Intervals
Names for these intervals come from their degree in the scale and how "bright" or "dark" they are. Degree comes from ordering the intervals by pitch after reducing by the period. Bright and dark compare the pitch of the interval. Bright intervals include major and augmented, as well as the "bright" perfect generator. The bright generator is the one which generates intervals higher in pitch; those intervals are bright. Conversely, dark intervals are lower in pitch than their bright counterparts and include minor and diminished and the "dark" perfect generator that generates them. An example for this naming system comes from the diatonic scale, MOS 5L 2s. The generators are the perfect fourth and perfect fifth, between 3\7 and 2\5, and 4\7 and 3\5, respectively, and the period is 2/1. Stacking perfect fifths creates a chain of bright intervals (P5 M2 M6 ...) and perfect fourths generate dark intervals (P4 m7 m3 ...). In 3L 4s, the perfect third between 1\3 and 2\7 is the bright generator and the perfect sixth between 2\3 and 5\7 is the dark generator. These are the intervals found in the 7 note MOS (LssLsLs and it's modes):
| "Bright" Intervals
|
Genchain Distance
|
"Dark" Intervals
|
Genchain Distance
|
| Perfect Unison
|
P1
|
0
|
Perfect Octave
|
P8
|
0
|
| Major Second
|
M2
|
+4
|
Minor Seventh
|
m7
|
-4
|
| Perfect Third
|
P3
|
+1
|
Perfect Sixth
|
P6
|
-1
|
| Major Fourth
|
M4
|
+5
|
Minor Fifth
|
m5
|
-5
|
| Major Fifth
|
M5
|
+2
|
Minor Fourth
|
m4
|
-2
|
| Augmented Sixth
|
A6
|
+6
|
Diminished Third
|
d3
|
-6
|
| Major 7th
|
M7
|
+3
|
Minor 2nd
|
m2
|
-3
|
The 10 note chromatic scales 3L 7s (LsssLssLss), 7L 3s (sLLLsLLsLL), and 10edo (equalized) add these intervals:
| "Bright" Intervals
|
Genchain Distance
|
"Dark" Intervals
|
Genchain Distance
|
| Augmented Unison
|
A1
|
+7
|
Diminished Octave
|
d8
|
-7
|
| Augmented Third
|
A3
|
+8
|
Diminished Sixth
|
d6
|
-8
|
| Augmented Fifth
|
A5
|
+9
|
Diminished Fourth
|
d4
|
-9
|
Modes
| Mode in step patterns
|
Mode in intervals
|
Modal UDP Notation
|
| LsLsLss
|
P1 M2 P3 M4 M5 A6 M7 P8
|
6|0
|
| LsLssLs
|
P1 M2 P3 M4 M5 P6 M7 P8
|
5|1
|
| LssLsLs
|
P1 M2 P3 m4 M5 P6 M7 P8
|
4|2
|
| sLsLsLs
|
P1 m2 P3 m4 M5 P6 M7 P8
|
3|3
|
| sLsLssL
|
P1 m2 P3 m4 M5 P6 m7 P8
|
2|4
|
| sLssLsL
|
P1 m2 P3 m4 m5 P6 m7 P8
|
1|5
|
| ssLsLsL
|
P1 m2 d3 m4 m5 P6 m7 P8
|
0|6
|
Nominals and Accidentals
3L 4s is heptatonic and can be notated using the familiar letters of A, B, C, D, E, F, and G. Sharps and flats can be used as accidentals to signify altering pitch. The amount that they alter can be defined as the difference between a minor interval and its major counterpart, also equal to a generator * the number of notes in the scale, reduced by the period, in this case stacking 7 P3's = a sharp and 7 P6's = a flat. In diatonic, CDEFGABC corresponds to LLsLLLs, P1 M2 M3 P4 P5 M6 M7 P8. Here, CDEFGABC in 3L 4s is used as LssLsLs, P1 M2 P3 m4 M5 P6 M7 P8. For example, if you wanted to write the mode LsLsLss on C, you'd write CDEF#GA#BC. sLsLsLs on C is CDbEFGABC. LssLsLs on B is BC#DEF#GA#B. This can also be used to write altered scales, such as <nowiki>4|2<nowiki> #6. The step pattern is LssLLss and on C that would be CDEFGA#BC. This approach is structurally similar to traditional notation, extrapolating familiar concepts to new applications. The differences are in note relations, as in a diatonic C-E is different than C-E in 3L 4s, so it must be specified what scales are being used in order to avoid confusion.
Chords
Naming tertian chords in 3L 4s can be done similarly to diatonic names for tertian chords. In both systems triads are made of a root, a major or minor interval, and a perfect or imperfect generator. The difference is that in diatonic the fifth is the generator whereas in 3L 4s the third is the generator. When a chord contains a perfect generator, it is named after the major or minor interval. When it's made up of an imperfect generator and an "agreeing" minor or major interval, you name it after the imperfect generator. Other chords are named as alterations of the prior types of chords described. Extensions can be named with the intervals used. Here are examples of naming triads, with diatonic as an example:
| Diatonic
|
3L 4s
|
| Natural Triads
|
| P1 M3 P5
|
Major
|
C E G
|
P1 P3 M5
|
Major
|
C E G
|
| P1 m3 P5
|
Minor
|
C Eb G
|
P1 P3 m5
|
Minor
|
C E Gb
|
| P1 m3 d5
|
Diminished
|
C Eb Gb
|
P1 d3 m5
|
Diminished
|
C Eb Gb
|
| Other Triads
|
| P1 M3 A5
|
Augmented
|
C E G#
|
P1 A3 M5
|
Augmented
|
C E# G
|
| P1 M3 d5
|
Major b5
|
C E Gb
|
P1 d3 M5
|
Major b3
|
C Eb G
|
| P1 m3 A5
|
Minor #5
|
C Eb G#
|
P1 A3 m5
|
Minor #3
|
C E# Gb
|
| Mode
|
Chords
|
| 6|0
|
I II III #iv V #viº VII
|
| 5|1
|
I ii III #ivº V VI VII
|
| 4|2
|
I iiº III IV V VI vii
|
| 3|3
|
I bII III IV v VI viiº
|
| 2|4
|
I bII iii IV vº VI bVII
|
| 1|5
|
i bII iiiº IV bV VI bVII
|
| 0|6
|
iº bII bIII IV bV vi bVII
|
Notating EDOs
This notation can be used for EDOs that support 3L 4s to assign every degree of the EDO can be assigned a nominal/nominal plus accidental and interval names relative to a root. When a sharp/flat is greater than one degree of the EDO, Ups and Downs Notation can be added to avoid using large amounts of sharps and flats. They can also be used when notating a multiple of an EDO that already supports 3L 4s as a superset of the smaller EDO. Here are examples of EDOs supporting basic, hard, and soft 3L 4s.
| #\10edo (basic)
|
Cents
|
Notation
|
| 0
|
0
|
Perfect Unison
|
P1
|
C
|
| 1
|
120
|
Minor Second
|
m2
|
Db
|
| 2
|
240
|
Major Second, Diminished Third
|
M2, d3
|
D, Eb
|
| 3
|
360
|
Perfect Third
|
P3
|
E
|
| 4
|
480
|
Minor Fourth
|
m4
|
F
|
| 5
|
600
|
Major Fourth, Minor Fifth
|
M4, m5
|
F#, Gb
|
| 6
|
720
|
Major Fifth
|
M5
|
G
|
| 7
|
840
|
Perfect Sixth
|
P6
|
A
|
| 8
|
960
|
Augmented Sixth, Minor Seventh
|
A6, m7
|
A#, Bb
|
| 9
|
1080
|
Major Seventh
|
M7
|
B
|
| 10
|
1200
|
Perfect Octave
|
P8
|
C
|
| #\13edo (hard)
|
Cents
|
Notation
|
| 0
|
0
|
Perfect Unison
|
P1
|
C
|
| 1
|
92.31
|
Minor Second
|
m2
|
Db
|
| 2
|
184.62
|
Diminished Third
|
d3
|
Eb
|
| 3
|
276.92
|
Major Second
|
M2
|
D
|
| 4
|
369.23
|
Perfect Third
|
P3
|
E
|
| 5
|
461.54
|
Minor Fourth
|
m4
|
F
|
| 6
|
553.85
|
Minor Fifth
|
m5
|
Gb
|
| 7
|
646.15
|
Major Fourth
|
M4
|
F#
|
| 8
|
738.46
|
Major Fifth
|
M5
|
G
|
| 9
|
830.77
|
Perfect Sixth
|
P6
|
A
|
| 10
|
923.08
|
Minor Seventh
|
m7
|
Bb
|
| 11
|
1015.38
|
Augmented Sixth
|
A6
|
A#
|
| 12
|
1107.69
|
Major Seventh
|
M7
|
B
|
| 13
|
1200
|
Perfect Octave
|
P8
|
C
|
| #\17edo (soft)
|
Cents
|
Notation
|
| 0
|
0
|
Perfect Unison
|
P1
|
C
|
| 1
|
70.59
|
Augmented Unison, Diminished Second
|
A1, d2
|
C#, Dbb
|
| 2
|
141.18
|
Minor 2nd
|
m2
|
Db
|
| 3
|
211.76
|
Major Second
|
M2
|
D
|
| 4
|
282.35
|
Diminished Third
|
d3
|
Eb
|
| 5
|
352.94
|
Perfect Third
|
P3
|
E
|
| 6
|
423.53
|
Augmented Third, Diminished Fourth
|
A3, d4
|
E#, Fb
|
| 7
|
494.12
|
Minor Fourth
|
m4
|
F
|
| 8
|
564.71
|
Major Fourth
|
M4
|
F#
|
| 9
|
635.29
|
Minor Fifth
|
m5
|
Gb
|
| 10
|
705.88
|
Major Fifth
|
M5
|
G
|
| 11
|
776.47
|
Augmented Fifth, Diminished Sixth
|
A5, d6
|
G#, Ab
|
| 12
|
847.06
|
Perfect Sixth
|
P6
|
A
|
| 13
|
917.65
|
Augmented Sixth
|
A6
|
A#
|
| 14
|
988.24
|
Minor 7th
|
m7
|
Bb
|
| 15
|
1058.82
|
Major Seventh
|
M7
|
B
|
| 16
|
1129.41
|
Augmented Seventh, Diminished Octave
|
A7, d8
|
B#, Cb
|
| 17
|
1200
|
Perfect Octave
|
P8
|
C
|
Key Signatures
Using key signatures first requires a chain of generators for adding sharps and flats. In diatonic, the chain of fifths for sharps is FCGDAEB and the chain of fourths for flats is BEADGCF. In 3L 4s, the chain of thirds for sharps is FACEGBD and the chain of sixths for flats is DBGECAF. I think is also helpful to designate one key signature as representing the natural scale for multiple different notes. For example, in diatonic a key signature corresponds to a major and a minor scale for some two notes. Major is ionian and minor is aeolian. This makes it easier to write scores in different modes without using large amount of accidentals; if a key signature was only for a minor scale, writing in lydian would require 4 alterations, with major and minor scales you can minimize the difference between the natural scale in the key signature and the mode you're writing in. Now, the example of lydian only requires 2 sharps. Here, I am assigning the modes LssLsLs to major and sLssLsL to minor. This makes C major the relative major key of B minor. Here is a table describing keys from Ab major to E# major. The amount of sharps or flats is found from the distance in generators from C major or B minor and which notes are altered is found by the chain of thirds/sixths.
| Number of sharps/flats
|
Which notes are altered relative to C/b
|
Relative major key
|
Relative minor key
|
| None
|
None
|
C major
|
B minor
|
| 1 sharp
|
F#
|
E major
|
D minor
|
| 1 flat
|
Db
|
A major
|
G minor
|
| 2 sharps
|
F#A#
|
G major
|
F# minor
|
| 2 flats
|
DbBb
|
F major
|
E minor
|
| 3 sharps
|
F#A#C#
|
B major
|
A# minor
|
| 3 flats
|
DbBbGb
|
Db major
|
C minor
|
| 4 sharps
|
F#A#C#E#
|
D major
|
C# minor
|
| 4 flats
|
DbBbGbEb
|
Bb major
|
A minor
|
| 5 sharps
|
F#A#C#E#G#
|
F# major
|
E# minor
|
| 5 flats
|
DbBbGbEbCb
|
Gb major
|
F minor
|
| 6 sharps
|
F#A#C#E#G#B#
|
A# major
|
G# minor
|
| 6 flats
|
DbBbGbEbCbAb
|
Eb major
|
Db minor
|
| 7 sharps
|
F#A#C#E#G#B#D#
|
C# major
|
B# minor
|
| 7 flats
|
DbBbGbEbCbAbFb
|
Cb major
|
Bb minor
|
| 8 sharps
|
FxA#C#E#G#B#D#
|
E# major
|
D# minor
|
| 8 flats
|
DbbBbGbEbCbAbFb
|
Ab major
|
Gb minor
|
To Do: add intro, EDOs notated using sharps/flats and up;s/downs, images for examples of key signatures