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→My approach to diatonic descendant mosses: Chroma/diesis rules generalized to any mos; 3rd descendants are wip
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For a mos xL ys, its two child mosses can be described using replacement rulesets: L->Ls and s->s, producing xL (x+y)s, and L->sL and s->s, producing (x+y)L xs. These two rulesets can be applied to the two child mosses again to produce further descendants. However, the resulting descendant mosses are denoted using L's and s's, whose relationship to the parent mos's step sizes may be unclear. This section describes how descendant mosses' step sizes relate back to the parent.
For a mos xL ys, its two child mosses can be described using replacement rulesets: L->Ls and s->s, producing xL (x+y)s, and L->sL and s->s, producing (x+y)L xs. These two rulesets can be applied to the two child mosses again to produce further descendants. However, the resulting descendant mosses are denoted using L's and s's, whose relationship to the parent mos's step sizes may be unclear. This section describes how descendant mosses' step sizes relate back to the parent.


=== 1st descendants ===
=== Replacement rules of descendants ===
Since replacement rules apply to any mos, it may be easier to consider the replacement rules themselves rather than use a specific mos.
Since replacement rules apply to any mos, it's easier to consider applying these rules to the mos 1L 1s, a single large and small step.


* L->Ls, s->s (ruleset 1); apply to a mos in its brightest mode to produce a child mos in its brightest mode
* L->Ls, s->s (ruleset 1); apply to a mos in its brightest mode to produce a child mos in its brightest mode
* L->sL, s->L (ruleset 2); apply to a mos in its brightest mode to produce a child mos in its darkest mode
* L->sL, s->L (ruleset 2); apply to a mos in its brightest mode to produce a child mos in its darkest mode
Applying these rules to the mos pattern produces its children, and applying them repeatedly produces further descendants; shown below is three applications for three generations of descendants.
{| class="wikitable"
|+
!Mos
!Step pattern
!Mos
!Step pattern
!Mos
!Step pattern
!Mos
!Step pattern
|-
| rowspan="8" |1L 1s
| rowspan="8" |L s
| rowspan="4" |1L 2s
| rowspan="4" |Ls s
| rowspan="2" |1L 3s
| rowspan="2" |Lss s
|1L 4s
|Lss s
|-
|4L 1s
|sLL L
|-
| rowspan="2" |3L 1s
| rowspan="2" |sLL L
|3L 4s
|ssLsL sL
|-
|4L 3s
|LLsLs Ls
|-
| rowspan="4" |2L 1s
| rowspan="4" |sL L
| rowspan="2" |2L 3s
| rowspan="2" |ssL sL
|2L 5s
|sssL ssL
|-
|5L 2s
|LLLs LLs
|-
| rowspan="2" |3L 2s
| rowspan="2" |LLs Ls
|3L 5s
|LsLss Lss
|-
|5L 3s
|sLsLL sLL
|}
Adding a separator shows what happens to a single L and s, effectively creates production rules for producing all 14 possible mos descendants.
{| class="wikitable"
! colspan="2" |Parent mos
! colspan="2" |1st descendants
! colspan="2" |2nd descendants
! colspan="2" |3rd descendants
|-
!Mos
!Production rules
!Mos
!Production rules
!Mos
!Production rules
!Mos
!Production rules
|-
| rowspan="8" |xL ys
| rowspan="8" |none
| rowspan="4" |xL (x+y)s
| rowspan="4" |L->Ls
s->s
| rowspan="2" |xL (2x+y)s
| rowspan="2" |L->Lss
s->s
|xL (3x+y)s
|L->Lss
s->s
|-
|(3x+y)L xs
|L->sLL
s->L
|-
| rowspan="2" |(2x+y)L xs
| rowspan="2" |L->sLL
s->L
|(2x+y)L (3x+y)s
|L->ssLsL
s->sL
|-
|(3x+y)L (2x+y)s
|L->LLsLs
s->Ls
|-
| rowspan="4" |(x+y)L xs
| rowspan="4" |L->sL
s->L
| rowspan="2" |(x+y)L (2x+y)s
| rowspan="2" |L->ssL
s->sL
|(x+y)L (3x+2y)s
|L->sssL
s->ssL
|-
|(3x+2y)L (x+y)s
|L->LLLs
s->LLs
|-
| rowspan="2" |(2x+y)L (x+y)s
| rowspan="2" |L->LLs
s->Ls
|(2x+y)L (3x+2y)s
|L->LsLss
s->Lss
|-
|(3x+2y)L (2x+y)s
|L->sLsLL
s->sLL
|}


This process can be visualized using a rectangular horogram. Since a chroma is defined as the difference between a large step and small step, or c = L - s, the replacement rules and horogram suggest the large step becomes a chroma and small step. However, whether the small step is larger than the chroma or vice-versa depends on the step ratio of L and s, so that distinction becomes unimportant, resulting in one replacement ruleset:
==== 1st descendants ====
Since a chroma is the difference between a large and small step, 1st descendants have the large step of the parent break into a chroma and small step. Recalling that this small step refers to the parent's rather than the descendant's, and that whichever is larger depends on how big L and s are, the two rulesets for 1st descendants can be condensed into one ruleset. Using a rectangular horogram helps to illustrate what happens.
{| class="wikitable"
{| class="wikitable"
|+
|+
!Mos
!Mos
!Relation
! colspan="3" |Step pattern
! colspan="3" |Step pattern
!Ruleset used
(1L 1s)
! colspan="3" |Step pattern with chromas
! colspan="3" |Step pattern with chromas
!Ruleset
!Final ruleset
|-
|-
|xL ys
|Parent
|Parent
| colspan="2" |L
| colspan="2" |L
|s
|s
|
| colspan="2" |L
| colspan="2" |L
|s
|s
|
|
|-
|-
|xL (x+y)s
|Child 1
|Child 1
|L
|L
|s
|s
|s
|s
|1
| rowspan="2" |c
| rowspan="2" |c
| rowspan="2" |s
| rowspan="2" |s
Line 294: Line 414:
| rowspan="2" |L->cs, s->s
| rowspan="2" |L->cs, s->s
|-
|-
|(x+y)L xs
|Child 2
|Child 2
|s
|s
|L
|L
|L
|L
|2
|}
|}


=== 2nd descendants ===
==== 2nd descendants ====
To get replacement rules for 2nd descendants, we can extend the horograms produced before. However, this results into two horograms since the step counts will be different. To deduce what these steps are in relation to the parent, we introduce a second-order chroma: a diesis, defined as d = | L - 2s |. For ruleset 1+1 and 1+2, a large step breaks down into two small steps and a diesis. Whichever is larger depends on the step ratio of the parent, condensing the rulesets 1+1 and 1+2 into one ruleset.
To get replacement rules for 2nd descendants, we can extend the horograms produced before. However, this results into two horograms since the step counts will be different. To deduce what these steps are in relation to the parent, we introduce a second-order chroma: a diesis, defined as d = | L - 2s |. For ruleset 1+1 and 1+2, a large step breaks down into two small steps and a diesis. Whichever is larger depends on the step ratio of the parent, condensing the rulesets 1+1 and 1+2 into one ruleset.
{| class="wikitable"
{| class="wikitable"
! rowspan="2" |Mos
! rowspan="2" |Mos
! colspan="6" |As L's and s's
! rowspan="2" |Relation
! colspan="5" |In terms of chromas and dieses
! colspan="8" |Step pattern
! rowspan="2" |Final rulesets
|-
|-
! colspan="4" |Step pattern
! colspan="4" |L's and s's
!Ruleset used
! colspan="4" |s's and c's
!Final rulesets
! colspan="4" |Step pattern
!Final rulesets
|-
|-
|xL ys
|Parent
|Parent
| colspan="3" |L
| colspan="3" |L
|s
|s
|
|
| colspan="3" |L
| colspan="3" |L
|s
|s
|
|
|-
|-
|xL (x+y)s
|Child 1
|Child 1
| colspan="2" |L
| colspan="2" |L
|s
|s
|s
|s
|1
|
| colspan="2" |c
| colspan="2" |c
|s
|s
Line 334: Line 450:
|
|
|-
|-
| rowspan="2" |Descendants of child 1
|xL (2x+y)s
|Grandchild 11
|L
|L
|s
|s
|s
|s
|s
|s
|1, then 1
|L->Lss, s->s
| rowspan="2" |c
| rowspan="2" |c
| rowspan="2" |s
| rowspan="2" |s
Line 347: Line 462:
| rowspan="2" |L->css, s->s
| rowspan="2" |L->css, s->s
|-
|-
|(2x+y)L xs
|Grandchild 12
|s
|s
|L
|L
|L
|L
|L
|L
|1, then 2
|L->sLL, s->L
|}
|}
For ruleset 2+1 and 2+2, the parent's small step itself breaks into smaller steps. Given a chroma is the difference between a large and small step, the horograms below show that two chromas fit within a large step, and one chroma within the small step. Again, whichever is larger depends on the step ratio of L and s.
 
{| class="wikitable"
{| class="wikitable"
! rowspan="2" |Mos
! rowspan="2" |Mos
! colspan="7" |As L's and s's
! rowspan="2" |Relation
! colspan="6" |In terms of chromas and dieses
! colspan="10" |Step pattern
! rowspan="2" |Final rulesets
|-
|-
! colspan="5" |Step pattern
! colspan="5" |L's and s's
!Ruleset used
! colspan="5" |c's and d's
!Final rulesets
! colspan="5" |Step pattern
!Final rulesets
|-
|-
|xL ys
|Parent
|Parent
| colspan="3" |L
| colspan="3" |L
| colspan="2" |s
| colspan="2" |s
|
|
| colspan="3" |L
| colspan="3" |L
| colspan="2" |s
| colspan="2" |s
|
|
|-
|-
|(x+y)L xs
|Child 2
|Child 2
|s
|s
| colspan="2" |L
| colspan="2" |L
| colspan="2" |L
| colspan="2" |L
|2
|
|c
|c
| colspan="2" |s
| colspan="2" |s
Line 386: Line 497:
|
|
|-
|-
| rowspan="2" |Descendants of child 2
|(x+y)L (2x+y)s
|Grandchild 21
|s
|s
|s
|s
Line 392: Line 504:
|s
|s
|L
|L
|2, then 1
|L->ssL, s->sL
| rowspan="2" |c
| rowspan="2" |c
| rowspan="2" |c
| rowspan="2" |c
Line 401: Line 511:
| rowspan="2" |L->ccd, s->cd
| rowspan="2" |L->ccd, s->cd
|-
|-
|(2x+y)L (x+y)s
|Grandchild 22
|L
|L
|L
|L
Line 406: Line 518:
|L
|L
|s
|s
|2, then 2
|L->LLs, s->Ls
|}
|}


=== 3rd descendants ===
==== 3rd descendants ====


* todo
* todo
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