music theories
sometimes incomplete thoughts about music
diatonic chord implications
rank-2 temperament moments of symmetry
3-limit diatonic / (3, 2) rank-2 temperament
- pattern : LLLsLLs
- large step : 9/8
- small step : 256/243
- hardness : 2187/2048
- pitches : F G A B C D E
[ 0 0> 1/1 F
[-1 1> 3/2 C
[-3 2> 9/8 G
[-4 3> 27/16 D
[-6 4> 81/64 A
[-7 5> 243/128 E
[-9 6> 729/512 B
[ 1 0> 2/1 F'
0 1 2 3 4 5 6 7 8 |
+------------------+
0 |F |
-1 | C |
-2 | |
-3 | G |
-4 | D |
-5 | |
-6 | A |
-7 | E |
-8 | |
-9 | B |
-10| |
+------------------+
3-limit "semi-diatonic" / (3, 3/2) rank-2 temperament
- pattern : sLLL
- large step : 9/8
- small step : 256/243
- hardness : 2187/2048
- pitches : B D E A
this moment-of-symmetry structure repeats at the ratio 3/2 instead
of the typical 2/1, but its shape closely mimics the diatonic scale
which repeats at 2/1
[ 0 0> 1/1 B
[ 2 -1> 4/3 E
[ 5 -3> 32/27 A
[ 8 -5> 256/243 D
[-1 1> 3/2 B'
+-------------------+
| |9
| D |8
| |7
| |6
| A |5
| |4
| |3
| E |2
| |1
| B|0
+-------------------+
|9 8 7 6 5 4 3 2 1 0|
if repeated each occurrence of ~2/1 instead of 3/2, as the diatonic,
we arrive at a collection of irregular scales with two step sizes:
- LLLsLLs (2/1 diatonic for reference)
- sLLLsLL
- LLLsLLL sharp pseudo-octave 2187/1024
- LLsLLLs
- LsLLLsL
this creates four "modes" with the following intervals
first mode: sLLLsLL "phrygian"
- 90c | semitone
- 294c | minor third
- 498c | perfect fourth
- 702c | perfect fifth
- 792c | minor sixth
- 996c | minor seventh
second mode: LLLsLLL "lydian"
- 204c | whole tone
- 408c | major third
- 612c | augmented fourth
- 702c | perfect fifth
- 906c | major sixth
- 1110c | major seventh
- 1313c | sharp pseudo octave
third mode: LLsLLLs "ionian"
- 204c | whole tone
- 408c | major third
- 498c | perfect fourth
- 702c | perfect fifth
- 906c | major sixth
- 1110c | major seventh
fourth mode: LsLLLsL "dorian"
- 204c | whole tone
- 294c | minor third
- 498c | perfect fourth
- 702c | perfect fifth
- 906c | major sixth
- 996c | minor seventh
3-limit "expanded semi-diatonic" / (3, 3/2) rank-2 temperament
- pattern : ssLsLsL
- large step : 2187/2048
- small step : 256/243
- hardness : 531441/524288
- pitches : B D F A C E G
[ 0 0> 1/1 B
[ 2 -1> 4/3 E
[ 5 -3> 32/27 A
[ 8 -5> 256/243 D
[10 -6> 1024/729 G
[13 -8> 8192/6561 C
[16 -10> 65536/59049 F
[-1 1> 3/2 B'
+---------------------+
|F |16
| |15
| |14
| C |13
| |12
| |11
| G |10
| |9
| D |8
| |7
| |6
| A |5
| |4
| |3
| E |2
| |1
| B|0
+---------------------+
|A 9 8 7 6 5 4 3 2 1 0|
irregular scales
5-limit diatonic
(one possible solution)
[ 0 0 0> 1/1 C
[-1 1 0> 3/2 G
[ 1 -2 1> 10/9 D
[ 0 -1 1> 5/3 A
[-2 0 1> 5/4 E
[-3 1 1> 15/8 B
[-1 -2 5> 4/3 F
[ 1 0 0> 2/1 C'