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:
- Replacement ruleset 1 (where L - s > s)
- L -> Ls
- s -> s
- 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:
- Replacement ruleset 1
- L -> Ls
- s-> s
- Replacement ruleset 2
- L -> sL
- s -> L
- Replacement ruleset 3
- L->Lss...ss (n s's)
- s->s
- Replacement ruleset 4
- L->sLL...LL (n L's)
- s->L
- Replacement ruleset 5
- L->Lss...ss (n+1 s's)
- s->Lss...s (n s's)
- 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 |
Extending TAMNAMS names (archived)
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 rather than note count. As of August 2022, much of this section is rendered unnecessary due to TAMNAMS names being reorganized in the following ways, hence this section is kept for archival purposes.
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.
| 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 | prototonic*
(currently monowood and trivial) |
1L 2s | antideuteric*
(currently antrial) |
1L 3s | antitetric*
(currently antetric) |
1L 4s | antimanic
(currently pedal) |
1L 5s | antimachinoid*
(currently antimachinoid) |
1L 6s | anti-archeotonic
(currently onyx) |
| 6L 1s | archeotonic | ||||||||||
| 5L 1s | machinoid | 5L 6s | |||||||||
| 6L 5s | |||||||||||
| 4L 1s | manual
(formerly manic) |
4L 5s | gramitonic
(formerly 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 | dicoid
(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 | deuteric*
(currently 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 | (formerly p-chromatic) | 5L 12s | s-enharmonic* | ||||||
| 12L 5s | p-enharmonic* | ||||||||||
| 7L 5s | (formerly m-chromatic) | 7L 12s | f-enharmonic* | ||||||||
| 12L 7s | m-enharmonic* | ||||||||||
| 3L 2s | antipentic | 3L 5s | checkertonic
(formerly 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 (archived)
As of August 14, 2022, all of these scales have been named. 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 and trivial | 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), and is used as a base for such scales. The name trivial comes from the fact that this is a trivial (octave-equivalent) scale, consisting of only its generators. |
1L 2s | antideuterotonic or antideuteric | antrial | 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 | 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 (now gramitonic) and semiquartal, the name tetric is assigned similarly to pentic being the parent of diatonic and antidiatonic. | ||||||||
| 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 (archived)
5L 2s has two children: 5L 7s and 7L 5s, or formerly p- and m-chromatic respectively. This section describes a scheme for naming the daughter and granddaughter scales for 5L 2s, which can be generalized to any parent mos; see User:Frostburn/TAMNAMS Extension or the following section for more information.
| 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 daughter and 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 | |||||||
Describing moschromatic and mosenharmonic scales
This is a system for describing scales beyond the set of named TAMNAMS scales. Although this is already described by User:Frostburn, this is how I would word it.
Although naming scales beyond the current cap of 10 notes is antithetical to the purpose of TAMNAMS, a general description can still be made without establishing concrete names, while using names for already named scales. Any mos will have 2 child scales, collectively referred to as moschromatic. Those child scales scales will have 2 child scales of their own, or 4 grandchild scales, collectively referred to as mosenharmonic. For describing the scales of a named mos, the prefix of mos- is removed and replaced with the mos's prefix instead; for example, the child and grandchild scales for the mos 5L 3s (oneirotonic, prefix oneiro-) are oneirochromatic and oneiroenharmonic respectively. Note that the absence of a prefix will specifically describe the chromatic and enharmonic scales of 5L 2s. If talking about a specific scale, p-, m-, s-, or f- can be added based on the step ratio of the parent mos.
| Parent scale | Moschromatic scales | Mosenharmonic scales | |||||
|---|---|---|---|---|---|---|---|
| Steps | Step ratio range
(hardest to softest) |
Steps | Specific name | Step ratio range
(hardest to softest) |
Steps | Specific name | Step ratio range
(hardest to softest) |
| xL ys | 1:0 to 1:1 | xL (x+y)s | p-moschromatic | 1:0 to 2:1 | xL (2x+y)s | s-mosenharmonic | 1:0 to 3:1 |
| (2x+y)L xs | p-mosenharmonic | 3:1 to 2:1 | |||||
| (x+y)L xs | m-moschromatic | 2:1 to 1:1 | (x+y)L (2x+y)s | f-mosenharmonic | 3:2 to 1:1 | ||
| (2x+y)L (x+y)s | m-mosenharmonic | 2:1 to 3:2 | |||||
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 | M-at | N | C | - | Eb | F | G | - | Bb |
| 2|2 | Pentatonic Dorian | J | K-at | L | M-at | N | C | D | - | F | G | - | Bb |
| 1|3 | Pentatonic Mixolydian | J | K-at | L | M-at | N-at | C | D | - | F | G | A | - |
| 0|4 | Pentatonic Ionian (major pentatonic) | J | K-at | L-at | M-at | C | D | E | - | G | A | ||
Note: the recommended TAMNAMS symbol to denote a downchroma (@) is replaced with the word "at" to prevent the note names from being parsed as email addresses.
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.