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

TAMNAMS-related notes

See User:Ganaram inukshuk/Notes/TAMNAMS.

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.

Proposal: Equave-agnostic mos names (work-in-progress)

See User:Ganaram inukshuk/TAMNAMS Extension

Other mos naming schemes

Names by large step count

Rather than name mosses related by the number of large steps they have, where the mosses are of the form xL (nx + y)s and relate back to a mos xL ys (n=0), these mosses can be described as members of a family. An example of such a family is the mos sequence 5L 2s, 5L 7s, 5L 12s, 5L 17s, etc, where each successive mos has 5 more small steps than the last. By extension, the mos 7L 5s (the sister of 5L 7s) is not seen as a member of this linear family even though it's part of the diatonic family as a whole, but rather as the start of its own linear family; put another way, the mosses 5L 2s, 5L 7s, 5L 12s, 5L 17s, etc are a subfamily within the larger diatonic family.

Mosses in a linear family are based on repeated applications of the replacement ruleset L->Ls and s->s on the initial mos, and reaching the nth member of a linear family requires the initial mos have a hard or pseudocollapsed step ratio. The child mos (x+y)L xs is the start of its own linear family, which relates back to the initial mos xL ys if the initial mos has a step ratio that is soft or pseudoequalized.

Names for these families describe a subset of a mos descendant family, and most mos families go by the name of (mos name) linear family or (mos-prefix)linear family.

Names of single-period mos linear families (work-in-progress)

Trivial families (names not based on "linear")
Mos	Name	Reasoning
1L (n+1)s	monolarge family	Represents an entire family of mosses formerly unnamed by TAMNAMS The name "monolarge" is chosen as it succinctly describes the only possible 1L family
2L (2n+1)s	bilarge family	Named analogously to the monolarge family
3L (3n+1)s	trilarge family	Named analogously to the monolarge family Prevents potential confusion with the name "tetralinear"
Families with 3 large steps
Mos	Name	Reasoning
3L (3n+2)s	apentilinear family	Named after anpentic
Families with 4 large steps
Mos	Name	Reasoning
4L (4n+1)s	manulinear family	Named after manual
4L (4n+3)s	smilinear family	Named after smitonic
Families with 5 large steps
Mos	Name	Reasoning
5L (5n+1)s	mechlinear family	Named after machinoid (prefix mech-)
5L (5n+2)s	p-linear family	Named after p-chromatic rather than diatonic, which has no prefix
5L (5n+3)s	oneirolinear family	Named after oneirotonic
5L (5n+4)s	chtonlinear family	Named after semiquartal (prefix chton-)
Families with 6 large steps
Mos	Name	Reasoning
6L (6n+1)s	archeolinear family	Named after archeotonic
6L (6n+5)s	xeimlinear family	Named after xeimtonic, a former name for 6L 5s
Families with 7 large steps
Mos	Name	Reasoning
7L (7n+1)s	pinelinear family	Named after pine
7L (7n+2)s	armlinear family	Named after superdiatonic (also called armotonic)
7L (7n+3)s	dicolinear family	Named after dicotonic
7L (7n+4)s	prasmilinear family	Named after a truncation of a former name for 7L 4s (suprasmitonic)
7L (7n+5)s	m-linear family	Named after m-chromaticralic (prefix blu-)
Families with 8 large steps
Mos	Name	Reasoning
8L (8n+3)s
8L (8n+5)s	petrlinear family	Named after petroid, a former name for 8L 5s
8L (8n+7)s
Families with 9 large steps
Mos	Name	Reasoning
9L (9n+1)s	sinalinear family	Named after sinatonic
9L (9n+2)s
9L (9n+4)s
9L (9n+5)s
9L (9n+7)s
9L (9n+8)s

Miscellaneous notation

Alternative UDP notation for filenames

UDP notation is currently notated as u|d for single-period mosses, and up|dp(p) for multi-period mosses. An alternative notation, intended for use for filenames since "|" cannot be used as part of a filename, is uU dD, or upU dpD.

Examples

Example mos	Standard UDP notation	Alternate notation
5L 2s	5|1 (ionian mode)	5U 1D
	3|3 (dorian mode)	3U 3D
3L 3s	3|0(3)	3U 0D

N(k) note name notation (work-in-progress)

Rather than using alphabetical names, notes of the form N(k) are used. These are used to indicate position on a staff, where N(0) is the root. These names serve as an alternative to using different notations for different scales, but may be interpreted as blanks for one to fill in with different, more specific notation. If k is unbounded, then this notation denotes position on a staff. However, k may be bounded within the range [0, n), where n is the note count, to indicate pitch classes.

For a given mos xL ys, note names are based on a mode u|p; the choice of mode is up to the user. Starting at the root of N(0), successive pitch classes are named N(1), N(2), and so on. If note names are given and assuming N(0) is the root, then N(k) can be thought of as a function that returns an unaltered note name corresponding to the k-mosdegree of a mos xL ys in the mode u|p. In standard notation, N(0) is C, N(1), is D, and so on. Since this is cyclical, N(7) and N(0) are both the same value of C.

If two pitches, reached by going up or down some quantity of mossteps, have the same remainder when divided by xL+ys (which is the same as octave-reducing), then they are in the same pitch class.

Example with standard notation (5L 2s, mode 5|1)

Mossteps from root	Substring	Mosstep sum	Standard note name	Nk note name
0	none	0	C	N(0)
1	L	L	D	N(1)
2	LL	2L	E	N(2)
3	LLs	2L+s	F	N(3)
4	LLsL	3L+s	G	N(4)
5	LLsLL	4L+s	A	N(5)
6	LLsLLL	5L+s	B	N(6)
7	LLsLLLs	5L+2s	C	N(7) (same as N(0))

Chromas are denoted using the letter c, and are expressed as a multiple of c being added (or subtracted) from a note N(k). Half-accidentals are denoted as fractions (such as c/2) or decimals (such as 0.5c). Dieses, if present, are expressed similarly using the letter d. If this notation denotes position on a staff, then chromas and dieses don't change position on a staff, but modify the pitch at that position. If this notation is treated as placeholders for more specific notation, then adding or subtracting c represents the use of sharp or flat (or equivalent) accidentals.

Since chromas and dieses can be expressed in terms of L and s – where a chroma is L - s and a diesis is the absolute value of L - 2s – modifying a note by a chroma or diesis can equivalently expressed as going up (or down) some interval iL+js. If, for a given step ratio L:s, two pitch classes Np and Nq are modified by different amounts of chromas uc and vc to produce pitch classes N(p)+uc and N(q)+vc, if dividing both by xL+ys produces the same remainder, then the two pitches are enharmonic equivalents.

As an example, the table below denotes diatonic (5L 2s) pitch classes as sums of L's and s's, and shows how different step ratios produce different enharmonic equivalences; namely, in 12edo, C# and Db are equivalent, but in 19edo, C# and Db are not equivalent but B# and Cb are equivalent.

Examples with standard diatonic notation

Note name	N(k) note name with chroma	Mosstep sum	Like terms combined	If L:s = 2:1	If L:s = 3:2
C	N(0)	0	0	0	0
C#	N(0)+c	L-s	L-s	1	1
Db	N(1)-c	L-(L-s)	s	1	2
D	N(1)	L	L	2	3
B	N(6)	5L+s	5L+s	11	17
B#	N(6)+c	5L+s+(L-s)	6L	12	18
Cb	N(7)-c	5L+2s-(L-s)	4L+3s	11	18
C (one octave up)	N(7) (same as N(0), as a pitch class)	5L+2s (reduced to 0 due to modular arithmetic)	5L+2s (reduced to 0)	12 (reduced to 0)	19 (reduced to 0)

N(k) notation can also be used to build a genchain that is agnostic of the size (in cents) of the generator and equave. For example, the genchain for standard notation can be written as N(0), N(4), N(8), N(12), N(16), N(20), N(24)+c, N(28)+c for the ascending chain. The descending chain can be written as N(0), N(3), N(6)-c, N(9)-c, N(12)-c, N(15)-c, N(18)-c, N(21)-c, or as N(0), N(-4), N(-8)-c, N(-12)-c, N(-16)-c, N(-20)-c, N(-24)-c, N(-28)-c. The value k isn't entered into the function, but rather its remainder when divided by the number of steps in the mos (modulo 7, for the case of standard notation), so N(8) is equivalent to N(1) for example.

Since the gamut on C is based on the ionian mode, or produced using 5 generators going up and 1 going down, the first note after N(20) has a chroma added, producing N(24)+c. Simply put, the first 5 notes after the root have zero chromas added, the next 6 after that have 1 chroma added, the next 6 have 2 chromas added, and so on. For the descending chain, accidentals are subtracted after the first note, and every 6 notes thereafter has one more chroma subtracted.

Ups and downs may also be represented, using the variable u. Up-C-sharp, or ^C#, is written as N(0)+c+u, where u is an edostep.

Chord notation using mossteps

For a chord built using stacked mossteps s1 and s2, the chord is referred to as an s1+s2 chord. The rules for classifying the shape of the chord are as follows:

If the interval s1 mossteps from the root is...	And the interval s2 mossteps from there is...	Then the overall chord is	Which, if s1 and s2 are diatonic or diatonic-like 3rds, is a(n)...
the large interval (eg, major)	the large interval (eg, minor)	Large symmetric	Augmented chord (M3+M3)
the large interval	the small interval	Major asymmetric	Major chord (M3+m3)
the small interval	the large interval	Minor asymmetric	Minor chord (m3+M3)
the small interval	the small interval	Small symmetric	Diminished chord (m3+m3)

If the quantities of mossteps s1 and s2 are different, then the symmetric chrods are quasisymmetric instead. The interval sizes don't need to be major or minor, either; they can also be augmented, perfect, or diminished if it's a generator.

Proposal (wip): strict and weak definitions for a chromatic pair

Strict definition

A chromatic pair is a pair of mosses zL ws and xL ys within some temperament, such that x = z + w and y = z, where zL ws is a haplotonic scale and xL ys is an albitonic scale. The large steps of the albitonic scale are such that haplotonic scale can be found within the large steps, forming a chromatic scale of either xL (x+y)s or (x+y)L xs, or more generally, xA (x+y)B.

Weak definition

A chromatic pair, under the weak definition, is a pair of mosses zL ws and xL ys, such that x = nz + w and y = z. The strict definition is such that n = 1. However, rather than the mosses zL ys and xL ys that form the chromatic scale of xA (x+y)B, it's the mosses zL ((n-1)z+w)s and xL ys that form the chromatic scale.

Things to consider

• A haplotonic scale's note count should be 4 or 5 notes, corresponding to the note counts of the grandchild mosses of 1L 1s: 2L 3s, 3L 2s, 1L 3s, and 3L 1s.
• An albitonic scale's note count should be around 7 notes.

Warped scales

A somewhat generalized notion of warping, described by the addition, removal, or substitution of a single step. The most common scales of 12edo are used as examples: 5L 2s, the whole-tone scale (effectively 6edo), the chromatic scale (effectively 12edo), and the diminished scale (4L 4s, hardness of 2).

The simplest ways to warp a scale are through the addition of a step and the removal of a step. Substitution of a step, where one step is changed for a step of a different size, can be thought of removing a step of one size and adding a step of a different size.

Warped 5L 2s

Small step changes	Large step changes
	-1L	+0L	+1L
-1s		5L 1s	6L 1s
+0s	5L 1s	5L 2s	6L 2s
+1s	4L 3s	5L 3s

Warped 6edo (equal-tempered whole-tone scale)

Small step changes	Large step changes
	-1L	+0L	+1L
-1s			1L 5s
+0s		6edo	1L 6s
+1s	5L 1s	6L 1s

Warped 12edo (equal-tempered chromatic scale)

Small step changes	Large step changes
	-1L	+0L	+1L
-1s			1L 11s
+0s		12edo	1L 12s
+1s	1L 11s	12L 1s

Warped 4L 4s

Small step changes	Large step changes
	-1L	+0L	+1L
-1s		4L 3s	5L 3s
+0s	3L 4s	4L 4s	5L 4s
+1s	3L 5s	4L 5s

EDO/ED classifications

• Deka-edo (deka-division): an equal division of the octave (or equave) where the number of divisions is in the tens.
• Hecto-edo (hecto-division): an equal division of the octave (or equave) where the number of divisions is in the hundreds.
• Kilo-edo (kilo-division): an equal division of the octave (or equave) where the number of divisions is in the thousands.
• Mega-edo (mega-division): an equal division of the octave (or equave) where the number of divisions is in the millions.
  • This term already exists to refer to a large edo, but how large is subjective. Since the terms deka-, hecto-, and kilo-edo (and deka-, hecto-, and kilo-division) explicitly refer to specific powers of 10 (specifically, tens, hundreds, and thousands), so should mega-edo and mega-division to refer to divisions in the millions.