User:Ganaram inukshuk/Notes

From Xenharmonic Wiki
< User:Ganaram inukshuk
Revision as of 09:28, 30 June 2022 by Ganaram inukshuk (talk | contribs) (→On various mosses, with extended scale names: Reformatted scale descriptions as a table)
Jump to navigation Jump to search

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, even for completion. This contains unofficial scale names that try to be as close to existing TAMNAMS names as possible. The following table shows single-period mosses sorted by generation, including TAMNAMS names, named monolarge (1L ns) scales (italicized), and extended names (marked with asterisks).

Mos Family Tree (single-period only), with TAMNAMS Names

italics denote 1L ns scales (named for completeness); asterisks denote non-official names (from my own notes)

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 prototonic* 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 sephiroid 3L 10s
10L 3s
7L 3s dicotonic 7L 10s
10L 7s
4L 3s smitonic 4L 7s kleistonic 4L 11s
11L 4s
7L 4s suprasmitonic 7L 11s
11L 7s
2L 1s deuteric* 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 p-superchromatic*
12L 5s p-suprachromatic*
7L 5s m-chromatic 7L 12s m-suprachromatic*
12L 7s m-superchromatic*
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

These three scales - 1L 1s, 1L 2s, and 2L 1s - have potentially too few notes to be considered true scales, hence these descriptions are for completeness. The 2nd-order children of 1L 1s include scales already described by TAMNAMS, but also includes a gap in the form of 3L 1s (and to an extent, 1L 3s).

Parent scale 1st-order child scales 2nd-order child scales
Steps Name Notes Steps Name Notes Steps Name Notes
1L 3s antitetric Monolarge scale. Similarly to 3L 1s with 1L 2s, 4L 1s may be worth considering.
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.

3L 1s tetric Parent scale to orwelloid and semiquartal, the name tetric is assigned similarly to pentic being the parent of diatonic and antidiatonic.
1L 1s prototonic or protic The progenitor scale of all single-period mosses.

Despite being a monolarge scale, it's also its own sister and is named regardless.

2L 3s pentic Already established name.
2L 1s deuterotonic or deuteric One of the child scales of 1L 1s.
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 p- and m-chromatic respectively. This section names the second-order children of 5L 2s.

Parent scale 1st-order child scales 2nd-order child scales
Steps Name Notes Steps Name Notes Steps Name Notes
5L 12s p-superchromatic Names are based on the relationship between a small step and diesis (|L-2s|); unlike the m-scales, the small step continues to be a prominent scale step

P-superchromatic is named for its diesis being larger than a small step, where the reverse is true for p-suprachromatic.

5L 7s p-chromatic Already established name.
12L 5s p-suprachromatic
5L 2s diatonic Already established name.
7L 5s m-suprachromatic Names are based on the relationship between a chroma (|L-s|) and diesis (|L-2s|) in meantone temperaments.

M-superchromatic is named for its diesis being smaller than a chroma, where the reverse is true for m-suprachromatic.

7L 5s m-chromatic Already established name.
5L 7s m-superchromatic
Retrieved from "https://en.xen.wiki/index.php?title=User:Ganaram_inukshuk/Notes&oldid=92826"