In middle eastern music, a maqam is a scale with a little extra structure. The structure tells you to play melodic phrases within certain subscales that mostly span a range of a perfect fourth, to ascend and descend at certain times, to dwell on certain notes, and to change certain scale degrees for flavor or to play them differently ascending and descending. The structure might also tell you when it's appropriate to modulate to another scale or scale fragment, which modulations are available to you, how to end a melodic phrases, which stock melodic phrases and rhythms are good to incorporate into your compositions, and other things like that. There's a lot of structure in a maqam, and a lot of it is fuzzy things you pick up by listening to songs written in that maqam, so it's easy to just pretend that a maqam is a scale or a scale and it's subscale structure, but that's not really correct. A maqam has a scale, but it is more than a scale.
The fundamental maqam of Arabic music is called maqam Rast. It has two microtones that can sound a little exotic to the western ear. And it's usually played monophonically (or in in octave in an ensemble, but still having a monophonic texture) or with a very simple harmony consisting of a drone against a monophonic texture.
Middle eastern music that is based on scales like the scale of maqam Rast don't have much microtonal polyphony and I want to do something about it - I want to figure out a way to make beautiful exotic microtonal polyphony. Admittedly, some turkish music has microtonal polyphony, but the dominant pedagogical paradigm for Turkish music is basically 5-limit just intonation, which is barely microtonal. Some Turkish music might be more microtonal than that, but it's hard to find sources about it from which to learn. Byzantine Liturgical chant, that was influenced by early Turkish/Ottoman music, also has microtonal polyphony, but no one seems to know a ton about it - I certainly don't - and that's a topic for a future post. What I want to do here is figure out how to harmonize music in the Arabic maqam Rast. I expect a lot of this work will translate to other Arabic maqamat.
Maqamat have highly variable intonation by region, time period, performer, and sometimes even by performance if the performer isn't that precise. As much as there is a standard intonation for maqam Rast, it's probably found on vynil records from the golden age of Egyptian music and cinema, centered on Cairo, Egypt between 1930 and 1970. There might be a somewhat precise intonation available if you do measurements of those records, but also the Cairo Congress of Arab Music in 1932 couldn't agree on any notational standard for intonation that was more precise than 24-EDO, so while there might be (or might have been) a precise intonation in practice, we can say that there is no theoretical, written pedagogical intonation with a precision below 50 cents.
All of that is just to say that 24-EDO might not be right, but it's hard for anyone to agree on anything better, and I won't feel too bad using 24-EDO intonation in this post. Let's use 24-EDO pitch notation for maqam Rast and see what chords we can build on it.
Here's our scale ascending:
[C3, D3, Ed3, F3, G3, A3, Bd3, C4]
The "d" accidental is a backward flat, which indicates a lowering by one step of 24-EDO. The chromatic pitches traditionally are considered to have a Pythagorean intonation, although the difference between Pythagorean 3-limit and just 5-limit intonation disappears in 24-EDO since that tuning system tempers out the acute unison. The "d" accidental is called a "half flat", and one step of 24-EDO is called a "quarter tone" since four of them make a "whole tone", i.e. a major second. I'll say that again: a half flat lowers a tuned pitch by a quarter tone. One music educator I like a lot on youtube is constantly saying "half flat or quarter flat" and, uh, those aren't the same thing, and it took me a long time to realize he was just using the wrong terms instead of talking about 48-EDO or 53-EDO or something similar that does have quarter flats. In addition to the half-flat accidental "d", there's also a half-sharp accidental "t", which raises the tuning of tuned pitches by one step of 24-EDO.
To start, let's look at which diatonic triads are available for harmonizing maqam Rast. On the first scale degree, ^1 at C natural, I think our best options are
[C, F, A] _ F.maj
[C, G, A] _ C.maj6(no 3), which could also be called A.m7(no 5).
These are not very good chords for a tonic, but they're what we have. I'd really like the tonic pitch of the scale to have a chord that is rooted on the tonic pitch, so I'm leaning toward C.maj6(no 3).
Scale degree two could be:
[D, F, A] _ D.m
or maybe
[D, F, G] _ G.7(no 3)
Scale degree three is hard. How do you harmonize the microtonal {Ed}? We've got {Bd} a perfect fifth over {Ed}, so that's a start.
[Ed, Bd] _ Ed.5
but it's not a triad, and in full generality I'd like to be able to do 4-voice and 5-voice harmony. If we want a minor triad on Ed, we need a tone that is one step of 24-EDO below {G}, i.e. {Gd}, and if we want a major triad rooted on {Ed}, then we need a tone that is one step of 24-EDO over {G}, i.e. {Gt}. But neither of these is in maqam Rast. We'll come back to it.
Scale degree four could be:
[F, A, C] _ F.maj
or maybe
[F, A, C] _ D.m
Scale degree five should probably be:
[G, D, F] _ G.7(no 3)
Scale degree six has a bunch of options:
[A, C, F] _ F.maj
[A, D, F] _ D.m
[A, C, G] _ A.m7(no 5), which could also be called C.maj6(no 3)
The {Bd} at scale degree 7 of maqam Rast only has {Ed} for an obvious harmonic friend:
[Bd, Ed] _ Ed.5 or Bd.4
which makes me a sad camper/panda.
Maqam Rast often has Bb when it descends, which gives us some interesting different options for triads, but it also makes harmonizing {Ed} more difficult. Before we get to the descending scale, let's see how we can mix microtones with chromatic tones.
If we consider Pythagorean tuning and 5-limit just intonation to be chromatic, then the first prime-limit that we could try using to build mixed microtonal+tonal chords is 7-limit. In 7-limit just intonation, the simplest frequency ratios are justly associated with intervals that have "sub-minor" and "super-major" qualities, which are respectively flatter than (5-limit or 3-limit) minor and sharper than (5-limit to 3-limit) major. This isn't a great model for middle eastern microtones, which are conventionally "neutral" in that they are tuned between minor and major intervals of the same ordinal (like a neutral third is between a minor third and a major third), but we can still do some work within the 7-limit system.
A subminor seventh over C is mapped to "Bbd" in 24-EDO, i.e. a step below Bb. If we want a 4-note chord tht has "Bd" for its 7th interval, we just have to raise our C chords up by an augmented unison and the Bbd will transform into a Bd. Here are two chordal options including {Bd}:
[C#, E#, G#, Bd] _ harmonic dominant seventh chord _ [P1, M3, P5, Sbm7]
[C#, E, G#, Bd] _ harmonic minor seventh chord _ [P1, m3, P5, Sbm7]
Now, we don't have most of those notes in Rast, but the E# is enharmonically equivalent to F, since 24-EDO tempers out the diminished second between them. F might be in Rast, but we already *had* a harmonic relationship between an {E} pitch in rast and a {B} pitch in Rast, so this isn't much progress. Still, it's good to know that C# and G# have strong relationship with Bd through 7-limit just intonation.
Another chord that sounds good in 7-limit, if not quite as good in a 24-EDO treatment of 7-limit is
[C#, Ed, G#, Bd] _ Subminor third chord with harmonic seventh _ [P1, Sbm3, P5, Sbm7]
And this nicely contains both Ed and Bd.
The analogous chords that have Ed as the pitch for the 7th interval are:
[F#, A#, C#, Ed] _ Harmonic dominant seventh
[F#, A, C#, Ed] _ Harmonic minor seventh
[F#, Ad, C#, Ed] _ Subminor third with harmonic seventh
the A# in the first chord is enharmonically equivalent to Bb in the descending form of Rast, since 24-EDO tempers out d2, and the A natural in the second chord is already in Rast. Nice.
I think it's good to have an option to harmonize the microtones of Rast with chromatic tones like F#, C#, G#, but it'll still take a little cleverness to use them since they're not in Rast. But now if we want to harmonize {Ed} and {Bd} we have options to introduce either microtonal notes like Gd and Gt or chromatic notes liek F#, C#, G#.
Another nice 7-limit chord is the subminor triad, [P1, Sbm3, P5]. We can reach our Rast microtones {Ed} and {Bd} by rooting this chord on C# and G# respectively:
[C#, Ed, G#]
[G#, Bd, D#]
And for the super-major triad, we have
[Cb, Ebt, Gb]
[Gb, Bbt, Db]
The Ebt is basically the same as Ed and Bbt is basically the same as Bd. How so? There are intervallic interpretations of 24-EDO pitch notation in which they're different - they would certainly be different in 7-limit just intonation, but if we're actually playing in 24-EDO, then they're tuned to the same same frequency ratio over C natural. These new chord pithces like D# and Gb are still not shared with the Rast scale, but they're giving us more options for harmonizing {Ed} and {Bd}.
Instead of rooting 7-limit chords on chromatic tones to hit microtones, let's root a 7-limit chord on a microtone and we'll catch a chromatic tone in the middle.
If we root, [P1, Sbm3, P5] on {Ed} we get [Ed, Gb, Bd].
If we root, [P1, SpM3, P5] on {Ed} we get [Ed, G#, Bd].
To recap: if you want to use a normal diatonic triad on Ed (like a major or minor triad) in order to reach both Ed and Bd in one chord, then you can use a microtonal third interval at Gd or Gt. And if you want to use a septimal triad rooted on Ed in order to reach Ed and Bd in one chord, then you can use a chromatic third interval at Gb or G#. Lots of options.
I'd say our best options for scale degree 3 are:
[Ed, G#, Bd] _ Ed.SpM3
[Ed, Gb, Bd] _ Ed.Sbm3
[Ed, G#, C#] _ C#.Sbm3
[Ed, G#, Bd, C#] _ C#.Sbm3,Sbm7
We'll have to think more about which of these we like.
In 18th century baroque musical practice, there was a thing called a partimento: a bass line with some extra notation about implied harmony that a student or musician could expand to make a full song, often using some partimento-specific rules for the expansion. To a first approximation, you'd harmonize the notes of the bass line according to "the rule of the octave" that describes what voiced chords work well on each bass note, ascending and descending, except in some well defined places you'd harmonize differently to make a strong, functional, chordal cadence to finish a musical phrase. Then, on top of that harmonic skeleton, you'd add melodic motives called diminutions. Now, I'm not a student of 18th century baroque practice, but I'm hoping we can do something similar to partimento practice for maqam Rast: we'll try to specify a "rule of the octave" for Rast, i.e. 3-voice or 4-voice chords for each scale degree ascending and descending, that will make for songs with good voice leading if used on top of (???normal???) bass lines. At the very least, I'm hoping that there will be good voice leading if we just play a bass line that moves up and down by 2nd intervals.
Let's go through triad options for the descending form of maqam Rast first and then we'll try to figure out which chords among those options we will actually use, and what voicings of those chords will make our chord progression sound as good/baroque as possible.
Maqam Rast often has a Bb instead of Bd "when the scale descends". This means 1) when the melodic motion played on top of the scale has an overall descending tendency, but especially this happens 2) after a song section where you had an ascending tendency, reaching a high note, at which point you emphasize that high note by playing riffs that repeat the note or dwell on the note or keep returning to the note, and then you want a contrasting musical phrase that moves down the scale. It's not that every moment you're moving up or down the scale and so you're constantly deciding between Bd and Bb - rather, the song has long broad phases, and you'll use Bb in a later phase of the song.
Here's the descending scale, but still written ascending: [C, D, Ed, F, G, A, Bb, C]
Here are some simple chord options:
^1: F.maj, D.m7, C.maj6(no 3)
^2: D.m, Bb.maj, G.m
^3: ???
^4: F.maj, D.m, Bb.maj
^5: G.m, C.maj6(no 3)
^6: F.maj, D.m, A.m7(no 5), C.maj6(no 3)
^7: G.m, Bb.maj,
We don't have {Bd} to harmonize with {Ed} any more. One option we discovered in our look at 7-limit harmony was
[F#, A#, C#, Ed] _ Harmonic dominant seventh
which has an A# that's enharmonic with Bb in 24-EDO.
if we drap the F# at just look at the top triad, that's a septimal sub-diminished chord,
[P1, m3, Sbd5]
which does indeed sound pretty good in just tuning, [1/1, 6/5, 7/5], and possibly also okay in 24-EDO. (The sub-minor sub-diminished triad, [P1, Sbm3, Sbd5] also sounds good in 7-limit, not that it gets us any closer to the Rast scale.)
So if you want to harmonize Ed in the descending form of Rast, then the best option I can see is probably [Ed, Bb, Db] which is like a Bb sub-diminished triad, although that would more properly be spelled [Bb, Db, Fbd], but we can kind of let it slide since {Fbd} and {Ed} are tuned the same by 24-EDO (and by other tuning systems that temper out d2, the diminished second).
Another option for harmonizing scale degree ^3 that we saw in the ascending form of Rast, and which is still available to us, is
[Ed, G#, C#] _ C#.Sbm3
Although this has two notes that aren't in Rast. If they were semitones below {C} and {G} of Rast, like {B} and {F#}, that feels like it would be good for voice leading, but {C#} and {G#} are kind of out of place, I'd say. You might be thinking "We can do a tritone substituton for G.7 rooted on Db which is like C#", but it doesn't work well in this case. So our best three options for harmonizing Ed in descending Rast are probably:
[Ed, F#, A#, C#] _ F#.maj3,Sbm7
[Fbd, Bb, Db] _ Bb.m3,Sbd5
[Ed, G#, C#] _ C#.Sbm3
We've got some options for chords now. I'm going to try to use these chord options to write a Rule Of The Octave for four voices that fits between [C3, E5], with the bass between C3 and C4 and the other three voices between C4 and E5. We'll see how many rules of voice leading I can follow.
...
Oh, interesting! When I listen to them, the chords I like most for harmonizing Ed and Bb ascending are just made from notes of maqam Rast
[Ed, G, Bd]
[Bd, Ed, G]
Obviously they have the same pitch classes and are inversions of the same chord, but it's a weird chord! It's not major, minor, super major, or subminor. It could be sub-major or super-minor, I guess? But actually, when I listened to the chords, I was listening to an 11-limit detempering in which the "d" accidental flattened things by 33/32 and the "t" accidental sharpens things by 33/32. So, since that chord sounds nice and is in fact closer to Arabic intonation than a septimal detempering, let's call the chord what it really is. For [Ed, G, Bd], the intervals and their just frequency ratios over the nearest C natural below are:
[DeAcM3, P5, DeAcM7] _ [27/22, 3/2, 81/44]
If we root that chord on Ed / DeAcM3 / 27/22, then we get this chord:
[P1, AsGrm3, P5] _ [1/1, 11/9, 3/2]
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