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).
Chroma-diesis model of mos child scales
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 both a temperament-related context and a temperament-agnostic context. (yay abuse of terminology)
7L 5s and 12L 7s (meantone temperament)
31edo is used as an arguably noteworthy example of an 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:
- 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.
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.
| 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 | |||
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 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.
| 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] | ||||||||||||||||||||||||
| c | s | c | s | s | c | s | c | s | c | s | s | 7L 5s | cs cs s cs cs cs s | m-chromatic | meantone[12] | |||||||||||||||||||
| 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 | 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 | |||
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.
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".
Including 7L 12s (flattone temperament)
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. However, it's still possible to describe 7L 12s in terms of chromas and dieses.
| Step Visualization (using ionian mode for comparison) | Mos | Step Pattern | TAMNAMS Name | Temperament | |||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| L | L | s | L | L | L | s | 5L 2s | LLsLLLs | diatonic | flattone[7] | |||||||||||||||||||
| c | s | c | s | s | c | s | c | s | c | s | s | 7L 5s | cs cs s cs cs cs s | m-chromatic | flattone[12] | ||||||||||||||
| 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 | unnamed | flattone[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 | 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.
Including 5L 7s, 5L 12s, and 12L 5s (pythagorean temperaments)
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 meantone and flattone temperaments), hard step ratios produce chromas that are larger than the small step.