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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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Interestingly, this definition of a default mode has some overlap with [[Naming Rank-2 Scales|Jake Freivald's method]] of enumerating a mos's modes.
Interestingly, this definition of a default mode has some overlap with [[Naming Rank-2 Scales|Jake Freivald's method]] of enumerating a mos's modes.


== My approach to diatonic descendant mosses ==
== My approach to descendant mosses ==
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 ===
Since replacement rules apply to any mos, it may be easier to consider the replacement rules themselves rather than use a specific mos.
 
* 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
 
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:
{| class="wikitable"
|+
!Mos
! colspan="3" |Step pattern
!Ruleset used
! colspan="3" |Step pattern with chromas
!Ruleset
|-
|Parent
| colspan="2" |L
|s
|
| colspan="2" |L
|s
|
|-
|Child 1
|L
|s
|s
|1
| rowspan="2" |c
| rowspan="2" |s
| rowspan="2" |s
| rowspan="2" |L->cs, s->s
|-
|Child 2
|s
|L
|L
|2
|}
 
=== 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.
{| class="wikitable"
! rowspan="2" |Mos
! colspan="6" |As L's and s's
! colspan="5" |In terms of chromas and dieses
|-
! colspan="4" |Step pattern
!Ruleset used
!Final rulesets
! colspan="4" |Step pattern
!Final rulesets
|-
|Parent
| colspan="3" |L
|s
|
|
| colspan="3" |L
|s
|
|-
|Child 1
| colspan="2" |L
|s
|s
|1
|
| colspan="2" |c
|s
|s
|
|-
| rowspan="2" |Descendants of child 1
|L
|s
|s
|s
|1, then 1
|L->Lss, s->s
| rowspan="2" |c
| rowspan="2" |s
| rowspan="2" |s
| rowspan="2" |s
| rowspan="2" |L->css, s->s
|-
|s
|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"
! rowspan="2" |Mos
! colspan="7" |As L's and s's
! colspan="6" |In terms of chromas and dieses
|-
! colspan="5" |Step pattern
!Ruleset used
!Final rulesets
! colspan="5" |Step pattern
!Final rulesets
|-
|Parent
| colspan="3" |L
| colspan="2" |s
|
|
| colspan="3" |L
| colspan="2" |s
|
|-
|Child 2
|s
| colspan="2" |L
| colspan="2" |L
|2
|
|c
| colspan="2" |s
| colspan="2" |s
|
|-
| rowspan="2" |Descendants of child 2
|s
|s
|L
|s
|L
|2, then 1
|L->ssL, s->sL
| rowspan="2" |c
| rowspan="2" |c
| rowspan="2" |d
| rowspan="2" |c
| rowspan="2" |d
| rowspan="2" |L->ccd, s->cd
|-
|L
|L
|s
|L
|s
|2, then 2
|L->LLs, s->Ls
|}
 
=== 3rd descendants ===
 
* todo
 
=== Examples with 5L 2s ===
My current approach to xenharmony, at least as it pertains to being both familiar and different, is to play with the familiar mos pattern of 5L 2s but to add additional notes in between. If restricted to 12edo, this means one of two things to me:
My current approach to xenharmony, at least as it pertains to being both familiar and different, is to play with the familiar mos pattern of 5L 2s but to add additional notes in between. If restricted to 12edo, this means one of two things to me:


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* '''45edo''': Like 19edo but more extreme. Whereas 19edo has Cx falling short of D but C#x being equivalent to D, in 45edo, C#x falls short of D, but the interval between C#x and D is its own smaller interval I'm dubbing the '''triesis''' for the purposes of this page.
* '''45edo''': Like 19edo but more extreme. Whereas 19edo has Cx falling short of D but C#x being equivalent to D, in 45edo, C#x falls short of D, but the interval between C#x and D is its own smaller interval I'm dubbing the '''triesis''' for the purposes of this page.


=== 7L 5s and 12L 7s ===
==== 7L 5s and 12L 7s ====
Given a chroma is the absolute difference between a large and small step (c = | L - s |), mathematically speaking, the large steps can be broken into a chroma and small step, the small and large steps of 7L 5s. Depending on the order, this produces two ways each of the seven modes can be broken down into the m-chromatic modes, with some overlap. Basically, a chroma and small step are not the same size, more commonly described with the terms "diatonic and chromatic semitones". This unevenness manifests itself as the m-chromatic scale, a scale I admittedly don't use as often compared to 12L 7s. I see this mos as an incomplete form of 12L 7s. Nonetheless, a mode table is still provided, providing a guide as to how these 12 modes relate back to the more familiar 7 modes.
Given a chroma is the absolute difference between a large and small step (c = | L - s |), mathematically speaking, the large steps can be broken into a chroma and small step, the small and large steps of 7L 5s. Depending on the order, this produces two ways each of the seven modes can be broken down into the m-chromatic modes, with some overlap. Basically, a chroma and small step are not the same size, more commonly described with the terms "diatonic and chromatic semitones". This unevenness manifests itself as the m-chromatic scale, a scale I admittedly don't use as often compared to 12L 7s. I see this mos as an incomplete form of 12L 7s. Nonetheless, a mode table is still provided, providing a guide as to how these 12 modes relate back to the more familiar 7 modes.
{| class="wikitable"
{| class="wikitable"
Line 1,130: Line 1,286:
todo:examples
todo:examples


=== 19L 7s ===
==== 19L 7s ====
Whereas a diesis is defined as d = | L - 2s |, a triesis is defined as t = | L - 3s |. As a summary, 19L 7s contains all the super-leading tone goodness I expect from 12L 7s, but with an additional step (literally) to get there.
Whereas a diesis is defined as d = | L - 2s |, a triesis is defined as t = | L - 3s |. As a summary, 19L 7s contains all the super-leading tone goodness I expect from 12L 7s, but with an additional step (literally) to get there.
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