Elves

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Definition of elves

An elf is a scale in a regular temperament which is tempered from a JI scale in the group of the temperament which is epimorphic via a val V which may not be, and characteristically is not, a val supporting the temperament. This allows the elf to have more freedom in scale size and structure but still to possess the coherence induced by the epimorphic mapping.

To construct an elf, take the intervals in the JI group of the temperament which lie within an octave and keep only the least complex (in terms of Benedetti height) representative for each corresponding interval of the temperament. Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity. For each integer value 1 ≤ i ≤ V(2), set the ith element of a transversal for the scale to be the first interval c in the listing such that V(c) = i; which is to say, the interval of least temperamental complexity with ties broken by Benedetti height. The tempering of this transversal by a tuning map for the temperament is the elf.

Rank two examples

13-limit leapday

elfleapday7

elfleapday8d

elfleapday9

elfleapday10

elfleapday12f

11-limit magic

elfmagic7

elfmagic8

elfmagic8d

elfmagic9

elfmagic10

elfmagic12

11-limit miracle

elfmiracle7

elfmiracle8d

elfmiracle9

elfmiracle10

elfmiracle12

13-limit myna

elfmyna7

elfmyna8d

elfmyna9

elfmyna10

elfmyna12f

13-limit octacot

elfoctacot7

elfoctacot8d

elfoctacot9

elfoctacot10

elfoctacot12f

13-limit qilin

elfqilin7

elfqilin8d

elfqilin9

elfqilin10

elfqilin12f

13-limit sensus

elfsensus7

elfsensus8d

elfsensus9

elfsensus10

elfsensus12

elfsensus12f

11-limit valentine

elfvalentine7

elfvalentine8d

elfvalentine9

elfvalentine10

elfvalentine12

Rank three examples

11-limit jove

elfjove7

elfjove8d

elfjove9

elfjove10

elfjove11c

elfjove12

13-limit madagascar

elfmadagascar7

elfmadagascar8d

elfmadagascar9

elfmadagascar10

elfmadagascar12f

elfmadagascar14c

elfmadagascar15

11-limit portent

elfportent9

elfportent10

elfportent11c

elfportent12

elfportent15

11-limit thrush

elfthrush7

elfthrush8d

elfthrush9

elfthrush10

elfthrush12

11-limit zeus

zeus7tri

elfzeus8

elfzeus9

elfzeus10

elfzeus12

Rank four examples

Keenanismic

elfkeenanismic7

elfkeenanismic8d

elfkeenanismic9

elfkeenanismic10

elfkeenanismic11c

elfkeenanismic12

elfkeenanismic19

Swetismic

elfswetismic8d

elfswetismic9

elfswetismic10

elfswetismic12

Valinorsmic

elfvalinorsmic7

elfvalinorsmic8d

elfvalinorsmic9

elfvalinorsmic10

elfvalinorsmic11c

elfvalinorsmic12