Hobbit: Difference between revisions

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A '''hobbit''', or '''hobbit scale''', is a generalization of [[mos scale]] for arbitrary [[regular temperament]]s which is a sort of cousin to [[dwarf|dwarf scales]]; examples may be found on the [[Scalesmith]] page. The idea is that mos scales give us a means of contructing scales for a [[rank-2 temperament]] which gives priority to the intervals of least complexity in that temperament, and so makes efficient use of it; a hobbit does the same in higher ranks, and so using them is one way to make higher ranks, including especially the interesting rank-3 case, accessible for musical purposes.

Given a regular temperament and an [[equal temperament]] ~~[[val]] ''V''~~ which [[support]]s ~~(or belongs to)~~ the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing ~~''v''1~~ notes to the ~~octave~~.

Given a regular temperament and an [[equal temperament]] which [[support]]s the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing the same number of notes as the equal temperament.

== 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 create a hobbit scale, we need to find the interval of least complexity in the regular temperament corresponding to each step of the equal temperament. The measure of complexity to be used is the "[[Tenney-Euclidean metrics #Octave-equivalent TE seminorm|octave equivalent Tenney-Euclidean seminorm]]", or OETES, here denoted T(x) where x is an interval. The OETES complexity of any comma of the temperament, and also of the octave, is 0, encoding [[Equivalence vs. tempering|the tempering of the commas and octave-equivalence]]. Note that this means that any given pitch class relative to the unison has a corresponding OETES complexity shared between all of its representative intervals, and additionally, that T(x) and T(2/x) are the same, where x and 2/x are octave complements.

~~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.~~

For each step within the octave in the equal temperament, we choose the simplest interval (by OETES complexity) in the temperament within the octave that maps to the scale step. For an edo with an odd number of notes, this process is unambiguous. However, when an edo is even and thus contains the perfect [[semioctave]], there is an ambiguity, and there are multiple options for the hobbit, differing by the central interval (this is similar to how 12-note Pythagorean tuning has no perfectly symmetrical mode; either the narrow or sharp tritone must be chosen).

~~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.

An alternative and equivalent approach is to work directly with the notes of the temperament, using the [[Tenney-Euclidean metrics #Temperamental complexity|temperamental norm]] defined on the note classes of the temperament modulo period (an octave or fraction of an octave) of the temperament.

== Example ==

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 A '''hobbit''', or '''hobbit scale''', is a generalization of [[mos scale]] for arbitrary [[regular temperament]]s which is a sort of cousin to [[dwarf|dwarf scales]]; examples may be found on the [[Scalesmith]] page. The idea is that mos scales give us a means of contructing scales for a [[rank-2 temperament]] which gives priority to the intervals of least complexity in that temperament, and so makes efficient use of it; a hobbit does the same in higher ranks, and so using them is one way to make higher ranks, including especially the interesting rank-3 case, accessible for musical purposes.
-Given a regular temperament and an [[equal temperament]] [[val]] ''V'' which [[support]]s (or belongs to) the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing ''v''<sub>1</sub> notes to the octave.
+Given a regular temperament and an [[equal temperament]] which [[support]]s the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing the same number of notes as the equal temperament.
 == 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 create a hobbit scale, we need to find the interval of least complexity in the regular temperament corresponding to each step of the equal temperament. The measure of complexity to be used is the "[[Tenney-Euclidean metrics #Octave-equivalent TE seminorm|octave equivalent Tenney-Euclidean seminorm]]", or OETES, here denoted T(x) where x is an interval. The OETES complexity of any comma of the temperament, and also of the octave, is 0, encoding [[Equivalence vs. tempering|the tempering of the commas and octave-equivalence]]. Note that this means that any given pitch class relative to the unison has a corresponding OETES complexity shared between all of its representative intervals, and additionally, that T(x) and T(2/x) are the same, where x and 2/x are octave complements.
-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.
+For each step within the octave in the equal temperament, we choose the simplest interval (by OETES complexity) in the temperament within the octave that maps to the scale step. For an edo with an odd number of notes, this process is unambiguous. However, when an edo is even and thus contains the perfect [[semioctave]], there is an ambiguity, and there are multiple options for the hobbit, differing by the central interval (this is similar to how 12-note Pythagorean tuning has no perfectly symmetrical mode; either the narrow or sharp tritone must be chosen).
-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.
-An alternative and equivalent approach is to work directly with the notes of the temperament, using the [[Tenney-Euclidean metrics #Temperamental complexity|temperamental norm]] defined on the note classes of the temperament modulo period (an octave or fraction of an octave) of the temperament.
 == Example ==