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.

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