Elf: Difference between revisions
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To construct an elf, the following steps are used: | To construct an elf, the following steps are used: | ||
# Take all intervals in the JI group of the temperament which lie within an octave. | # Take all intervals in the JI group of the temperament which lie within an octave. | ||
# For each interval of | # For each interval of the temperament, keep only the least complex (in terms of [[Benedetti height]]) JI interpretation of that interval. | ||
# Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity. | # Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity. | ||
# Construct a detempering of n-edo as follows: For each integer value 1 ≤ i ≤ n = V(2) = scale size, set the ith degree of the scale to be the least (according to the ordering in step 3) 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. | # Construct a detempering of n-edo as follows: For each integer value 1 ≤ i ≤ n = V(2) = scale size, set the ith degree of the scale to be the least (according to the ordering in step 3) 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. | ||
Revision as of 19:37, 28 March 2025
An elf is a scale in a regular temperament which is tempered from a just intonation (JI) scale in the group of the temperament which is a one-to-one detempering of n-edo 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, the following steps are used:
- Take all intervals in the JI group of the temperament which lie within an octave.
- For each interval of the temperament, keep only the least complex (in terms of Benedetti height) JI interpretation of that interval.
- Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity.
- Construct a detempering of n-edo as follows: For each integer value 1 ≤ i ≤ n = V(2) = scale size, set the ith degree of the scale to be the least (according to the ordering in step 3) 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.
- Temper this detempering of n-edo using a tuning map for the temperament.
The result is an elf.
Rank two examples
13-limit leapday
11-limit magic
11-limit miracle
13-limit myna
13-limit octacot
13-limit qilin
13-limit sensus
11-limit valentine
Rank three examples
11-limit jove
13-limit madagascar
- elfmadagascar7
- elfmadagascar8d
- elfmadagascar9
- elfmadagascar10
- elfmadagascar12f
- elfmadagascar14c
- elfmadagascar15
11-limit portent
11-limit thrush
11-limit zeus
Rank four examples
Keenanismic
- elfkeenanismic7
- elfkeenanismic8d
- elfkeenanismic9
- elfkeenanismic10
- elfkeenanismic11c
- elfkeenanismic12
- elfkeenanismic19