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 various mosses, with extended scale names

This is an attempt to describe various mosses that I feel are worth describing, based on experimenting with these scales or for completion. This contains unofficial scale names that try to be as close to existing names as possible and are not meant to be official or standard. The following table shows single-period mosses sorted by generation.

Extended names are denoted with an asterisk. Named 1L ns (monolarge) scales are denoted using italics and are based on its sister scale with the anti- prefix added.

This section may change over time due to TAMNAMS names being updated. Outdated proposals are kept here for archival purposes.

Mos Family Tree (single-period only), with TAMNAMS Names and extended names
Progenitor scale		1st-order child mosses		2nd-order child mosses		3rd-order child mosses		4th-order child mosses		5th-order child mosses
Steps	Scale name	Steps	Scale name	Steps	Scale name	Steps	Scale name	Steps	Scale name	Steps	Scale name
1L 1s	monowood	1L 2s	antideuteric*	1L 3s	antitetric*	1L 4s	antimanic	1L 5s	antimachinoid	1L 6s	anti-archeotonic
										6L 1s	archeotonic
								5L 1s	machinoid	5L 6s
										6L 5s
						4L 1s	manic	4L 5s	orwelloid	4L 9s
										9L 4s
								5L 4s	semiquartal	5L 9s
										9L 5s
				3L 1s	tetric	3L 4s	mosh	3L 7s	jenovic (formerly sephiroid)	3L 10s
										10L 3s
								7L 3s	neutertonic (formerly dicotonic)	7L 10s
										10L 7s
						4L 3s	smitonic	4L 7s	(formerly kleistonic)	4L 11s
										11L 4s
								7L 4s	(formerly suprasmitonic)	7L 11s
										11L 7s
		2L 1s	trial	2L 3s	pentic	2L 5s	antidiatonic	2L 7s	joanatonic	2L 9s
										9L 2s
								7L 2s	superdiatonic	7L 9s
										9L 7s
						5L 2s	diatonic	5L 7s	p-chromatic*	5L 12s	s-enharmonic*
										12L 5s	p-enharmonic*
								7L 5s	m-chromatic*	7L 12s	f-enharmonic*
										12L 7s	m-enharmonic*
				3L 2s	antipentic	3L 5s	sensoid	3L 8s		3L 11s
										11L 3s
								8L 3s		8L 11s
										11L 8s
						5L 3s	oneirotonic	5L 8s		5L 13s
										13L 5s
								8L 5s		8L 13s
										13L 8

1L 1s, 1L 2s, and 2L 1s, and their child mosses

As of August 14, 2022, all of these scales have been named, except for 1L 2s and 1L 3s as they are 1L ns. These descriptions are kept for archival purposes.

Parent scale				1st-order child scales				2nd-order child scales
Steps	Originally proposed name	Actual name	Notes	Steps	Originally proposed name	Actual name	Notes	Steps	Originally proposed name	Actual name	Notes
1L 1s	prototonic, protic, or monowood	monowood	The progenitor scale of all single-period mosses. Despite being a monolarge scale, it's also its own sister and is named regardless. The current name "monowood" comes from nL ns scales (such as pentawood for 5L 5s).	1L 2s	antideuterotonic or antideuteric	-	One of the child scales of 1L 1s. Being a monolarge scale, tetric (3L 1s) may be more worth considering as a parent scale.	1L 3s	antitetric	-	Monolarge scale. Similarly to 3L 1s with 1L 2s, 4L 1s may be worth considering as a parent scale.
								3L 1s	tetric	tetric	Parent scale to orwelloid and semiquartal, the name tetric is assigned similarly to pentic being the parent of diatonic and antidiatonic. (If the hand analogy for manic, or 4L 1s, still holds, consider that cartoon characters are usually depicted with 4 digits on their hands instead of 5.)
				2L 1s	deuterotonic or deuteric	trial	One of the child scales of 1L 1s.	2L 3s	-	pentic	Already established name.
								3L 2s	-	antipentic	Already established name.

5L 7s, 7L 5s, and their child mosses

5L 2s has two children: 5L 7s and 7L 5s, or formerly p- and m-chromatic respectively. This section names the second-order children (or granddaughters) of 5L 2s. This pattern may also be generalized to any parent mos; see User:Frostburn/TAMNAMS Extension.

Parent scale			1st-order child scales			2nd-order child scales
Steps	Name	Notes	Steps	Name	Notes	Steps	Name	Notes
5L 2s	diatonic	Already established name.	5L 7s	p-chromatic	Names are based on former names of these mosses: p-chromatic and m-chromatic. If the distinction between p- and m- isn't needed, both scales may collectively be referred to as "chromatic".	5L 12s	s-enharmonic	Names are based on discussions with xen discord members on attempting to name granddaughter scales in a systematic way. If the distinction between s-, p-, f-, and m- isn't needed, all four scales may collectively be referred to as "enharmonic".
						12L 5s	p-enharmonic
			7L 5s	m-chromatic		7L 12s	f-enharmonic
						12L 7s	m-enharmonic

Mode matrix, interval matrix, and degree matrix

Mode matrix

The notion of an interval matrix is already well-described, but not so much the idea of a mode matrix nor producing an interval matrix from a mode matrix. This is based on the idea of sorting the strings for a mos's modes in lexicographic order to equivalently sort its modes by modal brightness, so pulling from that section, we start with the modes of 5L 2s sorted by modal brightness as an example:

Binary	UDP	Mode name	Scale string
1110110	6|0	Lydian	LLLsLLs
1101110	5|1	Ionian	LLsLLLs
1101101	4|2	Mixolydian	LLsLLsL
1011101	3|3	Dorian	LsLLLsL
1011011	2|4	Aeolian	LsLLsLL
0111011	1|5	Phrygian	sLLLsLL
0110111	0|6	Locrian	sLLsLLL

A mode matrix for this is is a 7x7 matrix, consisting of only a single L or a single s in each entry, where each row vector corresponds to one of the mos's modes.

Scale string	Mode name	Step 1 (c1)	Step 2 (c2)	Step 3 (c3)	Step 4 (c4)	Step 5 (c5)	Step 6 (c6)	Step 7 (c7)
LLLsLLs	Lydian	L	L	L	s	L	L	s
LLsLLLs	Ionian	L	L	s	L	L	L	s
LLsLLsL	Mixolydian	L	L	s	L	L	s	L
LsLLLsL	Dorian	L	s	L	L	L	s	L
LsLLsLL	Aeolian	L	s	L	L	s	L	L
sLLLsLL	Phrygian	s	L	L	L	s	L	L
sLLsLLL	Locrian	s	L	L	s	L	L	L

Interval matrix

An interval matrix can be defined as the following: for an nxn mode matrix, its column matrix consists of n+1 columns and n rows. For our example, our interval matrix contains 8 columns and 7 rows. Recall that L and s not only stand for characters in a string, but are also in place for actual numbers. Each column vector in the interval matrix represents the sum of consecutive column vectors from the mode matrix; specifically, if the mode matrix's column vectors are enumerated as c1, c2, to cn, then the column vectors of the interval matrix are c1, c1+c2, c1+c2+c3, and so on to c1+c2+c3+...+cn.

An additional column is added before the column of seconds, as these are the roots of the scale. The last column represents an interval produced between the root an the same note one octave above, and all entries in this column are the same size.

For the mode matrix above, the interval matrix can then be calculated as this:

String	Mode	Unison (empty substring)	Second (c1)	Third (c1+c2)	Fourth (c1+c2+c3)	Fifth (c1+...+c4)	Sixth (c1+...+c5)	Seventh (c1+...+c6)	Octave (c1+...+c7)
LLLsLLs	Lydian	0	L	2L	3L	3L + s	4L + s	5L + s	5L + 2s
LLsLLLs	Ionian	0	L	2L	2L + s	3L + s	4L + s	5L + s	5L + 2s
LLsLLsL	Mixolydian	0	L	2L	2L + s	3L + s	4L + s	4L + 2s	5L + 2s
LsLLLsL	Dorian	0	L	L + s	2L + s	3L + s	4L + s	4L + 2s	5L + 2s
LsLLsLL	Aeolian	0	L	L + s	2L + s	3L + s	3L + 2s	4L + 2s	5L + 2s
sLLLsLL	Phrygian	0	s	L + s	2L + s	3L + s	3L + 2s	4L + 2s	5L + 2s
sLLsLLL	Locrian	0	s	L + s	2L + s	2L + 2s	3L + 2s	3L + 4s	5L + 2s

Degree matrix

Curiously, since the mode matrix consists of only two values, this makes it a logical (or binary) matrix. Likewise, the interval matrix can be converted into a logical interval matrix as such: for each column vector (except for the first and last), the larger of the two values is replaced with 1 and the smaller with 0. The first column vector is all zeros, and the last all ones (though this convention is arbitrary as these two columns are technically not needed). This in turn describes scale degrees as being major or minor, or in the case of the generating intervals, augmented, perfect, or diminished. (The unison and octave are both perfect.)

String	Mode	d0	d1	d2	d3	c4	c5	c6	c7
LLLsLLs	Lydian	0	1	1	1	1	1	1	1
LLsLLLs	Ionian	0	1	1	0	1	1	1	1
LLsLLsL	Mixolydian	0	1	1	0	1	1	0	1
LsLLLsL	Dorian	0	1	0	0	1	1	0	1
LsLLsLL	Aeolian	0	1	0	0	1	0	0	1
sLLLsLL	Phrygian	0	0	0	0	1	0	0	1
sLLsLLL	Locrian	0	0	0	0	0	0	0	1

This also describes a property of the generating intervals: they appear in one specific size in all but one mode; for 5L 2s's perfect 4th, it appears as the smaller size except in the lydian mode where it appears as an augmented 4th, and for the perfect 5th, it appears as the larger size except in the locrian mode where it's a diminished 5th.

Interpreting UDP as two mode enumeration methods

UDP notation is one of many mode notation systems that primarily focuses on how to organize the modes of a mos by modal brightness. This notation necessarily requires the notation to distinguish between the chroma-positive and chroma-negative generators of a mos. One issue with this focus on only its chroma-positive generator is that the generators may "flip". As an example, 5L 2s is said to have a perfect 5th as its generator, but although 2L 3s (the pentatonic scale) is said to have a perfect 4th as its chroma-positive generator, it's common to think of its generator as a perfect 5th regardless.

Modes of 5L 2s
UDP	Mode names	Scale degrees (starting at C)
		1st	2nd	3rd	4th	5th	6th	7th
6|0	Lydian	C	D	E	F#	G	A	B
5|1	Ionian	C	D	E	F	G	A	B
4|2	Mixolydian	C	D	E	F	G	A	Bb
3|3	Dorian	C	D	Eb	F	G	A	Bb
2|4	Aeolian	C	D	Eb	F	G	Ab	Bb
1|5	Phrygian	C	Db	Eb	F	G	Ab	Bb
0|6	Locrian	C	Db	Eb	F	Gb	Ab	Bb

Modes of 2L 3s
UDP	Mode "names"	Scale degrees (independent of 5L 2s)					Scale degrees (in relation to 5L 2s)
		0d	1d	2d	3d	4d	1st	2nd	3rd	4th	5th	6th	7th
4|0	Pentatonic Phrygian (default mode for sake of example)	J	K	L	M	N	C	-	Eb	F	-	Ab	Bb
3|1	Pentatonic Aeolian (minor pentatonic)	J	K	L	[email protected]	N	C	-	Eb	F	G	-	Bb
2|2	Pentatonic Dorian	J	[email protected]	L	[email protected]	N	C	D	-	F	G	-	Bb
1|3	Pentatonic Mixolydian	J	[email protected]	L	[email protected]	[email protected]	C	D	-	F	G	A	-
0|4	Pentatonic Ionian (major pentatonic)	J	[email protected]	[email protected]	[email protected]	[email protected]	C	D	E	-	G	A	-

This ironically means that major pentatonic is the darkest mode of 2L 3s, though this irony comes from specifying which generator is which.

UDP notation denotes how a scale is produced in terms of how many chroma-positive generators going up (u) and down (d) are needed, notated as "u|d". This can also be interpreted as how many chroma-negative generators are needed going down (d') and up (u'), where the notation is otherwise identical (since d' = u and u' = d). As of writing, TAMNAMS has a proposed mode-naming scheme that drops the number of generators going down, where modes are notated as "u|" instead. An equivalent system that favors a chroma-negative generator can thereby be notated as "|d". In relation to UDP, this is basically the notation of "u|d" separated into two: "u|" and "|d".

In the case of the modes of 2L 3s, even though the perfect 4th is the chroma-positive generator, enumerating modes either using standard UDP notation ("u|d") or the proposed TAMNAMS mode-naming scheme ("u|") and sorting by brightness results in mode 0|4 as being the "last" mode, whereas notating modes as "|d" notates mode 0|4 as the first mode.

This notion of favoring a generator can also extend to mosses that come after a specific mos, such as the chromatic mosses of 5L 7s and 7L 5s for 5L 2s, where the chroma-positive generators (relative to 5L 2s) are the perfect 5th and perfect 4th respectively, though it may be possible to think of the generator of either mos as being the perfect 5th regardless.