Hobbit: Difference between revisions

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== Definition ==

To define the hobbit scale we first define a particular [http://mathworld.wolfram.com/Seminorm.html seminorm] on interval space derived from a regular temperament, the [[Tenney-Euclidean metrics #Octave equivalent TE seminorm|octave equivalent Tenney-Euclidean seminorm]] or OETES. This seminorm applies to [[monzos and interval space|monzos]] and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity within the temperament of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OETES complexity for each scale step mapped to the integer ''i'' by the val ''V''.

To define the hobbit scale we first define a particular [http://mathworld.wolfram.com/Seminorm.html seminorm] on interval space derived from a regular temperament, the [[Tenney-Euclidean metrics #Octave-equivalent TE seminorm|octave equivalent Tenney-Euclidean seminorm]] or OETES. This seminorm applies to [[monzos and interval space|monzos]] and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity within the temperament of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OETES complexity for each scale step mapped to the integer ''i'' by the val ''V''.

Denoting the OETES for any element ''x'' of interval space by ''T''(''x''), we first define the hobbit of an odd-numbered scale; that is, a scale for which ''v''1 is an odd number. If ''v''1 is odd then for each integer ''j'', {{nowrap|0 < ''j'' ≤ ''v''1}}, we choose a corresponding monzo '''m''' such that {{nowrap|{{vmp|''V''|'''m'''}} {{=}} ''j''|0 < {{vmp| ''J''|'''m'''}} ≤ 1}} where ''J'' is the [[just tuning map]] {{val| log22 log23 … log2''p'' }}, and ''T''('''m''') is minimal.

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 == Definition ==
-To define the hobbit scale we first define a particular [http://mathworld.wolfram.com/Seminorm.html seminorm] on interval space derived from a regular temperament, the [[Tenney-Euclidean metrics #Octave equivalent TE seminorm|octave equivalent Tenney-Euclidean seminorm]] or OETES. This seminorm applies to [[monzos and interval space|monzos]] and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity within the temperament of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OETES complexity for each scale step mapped to the integer ''i'' by the val ''V''.
+To define the hobbit scale we first define a particular [http://mathworld.wolfram.com/Seminorm.html seminorm] on interval space derived from a regular temperament, the [[Tenney-Euclidean metrics #Octave-equivalent TE seminorm|octave equivalent Tenney-Euclidean seminorm]] or OETES. This seminorm applies to [[monzos and interval space|monzos]] and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity within the temperament of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OETES complexity for each scale step mapped to the integer ''i'' by the val ''V''.
 Denoting the OETES for any element ''x'' of interval space by ''T''(''x''), we first define the hobbit of an odd-numbered scale; that is, a scale for which ''v''<sub>1</sub> is an odd number. If ''v''<sub>1</sub> is odd then for each integer ''j'', {{nowrap|0 &lt; ''j'' &le; ''v''<sub>1</sub>}}, we choose a corresponding monzo '''m''' such that {{nowrap|{{vmp|''V''|'''m'''}} {{=}} ''j''|0 &lt; {{vmp|&#x200A;''J''|'''m'''}} &le; 1}} where ''J'' is the [[just tuning map]] {{val| log<sub>2</sub>2 log<sub>2</sub>3 … log<sub>2</sub>''p'' }}, and ''T''('''m''') is minimal.