User:Ganaram inukshuk/Notes

This page is for miscellaneous xen-related notes that I've written about but don't have an exact place elsewhere on the wiki (yet).

On the Origin of MOS Recursion

MOS recursion describes a set of properties that all moment-of-symmetry scales share that, among other things, allows us to create a few algorithms for determining whether an arbitrary scale of large and small steps has those properties.

The child scale of a MOS follows a distinct pattern in which the large step breaks up into the next large step and the next small step (in some order) and the small step becomes either the next large step or the next small step. As such, we can represent this as two sets of replacement rules:

1. Replacement ruleset 1 (where L - s > s)
   • L -> Ls
   • s -> s
2. Replacement ruleset 2 (where L - s < s)
   • L -> sL
   • s -> L

It should be noted that if the order of L's and s's is reversed (for example, L->sL and s->s for ruleset 1), the rulesets are still valid. The numbering of rulesets is also arbitrary. For explanation purposes, rulesets 1 and 2 and successive pairs of rulesets are denoted as though they were sisters of one another; this sistering process can be described with its own ruleset:

• L->s
• s->L

Applying ruleset 1 to itself n times produces ruleset 3, where L produces an L followed by n s's:

• L->Lss...ss (n s's)
• s->s

As such, applying ruleset 1 to itself n-1 times will result in L producing an L and n-1 s's. If ruleset 2 is applied to this, it creates ruleset 4, where L produces an s followed by n L's, the sister of ruleset 3:

• L->sLL...LL (n L's)
• s->L

Reversing the order of L's and s's of ruleset 2 produces this intermediate ruleset:

• L->Ls
• s->L

Applying ruleset 1 to the reversed form of ruleset 2 n times produces ruleset 5, where L produces an L followed by n+1 s's and s produces an L followed by n s's:

• L->Lss...ss (n+1 s's)
• s->Lss...s (n s's)

Applying ruleset 2 to ruleset 1 n times produces ruleset 6, the sister of ruleset 5 where L produces an s followed by n+1 L's and s produces an s followed by n times:

• L->sLL...LL (n+1 L's)
• s->sLL...L (n L's)

The final rulesets are as follows:

1. Replacement ruleset 1
   • L -> Ls
   • s-> s
2. Replacement ruleset 2
   • L -> sL
   • s -> L
3. Replacement ruleset 3
   • L->Lss...ss (n s's)
   • s->s
4. Replacement ruleset 4
   • L->sLL...LL (n L's)
   • s->L
5. Replacement ruleset 5
   • L->Lss...ss (n+1 s's)
   • s->Lss...s (n s's)
6. Replacement ruleset 6
   • L->sLL...LL (n+1 L's)
   • s->sLL...L (n L's)

The chunking operation is contingent on rulesets 5 and 6 since there must be two unique chunks whose string sizes differ by exactly one L or one s. Repeatedly applying these rules as reduction rules on a valid moment-of-symmetry scale will reduce the scale to a progenitor scale of either Ls or sL. The reduction rules can be denoted as such:

• Reduction ruleset 5
  • Lss...ss (n+1 s's) -> L
  • Lss...s (n s's) -> s
• Reduction ruleset 6
  • sLL...LL (n+1 L's) -> L
  • sLL...L (n L's) -> s

However, it may be the case that the reduced scale has only one L or one s, or that that the scale started out this way. In either case, rulesets 5 and 6 cannot be used, but rulesets 3 and 4 can be used instead:

• Reduction ruleset 3
  • Lss...ss (n s's) -> L
  • s -> s
• Reduction ruleset 4
  • L->sLL...LL (n L's) -> L
  • L -> s

For reduction ruleset 3, the entire scale except for one s is replaced with an L. For reduction ruleset 4, all but one L is replaced with an L with the remaining L replaced with an s. This can also be thought of using reduction ruleset 2 (sL -> L and L -> s) followed by reduction ruleset 3.

Using these rules as reduction rules allows for the scale to still be reduced back down to Ls or sL.

Since all MOSses must ultimately come from a pair of generators (represented in the progenitor scale as L and s), then this proves that if an arbitrary scale can be reduced to Ls or sL, then the scale itself must be a MOS.

Note that this only applies to single-period scales; for multi-period scales, such as LLsLsLLsLs, the rules must be applied individually to each period and the resulting progenitor scale will be either Ls or sL repeated multiple times, and it cannot be a mix of both Ls and sL.

On Modal Brightness and Numeric Encoding

Modal brightness typically refers to how "bright" or "dark" the usual diatonic modes are (lydian, ionian, mixolydian, dorian, aeolian, phrygian, locrian). Since diatonic (5L 2s) is one of many moment-of-symmetry scales, the idea of modal brightness can be generalized using UDP notation.

For example, the seven modes of diatonic can be encoded as LLLsLLs, LLsLLLs, LLsLLsL, LsLLLsL, LsLLsLL, sLLLsLL, and sLLsLLL, whose UDP are 6|0, 5|1, 4|2, 3|3, 2|4, 1|5, and 0|6 respectively. The scale codes can be interpreted as binary numbers (L = 1 and s = 0), producing 1110110, 1101110, 1101101, 1011101, 1011011, 0111011, and 0110111. Doing this provides a mathematical way of understanding how modal brightness works, since larger binary values mean brighter scales.

Scale code	Binary	Decimal	MOS	UDP	MOS name	Mode name
LLLsLLs	1110110	118	5L 2s	6|0	Diatonic	Lydian
LLsLLLs	1101110	110	5L 2s	5|1	Diatonic	Ionian
LLsLLsL	1101101	109	5L 2s	4|2	Diatonic	Mixolydian
LsLLLsL	1011101	93	5L 2s	3|3	Diatonic	Dorian
LsLLsLL	1011011	91	5L 2s	2|4	Diatonic	Aeolian
sLLLsLL	0111011	59	5L 2s	1|5	Diatonic	Phrygian
sLLsLLL	0110111	55	5L 2s	0|6	Diatonic	Locrian

Therefore, to produce the modes of a MOS in descending modal brightness, start with the scale code, produce all of its possible shifts, interpret them as binary numbers, and sort them in descending order. It should be noted that the characters "L" and "s", when sorted in lexicographic order (IE, alphabetical order), equivalently represent the binary representations in descending order, so the conversion to binary numbers is technically not necessary.

As an example, consider 3L 4s represented as sLsLsLs. Its six other shifts are LsLsLss, sLsLssL, LsLssLs, sLssLsL, LssLsLs, and ssLsLsL. Sorting them produces LsLsLss, LsLssLs, LssLsLs, sLsLsLs, sLsLssL, sLssLsL, and ssLsLsL, and are enumerated using UDP notation from 6|0 to 0|6 accordingly. Again, the binary representation (and decimal forms) gives an intuitive sense of what it means for a scale to be bright. As of writing, the article on 3L 4s is written using sLsLsLs (UDP 3|3) as the "default" mode, or the mode represented using middle C as the root (or TAMNAMS middle J); in comparison, the default mode for diatonic is ionian (UDP 5|1, or LLsLLLs). UDP notation gives a sense of how many modes are brighter or darker starting from the default mode, though these sortings (and thereby binary encodings) provide that sense without any notion of a "default" mode.

Scale code	Binary	Decimal	MOS	UDP	MOS name	Mode name
LsLsLss	1010100	84	3L 4s	6|0	Mosh	Dril
LsLssLs	1010010	82	3L 4s	5|1	Mosh	Gil
LssLsLs	1001010	74	3L 4s	4|2	Mosh	Kleeth
sLsLsLs	0101010	42	3L 4s	3|3	Mosh	Bish
sLsLssL	0101001	41	3L 4s	2|4	Mosh	Fish
sLssLsL	0100101	37	3L 4s	1|5	Mosh	Jwl
ssLsLsL	0010101	21	3L 4s	0|6	Mosh	Led

Side note: there is a concept known as "cyclic permutational order" that coincides with the notion of shifts, and the only reference to it anywhere on the wiki is this page on mavila temperament.

Including the Modes of More than One MOS

As a curiosity, there are 128 possible 7-bit numbers (0000000 to 1111111) representing the unsigned integer values of 0 to 127. Among the 6 possible heptatonic MOSses (1L 6s, 2L 5s, 4L 3s, 3L 4s, 5L 2s, and 6L 1s), there are therefore 42 modes total. For our purposes, we include equiheptatonic (7 equal divisions of the octave) as being represented by both 0000000 and 1111111 (or simultaneously being both 0L 7s and 7L 0s) for a total of 43 (or 44) scales.

Though modal brightness makes more sense when thinking about the modes of a single MOS, this is how the modes of all six MOSses are ordered when sorted from highest binary encoding to smallest binary encoding:

Scale code	Binary	Decimal	MOS	UDP	MOS name	Mode name
LLLLLLL	1111111	127	7L 0s	0|0	Equiheptatonic	Equiheptatonic
LLLLLLs	1111110	126	6L 1s	6|0	Archeotonic	Ryonian
LLLLLsL	1111101	125	6L 1s	5|1	Archeotonic	Karakalian
LLLLsLL	1111011	123	6L 1s	4|2	Archeotonic	Lobonian
LLLsLLL	1110111	119	6L 1s	3|3	Archeotonic	Horthathian
LLLsLLs	1110110	118	5L 2s	6|0	Diatonic	Lydian
LLsLLLL	1101111	111	6L 1s	2|4	Archeotonic	Oukranian
LLsLLLs	1101110	110	5L 2s	5|1	Diatonic	Ionian
LLsLLsL	1101101	109	5L 2s	4|2	Diatonic	Mixolydian
LLsLsLs	1101010	106	4L 3s	6|0	Smitonic	Nerevarine
LsLLLLL	1011111	95	6L 1s	1|5	Archeotonic	Tamashian
LsLLLsL	1011101	93	5L 2s	3|3	Diatonic	Dorian
LsLLsLL	1011011	91	5L 2s	2|4	Diatonic	Aeolian
LsLLsLs	1011010	90	4L 3s	5|1	Smitonic	Vivecan
LsLsLLs	1010110	86	4L 3s	4|2	Smitonic	Lorkhanic
LsLsLsL	1010101	85	4L 3s	3|3	Smitonic	Sothic
LsLsLss	1010100	84	3L 4s	6|0	Mosh	Dril
LsLssLs	1010010	82	3L 4s	5|1	Mosh	Gil
LssLsLs	1001010	74	3L 4s	4|2	Mosh	Kleeth
LssLsss	1001000	72	2L 5s	6|0	Antidiatonic	Antilocrian
LsssLss	1000100	68	2L 5s	5|1	Antidiatonic	Antiphrygian
Lssssss	1000000	64	1L 6s	6|0	Anti-archeotonic	Antizokalarian
sLLLLLL	0111111	63	6L 1s	0|6	Archeotonic	Zokalarian
sLLLsLL	0111011	59	5L 2s	1|5	Diatonic	Phrygian
sLLsLLL	0110111	55	5L 2s	0|6	Diatonic	Locrian
sLLsLsL	0110101	53	4L 3s	2|4	Smitonic	Kagrenacan
sLsLLsL	0101101	45	4L 3s	1|5	Smitonic	Almalexian
sLsLsLL	0101011	43	4L 3s	0|6	Smitonic	Dagothic
sLsLsLs	0101010	42	3L 4s	3|3	Mosh	Bish
sLsLssL	0101001	41	3L 4s	2|4	Mosh	Fish
sLssLsL	0100101	37	3L 4s	1|5	Mosh	Jwl
sLssLss	0100100	36	2L 5s	4|2	Antidiatonic	Anti-aeolian
sLsssLs	0100010	34	2L 5s	3|3	Antidiatonic	Antidorian
sLsssss	0100000	32	1L 6s	5|1	Anti-archeotonic	Antitamashian
ssLsLsL	0010101	21	3L 4s	0|6	Mosh	Led
ssLssLs	0010010	18	2L 5s	2|4	Antidiatonic	Antimixolydian
ssLsssL	0010001	17	2L 5s	1|5	Antidiatonic	Anti-ionian
ssLssss	0010000	16	1L 6s	4|2	Anti-archeotonic	Anti-oukranian
sssLssL	0001001	9	2L 5s	0|6	Antidiatonic	Antilydian
sssLsss	0001000	8	1L 6s	3|3	Anti-archeotonic	Antihorthathian
ssssLss	0000100	4	1L 6s	2|4	Anti-archeotonic	Antilobonian
sssssLs	0000010	2	1L 6s	1|5	Anti-archeotonic	Antikarakalian
ssssssL	0000001	1	1L 6s	0|6	Anti-archeotonic	Antiryonian
sssssss	0000000	0	0L 7s	0|0	Equiheptatonic	Equiheptatonic

Note that since both 0000000 and 1111111 both represent the same scale (equiheptatonic), this entire list is circular, so mathematically, there can't be a "globally" brightest mode. Also, this represents 44 out of 128 possible binary numbers, with the rest being MODMOSses of existing scales. Including all the MODMOSses based on just two step sizes (L and s) produces a diagram such as this by User:Xenoindex.

Including Assigned Values for L and s

So far, the previous table represented scales where the values for L and s are unassigned. However, a large enough edo can contain all six heptatonic MOSses with different step ratios. (TODO: expand)