User:Ganaram inukshuk/Methodologies

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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 descendant mosses

For a mos xL ys, its two child mosses can be described using replacement rulesets: L->Ls and s->s, producing xL (x+y)s, and L->sL and s->s, producing (x+y)L xs. These two rulesets can be applied to the two child mosses again to produce further descendants. However, the resulting descendant mosses are denoted using L's and s's, whose relationship to the parent mos's step sizes may be unclear. This section describes how descendant mosses' step sizes relate back to the parent.

Replacement rules of descendants

Since replacement rules apply to any mos, it's easier to consider applying these rules to the mos 1L 1s, a single large and small step.

  • L->Ls, s->s (ruleset 1); apply to a mos in its brightest mode to produce a child mos in its brightest mode
  • L->sL, s->L (ruleset 2); apply to a mos in its brightest mode to produce a child mos in its darkest mode

Applying these rules to the mos pattern produces its children, and applying them repeatedly produces further descendants; shown below is three applications for three generations of descendants.

Mos Step pattern Mos Step pattern Mos Step pattern Mos Step pattern
1L 1s L s 1L 2s Ls s 1L 3s Lss s 1L 4s Lss s
4L 1s sLL L
3L 1s sLL L 3L 4s ssLsL sL
4L 3s LLsLs Ls
2L 1s sL L 2L 3s ssL sL 2L 5s sssL ssL
5L 2s LLLs LLs
3L 2s LLs Ls 3L 5s LsLss Lss
5L 3s sLsLL sLL

Adding a separator shows what happens to a single L and s, effectively creates production rules for producing all 14 possible mos descendants.

Parent mos 1st descendants 2nd descendants 3rd descendants
Mos Production rules Mos Production rules Mos Production rules Mos Production rules
xL ys none xL (x+y)s L->Ls

s->s

xL (2x+y)s L->Lss

s->s

xL (3x+y)s L->Lss

s->s

(3x+y)L xs L->sLL

s->L

(2x+y)L xs L->sLL

s->L

(2x+y)L (3x+y)s L->ssLsL

s->sL

(3x+y)L (2x+y)s L->LLsLs

s->Ls

(x+y)L xs L->sL

s->L

(x+y)L (2x+y)s L->ssL

s->sL

(x+y)L (3x+2y)s L->sssL

s->ssL

(3x+2y)L (x+y)s L->LLLs

s->LLs

(2x+y)L (x+y)s L->LLs

s->Ls

(2x+y)L (3x+2y)s L->LsLss

s->Lss

(3x+2y)L (2x+y)s L->sLsLL

s->sLL

1st descendants

Since a chroma is the difference between a large and small step, 1st descendants have the large step of the parent break into a chroma and small step. Recalling that this small step refers to the parent's rather than the descendant's, and that whichever is larger depends on how big L and s are, the two rulesets for 1st descendants can be condensed into one ruleset. Using a rectangular horogram helps to illustrate what happens.

Mos Relation Step pattern

(1L 1s)

Step pattern with chromas Final ruleset
xL ys Parent L s L s
xL (x+y)s Child 1 L s s c s s L->cs, s->s
(x+y)L xs Child 2 s L L

2nd descendants

To get replacement rules for 2nd descendants, we can extend the horograms produced before. However, this results into two horograms since the step counts will be different. To deduce what these steps are in relation to the parent, we introduce a second-order chroma: a diesis, defined as d = | L - 2s |. For ruleset 1+1 and 1+2, a large step breaks down into two small steps and a diesis. Whichever is larger depends on the step ratio of the parent, condensing the rulesets 1+1 and 1+2 into one ruleset.

Mos Relation Step pattern Final rulesets
L's and s's s's and c's
xL ys Parent L s L s
xL (x+y)s Child 1 L s s c s s
xL (2x+y)s Grandchild 11 L s s s c s s s L->css, s->s
(2x+y)L xs Grandchild 12 s L L L
Mos Relation Step pattern Final rulesets
L's and s's c's and d's
xL ys Parent L s L s
(x+y)L xs Child 2 s L L c s s
(x+y)L (2x+y)s Grandchild 21 s s L s L c c d c d L->ccd, s->cd
(2x+y)L (x+y)s Grandchild 22 L L s L s

3rd descendants (wip)

To get replacement rules for 3rd descendants, the previous horograms can be extended as before, as well as adding a third-order chroma: a kleisma (or, for this page, an epsilon), defined as e = | L - 3s |.

Mos Relation Step pattern Final rulesets
L's and s's s's and c's
xL ys Parent L s L s
xL (x+y)s Child 1 L s s c s s
xL (2x+y)s Grandchild 11 L s s s c s s s
Mos Relation Step pattern Final rulesets
L's and s's s's and c's
xL ys Parent L s L s
xL (x+y)s Child 1 L s s c s s
(2x+y)L xs Grandchild 12 s L L L c s s s
Mos Relation Step pattern Final rulesets
L's and s's c's and d's
xL ys Parent L s L s
(x+y)L xs Child 2 s L L c s s
(x+y)L (2x+y)s Grandchild 21 s s L s L c c d c d
Mos Relation Step pattern Final rulesets
L's and s's c's and d's
xL ys Parent L s L s
(x+y)L xs Child 2 s L L c s s
(2x+y)L (x+y)s Grandchild 22 L L s L s c c d c d

Examples with 5L 2s

My current approach to xenharmony, at least as it pertains to being both familiar and different, is to play with the familiar mos pattern of 5L 2s but to add additional notes in between. If restricted to 12edo, this means one of two things to me:

  • Using the 12-note chromatic scale.
  • Using half-sharps and half-flats.

For me, this is antithetical to how I explore mosses, as well as being downright boring. Using the idea of mosses, an entire infinite family tree of possible scales can be explored, all containing the familiar 5L 2s as a subset. Some scales are even backwards-compatible with existing notation.

The following table is a table of all the mosdescendants of 5L 2s up to 3 generations. (Names are based on my version of TAMNAMS name extension; some names are not finalized). Mosses in bold denote mosses I've played enough with to make a fair amount of notes, at least enough to understand how they relate back to 5L 2s. Also listed are basic edos (the smallest edo needed to realize the given mos) and a few temperaments for reference.

Diatonic scale Chromatic scales Enharmonic scales "Schismic" scales
Steps Bright generator Basic edo Temperament Steps Specific name Bright generator

(relative to 5L 2s)

Basic edo Temperament Steps Specific name Bright generator

(relative to 5L 2s)

Basic edo Temperament Steps Specific name Bright generator

(relative to 5L 2s)

Basic edo Temperament
5L 2s Perfect 5th 12edo 5L 7s p-moschromatic Perfect 5th 17edo 5L 12s s-mosenharmonic Perfect 5th 22edo 5L 17s s-mosschismic Perfect 5th 27edo
17L 5s r-mosschismic Perfect 4th 39edo
12L 5s p-mosenharmonic Perfect 4th 29edo 12L 17s p-mosschismic Perfect 4th 41edo
17L 12s q-mosschismic Perfect 5th 46edo
7L 5s m-moschromatic Perfect 4th 19edo meantone[12] 7L 12s f-mosenharmonic Perfect 4th 26edo flattone[19] 7L 19s f-mosschismic Perfect 4th 33edo
19L 7s a-mosschismic Perfect 5th 45edo flattone[26]
12L 7s m-mosenharmonic Perfect 5th 31edo meantone[19] 12L 19s m-mosschismic Perfect 5th 43edo
19L 12s u-mosschismic Perfect 4th 50edo meantone[31]

The following list is some commentary on some of the basic edos shown in the table:

  • 19edo: I've seen one musician describe it as a baby version of 31edo, and I'm inclined to agree in that it's basically an equalized 12L 7s. It also introduces the idea that a note raised by two chromas can fall short of the next note one whole tone above it (for example, Cx falling short of D).
  • 31edo: This, and 50edo to an extent, is currently my go-to edo for meantone temperament. Generally speaking, I like the mellow sound of meantone, which explains why I do absolutely nothing to date with 17edo and friends. It's also compatible with half-sharp and half-flat notation, so "quartertone" compositions are possible.
  • 45edo: Like 19edo but more extreme. Whereas 19edo has Cx falling short of D but C#x being equivalent to D, in 45edo, C#x falls short of D, but the interval between C#x and D is its own smaller interval I'm dubbing the triesis for the purposes of this page.

7L 5s and 12L 7s

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. Basically, a chroma and small step are not the same size, more commonly described with the terms "diatonic and chromatic semitones". This unevenness manifests itself as the m-chromatic scale, a scale I admittedly don't use as often compared to 12L 7s. I see this mos as an incomplete form of 12L 7s. Nonetheless, a mode table is still provided, providing a guide as to how these 12 modes relate back to the more familiar 7 modes.

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 on top of the previous 12. In either case, it's easier to think of these modes as some sort of extension of the seven modes of 5L 2s. There are, for example, 13 modes of 12L 7s that contain ionian as a subset (UDPs 17|1 to 5|13), so it's confusing and unhelpful to say there are 13 ionian modes. Therefore, saying that the large steps break down in a specific order (chroma-chroma-diesis, chroma-diesis-chroma, or diesis-chroma-chroma, or simply CCD, CDC, and DCC) helps to narrow down ionian-containing modes to 3 specific 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

The resulting mos is dubbed m-enharmonic, or commonly (and simply) 12L 7s. Here, two chromas undershoot a whole tone, and the difference between two chromas and a whole tone is an even smaller interval: the diesis. This coincides with a lot of theory related to meantone, but my favorite use is to use it as a "super-leading" tone. In the key of C, B is considered the leading tone; in comparison, I define a "super-leading tone" as any note that's a diesis below another note. By that definition, every tone in 5L 2s has a super-leading tone.

CCD ionian is a prime example in that every tone in 5L 2s contains this "super-leading" tone. For example, Cx is the super-leading tone for D. This also works in reverse, as with falling from Dbb to C; a prime example for that is DCC ionian. Alternating between the two can achieve a sense of almost approaching something.

Of course, there is always the option of creating a melody using the "notes between the notes between the notes", creating something that sounds like it should work normally but is extremely constrained.

todo:examples

19L 7s

Whereas a diesis is defined as d = | L - 2s |, a triesis is defined as t = | L - 3s |. As a summary, 19L 7s contains all the super-leading tone goodness I expect from 12L 7s, but with an additional step (literally) to get there.

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