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m Correcting some errors; some more clarification
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I've been wanting to explain binary encodings as modal brightness for a while, and this is it. This also contains a curiosity I've had which involves including the modes of other scales with the same step count.
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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.
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 ==
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.
{| class="wikitable"
|'''Scale code'''
|'''Binary'''
|'''Decimal'''
|'''MOS'''
|'''UDP'''
|'''MOS name'''
|'''Mode name'''
|-
|LLLsLLs
|1110110
|118
|5L 2s
|<nowiki>6|0</nowiki>
|Diatonic
|Lydian
|-
|LLsLLLs
|1101110
|110
|5L 2s
|<nowiki>5|1</nowiki>
|Diatonic
|Ionian
|-
|LLsLLsL
|1101101
|109
|5L 2s
|<nowiki>4|2</nowiki>
|Diatonic
|Mixolydian
|-
|LsLLLsL
|1011101
|93
|5L 2s
|<nowiki>3|3</nowiki>
|Diatonic
|Dorian
|-
|LsLLsLL
|1011011
|91
|5L 2s
|<nowiki>2|4</nowiki>
|Diatonic
|Aeolian
|-
|sLLLsLL
|0111011
|59
|5L 2s
|<nowiki>1|5</nowiki>
|Diatonic
|Phrygian
|-
|sLLsLLL
|0110111
|55
|5L 2s
|<nowiki>0|6</nowiki>
|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.
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.
{| class="wikitable"
|'''Scale code'''
|'''Binary'''
|'''Decimal'''
|'''MOS'''
|'''UDP'''
|'''MOS name'''
|'''Mode name'''
|-
|LsLsLss
|1010100
|84
|3L 4s
|<nowiki>6|0</nowiki>
|Mosh
|Dril
|-
|LsLssLs
|1010010
|82
|3L 4s
|<nowiki>5|1</nowiki>
|Mosh
|Gil
|-
|LssLsLs
|1001010
|74
|3L 4s
|<nowiki>4|2</nowiki>
|Mosh
|Kleeth
|-
|sLsLsLs
|0101010
|42
|3L 4s
|<nowiki>3|3</nowiki>
|Mosh
|Bish
|-
|sLsLssL
|0101001
|41
|3L 4s
|<nowiki>2|4</nowiki>
|Mosh
|Fish
|-
|sLssLsL
|0100101
|37
|3L 4s
|<nowiki>1|5</nowiki>
|Mosh
|Jwl
|-
|ssLsLsL
|0010101
|21
|3L 4s
|<nowiki>0|6</nowiki>
|Mosh
|Led
|}
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 Modal Harmony|mavila temperament]].
=== 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:
{| class="wikitable"
|'''Scale code'''
|'''Binary'''
|'''Decimal'''
|'''MOS'''
|'''UDP'''
|'''MOS name'''
|'''Mode name'''
|-
|LLLLLLL
|1111111
|127
|7L 0s
|<nowiki>0|0</nowiki>
|Equiheptatonic
|Equiheptatonic
|-
|LLLLLLs
|1111110
|126
|6L 1s
|<nowiki>6|0</nowiki>
|Archeotonic
|Ryonian
|-
|LLLLLsL
|1111101
|125
|6L 1s
|<nowiki>5|1</nowiki>
|Archeotonic
|Karakalian
|-
|LLLLsLL
|1111011
|123
|6L 1s
|<nowiki>4|2</nowiki>
|Archeotonic
|Lobonian
|-
|LLLsLLL
|1110111
|119
|6L 1s
|<nowiki>3|3</nowiki>
|Archeotonic
|Horthathian
|-
|LLLsLLs
|1110110
|118
|5L 2s
|<nowiki>6|0</nowiki>
|Diatonic
|Lydian
|-
|LLsLLLL
|1101111
|111
|6L 1s
|<nowiki>2|4</nowiki>
|Archeotonic
|Oukranian
|-
|LLsLLLs
|1101110
|110
|5L 2s
|<nowiki>5|1</nowiki>
|Diatonic
|Ionian
|-
|LLsLLsL
|1101101
|109
|5L 2s
|<nowiki>4|2</nowiki>
|Diatonic
|Mixolydian
|-
|LLsLsLs
|1101010
|106
|4L 3s
|<nowiki>6|0</nowiki>
|Smitonic
|Nerevarine
|-
|LsLLLLL
|1011111
|95
|6L 1s
|<nowiki>1|5</nowiki>
|Archeotonic
|Tamashian
|-
|LsLLLsL
|1011101
|93
|5L 2s
|<nowiki>3|3</nowiki>
|Diatonic
|Dorian
|-
|LsLLsLL
|1011011
|91
|5L 2s
|<nowiki>2|4</nowiki>
|Diatonic
|Aeolian
|-
|LsLLsLs
|1011010
|90
|4L 3s
|<nowiki>5|1</nowiki>
|Smitonic
|Vivecan
|-
|LsLsLLs
|1010110
|86
|4L 3s
|<nowiki>4|2</nowiki>
|Smitonic
|Lorkhanic
|-
|LsLsLsL
|1010101
|85
|4L 3s
|<nowiki>3|3</nowiki>
|Smitonic
|Sothic
|-
|LsLsLss
|1010100
|84
|3L 4s
|<nowiki>6|0</nowiki>
|Mosh
|Dril
|-
|LsLssLs
|1010010
|82
|3L 4s
|<nowiki>5|1</nowiki>
|Mosh
|Gil
|-
|LssLsLs
|1001010
|74
|3L 4s
|<nowiki>4|2</nowiki>
|Mosh
|Kleeth
|-
|LssLsss
|1001000
|72
|2L 5s
|<nowiki>6|0</nowiki>
|Antidiatonic
|Antilocrian
|-
|LsssLss
|1000100
|68
|2L 5s
|<nowiki>5|1</nowiki>
|Antidiatonic
|Antiphrygian
|-
|Lssssss
|1000000
|64
|1L 6s
|<nowiki>6|0</nowiki>
|Anti-archeotonic
|Antizokalarian
|-
|sLLLLLL
|0111111
|63
|6L 1s
|<nowiki>0|6</nowiki>
|Archeotonic
|Zokalarian
|-
|sLLLsLL
|0111011
|59
|5L 2s
|<nowiki>1|5</nowiki>
|Diatonic
|Phrygian
|-
|sLLsLLL
|0110111
|55
|5L 2s
|<nowiki>0|6</nowiki>
|Diatonic
|Locrian
|-
|sLLsLsL
|0110101
|53
|4L 3s
|<nowiki>2|4</nowiki>
|Smitonic
|Kagrenacan
|-
|sLsLLsL
|0101101
|45
|4L 3s
|<nowiki>1|5</nowiki>
|Smitonic
|Almalexian
|-
|sLsLsLL
|0101011
|43
|4L 3s
|<nowiki>0|6</nowiki>
|Smitonic
|Dagothic
|-
|sLsLsLs
|0101010
|42
|3L 4s
|<nowiki>3|3</nowiki>
|Mosh
|Bish
|-
|sLsLssL
|0101001
|41
|3L 4s
|<nowiki>2|4</nowiki>
|Mosh
|Fish
|-
|sLssLsL
|0100101
|37
|3L 4s
|<nowiki>1|5</nowiki>
|Mosh
|Jwl
|-
|sLssLss
|0100100
|36
|2L 5s
|<nowiki>4|2</nowiki>
|Antidiatonic
|Anti-aeolian
|-
|sLsssLs
|0100010
|34
|2L 5s
|<nowiki>3|3</nowiki>
|Antidiatonic
|Antidorian
|-
|sLsssss
|0100000
|32
|1L 6s
|<nowiki>5|1</nowiki>
|Anti-archeotonic
|Antitamashian
|-
|ssLsLsL
|0010101
|21
|3L 4s
|<nowiki>0|6</nowiki>
|Mosh
|Led
|-
|ssLssLs
|0010010
|18
|2L 5s
|<nowiki>2|4</nowiki>
|Antidiatonic
|Antimixolydian
|-
|ssLsssL
|0010001
|17
|2L 5s
|<nowiki>1|5</nowiki>
|Antidiatonic
|Anti-ionian
|-
|ssLssss
|0010000
|16
|1L 6s
|<nowiki>4|2</nowiki>
|Anti-archeotonic
|Anti-oukranian
|-
|sssLssL
|0001001
|9
|2L 5s
|<nowiki>0|6</nowiki>
|Antidiatonic
|Antilydian
|-
|sssLsss
|0001000
|8
|1L 6s
|<nowiki>3|3</nowiki>
|Anti-archeotonic
|Antihorthathian
|-
|ssssLss
|0000100
|4
|1L 6s
|<nowiki>2|4</nowiki>
|Anti-archeotonic
|Antilobonian
|-
|sssssLs
|0000010
|2
|1L 6s
|<nowiki>1|5</nowiki>
|Anti-archeotonic
|Antikarakalian
|-
|ssssssL
|0000001
|1
|1L 6s
|<nowiki>0|6</nowiki>
|Anti-archeotonic
|Antiryonian
|-
|sssssss
|0000000
|0
|0L 7s
|<nowiki>0|0</nowiki>
|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 [[User:Xenoindex/Scale Codes|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. (TODO: expand)
Retrieved from "https://en.xen.wiki/w/User:Ganaram_inukshuk/Notes"