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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.
 
=== Replacement rules of descendants ===
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->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
|}
 
==== 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"
|+
!Mos
!Relation
! colspan="3" |Step pattern
(1L 1s)
! colspan="3" |Step pattern with chromas
!Final ruleset
|-
|xL ys
|Parent
| colspan="2" |L
|s
| colspan="2" |L
|s
|
|-
|xL (x+y)s
|Child 1
|L
|s
|s
| rowspan="2" |c
| rowspan="2" |s
| rowspan="2" |s
| rowspan="2" |L->cs, s->s
|-
|(x+y)L xs
|Child 2
|s
|L
|L
|}
 
==== 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
! rowspan="2" |Relation
! colspan="8" |Step pattern
! rowspan="2" |Final rulesets
|-
! colspan="4" |L's and s's
! colspan="4" |s's and c's
|-
|xL ys
|Parent
| colspan="3" |L
|s
| colspan="3" |L
|s
|
|-
|xL (x+y)s
|Child 1
| colspan="2" |L
|s
|s
| colspan="2" |c
|s
|s
|
|-
|xL (2x+y)s
|Grandchild 11
|L
|s
|s
|s
| rowspan="2" |c
| rowspan="2" |s
| rowspan="2" |s
| rowspan="2" |s
| rowspan="2" |L->css, s->s
|-
|(2x+y)L xs
|Grandchild 12
|s
|L
|L
|L
|}
 
{| class="wikitable"
! rowspan="2" |Mos
! rowspan="2" |Relation
! colspan="10" |Step pattern
! rowspan="2" |Final rulesets
|-
! colspan="5" |L's and s's
! colspan="5" |c's and d's
|-
|xL ys
|Parent
| colspan="3" |L
| colspan="2" |s
| colspan="3" |L
| colspan="2" |s
|
|-
|(x+y)L xs
|Child 2
|s
| colspan="2" |L
| colspan="2" |L
|c
| colspan="2" |s
| colspan="2" |s
|
|-
|(x+y)L (2x+y)s
|Grandchild 21
|s
|s
|L
|s
|L
| rowspan="2" |c
| rowspan="2" |c
| rowspan="2" |d
| rowspan="2" |c
| rowspan="2" |d
| rowspan="2" |L->ccd, s->cd
|-
|(2x+y)L (x+y)s
|Grandchild 22
|L
|L
|s
|L
|s
|}
 
==== 3rd descendants (wip) ====
To get replacement rules for 3rd descendants, the previous horograms can be extended as before, as well as adding a third-order chroma: a kleisma (or, for this page, an epsilon), defined as e = | L - 3s |.
{| class="wikitable"
! rowspan="2" |Mos
! rowspan="2" |Relation
! colspan="8" |Step pattern
! rowspan="2" |Final rulesets
|-
! colspan="4" |L's and s's
! colspan="4" |s's and c's
|-
|xL ys
|Parent
| colspan="3" |L
|s
| colspan="3" |L
|s
|
|-
|xL (x+y)s
|Child 1
| colspan="2" |L
|s
|s
| colspan="2" |c
|s
|s
|
|-
|xL (2x+y)s
|Grandchild 11
|L
|s
|s
|s
|c
|s
|s
|s
|
|}
 
{| class="wikitable"
! rowspan="2" |Mos
! rowspan="2" |Relation
! colspan="8" |Step pattern
! rowspan="2" |Final rulesets
|-
! colspan="4" |L's and s's
! colspan="4" |s's and c's
|-
|xL ys
|Parent
| colspan="3" |L
|s
| colspan="3" |L
|s
|
|-
|xL (x+y)s
|Child 1
| colspan="2" |L
|s
|s
| colspan="2" |c
|s
|s
|
|-
|(2x+y)L xs
|Grandchild 12
|s
|L
|L
|L
|c
|s
|s
|s
|
|}
 
{| class="wikitable"
! rowspan="2" |Mos
! rowspan="2" |Relation
! colspan="10" |Step pattern
! rowspan="2" |Final rulesets
|-
! colspan="5" |L's and s's
! colspan="5" |c's and d's
|-
|xL ys
|Parent
| colspan="3" |L
| colspan="2" |s
| colspan="3" |L
| colspan="2" |s
|
|-
|(x+y)L xs
|Child 2
|s
| colspan="2" |L
| colspan="2" |L
|c
| colspan="2" |s
| colspan="2" |s
|
|-
|(x+y)L (2x+y)s
|Grandchild 21
|s
|s
|L
|s
|L
|c
|c
|d
|c
|d
|
|}
 
{| class="wikitable"
! rowspan="2" |Mos
! rowspan="2" |Relation
! colspan="10" |Step pattern
! rowspan="2" |Final rulesets
|-
! colspan="5" |L's and s's
! colspan="5" |c's and d's
|-
|xL ys
|Parent
| colspan="3" |L
| colspan="2" |s
| colspan="3" |L
| colspan="2" |s
|
|-
|(x+y)L xs
|Child 2
|s
| colspan="2" |L
| colspan="2" |L
|c
| colspan="2" |s
| colspan="2" |s
|
|-
|(2x+y)L (x+y)s
|Grandchild 22
|L
|L
|s
|L
|s
|c
|c
|d
|c
|d
|
|}
 
=== 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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The following list is some commentary on some of the basic edos shown in the table:
The following list is some commentary on some of the basic edos shown in the table:


* '''19edo''': I've seen one musician describe it as a baby version of 31edo, and I'm inclined to agree in that it's basically an equalized 12L 7s.
* '''19edo''': I've seen one musician describe it as a baby version of 31edo, and I'm inclined to agree in that it's basically an equalized 12L 7s. It also introduces the idea that a note raised by two chromas can fall short of the next note one whole tone above it (for example, Cx falling short of D).
* '''31edo''': This, and 50edo to an extent, is currently my go-to edo for meantone temperament. Generally speaking, I like the mellow sound of meantone, which explains why I do absolutely nothing to date with 17edo and friends. It's also compatible with half-sharp and half-flat notation, so "quartertone" compositions are possible.
* '''31edo''': This, and 50edo to an extent, is currently my go-to edo for meantone temperament. Generally speaking, I like the mellow sound of meantone, which explains why I do absolutely nothing to date with 17edo and friends. It's also compatible with half-sharp and half-flat notation, so "quartertone" compositions are possible.
* '''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 ===
==== 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.
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"
!
!
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|SC Locrian
|SC Locrian
|}
|}
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. The mode "CS ionian" is basically an incomplete form of one of my favorite modes in 12L 7s.
This breakdown can be taken one step further by breaking down small steps into a chroma and diesis, where a diesis is defined as such: d = | L - 2s |. Ultimately, the large steps of 5L 2s are broken down into two chromas and a diesis, in some order. This produces the modes of 12L 7s and introduces an additional 7 modes on top of the previous 12. In either case, it's easier to think of these modes as some sort of extension of the seven modes of 5L 2s. There are, for example, 13 modes of 12L 7s that contain ionian as a subset (UDPs 17|1 to 5|13), so it's confusing and unhelpful to say there are 13 ionian modes. Therefore, saying that the large steps break down in a specific order (chroma-chroma-diesis, chroma-diesis-chroma, or diesis-chroma-chroma, or simply CCD, CDC, and DCC) helps to narrow down ionian-containing modes to 3 specific modes.
 
=== 12L 7s ===
This breakdown can be taken one step further by breaking down small steps into a chroma and diesis, where a diesis is defined as such: d = | L - 2s |. Ultimately, the large steps of 5L 2s are broken down into two chromas and a diesis, in some order. This produces the modes of 12L 7s and introduces an additional 7 modes. In either case, it's easier to think of these modes as some sort of extension of the seven modes of 5L 2s. The use of "CS/SC" and "CCD/CDC/DCC" is to enforce how these modes are broken down; there are 13 modes of 12L 7s that contain ionian as a subset (UDPs 17|1 to 5|13), so it's confusing to say there are 13 ionian modes.
{| class="wikitable"
{| class="wikitable"
!
!
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todo:examples
todo:examples
==== 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.
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