Hobbit: Difference between revisions

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A '''hobbit''', or '''hobbit scale''', is a generalization of [[MOS scale]] for arbitrary regular temperaments which is a sort of cousin to [[~~Dwarves~~|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 [[Regular Temperaments #Rank 2 (including "linear") temperaments|rank two regular 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 three case, accessible for musical purposes.

A '''hobbit''', or '''hobbit scale''', is a generalization of [[MOS scale]] for arbitrary regular temperaments 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 [[Regular Temperaments #Rank 2 (including "linear") temperaments|rank two regular 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 three case, accessible for musical purposes.

Given a regular temperament and an equal temperament val v which supports (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.

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After applying such a tuning, we discover than there seems to be a certain irregularity or inconsistency in action, in that some of the 11-limit intervals do not stem from the mapping for minerva, but represent additional temperings by 243/242 or 4000/3993. By adding one of these, we can flatten out the irregularity to a corresponding rank two temperament; by adding both, we obtain the rank one temperament with val {{val| 65 103 151 183 225 }}, giving a scale with steps 2433333242432424233333. Examples of this sort inconsistency seem to increase with increasing rank.

[[Category:~~Hobbits~~]]

[[Category:Hobbit]]

[[Category:Math]]

[[Category:Scale theory]]

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-A '''hobbit''', or '''hobbit scale''', is a generalization of [[MOS scale]] for arbitrary regular temperaments which is a sort of cousin to [[Dwarves|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 [[Regular Temperaments #Rank 2 (including "linear") temperaments|rank two regular 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 three case, accessible for musical purposes.
+A '''hobbit''', or '''hobbit scale''', is a generalization of [[MOS scale]] for arbitrary regular temperaments 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 [[Regular Temperaments #Rank 2 (including "linear") temperaments|rank two regular 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 three case, accessible for musical purposes.
 Given a regular temperament and an equal temperament val v which supports (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.
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 After applying such a tuning, we discover than there seems to be a certain irregularity or inconsistency in action, in that some of the 11-limit intervals do not stem from the mapping for minerva, but represent additional temperings by 243/242 or 4000/3993. By adding one of these, we can flatten out the irregularity to a corresponding rank two temperament; by adding both, we obtain the rank one temperament with val {{val| 65 103 151 183 225 }}, giving a scale with steps 2433333242432424233333. Examples of this sort inconsistency seem to increase with increasing rank.
-[[Category:Hobbits]]
+[[Category:Hobbit]]
 [[Category:Math]]
 [[Category:Scale theory]]