: History
The Byzantine empire existed for about a thousand years after the fall of unified Rome, from like 330 AD to 1453 AD. They had music influenced by the ancient Greeks and other cultures of the Hellenistic era. Eastern orthodox churches still have something that they call Byzantine chant. It's very plausibly descended from the music of the Byzantine empire, but has probably undergone some changes since then. The oldest theoretical treatise about it that I know of is Archbishop Chrysanthos of Matydos's "Theoretikon Mega Tes Mousikes" or "Great Theory of Music" published in 1832. That's 400 years later than we'd really like, but it's not too far off. Byzantine chant uses neutral tones and two-part polyphony - a melody and a drone. The drone note moves quite a bit, and the harmony ends up being a little richer than maqam music, though the scales are similar. Byzantine chant doesn't have chords and stacked triads, but it's still interesting.
Chrysanthos of Madytos described the tetrachords and scales of Byzantine music in 68-EDO. Within a few decades, the Eastern Orthodox church and its chanters decided that his work had too many errors, so a council of Byzantine musical experts convened and published new opinions about how Byzantine music should instead be understood in 72-EDO. But I think Chrysanthos's writing has a lot of merit, so we're going to start with that.
: The Byzantine Tetrachords Of Chrysanthos of Matydos
Chrysanthos had roughly four tetrachords, the Diatonic, the Hard Chromatic, the Soft Chromatic, and the Enharmonic. We'll address them in that order.
: The Diatonic
From the name, you might expect this to be a boring [M2, M2, m2] tetrachord, but it's actually more like jins Rast. Here are the steps:
[12, 9, 7] :: [0, 12, 21, 28]\64
Chrysanthos also gives a scale with this tetrachord repeated. He writes in fractions of a string, i.e. in wave space instead of in frequency space, but we just have to invert the fractions to make that look familiar. Here's the scale in absolute frequency ratios:
[1/1, 9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]
That's maqam rast, right? So the basic Diatonic scale of Byzantine chant is microtonal. It's great! We can now say that his Diatonic tetrachord corresponds exactly to this just intonation:
[AcM2, DeAcM2, AsGrm2] # [9/8, 12/11, 88/81] // Relative
[P1, AcM2, DeAcM3, P4] # [1/1, 9/8, 27/22, 4/3] // Absolute
which is a fine Zalzalian intonation of jins Rast. I've heard people say that Chrysanthos made lots of mathematical errors that undermine his whole theory and that we can only get hints and impressions from his writings about the truth of Byzantine music from him, but 68-EDO does in fact tune these intervals to the steps that Chrystanthos did, so that's fine. It all seems good and valid so far.
At first I only knew that (AcM2, DeAcM2, AsGrm2) were tuned to [12, 9, 7] commas or "moria" in his system, and I was supicious from other readings that I couldn't use 68-EDO to correctly and successfully analyze Chrysanthos's work. If we use only those numbers, [12, 9, 7] as quantities of some mysterious Greek musical unit that doesn't behave like steps of 68-EDO, we can still combine those intervals to get moria for a whole Pythagorean tonal sekeleton and some 11-limit neutral tones. Once I'd done this, I found that everything was still just validly tuned in 68-EDO, so I'm not going to pretend that Chrysanthos was wrong in that regard. Here's the table I generated with tempered tunings and just tunings for a few such intervals between P1 and P4:
0\68 ~ P1 # 1/1
3\68 ~ As1 # 33/32
4\68 ~ Grm2 # 256/243
7\68 ~ AsGrm2 # 88/81
8\68 ~ AcAcA1 # 2187/2048
9\68 ~ DeAcM2 # 12/11
12\68 ~ AcM2 # 9/8
16\68 ~ Grm3 # 32/27
19\68 ~ AsGrm3 # 11/9
21\68 ~ DeAcM3 # 27/22
24\68 ~ AcM3 # 81/64
28\68 ~ P4 # 4/3
If you want, you can use the term "moria" for microtonal divisions of Byzantine music. But it turns out that one "morio" (singular) is the same as one step of 68 EDO, so I think Chysanthos did a decent job with his math so far. It's a functional system.
: The Hard Chromatic
Next tetrachord! Here's a Hard Chromatic tetrachord from Chrysanthos:
[7, 18, 3]\68
He introduces the Hard and Soft Chromatic tetrachords in the same chapter, and I don't think he actually gives them separate names, but this is a fine name.
Using just the frequencies in the table above, the first relative interval (7 steps) can be detempered to 88/81, and last interval (3 steps) can be detempered to 33/32. We don't have an interval at 18\64 in our table above, but there's only one interval that can go here so the tetrachord outlines a perfect fourth. All together we get this just intonation:
[88/81, 144/121, 33/32] @ [143, 301, 53] c
So it's a neutral second, a thing that sounds like a minor third, and a quarter tone. In "Archbishop Chrysanthos' 64 And 68 EDO Music Theory" (2025), Antonije Tot instead says that the 3-step quarter tone at the end of the Hard Chromatic tetrachord should be detempered to 36/35. I've paged through the original Chrysanthos book and I didn't see that, but I also didn't follow all of Chrysanthos's wavespace calculation, so maybe Tot is right. He surely seems to have read the primary source closely. If we use Tot's quarter tone, then we get this just intonation of the Hard Chromatic tetrachord:
[88/81, 105/88, 36/35] @ [143, 306, 49] c
which is hardly aurally distinct from the previous one. So that's great, right? We can use either quarter tone and it still works.
I think these two tunings are good and plausible representations of the Byzantine Hard Chromatic tetrachord, but let's go on a tangent for a moment, because we now have an opportunity to connect Byzantine music theory to ancient Greek music theory!
Here are the 68-EDO tunigns of the Chromatic tetrachords of Ancient Greece that we've learned about in a previous chapter:
[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
We compare that to the steps of Chrysanthos's Hard Chromatic:
[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 in 68-EDO - Eratostheses Chromatic and Ptolemy's Chromatic Malakon both have steps of [3, 7, 18]\68. I'll write in the just tunings for them 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 two more somewhat plausible just intonations for the Hard Chromatic scale of Chrysanthos. All together now:
[88/81, 144/121, 33/32] @ [143, 301, 53] c # My version of the Hard Chromatic in the 2.3.11 just intonation subgroup
[88/81, 105/88, 36/35] @ [143, 306, 49] c # Hard Chromatic from 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
If you look at the cent values, you'll see that the first two intonations are much closer to each other than to the ancient Greek chromatic tetrachords, but it's nice to see a historic similarity.
: The Soft Chromatic
Our next tetrachord is a little tricky. This is the Soft Chromatic tetrachord of Chrysanthos:
[7, 12, 7]\68
It only reached 26 steps of 68-EDO, not a perfect fourth at 28 steps. This is why you'll sometimes see people claim that Chrysanthos's Soft Chromatic scale was specified in 64-EDO: it has two 26-step tetrachords separated by a 12-step acute major second:
[7, 12, 7] + [12] + [7, 12, 7]
[26 + 12 + 26] = 64 steps.
I don't know what other mathematical mistakes Chrysanthos made, but this sure seems like one.
We have a few options here. We could say that this is what Chrysanthios wrote and and what he meant, and accept it as 64-EDO. Unfortunately, 64-EDO tunes its AcM2 to 10 steps, not 12 steps, so this still doesn't get us a scale made of disjunct tetrachords spanning P4. We could instead edit the steps to reach P4 in 68-EDO. These are the most obvious choices for a corrected Soft Chromatic:
[7, 14, 7]
[8, 12, 8]
Tot says that subsequent writers went with [7, 14, 7], so let's try that. Here are some options for a just intonation of that tetrachord:
[15/14, 784/675, 15/14]
[14/13, 169/147, 14/13]
[15/14, 52/45, 14/13]
[88/81, 2187/1936, 88/81]
I think those are pretty ugly in terms of numerical aesthetics. More importantly, it's implausible to me that Chrysanthos would write 12 steps, like a Pythagorean major second, for an interval that was actually 14 steps in size. Chrysanthos would have known what a major second sounded like. The disjunct scale has one between the tetrachords - he could just listen to it there.
If we instead replace [7, 12, 7] with [8, 12, 8], then we're simply replacing a neutral 7-step interval with a neutral 8-step interval. And if we do this, then we're not saying that some of the 12 steps in the Soft Chromatic scale should be edited while the middle one should remain at 12 steps. And we're only correcting individual intervals by 1 step instead of 2 steps. This is simpler in so many ways.
Another benefit of usuing [8, 12, 8] for the Soft Chromatic tetraachord is that we get a nice Zalzalian tetrachord as an option for its just intonation:
[13/12, 9/8, 128/117]
Neutral intervals like 7 and 8 steps of 68-EDO are harder for a singer 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 9/8.
So I contend that the Soft Chromatic scale of Byzantine Chant is
[8, 12, 8]\68
Here's another possibility!
If the Hard Chromatic is [7, 18, 3], and the Soft Chromatic is [7, 12, 7], doesn't it kind of look like he's moving a few moria between the second and third relative intervals? If the second interval, from Hard to Soft, goes down by 6 steps, then the third interval should go up by 6 steps, giving us this tetrachord:
[7, 12, 9]\68
which is just a re-arrangement of the Diatonic genus. I don't think it's right, but it's not a terrible guess, right?
: The Enharmonic
The the chapter where Chrysanthos introduces the Enharmonic genus, he writes out five scales in moria that have one tetrachord or both being Enharmonic. Across the scales, he uses three different Enharmonic tetrachords with the same relative intervals in different arrangements. Here are the five scales:
[13, 3, 12] [12] [3, 13, 12]
[9, 7, 12] [12] [3, 13, 12]
[3, 13, 12] [12] [9, 7, 12]
[13, 3, 6] [18] [9, 7, 12]
[13, 12 3] [12] [9, 7, 12]
You may note that the fourth of these scales has an interval tuned to 18\68 steps instead of a major second between two tetrachords. Highly peculiar! We saw an 18-step interval in the Hard Chromatic genus:
[7, 18, 3] \68
Maybe that's a hint about what's going on in the fourth scale.
All together we have three proper Enharmonic genera with 3-step quartertones that span P4 and one improper one:
[13, 3, 12]
[3, 13, 12]
[13, 12, 3]
[13, 3, 6]
Despite the name, these enharmonic genera are not closely related to Ancient greek Enharmonic genera which had two quartertones instead of one.
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 for numerical aesthetics, but then I also like Archytas. I think the first two tetrachords here 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.
We're going to discuss the tetrachords of the Music Committee next, and then a few other sources: Konstantinos Pringos and Misael Misaelides. Once we have a firm foundation in the tetrachords, then we'll look at the scales and the harmonies that arise within them.
...
: Content That I'll Probably Discard
Chrysanthos introduces the Enharmoncic genus in the Third Book, Chapter VII. I didn't want to talk about scales until we understood tetrachords, but he doesn't give us much choice:
Enharmonic genus is the one in the scale of which are semitones, actually quarters of a major tone, either as hypheses or dieses or as both hypheses and dieses.
As hypheses thus:
Pa bou 𝈉 di ke zo ne pa.
As dieses thus:
Pa ♁ ga di ka zo ne pa.
As dieses and hypheses, thus:
Pa ♁ ga di ke 𝈉 ne pa.
The syllables are Greek solfege and the symbols [♁, 𝈉] I'm using to approximate his accidentals. For comparison, the diatonic scale is sung
Pa bou ga di ke zo ne pa.
So the accidentals are actually replacing solfege notes. In the first scale, ga is repalced with circle-down. In the next scale, bou is replaced with circle-up. These are all 7-note scales, but they're not tetrachords and I'm not sure why he's sharing them.
He actually capitalizes the octave, the second "pa" of each scale, and not the first "pa", but I write in English and want a capital letter at the start of a thought, not near the end. But if you see a Byzantine scale written in Greek letters and you're not sure which end is high and which is low, the capitalized letter is high.
As to his terminology, a "hyphesis" literally means a flattening of a sound and a "diesis" means a sharpening of a sound. Elsewhere in the book, he makes it clear that this is a sharpening or flattening by a semitone. A semitone could be
Grm2 # 256/243 ~ 4\68
AcAcA1 # 2187/2048 ~ 8\68
I think, unless we see otherwise, we can give him the benefit of the doubt and assume that he's usually sharpening intervals with an augmented unison. However, here he is not! Let's read it again.
Enharmonic genus is the one in the scale of which are semitones, actually quarters of a major tone, either as hypheses or dieses or as both hypheses and dieses.
So he's sharpening and flattening by quartertones.
The scale that he says has a hyphesis does indeed have a downward pointing circle symbol, and the diesis scale has an upward pointing circle, and the scale with both has both. That's all well and good. I'd guess that he's sharpening things relative to the Diatonic scale, not the Pythagorean major. Let's try to write his scales out in moria. Here's the diatonic scale for our base case:
[12, 9, 7] [12] [12, 9, 7] : [0, 12, 21, 28, 40, 52, 61, 68]\68
Our three enharmonic scales respectively flatten the third, raise the second, and raise the second while flattening the sixth. If we treat 3\68 as a quarter tone, then we get
[12, 6, 10] [12] [12, 9, 7] : [0, 12, 18, 28, 40, 52, 61, 68]\68 // Enharmonic 1
[15, 6, 7] [12] [12, 9, 7] : [0, 15, 21, 28, 40, 52, 61, 68]\68 // Enharmonic 2
[15, 6, 7] [12] [9, 12, 7] : [0, 15, 21, 28, 40, 49, 61, 68]\68 // Enharmonic 3
Note of these makes sense wih what he says later, but I sure tried to figure it out.
I should note, to avoid confusion, that some music theory sources that aren't Byzantine use "diesis" to mean a 5-limit diminished second
d2 # 128/125 @ 41 cents
I don't think I've ever seen it used to refer to the Pythagorean diminished second
GrGrGrd2 # 524288/531441 @ -23 cents
but I don't know why that wouldn't also be a diesis, unless people use the 5-limit d2 as a stand-in for "sharp by about a quarter tone".
....
Chrysanthos writes
"In Euclid’s times the enharmonic genus was sung in descent as ditone, diesis, diesis, or else, ditone, quarter, quarter. In ascent it was sung reversely, diesis, diesis, ditone, or else, quarter, quarter, ditone. But in our days melodies of such scales are not preserved. Instead, one enharmonic diesis and one enharmonic hyphesis exist in two tetrachords, not in one."
The term "ditone" means a major third, since it's twice as large as a tone
AcM2 + AcM2 = AcM3
(9/8) * (9/8) = 81/64
And indeed the Enharmonic genera of ancient Greece that we looked at before were sized like [Major third, quarter tone quarter tone]. Besides that, I don't really understand his comment.
...
Chrysanthos gives 4 scales in moria in the chapter where he introduced the chromatic genera:
[7, 12, 7] [12] [7, 12, 7]
[7, 18, 3] [12] [7, 18, 3]
[7, 18, 3] [12] [9, 7, 12]
[9, 7, 12] [12] [7, 18, 3]
The first two scales are purely (soft and hard) chromatic, while the second two scales are a mix of chromatic and diatonic tetrachords.
...
In the chapter on the Chromatic genus, Chrysanthos gives us some 7 note scales with hyphesis and diesis symbols. I don't have much hope of figuring anything out from them, but let's trascribe it anyway.
Chromatic genus is the one in the scale of which exist semitones derived from hypheses, or from dieses or from both hypheses and dieses. The scale with the hypheses is:
Ne 𝈉 bou ga di 𝈉 zo ne.
The one with the dieses is:
Pa bou ♁ di ke zo ♁ pa.
The scale with both dieses and hypheses is:
Pa ♁ 𝈉 di ke ♁ 𝈉 pa.
...
: Tetrachords Of The Patriarchal Committee
I'm trying to get through "Modern Theory And Notation Of Byzantine Chanting Tradition A Near-Eastern Musicological Perspective" by Markos Skoulios (2012), but it's not easy. See for example:
It was Chrysanthos again who introduced the arithmetic representation of intervals assigning 12 tmimata (lit. “parts”) to the major tone in accordance with Cleonides’ dodekatimoria 38. The rest of his choices though, such as that of the 68 tmimata for the octave, 9 for the 12/11 and 7 for the 88/81 neutral seconds, do not constitute a mathematically consistent system and were rejected by most subsequent scholars including the Patriarchical committee.
This makes my blood boil. Even if you didn't know how to define 68-EDO in the most natural and intervallically consistent way where intervals justly associated with prime harmonics are tuned to their closest step to the just tuning, any school child who can add integers and multiply fractions can figure out 68-EDO step values for intervals with just tunings in the 2.3.11 just intonation subgroup just from knowing the mapping
(9/8, 12/11, 88/81) -> (12, 9, 7) units
It's perfectly mathematically consistent. So not only was Chrysanthos obviously correct in that regard, but a whole council of his peers couldn't figure it out and shot down his work because of their stupidity, and 150 years later the consensus hasn't improved. I'll have to find the original sources to learn the full depths and true contours of their stupidity.
...
Anyway, this source tells us that the Musical Committe of 1881 used only even step sizes in their tetrachords, so it's more like 72/2 = 36-EDO, but they are written in 72 and we shall do the same.
The committe gives a "mild" diatonic genus as
[12, 10, 8]\72
and contrasts it with the Ussak genus of Turkish classical music, which they also call a Mild Diatonic and render in 72-EDO as:
[10, 8, 12]\72
I don't think they use that one in their music, they just wanted people to know that they were familiar with Ottoman classical music. Fine.
...