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.

--

I'm coming back to this after a while. Here's the scale data that I never shared from one or two microtonal scale websites that I probably shouldn't trust, but one has to start somewhere?

mode_to_edo_degrees = {

"Athanasopoulos' Byzantine Liturgical Chromatic, Dastgah-e Chahargah": [0, 3, 5, 2, 4, 3, 5, 2],

"Balzano Nine-tone, Hyperpentatonic, Score-9": [0, 2, 3, 2, 2, 2, 3, 2, 2, 2],

"Byzantine Palace mode": [0, 6, 20, 4, 12, 9, 10, 11],

"Chrysanthos 1st Byzantine Liturgical mode": [0, 9, 7, 12, 12, 9, 7, 12],

"Chrysanthos 3rd Byzantine Liturgical mode": [0, 12, 13, 3, 12, 12, 5, 11],

"Chrysanthos 4th Byzantine Liturgical mode": [0, 12, 9, 7, 12, 9, 7, 12],

"Chrysanthos Diatonic-Enharmonic Byzantine mode": [0, 9, 7, 12, 12, 3, 13, 12],

"Chrysanthos Enharmonic-Diatonic Byzantine mode": [0, 13, 12, 3, 12, 9, 7, 12],

"Chrysanthos Hard Chromatic 2nd plagal Byzantine mode": [0, 7, 18, 3, 12, 7, 18, 3],

"Chrysanthos Hard Chromatic/Diatonic Byzantine mode": [0, 7, 18, 3, 12, 9, 7, 12],

"Chrysanthos Soft Chromatic Byzantine mode": [0, 7, 12, 7, 12, 7, 12, 7],

"Double Harmonic Major, Major Gipsy, Bhairav That, Mela Mayamalavagaula, Raga Paraj, Kalingada (Kalingda), Gaulipantu, Lalitapancamam, Gandhakriya, Manjiri, Chromatic 2nd Byzantine Liturgical, Hitzazkiar: Greece, Maqam Zengule, Hijaz Kar, Suzidil": [0, 1, 3, 1, 2, 1, 3, 1],

"First plagal Byzantine Liturgical mode ascending": [0, 5, 4, 6, 6, 5, 4, 6],

"First plagal Byzantine Liturgical mode descending": [0, 5, 4, 6, 6, 3, 6, 6],

"Fokaeas 2nd plagal Byzantine Liturgical mode": [0, 9, 7, 12, 7, 18, 3, 12],

"Fourth authentic Byzantine Liturgical mode": [0, 4, 6, 6, 5, 4, 6, 5],

"Fourth plagal Byzantine Liturgical mode": [0, 6, 5, 4, 6, 6, 5, 4],

"G.Dorian, M.Phrygian, G.M.Hypoaeolian, Bhairavi That, Mela Hanumatodi, Raga Asavari (Asaveri), Bilashkhani Todi, Darjeeling, Ghanta, Makam Kurd, Gregorian nr.3, Escala Andaluza, In, Zokuso: Japan, Ousak: Greece, Major inverse": [0, 1, 2, 2, 2, 1, 2, 2],

"G.Hypophrygian, G.Ionian (Iastian), M.Mixolydian, G.M.Hypoionian, Hypermixolydian, Mischung 3, Khamaj That, Mela Harikambhoji, Raga Balahamsa, Bhim, Devamanohari, Gaoti, Harini, Janjhuti, Kaamaai, Kalashri, Khambhavati, Sahana, Sakh, Surati, Gregorian nr.7, Enharmonic Byzantine Liturgical, Rast descending: Greece, Ching, Shang: China": [0, 2, 2, 1, 2, 2, 1, 2],

"G.Lydian, M.Ionian, M.Hypolydian, Major, Bilaval That, Mela Shankarabharanam, Raga Atana, Begada, Kathanakuthuhalam, Ghana Heptatonic, Peruvian Major, Matzore, Rast ascending: Greece, 4th plagal Byzantine, Ararai: Ethiopia, Makam Cargah, Ajam Ashiran, Dastgah-e Mahur, Dastgah-e Rast Panjgah, Xin: China, DS2, Heptatonia prima": [0, 2, 2, 1, 2, 2, 2, 1],

"G.M.Hypodorian, G.M.Aeolian, G.Hyperphrygian, Natural Minor, Melodic Minor descending, Asavari That, Mela Natabhairavi, Raga Jaunpuri, Adana, Darbari, Dhanyasi, Jingla, Sampurna Malkauns, Gregorian nr.2, Makam Buselik, Nihavend, Peruvian Minor, Se, Chiao: China, Geez, Ezel: Ethiopia, Kiourdi descending: Greece, Cushak: Armenia": [0, 2, 1, 2, 2, 1, 2, 2],

"Harmonic Minor inverse, Mixolydian flat 2, Mela Cakravaka, Raga Ahir Bhairav, Bindumalini, Hevitri, Vegavahini, Makam Hicaz, Zanjaran": [0, 1, 3, 1, 2, 2, 1, 2],

"Konstantinos 3rd Byzantine Liturgical mode": [0, 12, 9, 7, 12, 12, 3, 13],

"Konstantinos 4th plagal Byzantine Liturgical mode": [0, 12, 4, 12, 9, 7, 12, 12],

"Major Pentatonic, Ryosen, Yona Nuki Major: Japan, Man Jue, Gong: China, Raga Bhopali (Bhup), Bilahari, Deskar, Kokila, Jait Kalyan, Mohanam, Peruvian Pentatonic 1, Ghana Pentatonic 2, Tizita Major: Ethiopia": [0, 2, 2, 3, 2, 3],

"Makam Gulizar, Beyati (Bayati), Karcigar": [0, 8, 5, 9, 5, 4, 4, 4, 5, 9],

"Maqam Bastanikar, Tarz Nuin (Tarznauyn)": [0, 3, 4, 3, 3, 2, 6, 2, 1],

"Misaelides 1st Byzantine Liturgical mode": [0, 11, 7, 12, 12, 11, 7, 12],

"Misaelides 1st plagal Byzantine Liturgical mode": [0, 11, 7, 12, 15, 3, 12, 12],

"Misaelides 2nd plagal Byzantine Liturgical mode": [0, 7, 20, 3, 12, 7, 20, 3],

"Misaelides 3rd Byzantine Liturgical mode": [0, 12, 12, 11, 7, 12, 11, 7],

"Misaelides 4th Byzantine Liturgical mode": [0, 7, 12, 12, 11, 7, 12, 11],

"Neapolitan Minor, Hungarian Gipsy, Mela Dhenuka, Raga Bhinnasadjam, Dhunibinnashadjam, Kirvanti, Takka, Maqam Shahnaz Kurdi": [0, 1, 2, 2, 2, 1, 3, 1],

"Neutral Dorian, Misaelides 2nd Byzantine mode, Iced Fridgian, Maqam Sikah Baladi, Maqamic-7, Ioniophrygian": [0, 3, 4, 3, 4, 3, 4, 3],

"Neutral Mixolydian, Iced Blizzard": [0, 3, 4, 3, 3, 4, 3, 4],

"Phrygian Dominant, Phrygian Major, Mela Vakulabharanam, Raga Ahiri, Basant Mukhari, Jogiya, Prabhakali, Vativasantabhairavi, Zilof, Ahava rabbah, Freygish: Jewish, Maqam Hijaz-Nahawand, Humayun, Spanish Gipsy, Dorico Flamenco, Frigio Flamenco: Spain, Hitzaz: Greece, Harmonic Major inverse, Mixolydian flat 2 flat 6": [0, 1, 3, 1, 2, 1, 2, 2],

"Savas Diatonic Byzantine Liturgical mode": [0, 4, 6, 5, 6, 4, 6, 5],

"Savas Enharmonic Byzantine Liturgical mode": [0, 4, 8, 3, 6, 4, 8, 3],

"Second authentic Byzantine Liturgical mode, Savas Soft Chromatic 2nd Byzantine mode": [0, 4, 7, 4, 6, 4, 7, 4],

"Second plagal Byzantine Liturgical mode": [0, 2, 7, 1, 4, 2, 7, 1],

"Second plagal Byzantine Liturgical mode, Hard Chromatic 2nd plagal Byzantine mode": [0, 3, 10, 2, 6, 3, 10, 2],

"Tiby 1st Byzantine Liturgical mode": [0, 12, 13, 3, 12, 12, 13, 3],

"Tiby 2nd Byzantine Liturgical mode": [0, 12, 5, 11, 12, 12, 5, 11],

"Tiby 4th Byzantine Liturgical mode": [0, 9, 12, 7, 12, 9, 12, 7],

"Tsiknopoulos 2nd Byzantine Liturgical mode": [0, 7, 14, 7, 12, 7, 14, 7],

"Tsiknopoulos 4th Byzantine Liturgical mode": [0, 7, 12, 12, 9, 7, 12, 9],

"Tsiknopoulos 4th plagal Byzantine Liturgical mode, Tiby 3rd Byzantine mode": [0, 12, 9, 7, 12, 12, 9, 7],

"Xenakis Byzantine Liturgical Chromatic": [0, 5, 19, 6, 12, 5, 19, 6],

"Xenakis Byzantine Liturgical Diatonic, Misaelides 4th plagal Byzantine": [0, 12, 11, 7, 12, 12, 11, 7],

"Xenakis Byzantine Liturgical Soft Chromatic": [0, 7, 16, 7, 12, 7, 16, 7],

# scales with disjust tetrachords:

"Octave natural diatonic scale": [0, 8, 10, 12, 12, 8, 10, 12],

"Hard chromatic scale": [0, 4, 20, 6, 12, 4, 20, 6],

"Soft chromatic scale": [0, 8, 14, 8, 12, 8, 14, 8],

"Scale of the grave diatonic mode": [0, 6, 16, 8, 12, 10, 12, 8],

"Scale of the grave enharmonic mode from Zo": [0, 6, 12, 12, 12, 6, 12, 12],

"Scale of the grave enharmonic mode from Ga": [0, 6, 12, 12, 6, 12, 12, 6, 12, 12],

I don't remember how much I processed those from the original text. Presumably they weren't comma sepated bracketed lists.

...

If we sum up the integers in each list and hope that the scales reach an octave, these scales would seem to be in a mix of EDOs, namely [12, 20, 24, 36, 53, 64, 68, 72, 90]-EDO. Quite a lot of variety.

Actually, looking through the names of those scales, a lot of them don't seem to have anything to do with Byzantine chant. 

Okay, new plan. Weed out the non-Byzantine scales, clean up the names of the rest, take a scale in 12 or 24 or 36 edo and multiply it by (6, 3, or 2) respectively to get a 72-EDO scale. Ignore the 64-EDO scale and the 68-EDO scales for now. See what we can do. Here we go! I did the multiplication by hand because I was too tired to code, hopefully I got it all correct:

[0, 4, 24, 30, 42, 46, 66, 72] _ Hard chromatic scale

[0, 5, 24, 30, 42, 47, 66, 72] _ Xenakis Byzantine Liturgical Chromatic

[0, 6, 18, 30, 42, 48, 60, 72] _ Scale of the grave enharmonic mode from Zo

[0, 6, 22, 30, 42, 52, 64, 72] _ Scale of the grave diatonic mode

[0, 6, 24, 30, 42, 48, 66, 72] _ Chromatic 2nd Byzantine Liturgical

[0, 6, 26, 30, 42, 48, 68, 72] _ Second plagal Byzantine Liturgical mode, Hard Chromatic 2nd plagal Byzantine mode

[0, 6, 26, 30, 42, 51, 61, 72] _ Byzantine Palace mode

[0, 6, 27, 30, 42, 48, 69, 72] _ Second plagal Byzantine Liturgical mode

[0, 7, 19, 31, 42, 49, 61, 72] _ Misaelides 4th Byzantine Liturgical mode

[0, 7, 23, 30, 42, 49, 65, 72] _ Xenakis Byzantine Liturgical Soft Chromatic

[0, 7, 27, 30, 42, 49, 69, 72] _ Misaelides 2nd plagal Byzantine Liturgical mode

[0, 8, 20, 30, 42, 50, 62, 72] _ Savas Diatonic Byzantine Liturgical mode

[0, 8, 20, 32, 42, 50, 62, 72] _ Fourth authentic Byzantine Liturgical mode

[0, 8, 22, 30, 42, 50, 62, 72] _ Second authentic Byzantine Liturgical mode, Savas Soft Chromatic 2nd Byzantine mode

[0, 8, 22, 30, 42, 50, 64, 72] _ Soft chromatic scale

[0, 8, 24, 30, 42, 50, 66, 72] _ Savas Enharmonic Byzantine Liturgical mode

[0, 9, 21, 30, 42, 51, 63, 72] _ Misaelides 2nd Byzantine mode

[0, 9, 24, 30, 42, 51, 66, 72] _ Athanasopoulos' Byzantine Liturgical Chromatic

[0, 10, 18, 30, 42, 48, 60, 72] _ First plagal Byzantine Liturgical mode descending

[0, 10, 18, 30, 42, 52, 60, 72] _ First plagal Byzantine Liturgical mode ascending

[0, 11, 18, 30, 42, 53, 60, 72] _ Misaelides 1st Byzantine Liturgical mode

[0, 11, 18, 30, 45, 48, 60, 72] _ Misaelides 1st plagal Byzantine Liturgical mode

[0, 12, 22, 30, 42, 54, 64, 72] _ Fourth plagal Byzantine Liturgical mode

[0, 12, 23, 30, 42, 54, 65, 72] _ Xenakis Byzantine Liturgical Diatonic, Misaelides 4th plagal Byzantine

[0, 12, 24, 30, 42, 54, 60, 72] _ Enharmonic Byzantine Liturgical

[0, 12, 24, 30, 42, 54, 66, 72] _ 4th plagal Byzantine

[0, 12, 24, 35, 42, 54, 65, 72] _ Misaelides 3rd Byzantine Liturgical mode

Nice.

Next let's try to figure out a just intonation interpretation. I'm going to assume that these people don't understand 72-EDO as an object that maps intervals associated with prime harmonics. They're just going on cent values. So if we have a 7th interval that's about X cents, we'll just substite a similar ratio that is a just tuning of a 7th interval and has a similar cent value, hopefully to the nearest 16 + 2/3 cents. I don't think these people will have much of a concept of which intervals have which ordinals either, but I want my inferred scales to  be spelled alphaetically, so what else am I to do?

Just not? Okay. I'll just find a close frequency ratio that is low complexity and call it that. See how it looks.

Here are the ratios I'll use. The ones with larger numerators are Pythagorean, they don't count as complex: "1/1, 2/1, 3/2, 4/3, 5/3, 5/4, 6/5, 7/4, 7/5, 7/6, 8/5, 8/7, 9/5, 9/7, 9/8, 10/7, 10/9, 11/6, 11/7, 11/8, 11/9, 11/10, 12/7, 12/11, 13/7, 13/8, 13/9, 13/10, 13/11, 13/12, 14/9, 14/11, 14/13, 15/8, 15/11, 15/13, 15/14, 16/9, 16/11, 16/13, 16/15, 17/9, 17/10, 17/11, 17/12, 17/13, 17/14, 17/15, 17/16, 18/11, 18/13, 18/17, 20/11, 20/13, 20/17, 21/11, 21/13, 21/16, 21/17, 21/20, 22/13, 22/15, 22/17, 22/21, 24/13, 24/17, 24/25, 25/12, 25/13, 25/14, 25/16, 25/17, 25/18, 25/21, 25/22, 25/24, 26/15, 26/17, 26/21, 26/25, 27/14, 27/16, 27/17, 27/20, 27/22, 27/25, 27/26, 28/15, 28/17, 28/25, 28/27, 30/17, 32/17, 32/21, 32/25, 32/27, 33/17, 33/20, 33/25, 33/26, 33/28, 33/32, 34/21, 34/25, 34/27, 34/33, 35/18, 35/22, 35/24, 35/26, 35/27, 35/32, 35/33, 35/34, 36/25, 36/35, 39/20, 39/22, 39/25, 39/28, 39/32, 39/34, 39/35, 40/21, 40/27, 40/33, 40/39, 42/25, 44/25, 44/27, 44/35, 44/39, 45/26, 45/28, 45/32, 45/34, 45/44, 48/25, 48/35, 49/25, 49/26, 49/27, 49/30, 49/32, 49/33, 49/34, 49/36, 49/39, 49/40, 49/44, 49/45, 49/48, 50/27, 50/33, 50/39, 50/49, 51/26, 51/28, 51/32, 51/35, 51/40, 51/44, 51/49, 51/50, 52/27, 52/33, 52/35, 52/45, 52/49, 52/51, 54/35, 54/49, 55/28, 55/32, 55/34, 55/36, 55/39, 55/42, 55/48, 55/49, 55/51, 55/52, 55/54, 56/33, 56/39, 56/45, 56/51, 56/55, 60/49, 63/32, 63/34, 63/40, 63/44, 63/50, 63/52, 63/55, 64/33, 64/35, 64/39, 64/45, 64/49, 64/51, 64/55, 64/63, 65/33, 65/34, 65/36, 65/42, 65/44, 65/48, 65/49, 65/51, 65/54, 65/56, 65/63, 65/64, 66/35, 66/49, 66/65, 68/35, 68/39, 68/45, 68/49, 68/55, 68/63, 68/65, 70/39, 70/51, 72/49, 72/55, 72/65, 75/44, 75/49, 75/52, 75/56, 75/64, 75/68, 77/39, 77/40, 77/45, 77/48, 77/50, 77/51, 77/52, 77/54, 77/60, 77/64, 77/65, 77/68, 77/72, 77/75, 78/49, 78/55, 78/77, 80/49, 80/51, 80/63, 80/77, 80/81, 81/40, 81/44, 81/49, 81/50, 81/52, 81/55, 81/56, 81/64, 81/65, 81/68, 81/70, 81/77, 81/80, 84/55, 84/65, 85/44, 85/48, 85/49, 85/52, 85/54, 85/56, 85/63, 85/64, 85/66, 85/72, 85/77, 85/78, 85/81, 85/84, 88/45, 88/49, 88/51, 88/63, 88/65, 88/75, 88/81, 88/85, 90/49, 90/77, 91/48, 91/50, 91/51, 91/54, 91/55, 91/60, 91/64, 91/66, 91/68, 91/72, 91/75, 91/80, 91/81, 91/85, 91/88, 91/90, 96/49, 96/55, 96/65, 96/77, 96/85, 96/91, 98/51, 98/55, 98/65, 98/75, 98/81, 98/85, 99/50, 99/52, 99/56, 99/64, 99/65, 99/68, 99/70, 99/80, 99/85, 99/91, 99/98, 100/51, 100/63, 100/77, 100/81, 100/91, 100/99, 102/55, 102/65, 102/77, 102/91, 104/55, 104/63, 104/75, 104/77, 104/81, 104/85, 104/99, 105/64, 105/68, 105/88, 105/104, 108/55, 108/65, 108/77, 108/85, 108/91, 110/63, 110/81, 110/91, 112/65, 112/75, 112/81, 112/85, 112/99, 117/64, 117/68, 117/70, 117/77, 117/80, 117/85, 117/88, 117/98, 117/112, 119/60, 119/64, 119/65, 119/66, 119/72, 119/75, 119/78, 119/80, 119/81, 119/88, 119/90, 119/96, 119/99, 120/77, 120/91, 121/63, 121/64, 121/65, 121/68, 121/70, 121/72, 121/75, 121/78, 121/80, 121/81, 121/84, 121/85, 121/90, 121/91, 121/96, 121/98, 121/117, 121/120, 125/63, 125/64, 125/66, 125/68, 125/72, 125/77, 125/78, 125/81, 125/84, 125/88, 125/91, 125/96, 125/98, 125/99, 125/108, 125/121, 126/65, 126/85, 128/65, 128/75, 128/77, 128/81, 128/85, 128/91, 128/99, 128/125, 128/135, 130/77, 130/81, 130/99, 132/85, 132/91, 135/64, 135/68, 135/77, 135/88, 135/91, 135/98, 135/128, 136/75, 136/77, 136/81, 136/91, 136/99, 140/81, 140/99, 143/72, 143/75, 143/80, 143/81, 143/84, 143/85, 143/90, 143/96, 143/98, 143/140, 144/77, 144/85, 144/91, 144/125, 144/143, 147/80, 147/85, 147/88, 150/77, 150/91, 150/143, 153/77, 153/80, 153/88, 153/91, 153/98, 154/81, 154/85, 156/85, 160/81, 160/91, 160/99, 162/85, 162/91, 162/125, 165/91, 165/98, 168/85, 169/85, 169/88, 169/90, 169/96, 169/98, 169/99, 169/168, 170/91, 170/99, 175/88, 175/96, 175/99, 175/169, 176/91, 176/169, 176/175, 180/91, 182/99, 187/96, 187/98, 192/125, 195/98, 196/99, 216/125, 225/128, 243/125, 243/128, 243/160, 243/200, 243/242, 243/250, 250/243, 256/135, 256/225, 256/243, 256/245, 320/243, 400/243, 405/256, 500/243, 512/405, 625/324, 625/486, 648/625, 675/512, 729/400, 729/500, 729/512, 729/625, 800/729, 972/625, 1000/729, 1024/675, 1024/729, 1250/729, 2025/1024, 2048/2025, 2187/1250, 2187/2048, 2500/2187, 4096/2187, 6561/4096, 8192/6561, 19683/16384, 32768/19683, 59049/32768, 65536/59049, 177147/131072, 262144/177147, 531441/524288, 1048576/531441"

Here are just intonation versions of the scales, using the lowest complexituy ratio in that list (the one with the smallest numerator) for each scale degree, provided that the just intonation cent value is within 16 cents of the 72-EDO cent value:

Hard chromatic scale [1/1, 22/21, 5/4, 4/3, 3/2, 11/7, 15/8, 2/1]

Xenakis Byzantine Liturgical Chromatic [1/1, 18/17, 5/4, 4/3, 3/2, 11/7, 15/8, 2/1]

Scale of the grave enharmonic mode from Zo [1/1, 16/15, 13/11, 4/3, 3/2, 8/5, 16/9, 2/1]

Scale of the grave diatonic mode [1/1, 16/15, 16/13, 4/3, 3/2, 18/11, 13/7, 2/1]

Chromatic 2nd Byzantine Liturgical [1/1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8, 2/1]

Second plagal Byzantine Liturgical mode, Hard Chromatic 2nd plagal Byzantine mode [1/1, 16/15, 9/7, 4/3, 3/2, 8/5, 21/11, 2/1]

Byzantine Palace mode [1/1, 16/15, 9/7, 4/3, 3/2, 13/8, 9/5, 2/1]

Second plagal Byzantine Liturgical mode [1/1, 16/15, 9/7, 4/3, 3/2, 8/5, 27/14, 2/1]

Misaelides 4th Byzantine Liturgical mode [1/1, 14/13, 6/5, 27/20, 3/2, 8/5, 9/5, 2/1]

Xenakis Byzantine Liturgical Soft Chromatic [1/1, 14/13, 5/4, 4/3, 3/2, 8/5, 13/7, 2/1]

Misaelides 2nd plagal Byzantine Liturgical mode [1/1, 14/13, 9/7, 4/3, 3/2, 8/5, 27/14, 2/1]

Savas Diatonic Byzantine Liturgical mode [1/1, 13/12, 11/9, 4/3, 3/2, 13/8, 9/5, 2/1]

Fourth authentic Byzantine Liturgical mode [1/1, 13/12, 11/9, 15/11, 3/2, 13/8, 9/5, 2/1]

Second authentic Byzantine Liturgical mode, Savas Soft Chromatic 2nd Byzantine mode [1/1, 13/12, 16/13, 4/3, 3/2, 13/8, 9/5, 2/1]

Soft chromatic scale [1/1, 13/12, 16/13, 4/3, 3/2, 13/8, 13/7, 2/1]

Savas Enharmonic Byzantine Liturgical mode [1/1, 13/12, 5/4, 4/3, 3/2, 13/8, 15/8, 2/1]

Misaelides 2nd Byzantine mode [1/1, 11/10, 11/9, 4/3, 3/2, 13/8, 11/6, 2/1]

Athanasopoulos' Byzantine Liturgical Chromatic [1/1, 11/10, 5/4, 4/3, 3/2, 13/8, 15/8, 2/1]

First plagal Byzantine Liturgical mode descending [1/1, 10/9, 13/11, 4/3, 3/2, 8/5, 16/9, 2/1]

First plagal Byzantine Liturgical mode ascending [1/1, 10/9, 13/11, 4/3, 3/2, 18/11, 16/9, 2/1]

Misaelides 1st Byzantine Liturgical mode [1/1, 10/9, 13/11, 4/3, 3/2, 5/3, 16/9, 2/1]

Misaelides 1st plagal Byzantine Liturgical mode [1/1, 10/9, 13/11, 4/3, 14/9, 8/5, 16/9, 2/1]

Fourth plagal Byzantine Liturgical mode [1/1, 9/8, 16/13, 4/3, 3/2, 22/13, 13/7, 2/1]

Xenakis Byzantine Liturgical Diatonic, Misaelides 4th plagal Byzantine [1/1, 9/8, 5/4, 4/3, 3/2, 22/13, 13/7, 2/1]

Enharmonic Byzantine Liturgical [1/1, 9/8, 5/4, 4/3, 3/2, 22/13, 16/9, 2/1]

4th plagal Byzantine [1/1, 9/8, 5/4, 4/3, 3/2, 22/13, 15/8, 2/1]

Misaelides 3rd Byzantine Liturgical mode [1/1, 9/8, 5/4, 7/5, 3/2, 22/13, 13/7, 2/1]

If we narrow that to 8 cents of difference acceptable, then basically every scale changes in at least one place. Here are tighter fitting scales with slightly more complex ratios:

Hard chromatic scale [1/1, 25/24, 34/27, 4/3, 3/2, 14/9, 17/9, 2/1]

Xenakis Byzantine Liturgical Chromatic [1/1, 21/20, 34/27, 4/3, 3/2, 11/7, 17/9, 2/1]

Scale of the grave enharmonic mode from Zo [1/1, 17/16, 25/21, 4/3, 3/2, 27/17, 16/9, 2/1]

Scale of the grave diatonic mode [1/1, 17/16, 16/13, 4/3, 3/2, 28/17, 13/7, 2/1]

Chromatic 2nd Byzantine Liturgical [1/1, 17/16, 34/27, 4/3, 3/2, 27/17, 17/9, 2/1]

Second plagal Byzantine Liturgical mode, Hard Chromatic 2nd plagal Byzantine mode [1/1, 17/16, 9/7, 4/3, 3/2, 27/17, 25/13, 2/1]

Byzantine Palace mode [1/1, 17/16, 9/7, 4/3, 3/2, 18/11, 9/5, 2/1]

Second plagal Byzantine Liturgical mode [1/1, 17/16, 13/10, 4/3, 3/2, 27/17, 33/17, 2/1]

Misaelides 4th Byzantine Liturgical mode [1/1, 15/14, 6/5, 27/20, 3/2, 8/5, 9/5, 2/1]

Xenakis Byzantine Liturgical Soft Chromatic [1/1, 15/14, 5/4, 4/3, 3/2, 8/5, 15/8, 2/1]

Misaelides 2nd plagal Byzantine Liturgical mode [1/1, 15/14, 13/10, 4/3, 3/2, 8/5, 33/17, 2/1]

Octave natural diatonic scale [1/1, 13/12, 25/21, 4/3, 3/2, 13/8, 16/9, 2/1]

Savas Diatonic Byzantine Liturgical mode [1/1, 13/12, 17/14, 4/3, 3/2, 13/8, 20/11, 2/1]

Fourth authentic Byzantine Liturgical mode [1/1, 13/12, 17/14, 15/11, 3/2, 13/8, 20/11, 2/1]

Second authentic Byzantine Liturgical mode, Savas Soft Chromatic 2nd Byzantine mode [1/1, 13/12, 16/13, 4/3, 3/2, 13/8, 20/11, 2/1]

Soft chromatic scale [1/1, 13/12, 16/13, 4/3, 3/2, 13/8, 13/7, 2/1]

Savas Enharmonic Byzantine Liturgical mode [1/1, 13/12, 34/27, 4/3, 3/2, 13/8, 17/9, 2/1]

Misaelides 2nd Byzantine mode [1/1, 12/11, 11/9, 4/3, 3/2, 18/11, 11/6, 2/1]

Athanasopoulos' Byzantine Liturgical Chromatic [1/1, 12/11, 34/27, 4/3, 3/2, 18/11, 17/9, 2/1]

First plagal Byzantine Liturgical mode descending [1/1, 11/10, 25/21, 4/3, 3/2, 27/17, 16/9, 2/1]

First plagal Byzantine Liturgical mode ascending [1/1, 11/10, 25/21, 4/3, 3/2, 28/17, 16/9, 2/1]

Misaelides 1st Byzantine Liturgical mode [1/1, 10/9, 25/21, 4/3, 3/2, 5/3, 16/9, 2/1]

Misaelides 1st plagal Byzantine Liturgical mode [1/1, 10/9, 25/21, 4/3, 17/11, 27/17, 16/9, 2/1]

Fourth plagal Byzantine Liturgical mode [1/1, 9/8, 16/13, 4/3, 3/2, 27/16, 13/7, 2/1]

Xenakis Byzantine Liturgical Diatonic, Misaelides 4th plagal Byzantine [1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1]

Enharmonic Byzantine Liturgical [1/1, 9/8, 34/27, 4/3, 3/2, 27/16, 16/9, 2/1]

4th plagal Byzantine [1/1, 9/8, 34/27, 4/3, 3/2, 27/16, 17/9, 2/1]

Misaelides 3rd Byzantine Liturgical mode [1/1, 9/8, 34/27, 7/5, 3/2, 27/16, 15/8, 2/1]

Still pretty simple overall. Are you surprised by how many 17-limit frequency ratios there are? Kind of cool.

...

Okay, I did the unpleasant thing, now let's do the pleasant thing. I want to find scales that are spelled correctly numerically in interval ordinals or alphabetically in pitches.

These scales (derived from the 8 cent deviation criterion) increase by 2nd intervals, as they should:

4th plagal Byzantine : [P1, AcM2, ExGrM3, P4, P5, AcM6, ExGrM7, P8] # [1/1, 9/8, 34/27, 4/3, 3/2, 27/16, 17/9, 2/1]

Athanasopoulos' Byzantine Liturgical Chromatic : [P1, DeAcM2, ExGrM3, P4, P5, DeAcM6, ExGrM7, P8] # [1/1, 12/11, 34/27, 4/3, 3/2, 18/11, 17/9, 2/1]

Enharmonic Byzantine Liturgical : [P1, AcM2, ExGrM3, P4, P5, AcM6, Grm7, P8] # [1/1, 9/8, 34/27, 4/3, 3/2, 27/16, 16/9, 2/1]

Fourth plagal Byzantine Liturgical mode : [P1, AcM2, ReM3, P4, P5, AcM6, PrSpGrm7, P8] # [1/1, 9/8, 16/13, 4/3, 3/2, 27/16, 13/7, 2/1]

Misaelides 2nd Byzantine mode : [P1, DeAcM2, AsGrm3, P4, P5, DeAcM6, AsGrm7, P8] # [1/1, 12/11, 11/9, 4/3, 3/2, 18/11, 11/6, 2/1]

Savas Enharmonic Byzantine Liturgical mode : [P1, Prm2, ExGrM3, P4, P5, Prm6, ExGrM7, P8] # [1/1, 13/12, 34/27, 4/3, 3/2, 13/8, 17/9, 2/1]

Second authentic Byzantine Liturgical mode, Savas Soft Chromatic 2nd Byzantine mode : [P1, Prm2, ReM3, P4, P5, Prm6, DeM7, P8] # [1/1, 13/12, 16/13, 4/3, 3/2, 13/8, 20/11, 2/1]

Soft chromatic scale : [P1, Prm2, ReM3, P4, P5, Prm6, PrSpGrm7, P8] # [1/1, 13/12, 16/13, 4/3, 3/2, 13/8, 13/7, 2/1]

Xenakis Byzantine Liturgical Diatonic, Misaelides 4th plagal Byzantine : [P1, AcM2, M3, P4, P5, AcM6, M7, P8] # [1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1]

These scales became alphabetical after I tightened the tolerance to 4 cents. I didn't expect that to happen, but I guess that just means that a correctly spelled interval for that position is better tuned but not simpler in frequency ratio than another one which is simpler but incorrectly spelled:

_Byzantine Palace mode : [P1, HbAcm2, SpM3, P4, P5, DeAcM6, m7, P8] # [1/1, 18/17, 9/7, 4/3, 3/2, 18/11, 9/5, 2/1]

_Chromatic 2nd Byzantine Liturgical : [P1, HbAcm2, ExGrM3, P4, P5, HbAcm6, ExGrM7, P8] # [1/1, 18/17, 34/27, 4/3, 3/2, 27/17, 17/9, 2/1]

_Hard chromatic scale : [P1, Prd2, ExGrM3, P4, P5, Sbm6, ExGrM7, P8] # [1/1, 26/25, 34/27, 4/3, 3/2, 14/9, 17/9, 2/1]

_Second plagal Byzantine Liturgical mode, Hard Chromatic 2nd plagal Byzantine mode : [P1, HbAcm2, SpM3, P4, P5, HbAcm6, ReA7, P8] # [1/1, 18/17, 9/7, 4/3, 3/2, 27/17, 25/13, 2/1]

These are spelled correctly if we can stand a discrepancy < 13 cents:

_First plagal Byzantine Liturgical mode descending : [P1, Asm2, PrDem3, P4, P5, HbAcm6, Grm7, P8] # [1/1, 11/10, 13/11, 4/3, 3/2, 27/17, 16/9, 2/1]

_Misaelides 1st Byzantine Liturgical mode : [P1, M2, PrDem3, P4, P5, M6, Grm7, P8] # [1/1, 10/9, 13/11, 4/3, 3/2, 5/3, 16/9, 2/1]

_Misaelides 1st plagal Byzantine Liturgical mode : [P1, M2, PrDem3, P4, ExDeA5, HbAcm6, Grm7, P8] # [1/1, 10/9, 13/11, 4/3, 17/11, 27/17, 16/9, 2/1]

_Misaelides 4th Byzantine Liturgical mode : [P1, ReSbAcM2, m3, Ac4, P5, m6, m7, P8] # [1/1, 14/13, 6/5, 27/20, 3/2, 8/5, 9/5, 2/1]

_Scale of the grave enharmonic mode from Zo : [P1, m2, PrDem3, P4, P5, HbAcm6, Grm7, P8] # [1/1, 16/15, 13/11, 4/3, 3/2, 27/17, 16/9, 2/1]

_Xenakis Byzantine Liturgical Soft Chromatic : [P1, ReSbAcM2, M3, P4, P5, m6, PrSpGrm7, P8] # [1/1, 14/13, 5/4, 4/3, 3/2, 8/5, 13/7, 2/1]

All of these make the cut if we allow 16 cents of discrepancy:

_First plagal Byzantine Liturgical mode ascending : [P1, M2, PrDem3, P4, P5, DeAcM6, Grm7, P8] # [1/1, 10/9, 13/11, 4/3, 3/2, 18/11, 16/9, 2/1]

_Fourth authentic Byzantine Liturgical mode : [P1, Prm2, AsGrm3, DeAcA4, P5, Prm6, m7, P8] # [1/1, 13/12, 11/9, 15/11, 3/2, 13/8, 9/5, 2/1]

_Misaelides 2nd plagal Byzantine Liturgical mode : [P1, ReSbAcM2, SpM3, P4, P5, m6, SpM7, P8] # [1/1, 14/13, 9/7, 4/3, 3/2, 8/5, 27/14, 2/1]

Misaelides 3rd Byzantine Liturgical mode : [P1, AcM2, M3, Sbd5, P5, ReAsM6, PrSpGrm7, P8] # [1/1, 9/8, 5/4, 7/5, 3/2, 22/13, 13/7, 2/1]

_Octave natural diatonic scale : [P1, Prm2, PrDem3, P4, P5, Prm6, Grm7, P8] # [1/1, 13/12, 13/11, 4/3, 3/2, 13/8, 16/9, 2/1]

_Savas Diatonic Byzantine Liturgical mode : [P1, Prm2, AsGrm3, P4, P5, Prm6, m7, P8] # [1/1, 13/12, 11/9, 4/3, 3/2, 13/8, 9/5, 2/1]

_Scale of the grave diatonic mode : [P1, m2, ReM3, P4, P5, DeAcM6, PrSpGrm7, P8] # [1/1, 16/15, 16/13, 4/3, 3/2, 18/11, 13/7, 2/1]

_Second plagal Byzantine Liturgical mode : [P1, m2, SpM3, P4, P5, m6, SpM7, P8] # [1/1, 16/15, 9/7, 4/3, 3/2, 8/5, 27/14, 2/1]

And that just leaves one scale left, "Xenakis Byzantine Liturgical Chromatic", which might have a tricky sixth interval? It really wants to be 11/7, which my system calls the just tuning of AsSpGr5. If we don't allow 11/7, then 25/16 becomes the best option, which is a 5-limit augmented fifth. If we don't allow that one either, then we finally a sixth interval of some kind, namely we get 39/25, a prominent diminished sixth, Prd6. If we rule that one out, we get PrDem6 at 52/33. This is only 4 cents off, it's not actually bad, but it's weird that it was so hard to find. Did I mess up a multiplication on that one?

The original data was [0, 5, 19, 6, 12, 5, 19, 6] which we can accumulate to get [0, 5, 24, 30, 42, 47, 66, 72]. And that's the only scale with a 47\72 absolute step. Steps of 48 to 54 are much more common in that position.

Anyway, I'd say we have pretty good just intonation scales at this point. What next? Do I listen to them a lot while looking at the names and try to figure out if the names tell you something about the music, instead of just the history? Do I just find resources that will teach me about these, instead of fitting fractions to what are probably incorrect scale steps to begin with? Do I try to find repeated tetrachords across different scales?

....

Looking through the scales, there's an interval of DeAcA4 tuned to 15/11 in "Fourth authentic Byzantine Liturgical mode" at 32\72 steps. That's pretty jank. I'd rather call that an acute fourth at 27/20 or an exalted fourth at 34/25 or a septimal super fourth at 48/35. Maybe I shouldn't lean so heavily on numerator magnitude as a measure of complextity.

...

I found a website with data, not just an old file on my hard drive: https://orthodoxwiki.org/Byzantine_Chant

It describes just four scales, which have their own modes/tonics. But at a high level:

    "Diatonic": [0, 12, 22, 30, 42, 54, 64, 72] : [12, 10, 8, 12, 12, 10, 8],

    "Enharmomnic": [0, 12, 24, 30, 42, 54, 66, 72] : [12, 12, 6, 12, 12, 12, 6],

    "Chromatic": [0, 8, 22, 30, 42, 50, 64, 72] : [8, 14, 8, 12, 8, 14, 8],

    "Hard Chromatic": [0, 6, 26, 30, 42, 48, 68, 72] : [6, 20, 4, 12, 6, 20, 4],

They just have relative steps. I accumulated them into absolute steps. All of these scales, in terms of steps, appeared among the previous scales, but with very different names. 

Like "Hard Chromatic" from the wiki was "#"Second plagal Byzantine Liturgical mode, Hard Chromatic 2nd plagal Byzantine mode". What the wiki calls "Enharmonic" was "4th plagal Byzantine" on the microtonal scale website. "Diatonic" was "Fourth plagal Byzantine Liturgical mode". I've got some learning to do.

I also need to check whether normal 3-limit and 5-limit ratios cover some of these scale steps adequately, because there's no reason to invoke 7, 11, 13, 17 limit ratios when those work.

... 

Here are alphabetical scales, with no more than 16 cents discrepancy from 72-EDO on each scale degree, in which I use a simple 5-limit ratio first if I can get away with it, and only bring in 7-, 11-, 13-, 17-limit ratios when that fails:

_4th plagal Byzantine : [P1, AcM2, M3, P4, P5, AcM6, M7, P8] # [1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1]

_Athanasopoulos' Byzantine Liturgical Chromatic : [P1, Asm2, M3, P4, P5, Prm6, M7, P8] # [1/1, 11/10, 5/4, 4/3, 3/2, 13/8, 15/8, 2/1]

_Byzantine Palace mode : [P1, m2, SpM3, P4, P5, Prm6, m7, P8] # [1/1, 16/15, 9/7, 4/3, 3/2, 13/8, 9/5, 2/1]

_Chromatic 2nd Byzantine Liturgical : [P1, m2, M3, P4, P5, m6, M7, P8] # [1/1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8, 2/1]

_Enharmonic Byzantine Liturgical : [P1, AcM2, M3, P4, P5, AcM6, Grm7, P8] # [1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 16/9, 2/1]

_First plagal Byzantine Liturgical mode ascending : [P1, M2, Grm3, P4, P5, DeAcM6, Grm7, P8] # [1/1, 10/9, 32/27, 4/3, 3/2, 18/11, 16/9, 2/1]

_First plagal Byzantine Liturgical mode descending : [P1, M2, Grm3, P4, P5, m6, Grm7, P8] # [1/1, 10/9, 32/27, 4/3, 3/2, 8/5, 16/9, 2/1]

_Fourth authentic Byzantine Liturgical mode : [P1, Prm2, AsGrm3, Ac4, P5, Prm6, m7, P8] # [1/1, 13/12, 11/9, 27/20, 3/2, 13/8, 9/5, 2/1]

_Fourth plagal Byzantine Liturgical mode : [P1, AcM2, ReM3, P4, P5, AcM6, PrSpGrm7, P8] # [1/1, 9/8, 16/13, 4/3, 3/2, 27/16, 13/7, 2/1]

_Misaelides 1st Byzantine Liturgical mode : [P1, M2, Grm3, P4, P5, M6, Grm7, P8] # [1/1, 10/9, 32/27, 4/3, 3/2, 5/3, 16/9, 2/1]

_Misaelides 2nd Byzantine mode : [P1, Asm2, AsGrm3, P4, P5, Prm6, AsGrm7, P8] # [1/1, 11/10, 11/9, 4/3, 3/2, 13/8, 11/6, 2/1]

_Misaelides 2nd plagal Byzantine Liturgical mode : [P1, m2, A3, P4, P5, m6, A7, P8] # [1/1, 16/15, 125/96, 4/3, 3/2, 8/5, 125/64, 2/1]

_Misaelides 3rd Byzantine Liturgical mode : [P1, AcM2, M3, A4, P5, AcM6, M7, P8] # [1/1, 9/8, 5/4, 25/18, 3/2, 27/16, 15/8, 2/1]

_Misaelides 4th Byzantine Liturgical mode : [P1, m2, m3, Ac4, P5, m6, m7, P8] # [1/1, 16/15, 6/5, 27/20, 3/2, 8/5, 9/5, 2/1]

_Octave natural diatonic scale : [P1, Prm2, Grm3, P4, P5, Prm6, Grm7, P8] # [1/1, 13/12, 32/27, 4/3, 3/2, 13/8, 16/9, 2/1]

_Savas Diatonic Byzantine Liturgical mode : [P1, Prm2, AsGrm3, P4, P5, Prm6, m7, P8] # [1/1, 13/12, 11/9, 4/3, 3/2, 13/8, 9/5, 2/1]

_Savas Enharmonic Byzantine Liturgical mode : [P1, Prm2, M3, P4, P5, Prm6, M7, P8] # [1/1, 13/12, 5/4, 4/3, 3/2, 13/8, 15/8, 2/1]

_Scale of the grave diatonic mode : [P1, m2, ReM3, P4, P5, DeAcM6, PrSpGrm7, P8] # [1/1, 16/15, 16/13, 4/3, 3/2, 18/11, 13/7, 2/1]

_Scale of the grave enharmonic mode from Zo : [P1, m2, Grm3, P4, P5, m6, Grm7, P8] # [1/1, 16/15, 32/27, 4/3, 3/2, 8/5, 16/9, 2/1]

_Second authentic Byzantine Liturgical mode, Savas Soft Chromatic 2nd Byzantine mode : [P1, Prm2, ReM3, P4, P5, Prm6, m7, P8] # [1/1, 13/12, 16/13, 4/3, 3/2, 13/8, 9/5, 2/1]

_Second plagal Byzantine Liturgical mode : [P1, m2, A3, P4, P5, m6, A7, P8] # [1/1, 16/15, 125/96, 4/3, 3/2, 8/5, 125/64, 2/1]

_Second plagal Byzantine Liturgical mode, Hard Chromatic 2nd plagal Byzantine mode : [P1, m2, SpM3, P4, P5, m6, ReA7, P8] # [1/1, 16/15, 9/7, 4/3, 3/2, 8/5, 25/13, 2/1]

_Soft chromatic scale : [P1, Prm2, ReM3, P4, P5, Prm6, PrSpGrm7, P8] # [1/1, 13/12, 16/13, 4/3, 3/2, 13/8, 13/7, 2/1]

_Xenakis Byzantine Liturgical Chromatic : [P1, Grm2, M3, P4, P5, Grm6, M7, P8] # [1/1, 256/243, 5/4, 4/3, 3/2, 128/81, 15/8, 2/1]

_Xenakis Byzantine Liturgical Diatonic, Misaelides 4th plagal Byzantine : [P1, AcM2, M3, P4, P5, AcM6, M7, P8] # [1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1]

_Xenakis Byzantine Liturgical Soft Chromatic : [P1, m2, M3, P4, P5, m6, M7, P8] # [1/1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8, 2/1]

_Hard chromatic scale : [P1, Prd2, M3, P4, P5, Prd6, M7, P8] # [1/1, 26/25, 5/4, 4/3, 3/2, 39/25, 15/8, 2/1]

_Misaelides 1st plagal Byzantine Liturgical mode : [P1, M2, Grm3, P4, ReA5, m6, Grm7, P8] # [1/1, 10/9, 32/27, 4/3, 20/13, 8/5, 16/9, 2/1]

Here's the best alphabetical fit at <8 cents of deviation from 72-EDO steps, considering natural 3-limit and 5-limit ratios simpler than all higher limit ratios:

_First plagal Byzantine Liturgical mode ascending : [P1, Asm2, Grm3, P4, P5, Asm6, Grm7, P8] # [1/1, 11/10, 32/27, 4/3, 3/2, 33/20, 16/9, 2/1]

_Fourth authentic Byzantine Liturgical mode : [P1, Prm2, DeM3, DeAcA4, P5, Prm6, DeM7, P8] # [1/1, 13/12, 40/33, 15/11, 3/2, 13/8, 20/11, 2/1]

_Misaelides 2nd plagal Byzantine Liturgical mode : [P1, m2, A3, P4, P5, m6, ExSpGrM7, P8] # [1/1, 16/15, 125/96, 4/3, 3/2, 8/5, 68/35, 2/1]

_Misaelides 3rd Byzantine Liturgical mode : [P1, AcM2, ExGrM3, AcA4, P5, AcM6, M7, P8] # [1/1, 9/8, 34/27, 45/32, 3/2, 27/16, 15/8, 2/1]

_Octave natural diatonic scale : [P1, Prm2, Grm3, P4, P5, Prm6, Grm7, P8] # [1/1, 13/12, 32/27, 4/3, 3/2, 13/8, 16/9, 2/1]

_Savas Diatonic Byzantine Liturgical mode : [P1, Prm2, DeM3, P4, P5, Prm6, DeM7, P8] # [1/1, 13/12, 40/33, 4/3, 3/2, 13/8, 20/11, 2/1]

_Scale of the grave diatonic mode : [P1, HbAcm2, ReM3, P4, P5, Asm6, PrSpGrm7, P8] # [1/1, 18/17, 16/13, 4/3, 3/2, 33/20, 13/7, 2/1]

_Second plagal Byzantine Liturgical mode : [P1, HbAcm2, A3, P4, P5, HbAcm6, ExSpGrM7, P8] # [1/1, 18/17, 125/96, 4/3, 3/2, 27/17, 68/35, 2/1]

_Xenakis Byzantine Liturgical Chromatic : [P1, Grm2, ExGrM3, P4, P5, PrDem6, ExGrM7, P8] # [1/1, 256/243, 34/27, 4/3, 3/2, 52/33, 17/9, 2/1]

_4th plagal Byzantine : [P1, AcM2, ExGrM3, P4, P5, AcM6, ExGrM7, P8] # [1/1, 9/8, 34/27, 4/3, 3/2, 27/16, 17/9, 2/1]

_Athanasopoulos' Byzantine Liturgical Chromatic : [P1, DeAcM2, ExGrM3, P4, P5, DeAcM6, ExGrM7, P8] # [1/1, 12/11, 34/27, 4/3, 3/2, 18/11, 17/9, 2/1]

_Byzantine Palace mode : [P1, HbAcm2, SpM3, P4, P5, DeAcM6, m7, P8] # [1/1, 18/17, 9/7, 4/3, 3/2, 18/11, 9/5, 2/1]

_Chromatic 2nd Byzantine Liturgical : [P1, HbAcm2, ExGrM3, P4, P5, HbAcm6, ExGrM7, P8] # [1/1, 18/17, 34/27, 4/3, 3/2, 27/17, 17/9, 2/1]

_Enharmonic Byzantine Liturgical : [P1, AcM2, ExGrM3, P4, P5, AcM6, Grm7, P8] # [1/1, 9/8, 34/27, 4/3, 3/2, 27/16, 16/9, 2/1]

_First plagal Byzantine Liturgical mode descending : [P1, Asm2, Grm3, P4, P5, HbAcm6, Grm7, P8] # [1/1, 11/10, 32/27, 4/3, 3/2, 27/17, 16/9, 2/1]

_Fourth plagal Byzantine Liturgical mode : [P1, AcM2, ReM3, P4, P5, AcM6, PrSpGrm7, P8] # [1/1, 9/8, 16/13, 4/3, 3/2, 27/16, 13/7, 2/1]

_Hard chromatic scale : [P1, Prd2, ExGrM3, P4, P5, Prd6, ExGrM7, P8] # [1/1, 26/25, 34/27, 4/3, 3/2, 39/25, 17/9, 2/1]

_Misaelides 1st Byzantine Liturgical mode : [P1, M2, Grm3, P4, P5, M6, Grm7, P8] # [1/1, 10/9, 32/27, 4/3, 3/2, 5/3, 16/9, 2/1]

_Misaelides 1st plagal Byzantine Liturgical mode : [P1, M2, Grm3, P4, ReA5, HbAcm6, Grm7, P8] # [1/1, 10/9, 32/27, 4/3, 20/13, 27/17, 16/9, 2/1]

_Misaelides 2nd Byzantine mode : [P1, DeAcM2, AsGrm3, P4, P5, DeAcM6, AsGrm7, P8] # [1/1, 12/11, 11/9, 4/3, 3/2, 18/11, 11/6, 2/1]

_Misaelides 4th Byzantine Liturgical mode : [P1, m2, m3, Ac4, P5, m6, m7, P8] # [1/1, 16/15, 6/5, 27/20, 3/2, 8/5, 9/5, 2/1]

_Savas Enharmonic Byzantine Liturgical mode : [P1, Prm2, ExGrM3, P4, P5, Prm6, ExGrM7, P8] # [1/1, 13/12, 34/27, 4/3, 3/2, 13/8, 17/9, 2/1]

_Scale of the grave enharmonic mode from Zo : [P1, HbAcm2, Grm3, P4, P5, HbAcm6, Grm7, P8] # [1/1, 18/17, 32/27, 4/3, 3/2, 27/17, 16/9, 2/1]

_Second authentic Byzantine Liturgical mode, Savas Soft Chromatic 2nd Byzantine mode : [P1, Prm2, ReM3, P4, P5, Prm6, DeM7, P8] # [1/1, 13/12, 16/13, 4/3, 3/2, 13/8, 20/11, 2/1]

_Second plagal Byzantine Liturgical mode, Hard Chromatic 2nd plagal Byzantine mode : [P1, HbAcm2, SpM3, P4, P5, HbAcm6, ReA7, P8] # [1/1, 18/17, 9/7, 4/3, 3/2, 27/17, 25/13, 2/1]

_Soft chromatic scale : [P1, Prm2, ReM3, P4, P5, Prm6, PrSpGrm7, P8] # [1/1, 13/12, 16/13, 4/3, 3/2, 13/8, 13/7, 2/1]

_Xenakis Byzantine Liturgical Diatonic, Misaelides 4th plagal Byzantine : [P1, AcM2, M3, P4, P5, AcM6, M7, P8] # [1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1]

_Xenakis Byzantine Liturgical Soft Chromatic : [P1, m2, M3, P4, P5, m6, M7, P8] # [1/1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8, 2/1]

It looks like these are related to each other "~" by cyclic permutations (at least in 72-EDO, if not in my determperings):

4th plagal Byzantine ~ Enharmonic Byzantine Liturgical

First plagal Byzantine Liturgical mode ascending ~ Fourth authentic Byzantine Liturgical mode

Misaelides 1st Byzantine Liturgical mode ~ Misaelides 3rd Byzantine Liturgical mode

Misaelides 4th Byzantine Liturgical mode ~ Xenakis Byzantine Liturgical Diatonic, Misaelides 4th plagal Byzantine

...

Ah, it looks like the scales came from https://www.huygens-fokker.org/docs/modename.html or maybe a clone of it. That helps.

...

Here's the data I should have started with:

* 12 tone modes:

[2, 2, 1, 2, 2, 2, 1]: "4th plagal Byzantine",

[2, 2, 1, 2, 2, 1, 2]: "Enharmonic Byzantine Liturgical",

[1, 3, 1, 2, 1, 3, 1]: "Chromatic 2nd Byzantine Liturgical",

* 24 tone modes:

[3, 5, 2, 4, 3, 5, 2]: "Athanasopoulos' Byzantine Liturgical Chromatic",

[3, 4, 3, 4, 3, 4, 3]: "Misaelides 2nd Byzantine mode",

[2, 7, 1, 4, 2, 7, 1]: "Second plagal Byzantine Liturgical mode",

* 64 tone equal modes:

[7, 12, 7, 12, 7, 12, 7]: "Chrysanthos Soft Chromatic Byzantine mode",

* 68 tone equal modes:

[9, 7, 12, 12, 9, 7, 12]: "Chrysanthos 1st Byzantine Liturgical mode",

[12, 13, 3, 12, 12, 5, 11]: "Chrysanthos 3rd Byzantine Liturgical mode",

[12, 9, 7, 12, 9, 7, 12]: "Chrysanthos 4th Byzantine Liturgical mode",

[7, 18, 3, 12, 7, 18, 3]: "Chrysanthos Hard Chromatic 2nd plagal Byzantine mode",

[7, 18, 3, 12, 9, 7, 12]: "Chrysanthos Hard Chromatic/Diatonic Byzantine mode",

[9, 7, 12, 12, 3, 13, 12]: "Chrysanthos Diatonic-Enharmonic Byzantine mode",

[13, 12, 3, 12, 9, 7, 12]: "Chrysanthos Enharmonic-Diatonic Byzantine mode",

[9, 7, 12, 7, 18, 3, 12]: "Fokaeas 2nd plagal Byzantine Liturgical mode",

[12, 9, 7, 12, 12, 3, 13]: "Konstantinos 3rd Byzantine Liturgical mode",

[12, 4, 12, 9, 7, 12, 12]: "Konstantinos 4th plagal Byzantine Liturgical mode",

[12, 13, 3, 12, 12, 13, 3]: "Tiby 1st Byzantine Liturgical mode",

[12, 5, 11, 12, 12, 5, 11]: "Tiby 2nd Byzantine Liturgical mode",

[9, 12, 7, 12, 9, 12, 7]: "Tiby 4th Byzantine Liturgical mode",

[7, 14, 7, 12, 7, 14, 7]: "Tsiknopoulos 2nd Byzantine Liturgical mode",

[7, 12, 12, 9, 7, 12, 9]: "Tsiknopoulos 4th Byzantine Liturgical mode",

[12, 9, 7, 12, 12, 9, 7]: "Tsiknopoulos 4th plagal Byzantine Liturgical mode, Tiby 3rd Byzantine mode",

* 72 tone equal modes:

[11, 7, 12, 12, 11, 7, 12]: "Misaelides 1st Byzantine Liturgical mode",

[12, 12, 11, 7, 12, 11, 7]: "Misaelides 3rd Byzantine Liturgical mode",

[7, 12, 12, 11, 7, 12, 11]: "Misaelides 4th Byzantine Liturgical mode",

[11, 7, 12, 15, 3, 12, 12]: "Misaelides 1st plagal Byzantine Liturgical mode",

[7, 20, 3, 12, 7, 20, 3]: "Misaelides 2nd plagal Byzantine Liturgical mode",

[5, 19, 6, 12, 5, 19, 6]: "Xenakis Byzantine Liturgical Chromatic",

[7, 16, 7, 12, 7, 16, 7]: "Xenakis Byzantine Liturgical Soft Chromatic",

[12, 11, 7, 12, 12, 11, 7]: "Xenakis Byzantine Liturgical Diatonic, Misaelides 4th plagal Byzantine",

[6, 20, 4, 12, 9, 10, 11]: "Byzantine Palace mode",

Looking into the names associated with these scales, we probably are talking about:

    George Athanasopoulos

    Iannis Xenakis

    Savas I. Savas or Savas J. Savas (same person)

    Misael Misaelides

    Andreas Tsiknopoulos

    Ottavio Tiby

    Konstantinos Psachos 

    Theodoros Fokaeas and/or Alexandros Fokaeas

    Chrysanthos of Madytos 

All of those are referred to by their last name except for Psachos, who is referred to by his first name. Odd.

...

Lol, oh no. These are so stupidly simple. Almost all of the scales have repeated tetrachords: 

* 12 tone modes:

[2, 2, 1] [2] [2, 2, 1]: "4th plagal Byzantine",

[2, 2, 1] [2, 2, 1] [2]: "Enharmonic Byzantine Liturgical",

[1, 3, 1] [2] [1, 3, 1]: "Chromatic 2nd Byzantine Liturgical",

* 24 tone modes:

[3, 5, 2] [4] [3, 5, 2]: "Athanasopoulos' Byzantine Liturgical Chromatic",

[3, 4, 3] [4] [3, 4, 3]: "Misaelides 2nd Byzantine mode",

[2, 7, 1] [4] [2, 7, 1]: "Second plagal Byzantine Liturgical mode",

* 64 tone equal modes:

[7, 12, 7] [12] [7, 12, 7]: "Chrysanthos Soft Chromatic Byzantine mode",

* 68 tone equal modes:

[9, 7, 12] [12] [9, 7, 12]: "Chrysanthos 1st Byzantine Liturgical mode",

[12, 9, 7] [12, 9, 7] [12]: "Chrysanthos 4th Byzantine Liturgical mode",

[7, 18, 3] [12] [7, 18, 3]: "Chrysanthos Hard Chromatic 2nd plagal Byzantine mode",

[12, 13, 3] [12] [12, 13, 3]: "Tiby 1st Byzantine Liturgical mode",

[12, 5, 11] [12] [12, 5, 11]: "Tiby 2nd Byzantine Liturgical mode",

[9, 12, 7] [12] [9, 12, 7]: "Tiby 4th Byzantine Liturgical mode",

[7, 14, 7] [12] [7, 14, 7]: "Tsiknopoulos 2nd Byzantine Liturgical mode",

[12, 9, 7] [12] [12, 9, 7]: "Tsiknopoulos 4th plagal Byzantine Liturgical mode, Tiby 3rd Byzantine mode",

* 72 tone equal modes:

[11, 7, 12] [12] [11, 7, 12]: "Misaelides 1st Byzantine Liturgical mode",

[12] [12, 11, 7] [12, 11, 7]: "Misaelides 3rd Byzantine Liturgical mode",

[... 7] [12] [12, 11, 7] [12, 11, ...]: "Misaelides 4th Byzantine Liturgical mode",

[7, 20, 3] [12] [7, 20, 3]: "Misaelides 2nd plagal Byzantine Liturgical mode",

[5, 19, 6] [12] [5, 19, 6]: "Xenakis Byzantine Liturgical Chromatic",

[7, 16, 7] [12] [7, 16, 7]: "Xenakis Byzantine Liturgical Soft Chromatic",

[12, 11, 7] [12] [12, 11, 7]: "Xenakis Byzantine Liturgical Diatonic, Misaelides 4th plagal Byzantine",

And most of the others have tetrachord structure, but not repeated.

* 68 tone equal modes:

[12, 13, 3] [12] [12, 5, 11]: "Chrysanthos 3rd Byzantine Liturgical mode",

[7, 18, 3] [12] [9, 7, 12]: "Chrysanthos Hard Chromatic/Diatonic Byzantine mode",

[9, 7, 12] [12] [3, 13, 12]: "Chrysanthos Diatonic-Enharmonic Byzantine mode",

[13, 12, 3] [12] [9, 7, 12]: "Chrysanthos Enharmonic-Diatonic Byzantine mode",

[9, 7, 12] [7, 18, 3] [12]: "Fokaeas 2nd plagal Byzantine Liturgical mode",

[12, 9, 7] [12] [12, 3, 13]: "Konstantinos 3rd Byzantine Liturgical mode",

[12, 4, 12] [9, 7, 12] [12]: "Konstantinos 4th plagal Byzantine Liturgical mode",

[, ...7, 12] [12] [9, 7, 12] [9, ...]: "Tsiknopoulos 4th Byzantine Liturgical mode",

These two are a little to weird for me to parse right away:

* 72 tone equal modes:

[11, 7, 12, 15, 3, 12, 12]: "Misaelides 1st plagal Byzantine Liturgical mode",

[6, 20, 4, 12, 9, 10, 11]: "Byzantine Palace mode",

Crazy, man.

Ah, I found one that I probably mis-parsed. I bet it's 

[... 12] [9, 7, 12] [12] [3, 13, ...]: "Konstantinos 3rd Byzantine Liturgical mode",

12-EDO tetrachords

Enharmonic: [2, 2, 1]\12

Chromatic: [1, 3, 1]\12

24-EDO tetrachords:

[3, 5, 2]\24, 

[3, 4, 3]\24, 

[2, 7, 1]\24

64-EDO tetrachords:

Soft chromatic: [7, 12, 7]\64

These are the tetrachords of 68-EDO:

Hard Chromatic: [7, 18, 3]\68

Diatonic: [9, 7, 12]\68

Enharmonic: [3, 13, 12]\68

?: [12, 4, 12]\68

72-EDO tetrachords:

[12, 11, 7]\72

[7, 20, 3]\72

[5, 19, 6]\72

[7, 16, 7]\72

We can solve this.

For the 72-EDO tetrachords, I propose:

    [12, 11, 7]\72 -> [P1, AcM2, M3, P4] # [1/1, 9/8, 5/4, 4/3]

    [7, 20, 3]\72 -> [P1, m2, A3, P4] # [1/1, 16/15, 125/96, 4/3]

    [5, 19, 6]\72 -> [P1, Grm2, AcM3, P4] # [1/1, 256/243, 81/64, 4/3]

    [7, 16, 7]\72 -> [P1, m2, M3, P4] # [1/1, 16/15, 5/4, 4/3]

I had to remove a bunch of simpler ratios before the program would say that 81/64 was a good fit in the third one of those. Maybe it's not so good of a fit? I think it's okay. 400 cents for the 72\EDO third and 408 for the Pythagorean one. Since a single step of 72-EDO is 16.7 cents, that's plenty close.

Here's an attempted detempering of the 68-EDO tetrachords:

    Hard Chromatic : [P1, m2, A3, P4] # [1/1, 16/15, 125/96, 4/3]

    Diatonic : [P1, Asm2, Grm3, P4] # [1/1, 11/10, 32/27, 4/3]

    Enharmonic : [P1, d2, Grm3, P4] # [1/1, 128/125, 32/27, 4/3]

    ? : [P1, AcM2, Grm3, P4] # [1/1, 9/8, 32/27, 4/3]

We've now seen this tetrachord twice:

    [P1, m2, A3, P4] # [1/1, 16/15, 125/96, 4/3]

Called the Hard Chromatic in 68-EDO and not given a name but represented by [7, 20, 3] in 72-EDO.

I think this is also a good detempering of the 68-EDO hard chromatic:

    [P1, ReSbAcM2, SpM3, P4] # [1/1, 14/13, 9/7, 4/3]

The other ones don't align so well. All the remaining 72-EDO tetrachords had major thirds, but all the 68-EDO ones have minir thirds. The 72-EDO tetrachords had seconds that were major, major, minor, whereas 68-EDO has major, neutral, diminished.

...

We have 12-EDO tetrachords named enharmonic and chromatic. Let's see what those look like. The enharmonic is just 

    [M2, M2, m2] :: [P1, M2, M3, P4]

And the chromatic is

    [m2, A2, m2] :: [P1, m2, M3, P4]

And those do match 72-EDO tetrachords. The enharmonic is

    [12, 11, 7]\72 -> [P1, AcM2, M3, P4] # [1/1, 9/8, 5/4, 4/3]

and the chromatic is one of these:

    [5, 19, 6]\72 -> [P1, Grm2, AcM3, P4] # [1/1, 256/243, 81/64, 4/3]

    [7, 16, 7]\72 -> [P1, m2, M3, P4] # [1/1, 16/15, 5/4, 4/3]

..

Okay, after a little scholarship and a little looking over the things I've already written about or taken notes about, I provide here the 72-EDO forms of four tetrachords of Byzantine music along with their names:

    Diatonic: [12, 10, 8]

    Enharmonic: [12, 12, 6]

    Soft chromatic: [8, 14, 8]

    Hard chromatic: [6, 20, 4]

Here are some plausible just intonations for those:

Diatonic: [12, 10, 8] - [Major, Major, neutral]

[9/8, 11/10, 320/297]

[9/8, 128/117, 13/12]

[9/8, 208/189, 14/13]

Enharmonic: [12, 12, 6] - [Major, Major, minor]

[9/8, 9/8, 256/243]

Soft chromatic: [8, 14, 8] - [neutral, Super major, neutral]

[13/12, 8/7, 14/13]

[14/13, 8/7, 13/12]

[13/12, 192/169, 13/12]

[14/13, 169/147, 14/13]

Hard Chromatic: [6, 20, 4] - [minor, Super augmented, subminor]

[256/243, 2187/1792, 28/27]

[256/243, 2025/1664, 26/25]

These just tunings are correct for 72-EDO interval mapping. I haven't checked how close the just tunings are numerically to the irrational tunings.

We've seen interesting variations on those tetrachords directly above. For example, in analyzing the old scale data, we've seen a tetrachord with these numbers

    [12, 11, 7]\72

which is halfway between the Diatonic and Enharmonic genera directly above. 

We've also seen tetrachords

 

    [7, 20, 3]\72

    [5, 19, 6]\72

which I think are both variations on the Hard chromatic genus above. We've also seen this tetrachord:

    [7, 16, 7]\72

which is a variation on the soft chromatic tetrachord. So we've got lots of vague agreement and very little precise agreement.

...

Chrysanthos of Madytos gave tetrachords in 68-EDO. Then people decided his math was wrong and redid it in 72-EDO. Let's look at his stuff.

His diatonic tetrachord was

    [12, 9, 7]\64 // Relative

    [0, 12, 21, 28]\64 // Absolute

  

Antonije Tot tells us in "64 EDO and 68 EDO in The Great Theory of Music of Archbishop Chrysanthos", that this corresponds exactly to 

 

    [AcM2, DeAcM2, AsGrm2] # [9/8, 12/11, 88/81] 

    [P1, AcM2, DeAcM3, P4] # [1/1, 9/8, 27/22, 4/3]

by which I think he means that Chysanthos himself gave this just intonation interpretation of his EDO steps. People say that Chrysanthos made lots of mathematical errors, but 68-EDO does tune these as Chrystanthos did, so that's fine. You'll notice that this diatonic tetrachord is a Zalzalian jins Rast.

Only knowing that (AcM2, DeAcM2, AsGrm2) were tuned to [12, 9, 7] of some kind of unit, we can combine those intervals to get steps for a whole Pythagorean tonal sekeleton with 11-limit neutral tones:

1/1 = 0\68

33/32 = 3\68

256/243 = 4\68

88/81 = 7\68

2187/2048 = 8\68

12/11 = 9\68

9/8 = 12\68

32/27 = 16\68

11/9 = 19\68

27/22 = 21\68

81/64 = 24\68

4/3 = 28\68

When I was making that table by hand, I been writing the units as [12, 9, 7] "moria", the term for microtonal divisions of Byzantine music. But it turns out that moria is the same as steps of 68 EDO for all the intervals justly associated with the ratios above, so I think Chysanthos did a decent job with his math so far. It's a functional system.

Chrysanthos gives 

    [7, 18, 3]\68 

as a hard chromatic scale. Using just the frequencies in the table above, you might think that 7 steps should be detempered to 88/81, and 3 steps should be detempered to 33/32. To reach the perfect fourth, this gives us a tetrachord of

    [88/81, 144/121, 33/32] @ [143, 301, 53] c

Antonije Tot instead says that the 3-step quarter tone relative interval at the end of the Hard Chromatic tetrachord should be detempered 36/35. I think he's reading a primary source of Chrysanthos and following his geometric calcualtions on the neck of a string to figure out this frequency ratio. It certainly can't be reached by combinations of 9/8, 12/11, and 88/81, since thosea are in the 2.3.11 J.I. subgroup. Anyway, if we use Tot's quarter tone, then we get a Hard Chromatic tetrachord of:

    [88/81, 105/88, 36/35] @ [143, 306, 49] c

which is hardly aurally distinct to the previous one. Nice. But wait! There's an ancient Greek chromatic tetrachord that matches this nicely also. 

First an aside: Chrysanthos lived from 1770 to1846, not all that long ago in the history of music, but Byzantine chant is supposed to trace its origins at least as far back as the Byzantine empire (which existed between 330 and 1453), and it was almost surely influenced by ancient Greek music theory. So it's not unusual to look at Greek music theory for inspiration.

Here are the 68-EDO tunigns of the Chromatic tetrachords of Ancient Greece that we've learned about:

[4, 8, 16]\68 # [Grm2, AcAcA1, Grm3]: Pythagorean Chromatic

[3, 9, 16]\68 # [Sbm2, SpAcA1, Grm3]: Chromatic of Archytas

[6, 4, 18]\68 # [m2, A1, m3]: Chromatic of Didymuys

[3, 7, 18]\68 # [FaA1, Rsm2, m3]: Chromatic of Eratosthenes

[3, 7, 18]\68 # [Sbm2, SpA1, m3]: Chromatic Malakon of Ptolemy

[4, 9, 15]\68 # [AsSpGr1, DeAcM2, Sbm3]: Chromatic Syntonon of Ptolemy

And compare that to

    [7, 18, 3]\68 : Hard Chromatic Of Chrysanthos

You can see that Chryanthos puts the m3 interval in the middle instead of at the end, but otherwise, we've actually got two historic Greek tetrachords that match Chrysanthos's relative intervals. I'll write in the just tunings now instead of the 68-EDO steps:

[20/19, 19/18, 6/5] # [FaA1, Rsm2, m3]: Chromatic of Eratosthenes

[28/27, 15/14, 6/5] # [Sbm2, SpA1, m3]: Chromatic Malakon of Ptolemy

So if we move the first relative interval (ther quartertone) to the end of each of those, then we get too more plausible just intonations for the Hard Chromatic scale of Chrysanthos.

All together now:

[88/81, 144/121, 33/32] @ [143, 301, 53] c # Hard Chromatic, 2.3.11

[88/81, 105/88, 36/35] @ [143, 306, 49] c # Hard Chromatic, Tot

[19/18, 6/5, 20/19] @ [93, 315, 89] c # Eratosthenian Hard Chromatic

[15/14, 6/5, 28/27] @ [119, 315, 63] c # Ptolemaic Hard Chromatic

I think the first two are more like what Chrysanthos wanted, maybe exactly what he wanted, but it's nice to see a historic similarity in the other two.

Now things get tricky. This is the soft chromatic tetrachord of Chrysanthos:

    [7, 12, 7]

It only reached 26 steps of 68-EDO, not a perfect fourth at 28 steps. This is why you'll sometimes see a soft chromatic scale as being 64-EDO: two 26-step tetrachords plus a 12-step acute major second:

    [26 + 12 + 26] = 64 steps.

I don't know what other mathematical mistakes Chrysanthos made, but this sure seems like one. I don't know what to do with it. Should the soft chromatic be [8, 12, 8] or [7, 14, 7] or something else? Tot says that subsequent writers went with [7, 14, 7], so let's try that. Some options:

    [15/14, 784/675, 15/14]

    [14/13, 169/147, 14/13]

    [15/14, 52/45, 14/13]

    [88/81, 2187/1936, 88/81]

Okay. If we go with [8, 12, 8], we get a nice Zalzalian tetrachord:

    [13/12, 9/8, 128/117]

I think that makes way more sense. Neutral intervals like 7 and 8 steps of 68-EDO are harder to hit precisely than a Pythagorean major second. It makes more sense that Chrysanthos would have gotten the neutral intervals wrong than that he would have written 12 steps of 68-EDO for something that is ~35 cents higher than a 9/8.

Tot doesn't describe the Enharmonic genus of Chrysanthos in 68-EDO - he only mentions that it has a 3-step quarter tone. From the the Huygens Fokker scale data, we can guess that it's some arrangement of [3, 13, 12], but those show up in multiple different orders in scales that have "Enharmonic" in the name. In fact all three possible orders show up across his scales:

    [3, 13, 12]

    [13, 3, 12]

    [13, 12, 3]

Though maybe I should be happy he only uses half of the possible orderings and avoids these:

 

    [3, 12, 13]

    [12, 3, 13]

    [12, 13, 3]

    

Despite the name, these enharmonic genera are not closely related to Ancient greek Enharmonic genera which had two quartertones instead of one.

Anyway, regardless of the order of steps, we can come up with some guesses for just intonation. Using the order [12, 3, 13], we could have

[9/8, 33/32 1024/891]

[9/8, 36/35, 280/243]

[9/8, 28/27, 8/7]

[9/8, 39/38, 1216/1053]

[9/8, 704/675, 25/22]

I like [9/8, 28/27, 8/7] a lot, but then I also like Archytas. I think the first two are probably closest to what Chrysanthos would like. Within those two, I would just reccomend being consistent with yourself: if you liked 33/32 as a quartertone in the Hard Chromatic genus, then you should use it again here. If you liked 36/35 in the Hard chromatic genus, then you should use that one here. They hardly sound different anyway.

Okay! That is a review of Chrysanthos's tetrachords. After Chrysanthos published, there was a lot of discussion in the world of Greek liturgical music, culimating in the Great Ecumenical Patriarchal Musical Committee in Constantinople in 1881 where they decided that their music should be represented in 72-EDO, not 68-EDO. Let's investigate Byzantine tetrachords according to people besided Chrysanthos. Definitely we can talk about the Musical Committee. Hopefully some other sources too. Konstantinos Pringos? Misael Misaelides?

...