Functional systems
The n-function systems (alternately functional systems[idiosyncratic term]) are defined by osmium as a systematic way to try to heuristically find a best guess to the answer of approximately most universal enumeration of relative interval qualities theoretically perceivable by a sufficiently trained xenmelodic listener.
The 50-function system
The derivation is not currently documented on the xen wiki, but the basic premise of n-function systems is defining interval classes that are ambiguous between more basic interval classes, where what is considered basic is defined derivative of superimposing all perfect Ringer scales to form a (or multiple) category system(s), chosen for their exceptional combination of properties as JI scales and relevance to perception of harmony. The main result — the 50-function system — primarily forms from 24-grid (and to a lesser extent 36-grid) defined indirectly via superimposing all the categorical alignments and misalignments of the 10-form and 14-form, themselves derived from reconciling the 5-form and 7-form (as having the most predictive power) with the more unusual 4-form, where the connection to Ringer scales is very explicit but "form" is used to describe a distinction from corresponding edos, as the optimal perceptual xenmelodic placement will be at least slightly off from perfect in some specific cases.
A starting point: The 12-form
Even outside of 12edo, the 12-form is a significant organising principle influencing the human perception of intervals.
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Todo: expand
Add detailed derivation of 50-function system |
| Function | Type | Interval region | Rough center |
|---|---|---|---|
| 0.0 | integer | unison | 0c |
| sp | - | comma | 10c |
| 0.3 | - | diesis | 33c |
| 0.5 | ambitonal | quarter-tone | 50c |
| 0.7 | semiambitonal | subminor second | 67c |
| 1 | integer | semitone | 100c |
| 1.3 | semiambitonal | supraminor second | 133c |
| 1.5 | ambitonal | neutral second | 150c |
| 1.7 | semiambitonal | submajor second | 167c |
| 2 | integer | major second | 200c |
| 2.3 | semiambitonal | supramajor second | 233c |
| 2.5 | ambitonal | second-third | 250c |
| 2.7 | semiambitonal | subminor third | 267c |
| 3 | integer | minor third | 300c |
| 3.3 | semiambitonal | supraminor third | 333c |
| 3.5 | ambitonal | neutral third | 350c |
| 3.7 | semiambitonal | submajor third | 367c |
| 4 | integer | major third | 400c |
| 4.3 | semiambitonal | supermajor third | 435c |
| 4.5 | ambitonal | third-fourth | 460c |
| 4.7 | semiambitonal | subfourth | 475c |
| 5 | integer | perfect fourth | 500c |
| 5.3 | semiambitonal | superfourth | 533c |
| 5.5 | ambitonal | neutral fourth | 550c |
| 5.7 | semiambitonal | subtritone | 567c |
| 6 | integer | tritone | 600c |