Hobbit: Difference between revisions

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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|{{~~vmprod~~|''V''|'''m'''}} {{=}} ''j''|0 < {{~~vmprod~~|''J''|'''m'''}} ≤ 1}} where ''J'' is the [[just tuning map]] {{val| log22 log23 … log2''p'' }}, and ''T''('''m''') is minimal.

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.

If ''v''1 is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least [[Benedetti height]]. Another approach adopted here is to choose a monzo '''u''' such that ''T''('''u''') is minimal under the condition that {{nowrap|''T''('''u''') > 0}}; in other words, '''u''' is a shortest positive length interval. Then for each integer ''j'', where {{nowrap|0 < ''j'' ≤ ''v''1}}, we choose a corresponding monzo '''m''' such that {{nowrap|{{~~vmprod~~|''V''|'''m'''}} {{=}} ''j''|0 < {{~~vmprod~~|''J''|'''m'''}} ≤ 1}}, and where {{nowrap|''T''(2'''m''' − '''u''')}} is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is fully equal as a scale to the canonical hobbit.

If ''v''1 is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least [[Benedetti height]]. Another approach adopted here is to choose a monzo '''u''' such that ''T''('''u''') is minimal under the condition that {{nowrap|''T''('''u''') > 0}}; in other words, '''u''' is a shortest positive length interval. Then for each integer ''j'', where {{nowrap|0 < ''j'' ≤ ''v''1}}, we choose a corresponding monzo '''m''' such that {{nowrap|{{vmp|''V''|'''m'''}} {{=}} ''j''|0 < {{vmp| ''J''|'''m'''}} ≤ 1}}, and where {{nowrap|''T''(2'''m''' − '''u''')}} is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is fully equal as a scale to the canonical hobbit.

The intervals selected by this process are a [[transversal]] of the scale, and we may now apply the chosen tuning to the monzos in the transversal, obtaining values (in cents or fractional monzos) defining a scale. The monzos in the transversal are defined only modulo the commas of the temperament, but since these are tempered out this does not affect the definition of the scale.

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 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|{{vmprod|''V''|'''m'''}} {{=}} ''j''|0 &lt; {{vmprod|''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.
+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.
-If ''v''<sub>1</sub> is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least [[Benedetti height]]. Another approach adopted here is to choose a monzo '''u''' such that ''T''('''u''') is minimal under the condition that {{nowrap|''T''('''u''') &gt; 0}}; in other words, '''u''' is a shortest positive length interval. Then for each integer ''j'', where {{nowrap|0 &lt; ''j'' &le; ''v''<sub>1</sub>}}, we choose a corresponding monzo '''m''' such that {{nowrap|{{vmprod|''V''|'''m'''}} {{=}} ''j''|0 &lt; {{vmprod|''J''|'''m'''}} &le; 1}}, and where {{nowrap|''T''(2'''m''' &minus; '''u''')}} is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is fully equal as a scale to the canonical hobbit.
+If ''v''<sub>1</sub> is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least [[Benedetti height]]. Another approach adopted here is to choose a monzo '''u''' such that ''T''('''u''') is minimal under the condition that {{nowrap|''T''('''u''') &gt; 0}}; in other words, '''u''' is a shortest positive length interval. Then for each integer ''j'', where {{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}}, and where {{nowrap|''T''(2'''m''' &minus; '''u''')}} is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is fully equal as a scale to the canonical hobbit.
 The intervals selected by this process are a [[transversal]] of the scale, and we may now apply the chosen tuning to the monzos in the transversal, obtaining values (in cents or fractional monzos) defining a scale. The monzos in the transversal are defined only modulo the commas of the temperament, but since these are tempered out this does not affect the definition of the scale.