the [[relative interval error #Linearity|relative error space]] is also linear with respect to ''V''.
-For example, the relative errors of 12ettoc5 (12et in 5-limit TOC) is
+For example, the relative errors of 12et in 5-limit TOC is
 $$ \mathcal{E}_\text {r}(12) = \val{-1.55\% & -4.42\% & +10.08\% } $$
-That of 19ettoc5 is
+That of 19et in this tuning is
 $$ \mathcal{E}_\text {r}(19) = \val{+4.08\% & -4.97\% & -2.19\% } $$
-As 31 = 12 + 19, the relative errors of 31ettoc5 is
+As 31 = 12 + 19, the relative errors of 31et in this tuning is
 $$
@@ Line 464: / Line 464: @@
 == Systematic name ==
-In [[D&D's guide|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]]''.
+In [[D&D's guide|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]]''.
 == Open problems ==