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 an regular temperament context and 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.

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".

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.

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.

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)

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.

Step Visualization (using ionian for comparison)																						Mos	Step Pattern	TAMNAMS Name	Temperament
L				L				s	L				L				L				s	5L 2s	LLsLLLs	diatonic	superpyth[7]
c			s	c			s	s	c			s	c			s	c			s	s	5L 7s	cs cs s cs cs cs s	p-chromatic	superpyth[12]
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	unnamed	superpyth[17]
1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	22edo

Step Visualization (using ionian for comparison)																													Mos	Step Pattern	TAMNAMS Name	Temperament
L					L					s		L					L					L					s		5L 2s	LLsLLLs	diatonic	leapfrog[7]
c			s		c			s		s		c			s		c			s		c			s		s		5L 7s	cs cs s cs cs cs s	p-chromatic	leapfrog[12]
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	unnamed	leapfrog[17]
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

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.

Parent scale		1st orderchild 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	17n-edo	sss sss s sss sss sss s	s = d
					12L 5s	dss dss s dss dss dss s	s > d
5L 2s	LLsLLLs	12n-edo	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	19n-edo	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 scale 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	11n-edo	sss s s s sss s s	s = d
					9L 2s	dss s s s dss s s	s > d
2L 5s	LsssLss	9n-edo	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	16n-edo	ccccc cc cc cc ccccc cc cc	c = d
					9L 7s	ccdcd cd cd cd ccdcd cd cd	c > d