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, 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.

Meantone example

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 temperaments, 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 (sL-sL-L-sL-sL-sL-L and LLs-LLs-Ls-LLs-LLs-LLs-Ls respectively), and can be seen in the table below.

Step Pattern in L's and s's (using the ionian mode for comparison)																															L:s Ratio	TAMNAMS Name	Temperament
L					L					s			L					L					L					s			5:3	diatonic	meantone[7]
s		L			s		L			L			s		L			s		L			s		L			L			3:2	m-chromatic	meantone[12]
L		L		s	L		L		s	L		s	L		L		s	L		L		s	L		L		s	L		s	2:1	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, s, c, and d, rather than immediately notating which is the larger and smaller intervals. In short, in meantone[12], large steps break apart into a chroma and small-step pair, and in meantone[19], large steps break up into chroma-chroma-diesis triplets and the small steps chroma-diesis pairs.

Step Pattern in L's and s's (using the ionian mode for comparison)																															Step Ratio	TAMNAMS Name	Temperament
L					L					s			L					L					L					s			L:s = 5:3	diatonic	meantone[7]
c		s			c		s			s			c		s			c		s			c		s			s			s:c = 3:2	m-chromatic	meantone[12]
c		c		d	c		c		d	c		d	c		c		d	c		c		d	c		c		d	c		d	c:d = 2:1	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