Byzantine Liturgical Music

 I've been thinking about getting into Byzantine liturgical music. I've got some descriptions of its scales in 68 EDO and 72 EDO.

The 72 EDO steps come from this set: [0, 5, 6, 7, 11, 12, 18, 19, 23, 24, 26, 27, 30, 31, 35, 42, 45, 47, 48, 49, 51, 53, 54, 60, 61, 65, 66, 69, 72]

. Most of these are the 72-EDO tunings of the chromatic 3-limit and 5-limit chromatic intervals:

0 : 1/1 _ P1
6 : 256/243 _ Grm2
7 : 16/15 _ m2
11 : 10/9 _ M2
12 : 9/8 _ AcM2
18 : 32/27 _ Grm3
19 : 6/5 _ m3
23 : 5/4 _ M3
24 : 81/64 _ AcM3
30 : 4/3 _ P4
42 : 3/2 _ P5
48 : 128/81 _ Grm6
49 : 8/5 _ m6
53 : 5/3 _ M6
54 : 27/16 _ AcM6
60 : 16/9 _ Grm7
61 : 9/5 _ m7
65 : 15/8 _ M7

66 : 243/128 _ AcM7
72 : 2/1 _ P8

The remaining steps are more mysterious. Let's investigate. The remaining steps are [5, 26, 27, 31, 35, 45, 47, 51, 69].

5\72 might be:

    5 : 21/20 _ SbAcm2

26\72 might be one of these:

26 : 9/7 _ SpM3

26 : 32/25 _ d4

27\72 might be one of these:

27 : 13/10 _ Prd4

27 : 35/27 _ Sb4

27 : 125/96 _ A3

27 : 162/125 _ Acd4

31\72 might be one of these:

31 : 27/20 _ Ac4

31 : 35/26 _ ReSbAcA4

31 : 65/48 _ Pr4

35\72 might be one of these:

35 : 7/5 _ Sbd5

35 : 45/32 _ AcA4

35 : 88/63 _ AsSpGr4

45\72 might be one of these:

45 : 20/13 _ ReA5

45 : 54/35 _ Sp5

45 : 65/42 _ PrSpGr5

45 : 77/50 _ AsSbd6

45 : 99/64 _ As5

45 : 125/81 _ GrA5

45 : 192/125 _ d6

47\72 might be one of these:

47 : 11/7 _ AsSpGr5

47 : 52/33 _ PrDem6

47 : 63/40 _ SbAcm6

47 : 405/256 _ AcA5

51\72 might be one of these:

51 : 18/11 _ DeAcM6

51 : 44/27 _ AsGrm6

51 : 105/64 _ SbAcM6

69\72 might be one of these:

69 : 35/18 _ Sb8

69 : 39/20 _ Prd8

69 : 64/33 _ De8

69 : 125/64 _ A7

69 : 243/125 _ Acd8

...

The 68-EDO scales have these steps: [0, 7, 9, 12, 13, 16, 17, 19, 21, 25, 28, 31, 35, 37, 40, 43, 44, 47, 49, 52, 53, 55, 56, 57, 59, 61, 65, 68]. This set is missing tunings for most of the chromatic rank-2 and rank-3 intervals. Pretty confusing.

...

I've discovered a different set of Byzantine scale description in 72-EDO. These ones have steps of [0, 4, 6, 8, 18, 22, 24, 30, 42, 46, 48, 50, 52, 60, 64, 66, 72]. The new steps here are [4, 8, 22, 46, 50, 52, 64]. Here are some guesses:

4 : 25/24 _ A1
4 : 26/25 _ Prd2
4 : 28/27 _ Sbm2

8 : 13/12 _ Prm2
8 : 27/25 _ Acm2

22 : 16/13 _ ReM3
22 : 26/21 _ PrSpGrm3
22 : 99/80 _ Asm3
22 : 100/81 _ GrM3

46 : 14/9 _ Sbm6
46 : 25/16 _ A5
46 : 39/25 _ Prd6

50 : 13/8 _ Prm6
50 : 21/13 _ ReSbAcM6
50 : 81/50 _ Acm6
50 : 160/99 _ DeM6

52 : 33/20 _ Asm6
52 : 64/39 _ ReM6
52 : 288/175 _ Spm6
52 : 400/243 _ GrM6

64 : 13/7 _ PrSpGrm7
64 : 24/13 _ ReM7
64 : 50/27 _ GrM7

Pretty confusing.