Tenney–Euclidean temperament measures: Difference between revisions

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We can consider '''TE error''' to be a weighted average of the error of each prime harmonics in TE tuning. Multiplying it by 1200, we get a figure with values in cents.

By Graham Breed's definition, TE error may be accessed via [[Tenney–Euclidean tuning|TE tuning map]]. If T is the tuning map, then the TE error ''G'' can be found by

By Graham Breed's definition, TE error may be accessed via [[Tenney–Euclidean tuning|TE tuning map]]. If ''T'' is the tuning map, then the TE error ''G'' can be found by

<math>\displaystyle

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By Gene Ward Smith's definition, the TE error is derived from the relationship of TE simple badness and TE complexity. We denote this definition of TE error Ψ.

From the ratio {{nowrap|(‖''J'' ∧ ''M''‖/‖''M''‖)<sup>2</sup>}} we obtain {{nowrap|{{sfrac|({{subsup||''r'' + 1|''n''}})|''n''({{subsup||''r''|''n''}})}} {{=}} {{sfrac|''n'' − ''r''|''n''(''r'' + 1)}}}}. If we take the ratio of this for rank one with this for rank ''r'', the ''n'' cancels, and we get {{nowrap|{{sfrac|''n'' − 1|2}} · {{sfrac|''r'' + 1|''n'' − ''r''}} {{=}} {{sfrac|(''r'' + 1)(''n'' − 1)|2(''n'' − ''r'')}}}}. It follows that dividing TE error by the square root of this ratio gives a constant of proportionality such that if Ψ is the TE error of a rank-''r'' temperament then

From the ratio {{nowrap|(‖''J'' ∧ ''M''‖ / ‖''M''‖)<sup>2</sup>}} we obtain {{nowrap|{{sfrac|({{subsup||''r'' + 1|''n''}})|''n''({{subsup||''r''|''n''}})}} {{=}} {{sfrac|''n'' − ''r''|''n''(''r'' + 1)}}}}. If we take the ratio of this for rank one with this for rank ''r'', the ''n'' cancels, and we get {{nowrap|{{sfrac|''n'' − 1|2}} · {{sfrac|''r'' + 1|''n'' − ''r''}} {{=}} {{sfrac|(''r'' + 1)(''n'' − 1)|2(''n'' − ''r'')}}}}. It follows that dividing TE error by the square root of this ratio gives a constant of proportionality such that if Ψ is the TE error of a rank-''r'' temperament then

<math>\displaystyle

@@ Line 87: / Line 87: @@
 We can consider '''TE error''' to be a weighted average of the error of each prime harmonics in TE tuning. Multiplying it by 1200, we get a figure with values in cents.
-By Graham Breed's definition, TE error may be accessed via [[Tenney–Euclidean tuning|TE tuning map]]. If T is the tuning map, then the TE error ''G'' can be found by
+By Graham Breed's definition, TE error may be accessed via [[Tenney–Euclidean tuning|TE tuning map]]. If ''T'' is the tuning map, then the TE error ''G'' can be found by
 <math>\displaystyle
@@ Line 103: / Line 103: @@
 By Gene Ward Smith's definition, the TE error is derived from the relationship of TE simple badness and TE complexity. We denote this definition of TE error Ψ.
-From the ratio {{nowrap|(‖''J'' ∧ ''M''‖/‖''M''‖)<sup>2</sup>}} we obtain {{nowrap|{{sfrac|({{subsup||''r'' + 1|''n''}})|''n''({{subsup||''r''|''n''}})}} {{=}} {{sfrac|''n'' − ''r''|''n''(''r'' + 1)}}}}. If we take the ratio of this for rank one with this for rank ''r'', the ''n'' cancels, and we get {{nowrap|{{sfrac|''n'' − 1|2}} · {{sfrac|''r'' + 1|''n'' − ''r''}} {{=}} {{sfrac|(''r'' + 1)(''n'' − 1)|2(''n'' − ''r'')}}}}. It follows that dividing TE error by the square root of this ratio gives a constant of proportionality such that if Ψ is the TE error of a rank-''r'' temperament then
+From the ratio {{nowrap|(‖''J'' ∧ ''M''‖ / ‖''M''‖)<sup>2</sup>}} we obtain {{nowrap|{{sfrac|({{subsup||''r'' + 1|''n''}})|''n''({{subsup||''r''|''n''}})}} {{=}} {{sfrac|''n'' − ''r''|''n''(''r'' + 1)}}}}. If we take the ratio of this for rank one with this for rank ''r'', the ''n'' cancels, and we get {{nowrap|{{sfrac|''n'' − 1|2}} · {{sfrac|''r'' + 1|''n'' − ''r''}} {{=}} {{sfrac|(''r'' + 1)(''n'' − 1)|2(''n'' − ''r'')}}}}. It follows that dividing TE error by the square root of this ratio gives a constant of proportionality such that if Ψ is the TE error of a rank-''r'' temperament then
 <math>\displaystyle