User:Ganaram inukshuk/Methodologies

This page is for describing how I approach various xen-related topics. It's neither definitive nor pedagogical, rather it's insight into how I understand certain concepts and put them to use. (Mileage may vary.)

The default mode of a mos

By convention, the two main scales that most musicians talk about are the major and minor scales, or the ionian and aeolian modes. This convention is not something most musicians may agree on, but it provides a starting point when talking about music in a more general sense or in a context that is not as technical. Thus, when looking at a mos that falls outside the familiar diatonic scale (5L 2s), can this notion of a major and minor scale be generalized? This section shows my attempt at answering this question, with justification as to why.

Examples using heptatonic mosses

As an example, let's use all six more heptatonic mosses: 1L 6s (onyx), 2L 5s (antidiatonic), 3L 4s (mosh), 4L 3s (smitonic), 5L 2s (diatonic), and 6L 1s (archeotonic) to get the table below. All modes are sorted by modal brightness. The familiar major and minor scales are shown in bold.

Onyx		Antidiatonic		Mosh		Smitonic		Diatonic		Archeotonic
Mode Name	Step Pattern	Mode Name	Step Pattern	Mode Name	Step Pattern	Mode Name	Step Pattern	Mode Name	Step Pattern	Mode Name	Step Pattern
Antizokalaraian	Lssssss	Antilocrian	LssLsss	Dril	LsLsLss	Nerevarine	LLsLsLs	Lydian	LLLsLLs	Ryonian	LLLLLLs
Antitamashian	sLsssss	Antiphrygian	LsssLss	Gil	LsLssLs	Vivecan	LsLLsLs	Ionian (major)	LLsLLLs	Karakalian	LLLLLsL
Anti-oukranian	ssLssss	Anti-aeolian	sLssLss	Kleeth	LssLsLs	Lorkhanic	LsLsLLs	Mixolydian	LLsLLsL	Lobonian	LLLLsLL
Anti-horthathian	sssLsss	Antidorian	sLsssLs	Bish	sLsLsLs	Sothic	LsLsLsL	Dorian	LsLLLsL	Horthathian	LLLsLLL
Antilobonian	ssssLss	Antimixolydian	ssLssLs	Fish	sLsLssL	Kagrenacan	sLLsLsL	Aeolian (minor)	LsLLsLL	Oukranian	LLsLLLL
Antikarakalian	sssssLs	Anti-ionian	ssLsssL	Jwl	sLssLsL	Almalexian	sLsLLsL	Phrygian	sLLLsLL	Tamashian	LsLLLLL
Anti-ryonian	ssssssL	Antilydian	sssLssL	Led	ssLsLsL	Dagothic	sLsLsLL	Locrian	sLLsLLL	Zokalarian	sLLLLLL

There are a few ways to answer the default mode question. Looking at the scale pattern for ionian may provide some clues. Since it's the second-darkest mode, an easy answer is that it's the second-brightest mode. However, a more interesting answer is to say that the default bright mode is the darkest mode whose step pattern starts with L and ends with s.

As of the default dark mode, a similar easy answer is to say it's the third-darkest mode. However, a more interesting answer is to say it's the mode that starts with an L and ends with an L. However, this definition does not work with scales that more s's than L's; 1L 6s, 2L 5s, and 3L 4s have no such modes that fit this description. This definition can be amended to say that the default dark mode is the darkest mode whose step pattern starts and ends with the same step size.

Using this assumption as a guide, we can then identify the corresponding major and minor modes of these mosses below in bold. Here, an emergent pattern starts to appear.

Onyx		Antidiatonic		Mosh		Smitonic		Diatonic		Archeotonic
Mode Name	Step Pattern	Mode Name	Step Pattern	Mode Name	Step Pattern	Mode Name	Step Pattern	Mode Name	Step Pattern	Mode Name	Step Pattern
Antizokalaraian	Lssssss	Antilocrian	LssLsss	Dril	LsLsLss	Nerevarine	LLsLsLs	Lydian	LLLsLLs	Ryonian	LLLLLLs
Antitamashian	sLsssss	Antiphrygian	LsssLss	Gil	LsLssLs	Vivecan	LsLLsLs	Ionian (major)	LLsLLLs	Karakalian	LLLLLsL
Anti-oukranian	ssLssss	Anti-aeolian	sLssLss	Kleeth	LssLsLs	Lorkhanic	LsLsLLs	Mixolydian	LLsLLsL	Lobonian	LLLLsLL
Anti-horthathian	sssLsss	Antidorian	sLsssLs	Bish	sLsLsLs	Sothic	LsLsLsL	Dorian	LsLLLsL	Horthathian	LLLsLLL
Antilobonian	ssssLss	Antimixolydian	ssLssLs	Fish	sLsLssL	Kagrenacan	sLLsLsL	Aeolian (minor)	LsLLsLL	Oukranian	LLsLLLL
Antikarakalian	sssssLs	Anti-ionian	ssLsssL	Jwl	sLssLsL	Almalexian	sLsLLsL	Phrygian	sLLLsLL	Tamashian	LsLLLLL
Anti-ryonian	ssssssL	Antilydian	sssLssL	Led	ssLsLsL	Dagothic	sLsLsLL	Locrian	sLLsLLL	Zokalarian	sLLLLLL

It's important to note that there are many ways to settle on a mos's default mode, and which mode is decided on may vary between musicians and their musical goals. This is just one way to answer the default mode question. (Also, that X-pattern.)

Example using a mos with far more notes

wip

Other notes

Justification

Personal experiments with 2L 5s had led me to believe that anti-aeolian is not the default bright mode, but rather antiphrygian. (wip: examples)

Additionally, 19edo (and 31edo) supports both the diatonic and mosh mosses, where the generating interval for 3L 4s is 5L 2s's major third. This means that 3L 4s is, in a sense, 19edo's equivalent of an augmented scale. Constructing a tertian chord with C as the root results in the chord C-E-G#, an augmented chord, but constructing a tertian chord where C is the highest note requires the notes Fb and Ab. This is reached by starting at C and going down two major thirds, which matches the description of mode 4|2 of mosh (up 4 generators and down 2 generators), which is the exact mode that is described as the default mode.

Biases

I'm admittedly biased at the idea that unfamiliar scales necessarily need some form of familiar ground. The diatonic major scale is said to contain a leading tone (the "s" at the end of its step pattern), hence preserving this property was deemed necessary for how I explore (and play with) unfamiliar mosses. For this reason, my notion of a mos's "default mode" is whatever its equivalent of a major scale is.

Teasing out the properties of the diatonic minor scale is admittedly harder to justify. The current assumption is that the leading tone is lost, but the interval between the first and second degrees is still in its large form as seen diatonic major. This property only works for mosses that have more L's than s's, hence the amended definition described above. This currently remains untested.

Other possible answers

It's also possible to default to a mos's brightest mode, or its middle mode (such as the dorian mode for diatonic). This doesn't work with mosses with an even number of steps, since there will be two middle modes.

Comparison with other methods

Interestingly, this definition of a default mode has some overlap with Jake Freivald's method of enumerating a mos's modes.

My approach to 7L 5s and 12L 7s

My approach at looking at diatonic's mosdescendants of 7L 5s starts at looking at how the modes of 5L 2s can be broken down. Given a chroma is the absolute difference between a large and small step (c = | L - s |), mathematically speaking, the large steps can be broken into a chroma and small step, the small and large steps of 7L 5s. Depending on the order, this produces two ways each of the seven modes can be broken down into the m-chromatic modes, with some overlap.

	Bright generator: 13/31													Dark generator: 18/31																		UDP	M-chromatic modes
C	Dbb	C#	Db	C##	D	Ebb	D#	Eb	D##	E	Fb	E#	F	Gbb	F#	Gb	F##	G	Abb	G#	Ab	G##	A	Bbb	A#	Bb	A##	B	Cb	B#	C		Mode (CS)	Mode (SC)
	2		3			2		3			2		3			3			2		3			2		3			3			0|11	CS Lydian
	2		3			2		3			3			2		3			2		3			2		3			3			1|10	CS Ionian
	2		3			2		3			3			2		3			2		3			3			2		3			2|9	CS Mixolydian
	2		3			3			2		3			2		3			2		3			3			2		3			3|8	CS Dorian
	2		3			3			2		3			2		3			3			2		3			2		3			4|7	CS Aeolian
	3			2		3			2		3			2		3			3			2		3			2		3			5|6	CS Phrygian	SC Lydian
	3			2		3			2		3			3			2		3			2		3			2		3			6|5	CS Locrian	SC Ionian
	3			2		3			2		3			3			2		3			2		3			3			2		7|4		SC Mixolydian
	3			2		3			3			2		3			2		3			2		3			3			2		8|3		SC Dorian
	3			2		3			3			2		3			2		3			3			2		3			2		9|2		SC Aeolian
	3			3			2		3			2		3			2		3			3			2		3			2		10|1		SC Phrygian
	3			3			2		3			2		3			3			2		3			2		3			2		11|0		SC Locrian

This breakdown can be taken one step further by breaking down small steps into a chroma and diesis, where a diesis is defined as such: d = | L - 2s |. Ultimately, the large steps of 5L 2s are broken down into two chromas and a diesis, in some order. This produces the modes of 12L 7s and introduces an additional 7 modes. In either case, it's easier to think of these modes as some sort of extension of the seven modes of 5L 2s. The use of "CS/SC" and "CCD/CDC/DCC" is to enforce how these modes are broken down; there are 13 modes of 12L 7s that contain ionian as a subset (UDPs 17|1 to 5|13), so it's confusing to say there are 13 ionian modes.

	Bright generator: 18/31																		Dark generator: 13/31													UDP	M-enharmonic modes
C	Dbb	C#	Db	C##	D	Ebb	D#	Eb	D##	E	Fb	E#	F	Gbb	F#	Gb	F##	G	Abb	G#	Ab	G##	A	Bbb	A#	Bb	A##	B	Cb	B#	C		Mode (CCD)	Mode (CDC)	Mode (DCC)
	2		2		1	2		2		1	2		2		1	2		1	2		2		1	2		2		1	2		1	18|0	CCD Lydian
	2		2		1	2		2		1	2		1	2		2		1	2		2		1	2		2		1	2		1	17|1	CCD Ionian
	2		2		1	2		2		1	2		1	2		2		1	2		2		1	2		1	2		2		1	16|2	CCD Mixolydian
	2		2		1	2		1	2		2		1	2		2		1	2		2		1	2		1	2		2		1	15|3	CCD Dorian
	2		2		1	2		1	2		2		1	2		2		1	2		1	2		2		1	2		2		1	14|4	CCD Aeolian
	2		1	2		2		1	2		2		1	2		2		1	2		1	2		2		1	2		2		1	13|5	CCD Phrygian	CDC Lydian
	2		1	2		2		1	2		2		1	2		1	2		2		1	2		2		1	2		2		1	12|6	CCD Locrian	CDC Ionian
	2		1	2		2		1	2		2		1	2		1	2		2		1	2		2		1	2		1	2		11|7		CDC Mixolydian
	2		1	2		2		1	2		1	2		2		1	2		2		1	2		2		1	2		1	2		10|8		CDC Dorian
	2		1	2		2		1	2		1	2		2		1	2		2		1	2		1	2		2		1	2		9|9		CDC Aeolian
	2		1	2		1	2		2		1	2		2		1	2		2		1	2		1	2		2		1	2		8|10		CDC Phrygian
	2		1	2		1	2		2		1	2		2		1	2		1	2		2		1	2		2		1	2		7|11		CDC Locrian
	1	2		2		1	2		2		1	2		2		1	2		1	2		2		1	2		2		1	2		6|12			DCC Lydian
	1	2		2		1	2		2		1	2		1	2		2		1	2		2		1	2		2		1	2		5|13			DCC Ionian
	1	2		2		1	2		2		1	2		1	2		2		1	2		2		1	2		1	2		2		4|14			DCC Mixolydian
	1	2		2		1	2		1	2		2		1	2		2		1	2		2		1	2		1	2		2		3|15			DCC Dorian
	1	2		2		1	2		1	2		2		1	2		2		1	2		1	2		2		1	2		2		2|16			DCC Aeolian
	1	2		1	2		2		1	2		2		1	2		2		1	2		1	2		2		1	2		2		1|17			DCC Phrygian
	1	2		1	2		2		1	2		2		1	2		1	2		2		1	2		2		1	2		2		0|18			DCC Locrian