User:Ganaram inukshuk/Models
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).
Needs reorganizing.
Chroma-diesis model of mos child scales (outdated)
Note: Much of the ideas here can be more 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.
Note: the idea of diatonic, chromatic, enharmonic, and subchromatic scales also applies to this, as well as the smaller intervals of a 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.
This is a description of how to look at the child scales of a 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 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.
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 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.
| Step Visualization (using ionian mode for comparison) | Mos | Step Pattern | TAMNAMS Name | Temperament | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| L | L | s | L | L | L | s | 5L 2s | LLsLLLs | diatonic | meantone[7] | ||||||||||||||||||||||||
| s | L | s | L | L | s | L | s | L | s | L | L | 7L 5s | sL sL L sL sL sL L | m-chromatic | meantone[12] | |||||||||||||||||||
| L | L | s | L | L | s | L | s | L | L | s | L | L | s | L | L | s | L | s | 12L 7s | LLs LLs Ls LLs LLs LLs Ls | unnamed | meantone[19] | ||||||||||||
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 31edo | |||
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 replacement rules can be used to generate more complex rules, this is basically equivalent to using L's and s's.
| Step Visualization (using ionian mode for comparison) | Mos | Step Pattern | Large step | Small step | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| L | L | s | L | L | L | s | 5L 2s | LLsLLLs | L | s | ||||||||||||||||||||||||
| c | s | c | s | s | c | s | c | s | c | s | s | 7L 5s | cs cs s cs cs cs s | s | c = L - s | |||||||||||||||||||
| c | c | d | c | c | d | c | d | c | c | d | c | c | d | c | c | d | c | d | 12L 7s | ccd ccd cd ccd ccd ccd cd | c = L - s | d = L - 2s | ||||||||||||
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 31edo | |||
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.
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".
5L 2s, 7L 5s, and 7L 12s
When considering the 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.
| Step Visualization (using ionian mode for comparison) | Mos | Step Pattern | Large step | Small step | |||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| L | L | s | L | L | L | s | 5L 2s | LLsLLLs | L | s | |||||||||||||||||||
| c | s | c | s | s | c | s | c | s | c | s | s | 7L 5s | cs cs s cs cs cs s | s | c = L - s | ||||||||||||||
| c | c | d | c | c | d | c | d | c | c | d | c | c | d | c | c | d | c | d | 7L 12s | ccd ccd cd ccd ccd ccd cd | d = L - 2s | c = L - s | |||||||
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 26edo | |||
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.
5L 7s, 5L 7s, 5L 12s, and 12L 5s
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.
| Step Visualization (using ionian for comparison) | Mos | Step Pattern | Large step | Small step | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| L | L | s | L | L | L | s | 5L 2s | LLsLLLs | L | s | |||||||||||||||
| c | s | c | s | s | c | s | c | s | c | s | s | 5L 7s | cs cs s cs cs cs s | c = L - s | s | ||||||||||
| d | s | s | d | s | s | s | d | s | s | d | s | s | d | s | s | s | 5L 12s | dss dss s dss dss dss s | d = L - 2s | s | |||||
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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.
| Step Visualization (using ionian for comparison) | Mos | Step Pattern | Large step | Small step | ||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| L | L | s | L | L | L | s | 5L 2s | LLsLLLs | L | s | ||||||||||||||||||||||
| c | s | c | s | s | c | s | c | s | c | s | s | 5L 7s | cs cs s cs cs cs s | c = L - s | s | |||||||||||||||||
| d | s | s | d | s | s | s | d | s | s | d | s | s | d | s | s | s | 12L 5s | dss dss s dss dss dss s | s | d = L - 2s | ||||||||||||
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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
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.
| Parent scale | 1st order child scales | 2nd order child scales | |||||
|---|---|---|---|---|---|---|---|
| Mos | Step pattern | Mos | Step pattern | Step condition | Mos | Step pattern | Step condition |
| 5L 12s | dss dss s dss dss dss s | s < d | |||||
| 5L 7s | cs cs s cs cs cs s | s < c | 17edo | sss sss s sss sss sss s | s = d | ||
| 12L 5s | dss dss s dss dss dss s | s > d | |||||
| 5L 2s | LLsLLLs | 12edo | ss ss s ss ss ss s | s = c | |||
| 7L 12s | ccd ccd cd ccd ccd ccd cd | c < d | |||||
| 7L 5s | cs cs s cs cs cs s | s > c | 19edo | ccc ccc cc ccc ccc ccc cc | c = d | ||
| 12L 7s | ccd ccd cd ccd ccd ccd cd | c > d | |||||
Combining all four tables into a scale tree reveals a few patterns:
- Sister scale pairs, such as 5L 7s and 7L 5s, are being described without a notion of which is the large or small step.
- Scales with a hard step ratio have 2nd-order child scales where the scales are described using dieses and small steps. Only the parent's small step persists as being one of the two step sizes.
- Scales with a soft step ratio have 2nd-order child scales where both the large and small step of the parent scale eventually break down into chromas and dieses. None of the parent's step sizes persist this far.
- Mos recursion becomes readily apparent, especially the chunking operation with the 2nd-generation children of a soft-step-ratio parent scale.
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.
| Parent scale | 1st orderchild scales | 2nd order child scales | |||||
|---|---|---|---|---|---|---|---|
| Mos | Step pattern | Mos | Step pattern | Step condition | Mos | Step pattern | Step condition |
| 2L 9s | dss s s s dss s s | s < d | |||||
| 2L 7s | cs s s s cs s s | s < c | 11edo | sss s s s sss s s | s = d | ||
| 9L 2s | dss s s s dss s s | s > d | |||||
| 2L 5s | LsssLss | 9edo | ss s s s ss s s | s = c | |||
| 7L 9s | ccdcd cd cd cd ccdcd cd cd | c < d | |||||
| 7L 2s | cs s s s cs s s | s > c | 16edo | ccccc cc cc cc ccccc cc cc | c = d | ||
| 9L 7s | ccdcd cd cd cd ccdcd cd cd | c > d | |||||