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This page is for miscellaneous xen-related models for describing some facet of xenharmonic music theory that I've written about but don't have an exact place elsewhere on the wiki (yet).
This page is for miscellaneous xen-related models for describing some facet of xenharmonic music theory that I've written about but don't have an exact place elsewhere on the wiki (yet).


== Chroma-diesis model of mos child scales ==
Needs reorganizing.
This is a description of how to look at the child scales of a [[MOS scale|mos]] by looking at only the large and small steps of its parent mos. (It's also not well refined or proofread, hence it's a subpage of my userpage.) The motivation behind this comes from the notion of a [[chroma]] -- the interval that is defined as the difference between a mos's large and small steps -- and the [[diesis]], which can be defined as the difference between C# and Db in meantone temperaments.


This section describes the notion of a generalized diesis in both an [[regular temperament]] context and a [[TAMNAMS|temperament-agnostic]] context. I developed this model because I kept looking at child scales two generations after a parent scale, specifically 5L 2s and its children, and I needed a way to justify notating harmonic-7th chords (in meantone temperaments) as sharp-6 chords.
== Chroma-diesis model of mos child scales (outdated) ==
''Note: Much of the ideas here can be more [[User:Ganaram inukshuk/Notes#Describing moschromatic and mosenharmonic scales|easily described]] without having to consider the size of a (mos)chroma or (mos)diesis, and was also conceived before I realized that TAMNAMS already described a mosdiesis. This description is left for archival purposes.''


=== 7L 5s and 12L 7s (meantone temperament) ===
''Note: the idea of [[diatonic, chromatic, enharmonic, subchromatic|diatonic, chromatic, enharmonic, and subchromatic scales]] also applies to this, as well as the smaller intervals of a [[Extended meantone notation|chroma, diesis, and kleisma]]. I'd like to try to resurrect the ideas I independently tried to describe here as a temperament-agnostic interpretation of the two ideas described someday.''
[[31edo]] is used as an arguably noteworthy example of an [[EDO|edo]] that supports [[meantone]] temperament. Here, the diatonic ([[5L 2s]]) scale structure can be represented as the following pattern of large and small steps: 5-5-3-5-5-5-3, where the large steps are of size 5 and the small steps of size 3.


By definition of a chroma, the size of a chroma is calculated as 5-3 = 2, hence sharps and flats must raise or lower notes by 2 edosteps. The diesis in 31edo can be defined as 1 edostep of 31edo, or 1\31. However, a generalized definition can be put forth:
This is a description of how to look at the child scales of a [[MOS scale|mos]] by looking at only the large and small steps of its parent mos. (It's also not well refined or proofread, hence it's a subpage of my userpage.) The motivation behind this comes from the notion of a [[chroma]] -- the interval that is defined as the difference between a mos's large and small steps -- and the [[diesis]], which can be defined as the difference between C# and Db in meantone temperaments.


* A diesis is the difference between a large step and two small steps, or d = L - 2s.
This section describes the notion of a generalized diesis in a temperament-agnostic context. I developed this model because I kept looking at child scales two generations after a parent scale, specifically 5L 2s and its children, and I needed a way to justify notating harmonic-7th chords (in meantone temperaments) as sharp-6 chords.
* A diesis is also the difference between a small step and a chroma, or d = c - s. This is because, by definition, a chroma is defined as L - s, so mathematically, L - 2s and c - s are equivalent.


In meantone temperament, the pattern of child scales continues from 5L 2s to [[7L 5s]] and [[12L 7s]]. Both can be described as patterns of large and small steps, and can be seen in the table below.
=== 5L 2s, 7L 5s, and 12L 7s ===
The notion of a diesis, as well as its use, is (for our purposes) related to [[meantone]] temperament. [[31edo]] is used as an arguably noteworthy example of an [[EDO|edo]] that supports this temperament. Meantone temperament also describes a chain of more chromatic child scales that come "after" 5L 2s: [[7L 5s]] and [[12L 7s]]. These scales, along with 5L 2s, are described in the table below.
{| class="wikitable"
{| class="wikitable"
! colspan="31" |Step Visualization (using ionian mode for comparison)
! colspan="31" |Step Visualization (using ionian mode for comparison)
Line 108: Line 107:
| colspan="4" |31edo
| colspan="4" |31edo
|}
|}
The chroma-diesis model describes large and small steps as chromas and dieses. In terms of replacement rules, it can be described as L->ccd and s->cd; considering how [[User:Ganaram inukshuk/Notes|replacement rules can be used to generate more complex rules]], this is basically equivalent to using L's and s's. However, the sizes of the chroma and dieses were all based on that from 5L 2s, so this model focuses on what happens to L and s of 5L 2s, rather than immediately notating which is the larger and smaller intervals for successive scales.
A chroma is defined as the difference between a large step and a small step (c = L - s), and is used to describe how many edosteps it takes to sharpen or flatten a note, such as raising C to C#, or lowering D to Db. In comparison, a diesis is used to describe something smaller. Since in meantone temperament, C double-sharp falls short of D, the difference between the two is the diesis. It can be defined more generally as such:
* A diesis is the difference between a large step and two small steps, or d = L - 2s.
* A diesis is also the difference between a small step and a chroma, or d = c - s. This is because, by definition, a chroma is defined as L - s, so mathematically, L - 2s and c - s are equivalent.
Instead of describing 7L 5s and 12L 7s in terms of large steps and small steps unique to each mos, an alternate description can be formed based on only the large and small steps of 5L 2s. In terms of replacement rules, it can be described as L->ccd and s->cd; considering how [[User:Ganaram inukshuk/Notes|replacement rules can be used to generate more complex rules]], this is basically equivalent to using L's and s's.
{| class="wikitable"
{| class="wikitable"
! colspan="31" |Step Visualization (using ionian mode for comparison)
! colspan="31" |Step Visualization (using ionian mode for comparison)
!Mos
!Mos
!Step Pattern
!Step Pattern
!TAMNAMS Name
!Large step
!Temperament
!Small step
|-
|-
| colspan="5" |L
| colspan="5" |L
Line 125: Line 127:
|5L 2s
|5L 2s
|LLsLLLs
|LLsLLLs
|diatonic
|L
|meantone[7]
|s
|-
|-
| colspan="2" |c
| colspan="2" |c
Line 142: Line 144:
|7L 5s
|7L 5s
|cs cs s cs cs cs s
|cs cs s cs cs cs s
|m-chromatic
|s
|meantone[12]
|c = L - s
|-
|-
| colspan="2" |c
| colspan="2" |c
Line 166: Line 168:
|12L 7s
|12L 7s
|ccd ccd cd ccd ccd ccd cd
|ccd ccd cd ccd ccd ccd cd
|unnamed
|c = L - s
|meantone[19]
|d = L - 2s
|-
|-
|1
|1
Line 202: Line 204:
| colspan="4" |31edo
| colspan="4" |31edo
|}
|}
In short, in meantone[12], large steps break apart into a chroma and small-step, and in meantone[19], large steps break up into chroma-chroma-diesis triplets and the small steps chroma-diesis pairs.
31edo is used as an example since it represents 12L 7s using a L:s ratio of 2:1. It should be noted that any edo can work as well, even those that aren't described by meantone temperament. Successive sections will look at other examples in a temperament-agnostic context, and successive example edos will represent second-order child scales with a step ratio of 2:1.


Note that this model looks at child scales two generations beyond the parent scale. It's possible to generalize this to even smaller intervals (perhaps using a "triesis" defined as L - 3s and a general "polyesis" or "n-esis" defined as L - ns), but since the chroma and diesis are both familiar intervals (at least in a xen context), the named steps are limited to such, hence the name "chroma-diesis model".
This description arbitrarily stops at two scales after the parent scale of 5L 2s. It's possible to generalize this to higher-order child scales with even smaller intervals (perhaps using a "triesis" defined as L - 3s and a general "polyesis" or "n-esis" defined as L - ns), but since the chroma and diesis are both familiar intervals (at least in a xen context), the named steps are limited to such, hence the name "chroma-diesis model".


Also note that the example of 31edo was chosen because its chroma-diesis ratio (its L:s ratio) is 2:1. Other edos can work as well, such as [[50edo]].
=== 5L 2s, 7L 5s, and 7L 12s ===
 
When considering the [[User:Ganaram inukshuk/Diagrams#Family Tree of MOSses|mos family tree]], it's immediately obvious that 12L 7s is not the only child scale of 7L 5s. In a meantone context, the notion of a diesis is that it's smaller than a chroma, so instead of a chain of scales described by meantone as being 5L 2s, 7L 5s, and 12L 7s, that chain instead describes a sequence of [[flattone]] scales that diverges with [[7L 12s]]. Curiously, this results in a diesis being larger than the chroma. 26edo is shown as an example, since it supports 7L 12s with a step ratio of 2:1.
=== Including 7L 12s (flattone temperament) ===
When considering the [[User:Ganaram inukshuk/Diagrams#Family Tree of MOSses|mos family tree]], it's immediately obvious that 12L 7s is not the only child scale of 7L 5s. In a meantone context, the notion of a diesis is that it's smaller than a chroma. However, it's still possible to describe [[7L 12s]] in terms of chromas and dieses.
{| class="wikitable"
{| class="wikitable"
! colspan="26" |Step Visualization (using ionian mode for comparison)
! colspan="26" |Step Visualization (using ionian mode for comparison)
!Mos
!Mos
!Step Pattern
!Step Pattern
!TAMNAMS Name
!Large step
!Temperament
!Small step
|-
|-
| colspan="4" |L
| colspan="4" |L
Line 226: Line 226:
|5L 2s
|5L 2s
|LLsLLLs
|LLsLLLs
|diatonic
|L
|flattone[7]
|s
|-
|-
|c
|c
Line 243: Line 243:
|7L 5s
|7L 5s
|cs cs s cs cs cs s
|cs cs s cs cs cs s
|m-chromatic
|s
|flattone[12]
|c = L - s
|-
|-
|c
|c
Line 267: Line 267:
|7L 12s
|7L 12s
|ccd ccd cd ccd ccd ccd cd
|ccd ccd cd ccd ccd ccd cd
|unnamed
|d = L - 2s
|flattone[19]
|c = L - s
|-
|-
|1
|1
Line 298: Line 298:
| colspan="4" |26edo
| colspan="4" |26edo
|}
|}
In [[flattone]] temperament, it may be said that the diesis, as the difference between a small step and a chroma, is larger than the chroma; in comparison to meantone temperament, the diesis is smaller than the chroma. In a temperament-agnostic perspective, this is equivalent to describing a mos (7a 12b) without specifying which steps are the large or small steps, and specifying which is which will necessarily identify which of the two child mosses -- 7L 12s or 12L 7s -- is being described.
In comparing this with 12L 7s, this is equivalent to describing a mos (7a 12b) without specifying which steps are the large or small steps; specifying which is which will necessarily identify which of the two mosses -- 7L 12s or 12L 7s -- is being described.
 
As with 31edo, 26edo was chosen because its L:s ratio is also 2:1.


=== Including 5L 7s, 5L 12s, and 12L 5s (Pythagorean-based temperaments) ===
=== 5L 7s, 5L 7s, 5L 12s, and 12L 5s ===
The notion of chromas also apply to [[5L 7s]], the child mos of 5L 2s given a hard step ratio. Compared to soft step ratios (or, when considering temperaments, meantone and flattone temperaments), hard step ratios produce chromas that are larger than the small step. Still, the notion of describing child scales as either chromas or dieses can still be done here. [[22edo]] and [[29edo]] are used as examples the same way 31edo and 26edo were used as examples: the child scales two generations after 5L 2s are of a step ratio of 2:1.
The rules described above apply regardless of the step ratio of 5L 2s. Compared to soft step ratios, hard step ratios produce chromas that are larger than the small step. Still, the notion of describing child scales using chromas or dieses can still be done here. First is the chain of scales that can be described by [[superpyth]] temperament: 5L 2s, 5L 7s, and 5L 12s. Here, the number of large steps stays constant, but shrink by a small step. Additionally, the size of the small step from 5L 2s is the same with successive scales. 22edo is used as an example below.
{| class="wikitable"
{| class="wikitable"
! colspan="22" |Step Visualization (using ionian for comparison)
! colspan="22" |Step Visualization (using ionian for comparison)
!Mos
!Mos
!Step Pattern
!Step Pattern
!TAMNAMS Name
!Large step
!Temperament
!Small step
|-
|-
| colspan="4" |L
| colspan="4" |L
Line 320: Line 318:
|5L 2s
|5L 2s
|LLsLLLs
|LLsLLLs
|diatonic
|L
|[[superpyth]][7]
|s
|-
|-
| colspan="3" |c
| colspan="3" |c
Line 337: Line 335:
|5L 7s
|5L 7s
|cs cs s cs cs cs s
|cs cs s cs cs cs s
|p-chromatic
|c = L - s
|superpyth[12]
|s
|-
|-
| colspan="2" |d
| colspan="2" |d
Line 359: Line 357:
|5L 12s
|5L 12s
|dss dss s dss dss dss s
|dss dss s dss dss dss s
|unnamed
|d = L - 2s
|superpyth[17]
|s
|-
|-
|1
|1
Line 386: Line 384:
| colspan="4" |22edo
| colspan="4" |22edo
|}
|}
Second is a chain of scales that can be described using [[leapfrog]] temperament: 5L 2s, 5L 7s, and 12L 5s. 29edo is used as an example below.
{| class="wikitable"
{| class="wikitable"
! colspan="29" |Step Visualization (using ionian for comparison)
! colspan="29" |Step Visualization (using ionian for comparison)
!Mos
!Mos
!Step Pattern
!Step Pattern
!TAMNAMS Name
!Large step
!Temperament
!Small step
|-
|-
| colspan="5" |L
| colspan="5" |L
Line 402: Line 401:
|5L 2s
|5L 2s
|LLsLLLs
|LLsLLLs
|diatonic
|L
|[[leapfrog]][7]
|s
|-
|-
| colspan="3" |c
| colspan="3" |c
Line 419: Line 418:
|5L 7s
|5L 7s
|cs cs s cs cs cs s
|cs cs s cs cs cs s
|p-chromatic
|c = L - s
|leapfrog[12]
|s
|-
|-
|d
|d
Line 441: Line 440:
|12L 5s
|12L 5s
|dss dss s dss dss dss s
|dss dss s dss dss dss s
|unnamed
|s
|leapfrog[17]
|d = L - 2s
|-
|-
|1
|1
Line 475: Line 474:
| colspan="4" |29edo
| colspan="4" |29edo
|}
|}
Just like the 31 and 26edo examples, the result is a pair of sister scales (5L 12s and 12L 5s) both being described as though it were one scale: 5a 12b.


=== Combined scale tree ===
=== Combined scale tree ===
The past few sections arbitrarily had 31edo, 26edo, 22edo, and 29edo selected for the sake of example. It should be noted that other edos could have worked for these example. Combining these into a scale tree removes the notion of being locked to a specific edo and reveals a more common pattern that's closer to the mos family tree. A few edos are included as being the equalized points of sister scale pairs.
The past few sections had 31edo, 26edo, 22edo, and 29edo selected for the sake of example. It should be noted that other edos could have worked for these examples. Combining these into a family tree removes the notion of being locked to a specific edo (or set of temperaments, for that matter) and reveals a more common pattern that's closer to the mos family tree. A few edos are included as being the equalized points of sister scale pairs, where the two step sizes are the same.
{| class="wikitable"
{| class="wikitable"
! colspan="2" |Parent scale
! colspan="2" |Parent scale
! colspan="3" |1st orderchild scales
! colspan="3" |1st order child scales
! colspan="3" |2nd order child scales
! colspan="3" |2nd order child scales
|-
|-
Line 501: Line 501:
|cs cs s cs cs cs s
|cs cs s cs cs cs s
|s < c
|s < c
|''17n-edo''
|''17edo''
|sss sss s sss sss sss s
|sss sss s sss sss sss s
|s = d
|s = d
Line 512: Line 512:
|'''5L 2s'''
|'''5L 2s'''
|LLsLLLs
|LLsLLLs
|''12n-edo''
|''12edo''
|ss ss s ss ss ss s  
|ss ss s ss ss ss s  
|s = c
|s = c
Line 526: Line 526:
|cs cs s cs cs cs s
|cs cs s cs cs cs s
|s > c
|s > c
|''19n-edo''
|''19edo''
|ccc ccc cc ccc ccc ccc cc
|ccc ccc cc ccc ccc ccc cc
|c = d
|c = d
Line 542: Line 542:
* Mos recursion becomes readily apparent, especially the chunking operation with the 2nd-generation children of a soft-step-ratio parent scale.
* Mos recursion becomes readily apparent, especially the chunking operation with the 2nd-generation children of a soft-step-ratio parent scale.


=== Generalized scale tree for nondiatonic (not 5L 2s) mosses ===
=== Generalized tree for nondiatonic (not 5L 2s) mosses ===
The chroma-diesis model also generalizes for nondiatonic mosses. Since these mosses are greatly underexplored (compared to diatonic), it's hard to generally know which scales have a similar status as diatonic (such as having a similar note count), and thus, it may be easier to describe such scales using only L's and s's and not chromas and dieses. Though a good place to start may be the sister mos of diatonic: antidiatonic, or [[2L 5s]]. The antiphrygian mode is assumed to be the "default" mode, the same way ionian is for diatonic.
The chroma-diesis model also generalizes for nondiatonic mosses. Since these mosses are greatly underexplored (compared to diatonic), it's hard to generally know which scales have a similar status as diatonic (such as having a similar note count), and thus, it may be easier to describe such scales using only L's and s's and not chromas and dieses. Though a good place to start may be the sister mos of diatonic: antidiatonic, or [[2L 5s]]. The antiphrygian mode is assumed to be the "default" mode, the same way ionian is for diatonic.
{| class="wikitable"
{| class="wikitable"
Line 567: Line 567:
|cs s s s cs s s
|cs s s s cs s s
|s < c
|s < c
|''11n-edo''
|''11edo''
|sss s s s sss s s
|sss s s s sss s s
|s = d
|s = d
Line 578: Line 578:
|'''2L 5s'''
|'''2L 5s'''
|LsssLss
|LsssLss
|''9n-edo''
|''9edo''
|ss s s s ss s s
|ss s s s ss s s
|s = c
|s = c
Line 592: Line 592:
|cs s s s cs s s
|cs s s s cs s s
|s > c
|s > c
|''16n-edo''
|''16edo''
|ccccc cc cc cc ccccc cc cc
|ccccc cc cc cc ccccc cc cc
|c = d
|c = d
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