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'