Hobbit: Difference between revisions

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For an example, consider the 22 note hobbit for minerva temperament, the 11-limit temperament tempering out 99/98 and 176/175. Here the val is {{val| 22 35 51 62 76 }}, and an interval of minimal nonzero size for the temperament is 16/15, with monzo {{monzo| 4 -1 -1 0 0 }}. From this we may find a transversal minimizing T (2m - {{monzo| 4 -1 -1 0 0 }}) for each scale step, namely 36/35, 15/14, 11/10, 8/7, 7/6, 40/33, 5/4, 9/7, 4/3, 48/35, 10/7, 22/15, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 64/35, 15/8, 64/33, 2/1. A tuning can be defined in various ways, for instance by approximating the above in [[53edo]], or by using the minimax tuning, which has eigenmonzos 2, 3, and 11.

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.

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 of inconsistency seem to increase with increasing rank.

[[Category:Hobbit| ]]

[[Category:Math]]

[[Category:Scale theory]]

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 For an example, consider the 22 note hobbit for minerva temperament, the 11-limit temperament tempering out 99/98 and 176/175. Here the val is {{val| 22 35 51 62 76 }}, and an interval of minimal nonzero size for the temperament is 16/15, with monzo {{monzo| 4 -1 -1 0 0 }}. From this we may find a transversal minimizing T (2m - {{monzo| 4 -1 -1 0 0 }}) for each scale step, namely 36/35, 15/14, 11/10, 8/7, 7/6, 40/33, 5/4, 9/7, 4/3, 48/35, 10/7, 22/15, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 64/35, 15/8, 64/33, 2/1. A tuning can be defined in various ways, for instance by approximating the above in [[53edo]], or by using the minimax tuning, which has eigenmonzos 2, 3, and 11.
-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.
+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 of inconsistency seem to increase with increasing rank.
 [[Category:Hobbit| ]] <!-- main page -->
 [[Category:Math]]
 [[Category:Scale theory]]