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 and Replacement Rules 1 and 2

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

Replacement Rules 3 and 4

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

Replacement Rules 5 and 6

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)

On the Chunking Operation

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

Using Scale Codes to Sort by Modal Brightness

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.

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.

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

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. 26edo, for example, contains 1L 6s, 2L 5s, 3L 4s, 4L 3s, 5L 2s, and 6L 1s with the L:s ratios of 8:3, 8:2, 6:2, 5:2, 4:3, and 4:2 respectively. Equiheptatonic isn't included here because 26 isn't divisible by 7, meaning this list can't be circular (though a very large edo that's divisible by 7 can theoretically include all the heptatonic MOSses and equiheptatonic). Here, instead of a scale code of L's and s's, it's a 7-digit number. The largest value of L across all L:s ratios is 8 and the smallest value of s across L:s ratios is 2. Brightness values are calculated by subtracting 2 from every digit of every scale code and interpreting the resulting number as a base-7 number.

It's important to note that the ordering will vary from edo to edo, since the step ratios will be different, and that these orderings will be different from the ordering of binary encodings.

Scale code	Base-7	Decimal	MOS	UDP	MOS name	Mode name
8333333	6111111	725502	1L 6s	6|0	Anti-archeotonic	Antizokalarian
8228222	6006000	707952	2L 5s	6|0	Antidiatonic	Antilocrian
8222822	6000600	706188	2L 5s	5|1	Antidiatonic	Antiphrygian
6262622	4040400	480396	3L 4s	6|0	Mosh	Dril
6262262	4040040	480228	3L 4s	5|1	Mosh	Gil
6226262	4004040	471996	3L 4s	4|2	Mosh	Kleeth
5525252	3303030	404418	4L 3s	6|0	Smitonic	Nerevarine
5255252	3033030	361200	4L 3s	5|1	Smitonic	Vivecan
5252552	3030330	360318	4L 3s	4|2	Smitonic	Lorkhanic
5252525	3030303	360300	4L 3s	3|3	Smitonic	Sothic
4444442	2222220	274512	6L 1s	6|0	Archeotonic	Ryonian
4444424	2222202	274500	6L 1s	5|1	Archeotonic	Karakalian
4444244	2222022	274416	6L 1s	4|2	Archeotonic	Lobonian
4443443	2221221	274170	5L 2s	6|0	Diatonic	Lydian
4442444	2220222	273828	6L 1s	3|3	Archeotonic	Horthathian
4434443	2212221	272112	5L 2s	5|1	Diatonic	Ionian
4434434	2212212	272106	5L 2s	4|2	Diatonic	Mixolydian
4424444	2202222	269712	6L 1s	2|4	Archeotonic	Oukranian
4344434	2122212	257700	5L 2s	3|3	Diatonic	Dorian
4344344	2122122	257658	5L 2s	2|4	Diatonic	Aeolian
4244444	2022222	240900	6L 1s	1|5	Archeotonic	Tamashian
3833333	1611111	221292	1L 6s	5|1	Anti-archeotonic	Antitamashian
3444344	1222122	156816	5L 2s	1|5	Diatonic	Phrygian
3443444	1221222	156522	5L 2s	0|6	Diatonic	Locrian
3383333	1161111	149262	1L 6s	4|2	Anti-archeotonic	Anti-oukranian
3338333	1116111	138972	1L 6s	3|3	Anti-archeotonic	Antihorthathian
3333833	1111611	137502	1L 6s	2|4	Anti-archeotonic	Antilobonian
3333383	1111161	137292	1L 6s	1|5	Anti-archeotonic	Antikarakalian
3333338	1111116	137262	1L 6s	0|6	Anti-archeotonic	Antiryonian
2822822	600600	101136	2L 5s	4|2	Antidiatonic	Anti-aeolian
2822282	600060	100884	2L 5s	3|3	Antidiatonic	Antidorian
2626262	404040	68628	3L 4s	3|3	Mosh	Bish
2626226	404004	68604	3L 4s	2|4	Mosh	Fish
2622626	400404	67428	3L 4s	1|5	Mosh	Jwl
2552525	330303	57774	4L 3s	2|4	Smitonic	Kagrenacan
2525525	303303	51600	4L 3s	1|5	Smitonic	Almalexian
2525255	303033	51474	4L 3s	0|6	Smitonic	Dagothic
2444444	222222	39216	6L 1s	0|6	Archeotonic	Zokalarian
2282282	60060	14448	2L 5s	2|4	Antidiatonic	Antimixolydian
2282228	60006	14412	2L 5s	1|5	Antidiatonic	Anti-ionian
2262626	40404	9804	3L 4s	0|6	Mosh	Led
2228228	6006	2064	2L 5s	0|6	Antidiatonic	Antilydian

On 1L 1s, 1L 2s, and 2L 1s

This is an attempt to describe 1L 1s (prototonic/protic), 1L 2s (antideuterotonic/antideuteric), and 2L 1s (deuterotonic/deuteric). These descriptions are only for completeness.

This includes non-official TAMNAMS names that are based on existing names and are used solely for completeness:

• 1L 1s - protic or prototonic, named such for being the progenitor scale for all single-period mosses
• 2L 1s - deuteric or deuterotonic, named such for being the children of 1L 1s, and thus being part of the second generation of scales; likewise, 1L 2s may be named antideuteric or antideuterotonic
• 3L 1s - tetric, named based on the names pentic and antipentic; likewise, 1L 3s may be named antitetric

1L 1s

1L 1s is the mos pattern Ls. Though not a scale in its own right, it has the unique designation of being the parent scale of all possible single-period mosses. Its daughter mosses are 1L 2s and 2L 1s, and it's through these that more familiar mosses start to take shape, especially the pentatonic mosses 2L 3s and 3L 2s. It's also the only (single-period) scale that is its own sister.

Modes

1L 1s only has two modes.

Step Pattern	UDP	0-step	1-step	2-step
Ls	1|0	0	L	L+s
sL	0|1	0	s	L+s

1L 2s

1L 2s is the mos pattern Lss. Though not a scale in its own right, it's the progenitor scale of two of the tetratonic mosses - 1L 3s and 3L 1s, or antitetric and tetric.

Modes

Step Pattern	UDP	0-step	1-step	2-step	3-step
Lss	2|0	0	L	L+s	L+2s
sLs	1|1	0	s	L+s	L+2s
ssL	0|2	0	s	2s	L+2s

2L 1s

2L 1s is the mos pattern LLs. Though not a scale in its own right, it's the progenitor scale of two of the pentatonic mosses - 2L 3s and 3L 2s, or pentic and antipentic.

Modes

Step Pattern	UDP	0-step	1-step	2-step	3-step
LLs	2|0	0	L	2L	2L+s
LsL	1|1	0	L	2L	2L+s
sLL	0|2	0	s	L+s	2L+s