Sequel to Rationalizing 53-EDO for Turkish Makam Analysis and Quartertones (For Arabic Maqam Analysis).
I really want to figure out middle-eastern microtonal music. My last post was getting kind of long and busy, so I'll keep going here.
I once posted about the 24-EDO ("quartertone" analysis of Arabic maqamat. Here's a summary below. Each line has a maqam name, a number of steps of 24-EDO for each scale degree, a pitch class for each scale degree, and short comment about microtones. I just got these by transcribing staff notation off Wikipedia:
Hijaz (Nahawand ending) [0, 2, 8, 10, 14, 16, 20, 24] [D, Eb, F#, G, A, Bb, C, D] # Tonal.
Nawa Athar [0, 4, 6, 12, 14, 16, 22, 24] [C, D, Eb, F#, G, Ab, B, C] # Tonal.
Shad 'Araban [0, 2, 8, 10, 14, 16, 22, 24] [G, Ab, B, C, D, Eb, F#, G] # Tonal.
Bayati [0, 3, 6, 10, 14, 16, 20, 24] [D, E-, F, G, A, Bb, C, D] # Has E-.
Jiharkah [0, 4, 8, 10, 14, 18, 21, 24] [F, G, A, Bb, C, D, E-, F] # Has E-.
Huzam [0, 3, 7, 9, 15, 17, 21, 24] [E-, F, G, Ab, B, C, D, E-]. # Has E-.
Rahat al-Arwah [0, 3, 7, 9, 15, 17, 21, 24] [B-, C, D, Eb, F#, G, A, B-] # Has B-.
Saba [0, 3, 6, 8, 14, 16, 20, 24] [D, E-, F, Gb, A, Bb, C, D] # Has E-.
Rast [0, 4, 7, 10, 14, 18, 21, 24] [C, D, E-, F, G, A, B-, C] # Has E- and B-.
Husayni 'Ushayran [0, 3, 6, 10, 13, 16, 20, 24] [A, B-, C, D, E-, F, G, A] # Has B- and E-.
Also, the maqamat 'Ajam, Nahawand, and Kurd are tonal, and correspond to western Major, Minor, and Phrygian modes. I don't know the traditional tonic pitches though, so they're not listed above.
Now, 24-EDO is kind of an ugly tuning system. There is no rank-2 interval that you can temper out to produce a 24-EDO tuning with the usual famous order of natural interval (P1 m2 M2 m3 M3 P4 P5 m6 M6 m7 M7 P8). That doesn't mean the scale doesn't exist: you can of course make music in it if you like, and other people can analyze it as being rank-3 or higher if they like. And honestly maqamat probably should be analyzed with 5-limit or higher prime frequency ratios. But I still think we can do better than 24-EDO as an analysis framework. Partly I think we can do better because Arabic, Turkish, and Persian maqamat/makams are supposedly developed from ancient Greek Pythagorean tuning, which had rank-2 microtones. And Turkish makams are analyzed with 53-EDO which is basically Pythagorean tuning, so rank-2 microtones not only should be enough theoretically, but they seem to be enough in practice. And I'm told that Arabic music isn't even played as 24-EDO, that's just a short-hand for notation which doesn't match the measured frequencies. ... So it's great that most of the pitches of the maqamat above are also natural pitches found in 12-EDO. Among all the maqamat above, we only have E- and B- as microtones. So let's try to analyze those in the same 53-EDO framework as the Turkish makams, rather than 24-EDO, which I dodn't care for. And then we'll compare the Arabic maqamat in this post to the Turkish makams in the previous post. And hopefully everything will come together beautifully. The information I have from one musical tradition will clear up the confusions I have about the other, and vice versa.
Let's start by writing the maqams above in terms of intervals, rather than tuned steps of an EDO or pitch classes.
Here's the Arabic version of Rast:
['C', 'D', 'E-', 'F', 'G', 'A', 'B-', 'C']
.
The Turkish version had scale degrees on [0, 9, 17, 22, 31, 40, 48, 53] of 53-EDO, which correspond to
[P1, M2, d4, P4, P5, M6, d8, P8]
as simple intervals.
That scale is malformed numerically/alphabetically: we have a small fourth instead of a third interval, and a small octave instead of a seventh interval. But the scale is still nice in that it has diminished intervals where the Arabic notation suggests a microtone. Let's spell it in rank-2 pitches instead of using those weird +/- a quarter tone accidentals:
[C, D, Fb, F, G, A, Cb, C]
In Pythagorean tuning, a {Cb} is a little bit below a {B}, so don't go thinking "Oh, Cb = B, but Rast has a B half flat, so this isn't flat enough." I'm here to tell you, {Cb} does not equal a {B}, and this has consequences for microtonal music.
If we go up to a rank-3 analysis, then Rast is
[P1, AcM2, M3, P4, P5, AcM6, M7, P8]
This is well-formed as a scale, but corresponds less well to the Arabic staff notation.
Let's try to find another microtonal maqam where we also have an analysis for a Turkish makam with approximately the same name.
Arabic Huzam and Turkish Huzzam are clearly a pair, but I was confused about Huzzam, so that's not a good test case for confirmation. I need to cement my understanding of Arabic maqamat before I use it to fix my understanding of Turkish makams.
Let's do Arabic Bayati versus Turkish Beyâti!
The ascending Turkish Beyâti is defined by these "simge":
[K S T T B T T]
which correspond to these numbers of "commas" or steps of 53-EDO:
[8, 5, 9, 9, 4, 9, 9]
which we can accumulate in a running sum to get the tuned step sizes for each Beyâti scale degree:
[0, 8, 13, 22, 31, 35, 44, 53]
which correspond to these simple intervals:
[P1, d3, m3, P4, P5, m6, m7, P8]
.
We compare to the Arabic version above....
[D, E-, F, G, A, Bb, C, D]
And it all works again! We have a diminished third instead of a second interval, as might have been predicted from the case with Rast, and all of the other pitch classes match what they should be: i.e. Bb really is a minor sixth above D. I'm so happy! The simple rank-2 analysis is working really well. I believe the Arabic Bayati here is specifically "Bayati with Nahawand ending" in contrast to "Bayati with Rast ending".
Now let's try using the Arabic Huzam and Saba to figure out the Turkish versions! Because I was super confused about those. The Arabic Saba is really close to the Arabic Bayati: the G on the fourth scale degree just becomes a Gb. And a Gb is a diminished fourth above D, so we have:
Arabic Saba: [P1, d3, m3, d4, P5, m6, m7, P8]
If we tune those rank-2 intervals in 53-EDO, we get these as the tuned steps of (Arabic) Saba:
[0, 8, 13, 17, 31, 35, 44, 53]
.
This almost matches a description of Saba from the 53-EDO / Turkish maqam post: Ali C. Gedik gave this description of Saba in a dissertation:
[0, 8, 13, 18, 31, 35, 44, 49]
This differs on the fourth scale degree (where his written 18 is the number of 53-EDO steps for the tuned value for M3 not d4), and on the 8th scale degree (where his written 49 steps corresponds to a M7).
The 18 step 4th scale degree makes more sense in Turkish music theory than a step at 17 would: If we use 18, then
31 - 18 = 13
P5 - M3 = m3
we have a normal interval with a named simge ("A13") between the fourth and fifth scale degrees. If we use the 17, then the difference is 14 steps of 53-EDO, which Turkish music theory doesn't have a simge for.
Let's look at some different source for both the Arabic and the Turkish Saba. For Arabic, maqamworld.com gives Saba as either:
Saba with 'Ajam ending: [D, Eb-, F, Gb, A, Bb, C, D]
Saba with Nikriz ending: [D, Eb-, F, Gb, A, Bb, C, Db, E, F]
Excellent. I've seen some Turkish sources give an extended Saba that misses and goes over the octave as:
[K S S A B T S A S]
The "A"s are a little ambiguous between 12 and 13 steps of 53-EDO in Turkish music theory, but it won't be a problem here: We'll use 13 steps for the first A, just like Gedik. Next, we're told from the Arabic Saba page that the extended form of Saba has a scale fragment (or "jins" (plural "ajnas") called Nikriz built on the sixth scale degree. The 5-note Nikriz pentachord spans a perfect fifth, and we need 12 steps (of Turkish 53-EDO) on the {A} for that to work out. Here's the Nikriz pentachord in simge:
[T, S, A12, S]
and again in relative EDO steps from the previous scale degree
[0, 9, 5, 12, 5]
and here are the running totals:
[0, 9, 14, 26, 31]
and how about the rank-2 intervals, why not?
[P1, M2, A2, d5, P5]
With 12 steps on the second {A} simge, here are all the scale degrees in 53-EDO steps:
[0, 8, 13, 18, 31, 35, 44, 49, 61, 66]
which correspond to these simple intervals:
[P1, d3, m3, M3, P5, m6, m7, M7, d10, m10]
This is an extended Turkish Saba, perhaps, if I did everything right. And it doesn't match the Arabic version. The fourth scale degree is still mismatched, like before (18 for Arabic versus 17 for Turkish). But also the Arabic one ends "C, Db, E, F" and this guy above with the "d10" ends "C, C#, Fb, F" if you spell it out. Those Arabic pitches are the sorts of substitutions you might make from the Turkish scale if you work in 12-EDO or 24-EDO. But I wouldn't have minded if the two intervallic analyses had been identical.
I know that was a lot. But this is extended Saba as far as I can tell from Turkish notes on sigme:
[P1, d3, m3, M3, P5, m6, m7, M7, d10, m10]
And this is extended Saba as far as I can tell from Arabic staff notation:
[P1, d3, m3, d4, P5, m6, m7, d8, m9, m10]
Maybe the cultures actually have slightly different scales, or maybe the differences of staff notation and the EDOs they use for analysis are introducing noise into my intervallic analysis. I'm not sure. I think because Turkish music has a finer grained analysis, I tend to trust it more. And also, Arabic 24-EDO collapses lots of enharmonic interval distinctions, in addition to being coarser grained. So I'm leaning toward trusting the Turkish analysis, in so far as the two musical traditions share one object called Saba.
I was kind of hoping that any time an Nth scale degree had a diminished (N+1)th interval in the simge analysis, that the Arabic staff notation would notate a half-flat microtone, and vice versa. But the d10 at the end of the extended Turkish Saba didn't bear that out. So I don't have a general principle for converting between 24-EDO and 53-EDO notations.
Still this is good progress: in the last post, I was considering the possibility that the first two simge of the extended Turkish Saba were ornaments, and that [S A13 B T S A12 S] was the version that went from P1 to P8. Ridiculous! I'm learning a lot.
Ready to analyze Huzam/Huzzam? I know I am.
The Arabic Huzam goes
[E-, F, G, Ab, B, C, D, E-]
The Turkish Hüzzâm is notated
[S T S A S A B]
in simge. The {A} simge is ambiguous between 12 and 13 microtones, but the assignment that makes the most sense is
[S T S A12 S A13 B]
which becomes this
[0, 5, 14, 19, 31, 36, 49, 53]
if you convert simge to EDO steps and take a running sum. Ali C. Gedik also gives those numbers exactly in his dissertation, so I've got multiple directions of confirmation that this is the Turkish Huzzam.
This one frustrated me because I'd heard that Turkish music only uses intervals tuned to these steps
[0, 4, 5, 8, 9, 13, 14, 17, 18, 22, 23, 26, 27, 30, 31, 35, 36, 39, 40, 44, 45, 48, 49]
of 53-EDO, which doesn't include 19. Those scale degrees [0, 4, 5, 8, ...] also correspond to natural and once-modified intervals, whereas the simplest rank-2 interval that 53-EDO tunes to 19 steps is a twice augmented second, AA2. So 19 steps is a weird duck.
If we convert all of the Huzzam steps to simple rank-2 intervals, we get:
[P1, A1, A2, AA2, P5, A5, M7, P8]
Pretty weird. In the quartertone post, I was tempted to say, "Maybe E- isn't the actual tonic. What happens if we permute the scale and pretend it starts on C?". I'm not sure if I'll do that here, but I'll remark that Huzam is not looking any less weird in Turkey than it did in Arabia.
If we convert the Huzzam intervals to pitches rooted on C (not permuting the scale to start on the sixth scale degree, just taking the interval [P1, A1, ...] and stacking them on top of a C), then we get
[C, C#, D#, D##, G, G#, B, C]
which you might be tempted to spell enharmonically as
[C, Db, Eb, Fb, G, Ab, B, C]
Even though that changes and ruins all the interval and only works in 12-TET. This is almost the form we see on Piano Encyclopedia's page on Huzzam in C:
[C, Db, Eb, Fb, G, Ab, Bb, C]
It only differs in making the {B} into a {Bb}. But multiple sources confirmed the intervals of Turkish Huzzam for me, so I think I trust my own {B} more than the Piano Encyclopedia's {Bb}. Although maybe they're giving me Arabic Huzam? Because they spell it "maqam" rather than "makam" on the page. I don't know, man.
The Arabic "Rahat al-Arwah" maqam listed at the start of the post has the same intervals as Huzam but it's rooted on B- instead of E-. They seems pretty serious over there in the middle east about liking this weird scale, and about rooting it on a microtone, even though they perhaps only have two microtones in total. But also, B- and E- aren't real pitches, they're just weird accidentals made up for analyzing things in 24-EDO, and I want to connect them to real rank-2 intervals and rank-2 pitches.
So here's what we're going to do: take the Turkish Huzzam and transpose it so that the fifth scale degree, which is P5 over the tonic in Turkey, matches the fifth scale degree of the Arabic Huzam rooted on E-. We'll do that transposition, and then we'll compare pitches.
The Arabic Huzam has a B natural for its fifth scale degree. Therefore we'll root the Turkish Huzam on E natural, since B is a perfect fifth above E natural. Our old wonky friend, the Turkish Huzzam,
[P1, A1, A2, AA2, P5, A5, M7, P8]
becomes
[E, E#, F##, F###, B, B#, D#, E]
which we shouldn't simplify by 12-EDO enharmonic respelling, but let's see how it goes anyway:
[E, F, G, Ab, B, C, D#, E]
and now compare to the Arabic Huzam:
[E-, F, G, Ab, B, C, D, E-]
The half flat microtones are obviously missing, which is sad. And also, this Huzam has a D instead of a D#, just like how Piano Encyclopedia used a m7 instead of a M7.
I'm getting sad. I want a professor of middle eastern music theory to explain it all away.
...
There aren't very many pitches in the maqamat I got from wikipedia. Saba is the only one with a Gb, and besides that there are only 12 pitch classes, not a full 24. This is the full set of 13:
[A, Bb, B-, B, C, D, Eb, E-, F, F#, Gb, G, Ab]
and Gb is tuned the same as F# for them, so it's really just 12. They operate on this very cute little subspace of what their music theory allows.
...
Oh! Yeah, the respelling is what made me sad. The first interval of Huzam, at least in a rank-2 intervallic analysis, is an augmented unison. That's what it is in Turkey and Turks got the math right, or at least right enough for a very productive Pythagorean analysis. It makes sense that anyone who analyzes it with 24-EDO would spell it wrong. It's still a mystery to me why Arabic maqam seems to have a m7 and Turkish seems to have a M7, but maybe that's because they play different scales.
Anyway! When we compared the Arabic and Turkish descriptions of Rast, we concluded that {E-} is really an {Fb}. So let's spell Huzzam with an Fb for the root!
[Fb, F, G, G#, Cb, C, Eb, Fb]
I think this significantly better than starting on E. It only introduces one new pitch class, G#, relative to the 13 pitch classes in Arabic maqamat (when they're spelled in the Pythagorean way with Cb and Fb instead of B- and E-). It's the way you'd spell you'd spell your maqam if you already had a 12-EDO lute with extra fretlets added in for Cb and Fb. And maybe Arabs play D instead of Eb for the 7th scale degree, I don't know.
Okay, what's left to analyze of the Arabic maqamat? Hijaz, Shad Araban, and Nawa Athar are all tonal. Here are some cute facts about them from the Quartertones post in the meantime:
* If you spell Hijaz by thirds instead of by step, it's a D.11b9b13 chord, aka Phrygian with a major 3rd.
* If you spell Shad Araban by thirds, it's a G.Maj11b9b13 as a chord. Also it's a permutation of Nawa Athar.
* If you spell Nawa Athar by thirds, it's a C.minor-major9#11b13 chord.
I've heard that Jiharkah is uncommon in Arabic music. I don't know if it has a counterpart in Turkish music. And I still don't really have a general procedure for figuring out the actual rank-2 intervals that Arabic maqams notated in 24-EDO are made of.
Might the Arabic maqam Husayni Ushayran correspond to the Turkish makam Hüseynî? Let's find out!
The simge for Hüseynî are:
[K S T T K S T]
which in commas can be written
[8, 5, 9, 9, 8, 5, 9]
Here are the running sums:
[0, 8, 13, 22, 31, 39, 44, 53]
And here are simple corresponding intervals:
[P1, d3, m3, P4, P5, d7, m7, P8]
Rooted on {A}, this becomes:
[A, Cb, C, D, E, Gb, G, A]
So that's the spelling of the Turkish makam Hüseynî. Here's the Arabic maqam again, Husayni Ushayran, ([A, B-, C, D, E-, F, G, A]), spelled with rank-2 pitches:
[A, Cb, C, D, Fb, F, G, A]
Close, but no cigar. It's cool that the Arabic one has a diminished sixth instead of a perfect fifth.
Oh, weird. Alsiadi.com gives a description of Maqam Husayni (the Arabic spelling), but relates the intervals by commas. But also, the commas don't make sense? There are 6s and 7s, which don't have simge in Turkish music theory. But also interestingly, the scale has the same accidentals as wikipedia's Husayni Ushayran, but roots the scale on D:
[D, Fb, F, G, A, Cb, C, D]
and this is exactly the Turkish Huseyni transposed to D. So... the wikipedia Husayni is a permutation of the Alsiadi.com Husayni, which is incorrectly notated with commas, but if we ignore them then it's spelled the same as the Turkish makam Hüseynî.
I wish these things made more sense. I think I've also heard that Turkish music is played like P4 lower than notated? And {D} is P4 above {A}. So there.
The site "learnarabicmusic.com" also starts on D, with half flats on E (=Fb) and B (=Cb). So that's encouraging. I think wikipedia is just wrong.
...
I transcribed all but one of the maqamat on maqamworld.com. Some of them have notes about the ajnas (trichords, tetrachords, pentachords, hexachords) from which they're built. Maqamworld.com has more information; I just wasn't very thorough with it.
:: Arabic Maqamat:
'Ajam Family:
'Ajam (Upper Ajam Ending): [C, D, E, F, G, A, B, C] # 'Ajam pentachord + upper 'Ajam tetrachord. Major scale.
'Ajam (Nahawand Ending): [C, D, E, F, G, A, Bb, C] # 'Ajam pentachord + Nahawand tetrachord.
'Ajam 'Ushayran (descends): [Bb, A, G, F, Eb, D, C, Bb] # Nahawand trichord down to Kurd tetrachord down to 'Ajam tetrachord.
Shawq Afza: [C, D, E, F, G, Ab, B, C] # 'Ajam pentachord + Hijaz tetrachord
.
Bayati Family:
Bayati (Nahawand Ending): [D, E-, F, G, A, Bb, C, D]
Bayati (Rast Ending): [D, E-, F, G, A, B-, C, D]
Bayati Shuri: [D, E-, F, G, Ab, B, C, D]
Husayni: [D, E-, F, G, A, B- or Bb, C, D] # This has both descending and ascending parts, as written on MaqamWorld, and it has multiple sixth scale degrees. I don't get it. I've just written it ascending.
Muhayyar: Bayati (Rast Ending) but then you emphasize Jins Bayati on the octave?
.
Hijaz Family:
Hijaz (Nahawand Ending): [D, Eb, F#, G, A, Bb, C, D]
Hijaz (Rast Ending): [D, Eb, F#, G, A, B-, C, D]
Hijazkar (descends) : [E, Db, C, B, Ab, G, F, E, Db, C] # Also called Shadd 'Araban or Suzidil or Shahnaz
Zanjaran (descends): [C, Bb, A, G, F, E, Db, C]
.
Kurd Family:
Kurd: [D, Eb, F, G, A, Bb, C, D]
Hijazkar Kurd (descends): [E, Db, C, B or Bb, Ab, G, F, Eb, Db, C]
.
Nahawand Family:
Nahawand (Hijaz Ending): [C, D, Eb, F, G, Ab, B, C] # Nahawand pentachord + Hijaz tetrachord. Harmonic minor scale.
Nahawand (Kurd Ending): [C, D, Eb, F, G, Ab, Bb, C] # Nahawand pentachord + Kurd tetrachord. Natural minor scale.
Farahfaza: (Nahawand transposed to start on G.)
Nahawand Murassa': [C, D, Eb, F, Gb, A, Bb, C] # # Nahawand Murassa pentachord on the tonic, overlapped by Hijaz tetrachord on the 4th degree. And then add on the octave.
'Ushaq Masri: [D, E, F, G, A, B-, C, D] # Nahawand pentachord + Bayati tetrachord
.
Nikriz Family:
Nikriz (descends from 9 rather than octave): [D, C, Bb, A, G, F#, Eb, D, C]. Ascending, Nikriz pentachord + Nahawand pentachord.
Nawa Athar: [C, D, Eb, F, G, Ab, B, C] # Nikriz pentachord on tonic, overlapping with a Hijazkar Hexachord starting on third degree (centered on fifth degree)
Athar Kurd: [C, Db, Eb, F#, G, Ab, B, C]
.
Rast Family:
Rast (Upper Rast ending): [C, D, E-, F, G, A, B-, C]
Rast (Nahawand ending): [C, D, E-, F, G, A, Bb, C]
Kirdan: (descending Rast with upper rast ending)
Sazkar (descends): [C, B-, A, G, F, E-, D#, C]
Suznak: [C, D, E-, F, G, Ab, B, C] # Rast pentachord + Hijaz tetrachord.
Nairuz: [C, D, E-, F, G, A-, Bb, C] # Modern transposed maqam Yakah. Weird in that it has A half flat, which none of the other scales do.
Yakah: [G, A, B-, C, D, E-, F, G] # Older less common version of maqam Nairuz. Normal in that it has B- and E-, which many of the other scales do. Go team Yakah.
Dalanshin (descends): [E, Db, C, B-, A, G, F, E-, D, C]
Suzdalara (descends): [C, Bb, A, G, F, E-, D, C] # Just the descending form of rast with Nahawand ending.
Mahur: [C, D, E-, F, G, A, B, C]
.
Sikah family:
Sikah: [E-, F, G, A, B-, C, D, E-] # Sikah trichord + Upper Rast tetrachord + Rast trichord
Huzam: [E-, F, G, Ab, B, C, D, E-] # Sikah trichord + Hijaz tetrachord + rast trichord
Maqam Rahat al-Arwah: (Huzzam rooted on B-)
'Iraq: [B-, C, D, E-, F, G, A, B-] # Sika trichord + Bayati tetrachord + rast trichord
Awj ‘Iraq (descends): [D, C, B-, A#, G, F#, Eb, D, C, B-]
Bastanikar: [B-, C, D, E-, F, Gb, A, Bb, C, Db, E, F] # "Maqam Bastanikar is effectively Jins Sikah followed by Maqam Saba. Its scale starts with the root Jins Sikah on the tonic, then Jins Saba on the 3rd degree, an overlapping Jins Hijaz on the 5th degree, and finally Jins Nikriz on the octave."
Musta'ar: [E-, F#, G, A, Bb, C, D, E-]
.
No family:
Jiharkah: [-E, F, G, A, B-, C, D, E-, F] # F is the tonic, I think, not E-. As much as Maqams have tonics, i.e. final notes. Jiharkah hexachord + Upper Rast tetrachord.
Lami: [D, Eb, F, G, Ab, Bb, C, D] # Lami pentachord, overlapped with kurd tetrachord on the fourth, then add on the octave.
Saba ('Ajam ending): [D, E-, F, Gb, A, Bb, C, D]
Saba (Nikriz ending): [D, E-, F, Gb, A, Bb, C, Db, E, F]
Saba Zamzam ('Ajam ending): [D, Eb, F, Gb, A, Bb, C, D]
Saba Zamzam (Nikriz ending): [D, Eb, F, Gb, A, Bb, C, Db, E, F]
.
The one maqam I didn't transcribe, Sikah Baladi, has weird accidentals that I don't know how to interpret. I think its the only one with weird intervals? There's a small chance I saw other weird flats and rounded them off in my head to "the half flat accidental".
Here's the comment on Sikah Baladi from maqam world:
"Maqam Sikah Baladi is arguably the most challenging Arabic maqam. Its scale (and sayr) is something of a hybrid between a transposition of Maqam Huzam to an ordinary non-Sikah note, and Maqam Hijazkar – the intervals are not quite the same as either, but it sounds a bit like both. None of its intervals match either just or equal-tempered intonation, making it impossible to reproduce on anything but the voice and traditional Arabic instruments.""
I don't believe for a fraction of a second that the intervals are neither just nor equally tempered. But it's cool that they recognize it's super duper not 24-EDO. Without knowing the staff notation, my attempt at a transcription would be something like:
Sikah Baladi (descends): [Cdown, Bup-, Adown-, G, Fup++, Edown-, D, C#, Cdown, Bup-, Adownb, G] "
I'll look into it and do it better. Or I'll find a different source on Sikah Baladi and talk about that characterization.
Offtonic.com says: "An especially interesting variant, however, is Sikah Baladi. It results from exaggerated tuning of the Hijaz jins, so 1 b2 3 4 becomes 1 d2 d3 4. Instead of 5 - 12 - 5 commas, I used 6 - 10 - 6 commas in Sikah Baladi (53-TET) and 7 - 8 - 7 commas in Sikah Baladi X (53-TET), both in the Offtonic Scale Keyboard. Interestingly enough, the 24-TET values are sort of in between."
Eventually, I'll get all of the ajnas (trichords, tetrachords, pentachords, hexachords) written in, and that will speed along the process of figuring out fine-grained rank-2 intervals. But for now, I was really enjoying comparing the Turkish and Arabic relatives, and I might just find a few pairs to compare again.
I should also compare the MaqamWorld maqamat to the ones transcribed from wikipedia in case there are interesting differences. I've also found pitch classes for some Turkish makams on the Xenharmonic wiki. We'll synthesize and compare it all. Later.
From the names, I'm hopeful that we'll get at least 7 corresponding scales between the Arabic and the Turkish. And then maybe that will also give us Turkish intervals for some of the jins that make up the remaining Arabic scales.
:: Quick comparisons
: Segah with Sikah
The Turkish makam Segâh is defined by these simge:
[S, T, K, T, S, A13, B] : [5, 9, 8, 9, 5, 13, 4]
which we can accumulate to find these intervals:
[0, 5, 14, 22, 31, 36, 49, 53] :: [P1, A1, A2, P4, P5, A5, M7, P8]
Maybe Turkish Segâh is related to Arabic Sikah? Let's see what the Segah pitch-classes look like rooted on Fb, as the Arabic Sikah is rooted on E-:
[Fb, F, G, Bbb, Cb, C, Eb, Fb]
and now compare to Sikah:
Sikah: [E-, F, G, A, B-, C, D, E-]
Hm! The substitution of Turkish {Bbb} for Arabic {A} isn't too weird, with Arabs/24-EDO ignoring enharmonic differences. The replacement of {Cb} with {B-} is as expected. But we have a real difference with the Turks using {Eb} for the 7th scale degree and the Arabs using {D}.
: Nikriz with Nikriz
Turkish Nikriz is defined by these simge:
[T, S, A12, S, T, K, S] : [9, 5, 12, 5, 9, 8, 5]
which we can accumulate to get these steps and simple intervals:
[0, 9, 14, 26, 31, 40, 48, 53] :: [P1, M2, A2, d5, P5, M6, d8, P8]
The Arabic Nikriz has its tonic on C but descends from a D above the octave. Let's knock off the high D and write the whole arabic maqam as ascending, like the Turkish one:
[C, D, Eb, F#, G, A, Bb, C]
If we root the Turkish Nikriz intervals on C, we get
[C, D, D#, Gb, G, A, Cb, C]
The differences of Eb/D# and F#/Gb are to be expected enharmonically. The difference on the seventh scale degree seems genuine. Either I'm doing something wrong, or the two traditions use similar scales with differing seventh degrees. Segah/Sikah also differed on the seventh scale degree. Hm.
: Irak with 'Iraq
Turkish Irak is defined by these simge and relative steps:
[S, T, K, S, T, T, K] : [5, 9, 8, 5, 9, 9, 8]
which we can accumulate to get these scale degree steps and simple intervals:
[0, 5, 14, 22, 27, 36, 45, 53] :: [P1, A1, A2, P4, A4, A5, A6, P8]
.
Arabic 'Iraq is rooted on B half flat,
[B-, C, D, E-, F, G, A, B-]
which in my rank-2 analysis is called Cb. If we root the Turkish Irak on Cb, we get:
[Cb, C, D, Fb, F, G, A, Cb]
Perfect agreement! I was starting to doubt myself.
: Hüseynî with Husayni
In my analysis of Turkish makams, I was under the impression that both Nevâ and Hüseynî were defined by these simge and relative steps in 53 EDO:
[K, S, T, T, K, S, T] : [8, 5, 9, 9, 8, 5, 9]
which can be accumulated to get these scale degree steps and simple intervals:
[0, 8, 13, 22, 31, 39, 44, 53] :: [P1, d3, m3, P4, P5, d7, m7, P8]
Let's root the Turkish scale on D, like the Arabic maqam Husayni:
[D, Fb, F, G, A, Cb, C, D]
and now we'll compare to the Arabic:
[D, E-, F, G, A, B- or Bb, C, D]
Good agreement! On MaqamWorld, multiple pitches were listed for the 6th scale degrees of this maqam (B- or Bb). It seems Turkish music theory goes with the B- (i.e. Cb). The site alsiadi.com (with arabic maqam names ad pseudo-turkish commas) also lists B-.
: Sûzinâk with Suznak
The Turkish makam Basit Sûzinâk is defined by these simge and relative steps:
[T, K, S, T, S, A12, S] : [9, 8, 5, 9, 5, 12, 5]
We can accumulate these give to give scale degree steps in 53-EDO, which are the tuned values for these simple rank-2 intervals:
[0, 9, 17, 22, 31, 36, 48, 53] :: [P1, M2, d4, P4, P5, A5, d8, P8]
The Arabic maqam Suznak is notated:
[C, D, E-, F, G, Ab, B, C]
If we root the Turkish intervals on C, we get these pitch classes:
[C, D, Fb, F, G, G#, Cb, C]
The differences on the third and sixth scale degrees are enharmonic. The seventh scale degree seems to be a real difference, as far as I can tell from notation. Arabs are playing B natural, and Turks are playing what Arabs would call B half-flat.
: Acem and Uşşak versus Ajam and 'Ushaq Masri?
As far as I could tell in my reading of Turkish makams, all of Acem and Uşşak and Beyâti were defined by the same simge. And we've already successfully analyzed Beyâti. But them Turkish "Acem" kind of sounds like Arabic "Ajam" and Turkish "Ussak" kind of sounds like arabic "'Ushaq". I'm not sure what do about those. I guess I'll have to figure out the Turkish intervals for the scale fragments (jins) that make up the Arabic maqams to find out the rank-2 intervals, because I don't have a full Turkish scale with which to compare. Maybe I could compare against theoretical Turkish scales, i.e. those made out of simge and tuned in 53-EDO with natural intervals, and see if any of them can be enharmonically respelled to match Arabic maqamat like 'Ushaq.
I think in Arabic, 'Ajam means "mute", and is used to refer to people who don't speak their mother tongue, i.e. it's a pejoratives for foreigners. And in lots of other languages, including Turkish, it refers to Iranians. So maybe 'Ajam is a foreign scale for the Arabs and a native scale for the Turks or Persians with a different name. But ....
: Çârgâh versus 'Ajam
If I just look at the pitches / intervals and not the makam names, the Turkish makam Çârgâh and the Arabic maqam 'Ajam are just the C major scale. Not even worth comparing. Interesting though that the Arabic major scale is called "damn foreigner". It makes me wonder if maqam Rast came first for them and how that happened and what came next.
If I had just been looking at the names of maqamat, I might have tried comparing Çârgâh to Jiharkah. No need for that.
: Nivahend versus Nahawand
In Turkish music, makam Nivahend and makam Buselik have the same intervals, just a different root position. The ascending form of the scale is defined in term of simge and relative steps in 53-EDO as:
[T, B, T, T, B, T, T] : [9, 4, 9, 9, 4, 9, 9]
We can accumulate the steps to get absolute steps for each scale degree relative to the tonic. Here are also some simple intervals that 53-EDO tunes to each of those scale degrees:
[0, 9, 13, 22, 31, 35, 44, 53] :: [P1, M2, m3, P4, P5, m6, m7, P8]
If we root those intervals on the pitch class C, then we get this scale of pitch classes:
[C, D, Eb, F, G, Ab, Bb, C]
which exactly matches the Arabic maqam called Nahawand with Kurd ending. It's composed of a Nahawand pentachord and a Kurd tetrachord. It's also called the natural minor scale in western music theory. It's the type of minor scale that's a cyclic permutation of the major scale. Also called the Aeolian mode.
The harmonic minor scale
[C, D, Eb, F, G, Ab, B, C]
has a related name in Arabic muisc theory: it's called Nahawand with Hijaz ending. The Nahawand maqam intervals rooted on {G} instead of {C} are called maqam Farahfaza, and I think again you can use either the harmonic minor ending (Hijaz tetrachord) or the natural minor ending (Kurd tetrachord).
...
: Mahur with Mahur
Mahur on Turkish music theory sites has the same commas as a major scale
[T T B T T T B]
and is rooted on G and is written descending. But the staff notation has a weird somewhat-sharp accidental (Küçük mücenneb?) on the F, but I don't even like Western staff notation and I can't even. I think it corresponds to the simge S, i.e. 5 commas sharp. But ...then why are the intervals just for a major scale? I don't know man. Mahur on MaqamWorld is like a major scale but with E half flat:
[C, D, E-, F, G, A, B, C]
So maybe Mahur is a major scale with either a half flat third scale degree or a 5-comma sharp 7th scale degree.
:: Differences between Wikipedia's Maqamat and those of Maqam world:
: Nawa
Nawa Athar from wikipedia was:
[C, D, Eb, F#, G, Ab, B, C]
Nawa Athar from MaqamWorld:
[C, D, Eb, F, G, Ab, B, C]
They differ on the fourth scale degree. I trust the second source way more, but this is still something to investigate.
: Shadd
Shad 'Araban on Wikipedia was rooted on G and ascends:
[G, Ab, B, C, D, Eb, F#, G]
Shadd 'Araban on MaqamWorld (also called Hijazkar or Suzidil or Shahnaz) was rooted on C and descends, but it descends from two scale degrees above the tonic:
[E, Db, C, B, Ab, G, F, E, Db, C]
If we reverse that and start limit it to an octave,
[C, Db, E, F, G, Ab, B, C]
Then .... yeah, those are the same intervals as on Wikipedia.
[P1, m2, M3, P4, P5, m6, M7, P8]
Nice.
: Jiharkah
Jiharkah on Wikipedia is
[F, G, A, Bb, C, D, E-, F]
while on MaqamWorld it is
[-E, F, G, A, B-, C, D, E-, F]
with whole notes on the Fs, suggesting to me that those have a tonic function. There's a difference of Bb versus B- on the fourth scale degree.
: Husayni 'Ushayran
I think the last difference is that MaqamWorld doesn't have Husayni 'Ushayran at all. The website alsiadi.com gives the same ascending version as Wikipedia, namely
[A, B-, C, D, E-, F, G, A]
but then also lists a descending form, which I'll write ascending for ease of comparison:
[A, B-, C, D, Eb, F#, G, A]
.
That's it for differences from Wikipedia.
I think next I want to ...do a 5-limit analysis of all the scales real quick? Yes, I do.
Ah, but alsiadi.com has maqamat that MaqamWorld doesn't. Maybe I should gobble those up real quick. It seems like the sort of site that could go down any minute. Also, he has weird 53-EDO steps that don't make sense in Turkish analysis. Maybe they're clues to the fine-grained differences in performance of Arab music relative to Turkish, clues we can't get from 24-EDO analysis or notation. And then, uh, there's this great youtube channel called Oud For Guitarists. I'll check if his descriptions of any of the makams differ. And then I'll have a really strong foundation in this stuff. Many sources, multiple cultures, independent partial confirmations or interesting contradictions.
And then I'll do a five-limit analysis. And then I'll look at Ozan Yarman's weird, like, 23-limit frequency ratios, which I expect to be good (in their match to measured performances), even though the notated differences between frequency ratios are all sadly mathematically wrong. And I shouldn't be surprised, because anyone who spends too much time on the Xenharmonic wiki forgets how to do arithmetic.
...
Okay, from Mr. Alsiadi's site:
Rast ascending is the usual [C, D, E-, F, G, A, B-, C], i.e. the Upper Rast ending, but it's notated with commas [9, 7, 6, 9, 9, 7, 6]. # Rast tetrachord + 9/8 + Rast tetrachord.
Rast ascending for the Turks is analyzed with commas
[T, K, S, T, T, K, S] : [9, 8, 5, 9, 9, 8, 5]
Relative to them, Alsiadi is flattening each the E and the B by an extra comma. I guess this is more...foreign, detuned, microtonal that I'm used to, so maybe that's good? It's weird at least. If you want Rast to sound weird, this guy's commas are for you. None of this "basically a major scale if you're tone deaf" crap. This one stabs.
Rast descending has the Nahawand ending: [C, D, E-, F, G, A, Bb, C]. Here are his commas, written ascending also: [9, 7, 6, 9, 9, 4, 9]. # Rast tetrachord + 9/8 + Nahawand tetrachord.
Maqam Basandida (ascending): [C, D, Eb, F#, G, A, Bb, C] : [9, 4, 14, 4, 9, 4, 9] # Nakriz or Nawa Athar pentachord + busalik tetrachord
Maqam Basandida (descending): [C, D, E-, F, G, A, B-, C] : [9, 7, 6, 9, 9, 7, 6] # Rast tetrachord + 9/8 + Rast tetrachord. These are the same pitch classes that Rast uses to ascend.
Maqam Dalansheen (ascending): [C, D, E-, F, G, A, B-, C, Db, E, F] : [9, 7, 6, 9, 9, 7, 6, 4, 14, 4]. # Compound maqam. It's like Rast with an ornamentation on top that derives from an overlap with Saba (starting on the A).
Maqam Dalansheen (descending): [C, D, E-, F, G, A, Bb, C, D, E-, F] : [9, 7, 6, 9, 9, 4, 9, 9, 7, 6].
Mahur (ascending): [C, D, E, F, G, A, B, C] : [9, 9, 4, 9, 9, 9, 4]
Mahur (descending): [C, D, E-, F, G, A, Bb, C] : [9, 7, 6, 9, 9, 4, 9].
The ascending form matches my understanding of Turkish Mahur as basically a major scale (but also they put a kücük mücenneb accidental on the 7th scale degree in the staff notation, which doesn't even show up in their comm-based intervallic analysis).
Neither of Alsiadi's Mahur maqamat match those matches the Mahur from MaqamWorld, which was [C, D, E-, F, G, A, B, C]), but Alsiadi's descending form is closer and MaqamWorld's mahur descended, so we're making progress. I'm glad I'm going through this site. This is pretty interesting. I might never nail this stuff down.
Maqam Nishaburk (ascending): [D, E, F+, G, A, B-, C, D] : [9, 7, 6, 9, 6, 7, 9].
Maqam Nishaburk (descending): [D, E, F+, G, A, Bb, C, D] : [9, 7, 6, 9, 4, 9, 9].
To be clear, I'm just reproducing verbatim the commas that Alsiadi lists. I don't think for a second that they actually produce the pitches of the scale.
I've read (in the table of contents for "Harmonic Secrets of Arabic Music Scales" by Cameron Powers) that Nishaburk is Nayruz/Nairuz rooted on G. In fact, let's look at all the transposed maqam names that Powers gives:
Nishabur: (Ajam on D with Nahawand)
Bayati Ushayran: (Bayatayn on A)
Busalik Ushayran: (Bayati on A)
Midmi: (C Hijaz Kar 2nd becomes Tonic on Db)
Zanjaran: (Hijaz on C with Ajam)
Nuhuft: (Huseyni on A)
Irak: (Huzam on B half-flat with Bayati)
Bastanikar: (Huzam on B half-flat with Saba)
Rahat el Arwah: (Huzam on B half-flat)
Zirgulah: (Jaharka on D)
Shawki Tarab: (Kurd on A with Saba)
Tarz Nawin: (Kurd on C with Hijaz)
Hijaz Kar Kurd: (Kurd on C)
Farahfaza: (Nahawand 1 on G)
Sultani Yaka: (Nahawand 2 on G)
Ushaq Masri: (Nahawand on D with Bayati)
Nahawand Kurdi: (Nahawand on D)
Shiar: (Nahawand on E with Bayati)
Busalik: (Nahawand on E)
Dilkashidah: (Nahawand on G with Bayati)
Hisar: (Nawa Athar on D)
Nishaburk: (Nayruz on D)
Yekah: (Nayruz on G)
Yak-Gah (Rast on G): (Rast Nawa)
Qatar: (Saba Zamzamah on E)
Suznal: (Shawq Afza on C)
Panjigah: (Shawqi Awir on C)
Suzidil: (Shehnaz on A)
Jaharka Turki: (Shehnaz on F)
Sikah Balady: (Shehnaz on G with the “old intervals”)
Shad Araban: (Shehnaz on G)
Hijazi Ushayran: (Shuri on A)
Farahnak: (Sikah on B half-flat)
.
Nice list.
Okay, back to Alsiadi:
Maqam Suzdilara ascending (Jaharka ending): [C, D, E, F, G, A, Bb, C] : [9, 9, 4, 9, 9, 4, 9]
Maqam Suzdilara ascending (Bayati ending): [C, D, E, F, G, A B-, C] : [9, 9, 4, 9, 9, 6, 7]
Maqam Suzdilara descending: [C, D, E-, F, G, A, Bb, C] : [9, 7, 6, 9, 9, 4, 9] # The scale has Bb written, but then there are accidentals after the scale in the style of a key signature which has a half flat on the B line. So maybe you can do Bb or B- descending? But the notated jins above the scale says Jaharkah, which wouldn't gvie a microtone. Whatever. These scales are all the same. This is just Rast descending with the Nahawand ending.
Maqam Suznak ascending: [C, D, E-, F, G, A-, B, C] : [9, 7, 6, 9, 5, 13, 4]
Maqam Suznak descending: [C, D, E-, F, G, A, Bb, C] : [9, 7, 6, 9, 9, 4, 9] # Still just Rast descending with Nahawand ending.
Maqam Yakah: [G, A, B-, C, D, E-, F, G] : [9, 7, 6, 9, 6, 7, 9]
Maqam Nahawand (ascending): [C, D, Eb, F, G, Ab, Bb, C] : [9, 4, 9, 9, 4, 9, 9]
Maqam Nahawand (descending): [C, D, Eb, F, G, Ab, B, C] : [9, 4, 9, 9, 4, 14, 4] # this is the Hijaz ending for Nahawand per MaqamWorld
Maqam Al-Sinbulah (ascending) / Nahawand Murassah (ascending): [C, D, Eb, F, Gb, A, Bb, C] : [9, 4, 9, 4, 14, 4, 9]
Maqam Al-Sinbulah (descending) / Nahawand Murassah (descending): [C, D, Eb, F, G, Ab, B, C] : [9, 4, 9, 9, 4, 14, 4]
Maqam Farah Fazah (ascending): [G, A, Bb, C, D, Eb, F, G] : [9, 4, 9, 9, 4, 9, 9]
Maqam Farah Fazah (descending): [G, A, Bb, C, D, Eb, F#, G] : [9, 4, 9, 9, 4, 14, 4]
Maqam Hisar: [D, E-, F, G#, A, Bb, C#, D] : [6, 7, 14, 4, 4, 14, 4]
Maqam Nahawand Kabir (ascending): [C, D, Eb, F, G, Ab, B, C] : [9, 4, 9, 9, 4, 14, 4]
Maqam Nahawand Kabir (descending): [C, D, Eb, F, G, A, Bb, C] : [9, 4, 9, 9, 9, 4, 9]
Maqam Nakriz (ascending): [C, D, Eb, F#, G, A, Bb, C] : [9, 4, 14, 4, 9, 4, 9]
Maqam Nakriz (descending): [C, D, Eb, F#, G, A, B-, C] : [9, 4, 14, 4, 9, 7, 6]
Maqam Nawa Athar (ascending): [C, D, Eb, F#, G, Ab, B, C] : [9, 4, 14, 4, 4, 14, 4]
Maqam Nawa Athar (descending): [C, D, Eb, F#, G, Ab, Bb, C] : [9, 4, 14, 4, 4, 9, 9]
Maqam Sultani Yakah: [G, A, Bb, C, D, Eb, F#, G] : [9, 4, 9, 9, 4, 14, 4]
Maqam Hijaz Kar Kurd: [C, Db, Eb, F, G, Ab, Bb, C] : [4, 9, 9, 9, 4, 9, 9]
Maqam Ajam Kurdi: (page under construction)
Maqam Shawq Tarab (ascending): [A, Bb, C, D, E-, F, Gb, A] : [4, 9, 9, 6, 7, 4, 14]
Maqam Shawq Tarab (descending): [A, Bb, C, D, Eb, F, G, A] : [4, 9, 9, 4, 9, 9, 9]
Maqam Kurdi: [D, Eb, F, G, A, Bb, C, D] : [4, 9, 9, 9, 4, 9, 9]
Maqam Tarz Nawayn: [C, Db, Eb, F, Gb, A, Bb, C] : [4, 9, 9, 4, 14, 4, 9]
Maqam Hijaz (ascending): [D, E-, F#, G, A, B-, C, D] : [5, 13, 4, 9, 7, 6, 9] # Hijaz tetrachord + Rast pentachord
Maqam Hijaz (descending): [D, E-, F#, G, A, Bb, C#, D] : [5, 13, 4, 9, 4, 14, 4] # Hijaz tetrachord + 9/8 + Hijaz tetrachord
Maqam Hijaz Kar (ascending): [C, Db, E, F, G, Ab, B, C] : [4, 14, 4, 9, 4, 14, 4] # Hijaz tetrachord + 9/8 + Hijaz tetrachord
Maqam Hijaz Kar (descending): [C, Db, E, F, G, Ab, Bb, C] : [4, 14, 4, 9, 4, 9, 9] # Hijaz tetrachord + 9/8 + Kurd tetrachord. Alsiadi h as "Busalik tetrachord + 9/8 + Kurd tetrachord" notated in text, but the commas and pitch classes do not support that.
Maqam Shad Araban (ascending): [G, A-, B, C, D, Eb, F#, G] : [5, 13, 4, 9, 4, 14, 4] # Hijaz tetrachord + Nakriz pentachord. This one is really weird. The pitch classes suggest a Nakriz pentachord [9, 4, 14, 4], but the commas written in originally were [9, 7, 6, 9] for the pentachord, which is Rast pentachord. Since the pitch classes and the pentachord label match each other, and the commas don't have any corroboration, I changed the commas for consistency. My version with the changed commas has the same intervals as Maqam Hijaz, but rooted on G instead of D. I've read on Oud For Guitarists that "maqams Hijaz kar, Shad Araban, Suzidil, and Shahnaz" have the same intervals, so maybe that's encouraging.
Maqam Shad Araban (descending): [G, A-, B, C, D, Eb, F, G] : [5, 13, 4, 9, 4, 9, 9] # Hijaz tetrachord + 9/8 + Nahawand tetrachord
Maqam Shahnaz: [D, Eb, F#, G, A, Bb, C#, D] : [4, 14, 4, 9, 4, 14, 4] # Hijaz tetrachord + 9/8 + Hijaz tetrachord
Maqam Suzdal: [A, Bb, C#, D, E, F, G#, A] : [4, 14, 4, 9, 4, 14, 4] # Hijaz tetrachord + Nakriz pentachord
Maqam Zinkulah: [C, Db, E, F, G, A, Bb, C] : [4, 14, 4, 9, 9, 4, 9] # Hijaz tetrachord + Ajam pentachord
...
I got bored with that an transcribed another source. These are
"The modal system of Arabian and Persian music" by Owen Wright, which mostly translates and analyzes the medieval music theory treatise Kitab al-Adwār by Safi al-Din al-Urmawi a.k.a Safiaddin Ormavi.
: maqmat
rahawi: [G, A-, B(-), C, D-, Eb, F, G]
'ushshaq : [G, A, B, C, D, E, F, G]
busalik: [G, Ab, Bb, C, Db, Eb, F, G]
nawa: [G, A, Bb, C, D, Eb, F, G]
rast: [G, A, B-, C, D, E-, F, G]
hijazi1: [G, A, B- or B(-), C, D-, E(-), F, G]
hijazi2: [G, A-, Bb, C, D-, E(-), F, G]
'iraq: [G, A-, B-, C, D-, E-, F, F#, G]
husayni: [G, A-, Bb, C, D-, Eb, F, G]
kardaniya: [G, A, B-, C, C#, D, E, F+, G]
buzurg: [G, A-, B(-), C, C#, D, (E, F+, G)]
zankula: [G, A, B-, C, D-, E(-), F, ((F#), G)]
zirafkand: [G, A-, Bb, C, D-, Eb, E-, F#, G]
kawasht: [G, A-, B-, C, D-, Eb, E-, (F#, G)]
muhayyir husayni: [G, A-, Bb, C, D, E-, F, G]
nihuft: [G, A-, B(-), C, D, E-, F, G]
ishfahan: [G, A, B-, C, D, E-, F, F#, G]
I don't know what the parentheses mean. I missed that when skimming the text on multiple passes. I think some of those might actually be medieval forms of the scales which have since been modified by history.
: ajnas
The three diatonic tetrachords:
/1 2 3 4/ # 'ushshaq
/1 2 3b 4/ # nawa
/1 2b 3b 4/ # busalik
And the three four-note zalzalian tetrachords:
/1 2 3- 4/ # rast
/1 2- 3- 4/ # iraq
/1 2- 3b 4/ # nawruz
Main form of hijazi tetrachord:
/1 2- 3 4/
Alternative form of hijazi tetrachord:
/1 2b 3 4/
Maybe also a possible alternative form of hijazi tetrachord?:
/1 2b 3- 4/
.
Normal pentachords:
/1 2- 3b 4 5/ # nawruz tetrachord + whole tone = husayni pentachord
/1 2 3b 4 5/ # rast tetrachord + whole tone = rast pentachord
/1 2- 3 4 5/ # hijazi tetrachord + whole tone = 'uzzal pentachord
/1 2b 3b 4 5/ # busalik tetrachord + whole tone = busalik pentachord
/1 2- 3- 4 5 / # iraq tetrachord + whole tone = iraq pentachord
/1 2 3 4 5 / # 'ushshaq tetrachord + whole tone = 'ushshaq pentachord
Weird ajnas, some of which are pentachords:
/1 2- 3b 3 4/ # isfahan
/1 2- 3 4 4# 5/ # buzurg
/1 2- 3b 3-/ # kuchek
/1 2- 3b 3- 4# 5/ # hisar
/1 2 3b 4 5/ # nawa tetrachord + whole tone = nawa pentachord
/1 2- 3b 3/ # rahawi
Hijazi might have been varied ascending and descending? Like this is a possibility
hijazi ascending:
/1 2- 3 4/
hijazi descending:
/4 3- 2b 1/
Okay, back to alsiadi:
...
Maqam Bayati ascending: [D, E-, F, G, A, B-, C, D] : [6, 7, 9, 9, 6, 7, 9] # Bayati tetrachord + 9/8 + Bayati tetrachord
Maqam Bayati descending: [D, E-, F, G, A, Bb, C, D] : [6, 7, 9, 9, 4, 9, 9] # Bayati tetrachord + 9/8 + Kurd tetrachord
Maqam Qarjighar (Bayati Shuri) (ascending): [D, E-, F, G, Ab, B, C, D] : [6, 7, 9, 4, 14, 4, 9] # Bayati tetrachord + hijaz pentachord
Maqam Qarjighar (Bayati Shuri) (descending): [D, E-, F, G, A, Bb, C, D] : [6, 7, 9, 9, 4, 9, 9] # Bayati tetrachord + busalik pentachord
Maqam Husayni (ascending): [D, E-, F, G, A, B-, C, D] : [6, 7, 9, 9, 6, 7, 9] # Bayati pentachord + Bayati tetrachord
Maqam Husayni (descending): [D, E-, F, G, A, Bb, C, D] : [6, 7, 9, 9, 4, 9, 9] # Bayati pentachord + Kurd tetrachord
Maqam Bayati Ushayran (ascending): [A, B-, C, D, E-, F, G, A] : [6, 7, 9, 6, 7, 9, 9] # Bayati tetrachord + Bayati pentachord
Maqam Bayati Ushayran (descending): [A, B-, C, D, E, F, G, A] : [6, 7, 9, 9, 4, 9, 9] # Bayati tetrachord + Busalik pentachord
Maqam Husayni Ushayran (ascending): [A, B-, C, D, E-, F, G, A] : [6, 7, 9, 6, 7, 9, 9] # Bayati tetrachord + Bayati pentachord
Maqam Husayni Ushayran (descending): [A, B-, C, D, Eb, F#, G, A] : [6, 7, 9, 4, 14, 4, 9] # Bayati tetrachord overlapping with Nakriz hexachord
Maqam Saba (ascending): [D, E-, F, Gb, A, Bb, C, Db] : [6, 7, 4, 14, 4, 9, 4] # Alsiadi describes this as having a Bayati trichord on "Re", a Hijaz tetrachord starting on "Fa", and a Hijaz trichord starting on the high "Do". As notated, the "Re" is the low "D" note though, as though C is the tonic of the scale. Also there is no third note written above the Db to complete the Hijaz trichord: there's just blank space. I don't know what a Hijaz trichord is in order to infer the pitch above "Db" or the simge-comma-integer after "4". Perhaps 14?
Maqam Saba (descending): [D, E-, F, Gb, A, Bb, C, D] : [6, 7, 4, 14, 4, 9, 9] # Bayati trichord + Hijaz tetrachord + Ajam trichord.
Maqam Saba Zamzamah: [D, Eb, F, Gb, A, Bb, C, D] : [4, 9, 4, 14, 4, 9, 9] # Kurd trichord + Hijaz tetrachord + Ajam trichord
Maqam Sikah (ascending): [E-, F, G, A, B-, C, D, E-] : [6, 9, 9, 7, 6, 9, 7] # Sikah trichord + Rast tetrachord + Rast trichord
Maqam Sikah (descending): [E-, F, G, A, Bb, C, D, E-] : [6, 9, 9, 4, 9, 9, 7] # Sikah trichord + Nahawand tetrachord + Rast trichord. Alsiadi's comments make it seem like he thinks the tonic is C rather than E-. That's fine. He can think what he likes.
Maqam Huzam: [E-, F, G, Ab, B, C, D, E-] : [6, 9, 4, 14, 4, 9, 7] # Sikah trichord + Hijaz tetrachord + Rast trichord.
Maqam Mayah: [E-, F, G, A, Bb, C, D, E-] : [6, 9, 9, 4, 9, 9, 7] # Sikah trichord + Nahawand tetrachord + Rast trichord
Maqam Musta'ar: [E-, F#, G, A, Bb, C, D, E-] : [11, 4, 9, 4, 9, 9, 7] # Musta'ar trichord + Nahawand tetrachord + Rast trichord
Maqam Jaharkah: [F, G, A, Bb, C, D, E-, F] : [9, 9, 4, 9, 9, 7, 6] # Jaharkah tetrachord + whole tone + Rast tetrachord
Maqam Ajam Ushayran: [Bb, C, D, Eb, F, G, A, Bb] : [9, 9, 4, 9, 9, 9, 4] # Ajam tetrachord + whole tone + Ajam tetrachord
Maqam Shawq Afza (ascending): [Bb, C, D, E-, F, Gb, A, Bb] : [9, 9, 6, 7, 4, 14, 4] # Ajam trichord (9, 9) + Saba pentachord (6, 7, 4, 14) overlapping with Hijaz tetrachord (4, 14, 4).
Maqam Shawq Afza (descending): [Bb, C, D, Eb, F, G, A, Bb] : [9, 9, 4, 9, 9, 9, 4] # Ajam tetrachord (9, 9, 4) + whole tone + Jaharkah tetrachord (9, 9, 4).
Maqam Iraq (ascending): [B-, C, D, E-, F, G, A, B-] : [6, 9, 6, 7, 9, 9, 7] # Iraq trichord (6, 9) + Bayati tetrachord (6, 7, 9) + Rast trichord (9, 7)
Maqam Iraq (descending): [B-, C, D, E-, F, G, A, Bb] : [6, 9, 6, 7, 9, 9, 4] # Iraq trichord (6, 9) + Bayati tetrachord (6, 7, 9) + Busalik trichord (9, 4). This one has B- at the bottom and Bb at the top, and it doesn't form an octave. Really weird.
Maqam Awj Ara: [B-, C, D#, E-, F#, G, A#, B-] : [6, 14, 2, 11, 4, 14, 2] # Awj tetrachord (6, 14, 2) + Mustaar pentachord (11, 4, 14, 2)
Maqam Bastah Nikar: [B-, C, D, E-, F, Gb, A, Bb] : [7, 9, 7, 7, 4, 14, 4] # Iraq trichord (7, 9) + Saba tetrachord (7, 7, 4) + 14 commas + Kurd dichord (4).
Maqam Farahnak (ascending): [B-, C, D, E, F+, G, A, B-] : [6, 9, 9, 7, 6, 9, 7] # Sikah trichord (6, 9) + Rast tetrachord (9, 7, 6) + Rast trichord (9, 7)
Maqam Farahnak (descending): [B-, C, D, E, F+, G, A, Bb] : [6, 9, 6, 7, 6, 9, 4] # Sikah trichord (6, 9) + Rast tetrachord (9, 7, 6) + Busalik trichord (9, 4). This one has both B- and Bb and it doesn't form an octave. Weird.
Maqam Rahit Al Arwah (ascending): [B-, C, D, Eb, F#, G, A, B-] : [6, 9, 4, 14, 4, 9, 7] # Iraq trichord (6, 9) + Hijaz tetrachord (4, 14, 4) + Rast Trichord (9, 7)
Maqam Rahit Al Arwah (descending): [B-, C, D, Eb, F#, G, A, Bb] : [6, 9, 4, 14, 6, 9, 4] # Iraq trichord (6, 9) + Hijaz tetrachord (4, 14, 4) + Busalik Trichord (9, 4). Alsiadi has the Hijaz tetrachord annotated as [4, 14, 6] for the descending version, but that's not consistent with his pitch classes or how he quantifies Hijaz anywhere else, so I fixed it.
...
Done! Whew. That took several weeks because I didn't enjoy it. 74 maqamat and one "404, file not found". I bet I made a few error of transcription. Oh well! Time to analyze.
There are actually only unique 48 scales in terms of commas! I bet some are transpositions of each other and some descend the same way but ascend differently. If I remove the ones that say "(descending)", then there are still only 33 unique scales. 13 ascending scales have repeated commas, I guess.
Almost all of them sum to an octave with 53 commas. The ones that don't are:
75 commas - Maqam Dalansheen (ascending)
75 commas - Maqam Dalansheen (descending)
48 commas - Maqam Saba (ascending)
57 commas - Maqam Saba Zamzamah
50 commas - Maqam Iraq (descending)
52 commas - Maqam Bastah Nikar
47 commas - Maqam Farahnak (descending)
52 commas - Maqam Rahit Al Arwah (descending)
Bastah Nikar is only off by one comma. I double checked the text. It wasn't my mistake. I'd already commented about "Farahnak (descending)" and "Rahit Al Arwah (descending)" and "Iraq (descending)" not being octaves. Saba and Dalansheen are weird compound scales, no surprise there. I only found one typo this way, and I've already fixed it above. You'll never know what I got wrong.
Maqam Musta'ar and Maqam Awj Ara are the only ones with intervals of 11 commas.
If I look at cumulative intervals relative to the tonic, Maqam Bastah Nikar is the only one that hits 23 commas (though 22 and 24 are multiply attested across other maqamat, so I expect an off-by-one error somewhere in alsiadi's commas). Bastah Nikar is also the only one that reaches 30 commas (though 31 is dirty common). And Maqam Bastah Nikar is the only maqam to hit 7 commas. If we compare to other sources, "Bastanikar" on MaqamWorld is clearly a non-octave compound scale, so the "52" could be right. I don't know about the other relative and cumulative commas though. Still seems kind of suspicious.
"Maqam Suznak (ascending)" is the only maqam that reaches 36 commas, while 35 commas is really common. I think it might be okay though.
"Maqam Farahnak (descending)" is the only one that reaches 43 commas (and 44 is very common and 42 is unheard of). I tried a description of Farahnak from another source and got 1) a huge pdf of turkish and Arabic makams 2) nothing else. In a pdf with the file name "Music Theory Of Makams" and an internally listed title of "Turkish & Arab Makams: Music Theory For Oud", Farahnak is listed as ascending:
[B-, C, D, E, F#, G, A, B-, C, D, E-, F, G, A].
I believe the pdf was written by David Parfitt and originally hosted on oud.eclipse.co.uk. Anyway, Farahnak in this source is made of Sikah tetrachord overlapping by one note with Ajam pentachord overlapping by one note with Rast pentachord + Bayati pentachord. *This* is the sort of thing I expect from real maqam music. There's no such thing as trichords and dichords. This is a Farahnak I can believe in. The same source descends as:
[B-, C, D, E, F#, G, A, Bb, C, D, E-, F, G, A]
i.e. one B- becomes a Bb in the middle. This lets you have a nihawand pentachord over (G, A, Bb, C, D). The same source lists two alternative forms of descending including the Rast pentachord.
[..., G, A, B-, C, D, ...]
or
[..., D, E, F+, G, A, ...]
.
This an amazing book and I'm really sad I wasted so much time learning about makams from bad sources when this exists.
"Maqam Iraq (descending)" is the only maqam of Alsiadi's to hit 50 commas (and it lands on 50 commas).
First I looked at the scales in a rank-2 analysis. And specifically at the ones that were enumerated / spelled correctly (in the sense of having a 1st interval, a 2nd interval, a 3rd interval, ...).
I found that "Ajam Ushayran" and "Mahur (ascending)" were both just the major scale: [P1, M2, M3, P4, P5, M6, M7, P8]
I found that "Suzdilara ascending (Jaharka ending)" is a cyclic permuation of the major scale, namely the Mixolydian mode: [P1, M2, M3, P4, P5, M6, m7, P8]
"Basandida (ascending)" and "Nakriz (ascending)" have the same intervals: [P1, M2, m3, A4, P5, M6, m7, P8] ## Nevermind, I had a typo. I don't know what Basandida (ascending) is.
Maqam Nawa Athar (ascending) and (descending) are both tonal
[P1, M2, m3, A4, P5, m6, M7, P8] (ascending)
[P1, M2, m3, A4, P5, m6, m7, P8] (descending)
"Nahawand Kabir (descending)" is the Dorian mode: [P1, M2, m3, P4, P5, M6, m7, P8]
Five different maqamat are just the harmonic minor scale (i.e. the Aeolian mode with a major seventh instead of a minor seventh). They are
"Al-Sinbulah (descending) / Nahawand Murassah (descending)",
"Farah Fazah (descending)",
"Nahawand (descending)",
"Nahawand Kabir (ascending)",
"Sultani Yakah".
And the intervals are: [P1, M2, m3, P4, P5, m6, M7, P8]
"Farah Fazah (ascending)" and "Nahawand (ascending)" are just the Aeolian mode: [P1, M2, m3, P4, P5, m6, m7, P8] .
"Hijaz Kar Kurd" and "Kurdi" are both Phrygian mode: [P1, m2, m3, P4, P5, m6, m7, P8]
"Shawq Tarab (descending)" is the Locrian mode: [P1, m2, m3, P4, d5, m6, m7, P8]
The Lydian mode is not represented.
These three have the same intervals: "Maqam Hijaz Kar (ascending)", "Shahnaz" and "Suzdal". I don't know a Western name for the scale, but it is: [P1, m2, M3, P4, P5, m6, M7, P8]
And these are all distinct and I don't know Western names for the scales:
[P1, M2, m3, P4, d5, M6, m7, P8] Maqam Al-Sinbulah (ascending) / Nahawand Murassah (ascending) # Disregard this. I had a typo. I don't know what Al-Sibulah (ascending) is.
[P1, m2, M3, P4, P5, M6, m7, P8] Maqam Zinkulah
[P1, m2, M3, P4, P5, m6, m7, P8] Maqam Hijaz Kar (descending)
[P1, m2, m3, P4, d5, M6, m7, P8] Maqam Tarz Nawayn
.
I don't think there's a way to specify a scale in terms of steps of 53-EDO and not have it possible to be enumerated / spelled correctly with rank 3 intervals. There is some rank3 1st interval on each step and some rank3 2nd interval on each step and so on. They might have very complex names, but there will be some analysis available.
I started out with a very simple analysis wherein every step of 53-EDO had at most two very simple candidate intervallic interpretations. This did well for most of the maqamat, but struggled to analyze
* 17 steps as a kind of 4th interval (in Saba). Alsiadi's letter names indicate Gb (a diminished fourth above the tonic of D) but the rank-3 version of d4 is tuned to 19 steps in 53-EDO, not 17 steps. 17 steps is a major third and I didn't feel the need to consider any other interpretations.
* 27 steps as a kind of 4th interval (in basandida ascending, and in Hisar, and in Nakriz ascending and descending, and in Nawa Athar ascending and descending). Maybe I could revisit this one. I considered 27-steps to be a Grd5. Here are all the weird 4th intervals that were nearby in my rank-3 analysis:
18: ["Grd4"],
19: ["d4"],
20: ["Acd4"],
21: ["Gr4"],
22: ["P4"],
23: ["Ac4"],
24: ["GrA4"],
25: ["A4"],
26: ["AcA4"],
So none of those interval names are options and I'd have to find something even weirder that's tuned to 27-steps. GrAA4?
* 26 steps as a kind of 5th interval (in Al-Sinbulah/Murassah ascending, and Shaq Tarab descending and Tarz Nawayn and Qarjighar (Bayati Shuri) (ascending) and Husayni Ushayran (descending).
* 39 steps as a kind of 7th interval in Shawq Tarab (ascending).
...
Interesting news! At least among maqamat starting on C, Alsiadi seems to assign pitch classes to 53-EDO steps pretty consistently.
C: P1 # 1/1
Db: Grm2 # 256/243
D: AcM2 # 9/8
Eb: Grm3 # 32/27
E-: GrM3 # 100/81
E: AcM3 # 81/64
F: P4 # 4/3
Gb: 26 steps of 53-EDO
F#: 27 steps of 53-EDO
G: P5 # 3/2
Ab: Grm6 # 128/81
A: AcM6 # 27/16
Bb: Grm7 # 16/9
B-: GrM7 # 50/27
B: AcM7 # 243/128
C: P8 # 2/1
I've written in 5-limit frequency ratios that are justly associated with the rank-3 intervals I'm analyzing his scale steps as being. Using those intervals doesn't produce the letter names that Alsiadi gives, so the table above is a portrait of his letter assignment, not a fact of 5-limit just intonation rooted on C. The frequency ratios are all Pythagorean except for the half flat tones:
E-: GrM3 # 100/81
B-: GrM7 # 50/27
and the two scale degrees that I didn't have an analysis for:
Gb: 26 steps of 53-EDO
F#: 27 steps of 53-EDO
Alsiadi's Mahur (ascending) was just a C major scale without microtones. I wonder why that didn't show up in the rank-2 analysis if all its frequency ratios are Pythagorean.
Let's look a little more at the half-flat tones. They are each two commas flatter than their natural pitch classes, E and B respectively. In a rank-2 analysis, E is 18 steps of 53-EDO over C and the interval is called M3. That's what Alsiadi appears to be doing. Alsiadi assigns the pitch class E- to 16 steps of 53-EDO, which in a rank-2 analysis is a ddd5. That has a pitch class of Gbbb over C and a Pythagorean frequency ratio of *takes a deep breath* 4294967296/3486784401. I'd previously argued that the rank-2 way to analyze E half flat was as an Fb, i.e. a diminished fourth over C, with t(d4) = 8192/6561. Alsiadi goes one comma flatter than that. Good to know!
So start with a Pythagorean scale, and then flatten by two commas to get Alsiadi's microtones. And you can kind of decide for yourself whether that comma is the Pythagorean comma,
t(A0) = 531441/524288
or the syntonic comma,
t(Ac1) = 81/80
The first one is probably the historical origin of the microtones and the second one is a reinterpretation that keeps things spelled correctly and keeps the frequency ratios simpler.
Two stacked Pythagorean commas come to 46.9 cents, and two Syntonic commas come to 43 cents, and humans can only hear about 5 cents difference, so there isn't much that you can do to analyze actual performed music to decide between these as "the correct comma analysis".
Okay, I found some transcription errors on my part when doing the intervallic analysis of Alsiadi. I'm going to continue trying to find and fix those, and then we'll do a grand comparison of Alsiadi with maqamwolrd with the Turkish sources, and maybe with the ?oud.eclipse.co.uk? maqam bible PDF that might be the only source I really believe.
...
Okay, I've got all the obvious transcriptions errors cleaned up for maqamat starting on C, save for one. In Suzdilara (ascending) (Bayati ending), the B half flat is at 46 steps of 53-EDO (a rank-2 dddd10 over C or a rank-3 Acm7 over C) whereas it is at 47 steps of 53-EDO for the other two maqamat that start on C and have a B half flat (with 47 steps being a rank-2 dd9 over C or a rank-3 GrM7 over C), the other maqamat being Basandida (descending) and Nakriz (descending) and Rast (Ascending). I don't know what to do about this.
...
Woo! More progress. More transcription errors corrected. I've done a rank-2 analysis, and almost all of Alsiadi's pitches match things that I can derive from his commas. There are a few seemingly unresolvable differences though.
First up: "E-" is sometimes 15 steps of 53-EDO over C (i.e. a rank-2 dddd6, which is an Abbbbb over C) and sometimes it's 16 steps of 53-EDO (i.e. a rank-2 ddd5, which is a Gbbb over C). So it can be two or three commas flat relative to E natural.
The "B-" is also inconsistent: sometimes it's 46 steps of 53-EDO over C (i.e. a rank-2 dddd10, which is Ebbbb over C) and sometimes it's 47 steps over C (i.e. a rank-2 dd9 or Dbbb over C). Again, two or three commas flat (relative to B natural).
Now for some bigger problems: Suznak (ascending) is supposed to have an A half flat, apparently. But it happens at 36 steps over C, which is a rank-2 augmented fifth over C, better known as G#. If we follow in the footsteps of E- and B-, we'd expect it to be two or three steps flat relative to A natural, i.e. 37 or 38 steps over C (dddd9 or ddd8). MaqamWorld says that Suznak has an Ab, not an A-, but we don't have that either, we have a G# based on the commas.
Maqam Suznak (ascending) is notated as ending with a Hijaz tetrachord. And looking into this, I found a different inconsistency in Alsiadi's writing: A hijaz tetrachord is usually [4, 14, 4] (in like 35 spots), but it's occasionally [5, 13, 4] (in about 5 spots). And actually, in Hijaz (descending), we get both varieties!
"Hijaz (descending)": [5, 13, 4, 9, 4, 14, 4], # Hijaz tetrachord + 9 + Hijaz tetrachord
Oh shit, it gets even worse. Here are the pitch classes for Hijaz (descending):
"Hijaz (descending)": "D, E-, F#, G, A, Bb, C#, D",
It's microtonal on the first tetrachord and it's tonal on the second?
Anyway, if you take Suznak (ascending)
"Suznak (ascending)": [9, 7, 6, 9, 5, 13, 4],
and replace the uncommon Hijaz with the common one,
"Suznak (ascending)": [9, 7, 6, 9, 4, 14, 4],
then we get the pitch classes from Maqam world
[C, D, E-, F, G, Ab, B, C]
with the Ab, but still no A-. If we suppose that A- is 2 or 3 commas below A natural, then we'd need a 6 or a 7 as the value for the fifth comma in the list, since A natural is 9 steps of 53-EDO above G natural. But I don't think [7, 11, 4] or [6, 12, 4] really deserve to be called a Hijaz tetrachord. I wan to change it to [4, 14, 4], but I'm feeling too lost at the moment. I'll figure it out soon enough though.
It shouldn't be this irregular. Pythagorean tuning and 53-EDO are very regular. All the major seconds are 9 steps, all the minor seconds are 4 steps, all the sharps are raised by 5 steps, all the flats are lowered by 5 steps. *with conviction* I'll figure it out soon enough though.
Next up, Hijaz ascending and descending are supposed to have E-, but they both occur just five commas above the tonic of D. However, 5 steps of 53-EDO is an augmented unison, so 5 steps over D is just a D#. This is four commas flat of an E natural, not two or three. Too flat, I say! It is a comma higher than an Eb though, so we're still in "the range between Eb and E natural" I guess.
Shad Araban (ascending) and (descending) only differ on the commas surrounding the "E" note. The staff notation says that they should both be Eb, but the commas say that it should be E- ascending and Eb descending. I'd like to clear this up by appealing to a different source, but I first looked at MaqamWorld and it's totally different. The MaqamWorld "Maqam Shadd ‘Araban" has a tonic on C instead of G, and it has no microtones (whereas Alsiadi's Shad has an A- regardless of the E-/Eb choice), and it doesn't have different ascending and descending versions. Alsiadi's ascending Shad and Maqam world's ascending Shadd are both notated with a Nakriz/Nikriz pentachord over a Hijaz tetrachord, but then the MaqamWorld version ascends over the octave with a Hijazkar pentachord that overlaps the Nikriz by three notes.
I also want to address issues with
* Bastah Nikar
* Farahnak (descending)
* Rahit Al Arwah (descending)
All of these three start on B-. And it seems that B- can be 46 or 47 steps of 53-EDo in Alsiadi's analysis. So when the pitches classes I derived for the maqamat using Alsiadi's commas (and a B- of 47 steps) didn't match the pitch classes he presented, I naturally tired a B- of 46 steps. It turns out, that doesn't solve any of the three and also causes lots of other maqamat to be misspelled.
With a B- at 47 steps, Farahnak (ascending) is spelled correctly but descending it looks like this:
Farahnak (descending): [6, 9, 6, 7, 6, 9, 4]
Alsiadi: [B-, C, D, E, F+, G, A, Bb]
Derived: [B-, C, D, E-, F, E##, Dbbbbb, G##]
The steps that actually produce Alsiadi's pitch classes are:
[6, 9, 9, 7, 6, 9, 4]
which only differs in turning a 6 into a 9 at the third place. I'm strongly inclined to call this a typo on his part.
Next, "Bastah Nikar". It's written the same ascending and descnding, which makes it a little less likely that my analysis is failing because of a typo on his part, but still possible.
With a B- at 47 steps, we get
Bastah Nikar: [7, 9, 7, 7, 4, 14, 4],
Alsiadi: [B-, C, D, E-, F, Gb, A, Bb]
Derived: [B-, B#, C##, Fb, Bbbbbb, E##, Dbbbb, B-]
The commas which produce alsiadi's pitch classes are:
[6, 9, 6, 7, 4, 14, 4]
or
[6, 9, 7, 6, 4, 14, 4]
since there's a choice of E- (15 or 16 steps of 53-EDO).
If we use a B- at 46 steps, we get:
Bastah Nikar: [7, 9, 7, 7, 4, 14, 4],
Alsiadi: ['B-', 'C', 'D', 'E-', 'F', 'Gb', 'A', 'Bb']
Derived: ['B-', 'C', 'D', 'E-', 'E#', 'F#', 'G##', 'A#']
and the commas which produce alsiadi's pitch classes are:
[7, 9, 7, 6, 4, 14, 4],
or
[7, 9, 6, 7, 4, 14, 4],
since again there's a choice of where to put E- (at 15 or 16 steps of 53-EDO).
So, .... those are all facts. Since D-natural is at 9 commas over C (a Pythagorean major second) and F natural is at 22 commas over C (a Pythagorean perfect fourth), the commas between them have to sum to 13 steps, not 14. I don't know how to fix this without appealing to other sources. We'll do that later.
Let's talk about Rahit Al Arwah (descending). This one is pretty obviously a typo on Alsiadi's part. He has written [6, 9, 4, 14, 6, 9, 4], but it should be [6, 9, 4, 14, 4, 9, 4]. The ascending form has the same pitch classes notated for all but the highest note (a Bb descending, a B- ascending), and also the same first two ajnas, so the two maqamat should should have the same lower steps/commas. Two lines of evidence that agree, so the third datum is probably in error. Also, between F# and A, the only options for a kind of G in 53-EDO or Pythagorean tuning are G natural or G sharp, which aren't found at 6 steps over F# (they're at 4 commas and 9 commas respectively), so the 6 steps thing isn't really an option, even if you ignore the ajnas and notated pitch classes and the ascending form.
Okay, that's it for "analyzing Alsiadi by comparing his pitch classes to his tetrachords". Time to get really really serious and analyze everything at once. One maqam at a time, all the sources, all the information, all the logic. We'll figure out the tetrachords too.
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In fact, let's start with the tetrachords. Let's think of them as roughly Pythagorean. What should the neutral intervals be? Let's think of the neutral third to start. I hear Arabic music puts them rather close to the 24-EDO quarter tone, 350 cents, while Turkish music theory puts them closer to the justly tuned 5-limit rank-3 major third, t(M3) = 5/4 ~= 386 cents. For comparison, the 12-EDO major third is 400 cents and the Pythagorean major third is t(m3) = 81/64 ~= 408 cents. And for refresher, the just M3 is one justly tuned syntonic comma (t(Ac1) = 81/80 ~= 22 cents) lower than the Pythagorean one, and 53-EDO tunes both the syntonic comma and the Pythagorean comma (t(A0) = 531441/524288 ~= 23 cents) to 1 step, so we can pretty freely talk about lowering rank-2 intervals by A0 to get similar tuned frequencies as we would if we lowered them by a syntonic comma, Ac1.
If we lower the Pythagorean major third by one (Pythagorean) comma, we get a diminished fourth, d4, at 384 cents. Let's call that a Turkish neutral third in a rank-2 analysis. This was also a historically used value for analyzing the neutral third for Arabic music, e.g. the
medieval Arabic music theorist Safi al-Din al-Urmawi outlined a 17-tone Pythagorean analysis of maqam pitches made by an extended spiral of perfect fifths, and the pitch between m3 and M3 was t(d4) = 8192/6561 ~= 384 cents.
Using this, we can give the steps between intervals of the Rast tetrachord as:
[M2, d3, A1]
Or accumulative these to give intervals for each step relative to the tonic:
[P1, M2, d4, P4]
If we lower the Pythagorean major third by two (Pythagorean) commas, we get a three-times diminished fifth, ddd5, tuned to 361 cents. Alsiadi uses intervals that are 2 or sometimes 3 commas flat from Pythagorean major to represent neutral intervals, so this is an option for Arabic rank-2 anslysis.
There are some other options we could consider for the representation of neutral thirds. Ancient music theorists thinking about arithmetic divisions of the neck of a string instrument (or, equivalently, harmonic means of frequency ratios) found that
(8/7) * (14/13) = 16/13 ~= 359 cents
(8/7) * (13/12) = 26/21 ~= 370 cents
are easily constructed neutral thirds. The construction is basically starting with a string of length 16 units and dividing it in half, and then repeatedly dividing new segments in half:
8:16 # 2/1
8:12:16 # 3/2, 4/3
8:12:14:16 # 3/2, 7/6, 8/7
8:12:13:14:16 # 3/2, 13/12, 14/13, 8/7
And there's an interesting thing where even divisions of the string result in uneven divisions of the frequency ratios.
I think the former neutral third, (8/7) * (14/13) = 16/13, is more natural, but the two are pretty similar. According to Margo Schulter, modern Syrian and historic Ottoman maqam music still use/used to use neutral thirds in this range.
I know I didn't explain that construction in detail, but perhaps you can trust me or infer how it is that by dividing a Pythagorean scale with harmonic means in a similar way, we can get a neutral third of
(81/68) * (34/33) = 27/22 ~= 355 cents
And by taking certain harmonic divisions of the 5-limit m3 and M3 and combining them (or by just taking the mediant of 6/5 and 5/3), we can get a nice neutral third at
(11/10) * (10/8) = 11/9 ~= 347 cents
So there are lots and lots of justifiable options available to use for a tuned neutral third if we skip the edifice of regular intervallic theory. Those all have factors of 11 or 13. If we just want to look at prime factors up of 7, then Ben Johnston's septimal super unison, t(Sp1) = 36/35 ~= 49 cents is a very good quarter tone, so adjusting the Pythagorean m3 and M3 by that gives us good neutral thirds of:
t(SbGrm3) = 128/105 = (32/27) * (36/35) ~= 343 cents
t(SpAcM3) = 315/256 = (81/64) / (36/35) ~= 359 cents
not that middle eastern theorists ever use these.
If Turkish and Arabic neutral thirds differ 30 cents or whatever - by more than a comma - I feel like I should be notating them differently/separately. But if they're both consistently using only one value for "neutral third", then it also feels like I should have a tuning-agnostic intervallic representation that shows how they're basically using the same scales, in so far as they are using the same scales. Idk what to do about cultural intonation.
Honestly, Alsiadi's two-comma-flat ddd5 at 361 cents as a neutral third is a nice compromise. It's pretty squarely between the modern arabic 24-EDO value of 350 cents and the maybe-Turkish 5-limit value at 386 cents. It's a modern fine-grained division produced by someone familiar with both Arabic and Turkish makams. One comma flat honestly doesn't sound very middle eastern; it's no more exotic than a five-limit just intonation scale, which music theory nerds frequently try to push as sounding more harmonious than 12-EDO. Two commas flat, in contrast, is clearly a different tone and makes it so that Rast is not just a major scale with a weird tuning, but it's own scale worth teaching. At 16 steps of 53-EDO, the two-commas flat neutral third also has a nice short name in 5-limit just intonation that can be spelled/enumerated correctly as a third: it's a Grave major third, GrM3, two syntonic commas below the Pythagorean M3, which is called an AcM3 in 5-limit just intonation. So two commas flat has a lot going for it.
That in mind, let's notate some ajnas using the idea that a half flat pitch class is two-commas flat from the natural version / the major Nth.
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