Makams and Maqamat
:: The Motivation: You know who really loves microtonal music?
The most widely practiced and loved microtonal music scene in the modern era is in the Middle East, the Maghreb, balkans, and the Mediterranean. Lots of musical traditions in these areas are based on basically the same microtonal modal scales. In Arabic-speaking nations, such a scale is called a maqam. In Turkey, it's a makam. The Turkish Uyghurs in China call it a Muqam. In Azerbaijan, it's a mugham. In Uzbekistan and Tajikistan, it's shashmaqam. In Greek Rebetiko music, which was influenced by the music of the Ottoman empire, such a scale is called an adromoi. In the musical tradition of Nubah from Morocco, Algeria, Tunisia, and Libya, such a scale is called a tab'. And all of these scales are derived from the ancient Persian modal system, where a modal scale is called a dastgah.
These musical traditions almost never use rich harmony: a monophonic instrument wanders within one part of the scale and then to another, with expert dynamic and rhythmic expressivity, and the melodies are mostly just accompanied by percussion, or constant note drones for harmony, or sometimes by other instruments playing together at the unison.
The Turkish and the Arabic musical traditions are the best documented online, so I'll focus on those. For the Arabic scales I'll say "a maqam" or multiple "maqamat". For Turkish scales, I'll say makam and makams. The actual plural in Tukish is "makamlar", depending on the grammatical case, but Turkish people writing in English say "makams", so I will too.
Anyway, in a book about microtonal music theory, I'm definitely going to talk makams/maqamat - the building blocks of the most widely practiced and loved microtonal music tradition in the world. I'm going to start by giving lots of raw data about the scales. I'll describe the scales as I've learned about them from multiple sources, basically in the order that I found them: Wikipedia, The Turkish Sources (including Esendere Kültür Sanat Derneği, and the Website of Dr. Hazim Gökçen, and the publications of Dr. Ozan Yarman, and some other things I've forgetten), then MaqamWorld, and Alsiadi, and the Kitāb al-Adwār by Safi al-Din al-Urmawi (as related in English by Owen Wright), and maybe some data from Oud For Guitarists and makamlar.net. Once I've shown all the scales, I'll do a lot of analysis to point out inconsistencies of description within a musical tradition, and to point out which scales are related across musical traditions and how they differ, all in terms of intervals - with at least rank-2, rank-3, and rank-4 analysis of everything, maybe higher rank interval analysis and higher prime limit frequency ratio analysis if I'm feeling sassy.
It's a little frustrating to try to characterize makams and maqamat, since they vary by region and historic period and musician. It's a little bit like trying to construct the platonic ideal recipe for pie - not the best pie, but the one fundamental pie as it truly is, expressing itself in the recipes of all lands. You can't really do it, but you can gather lots of pie recipes and talk a little bit about the differences. That's what I'm here to do.
:: The Data
: Arabic Maqamat And 24-EDO (As Related By Wikipedia)
Arabic maqamat have a few "half-flat" notes, like "G half flat". A G half flat sounds somewhere between a G natural and a G flat. For notation, I'll use "d" to indicate a half-flat accidental, and "t" to indicate the a half-sharp accidental. Some other decent options in Unicode are ᵈ, as in Gᵈ, and ѣ as in Gѣ.
In 1932, the influential Cairo Congress of Arab Music decided that maqamat should be notated in 24-EDO, if not necessarily played that way. If you think of maqamat as tuned-scales within 24-EDO, then a half-flat accidental corresponds to flattening of pitch down by one step of 24-EDO, and a half-sharp means raising by one step of 24-EDO. Easy enough.
I'm pretty strongly opposed to representing scales in 24-EDO, for a bunch of reasons. One reason is that maqamat clearly have intervallic interpretations: almost all of them have seven notes and then reach the octave, like western diatonic scales and modes, suggesting that we should be able to spell them with ordinally increasing pitches, like [C, D, E, F, G, A, B] with various added accidentals. Another reason to eschew 24-EDO for maqam analysis is that we have hundreds of years treatises and artefacts giving us precise information on non-24 EDO intonations that were used for the maqam scales. It's an insult to the rich history of Arab music - to the historic work of Arabic music theorists and to the virtuosic mastery of Arab musicians - to pretend that Arabic music is just 24-EDO. Like, there's a thousand year old Arabic music theory text, the Kitab al-Musiqa al-Kabir by al-Farabi, which distinguishes a lute fingering that the author geometrically constructs to have a frequency ratio of 162/149 (@ 145 cents) and another with frequency ratio 54/49 (@ 168 cents), and pretending that these are both "3 steps of 24 EDO", when we're explicitly told to treat them differently by the master who has has honed his craft until he can hear the difference and perform the difference, is simply ignoring what the music really is and replacing it with a fantasy. And the history goes back way farther than that: there's a Persian flute called a ney at the University of Philadelphia Museum which is estimated to be from 3000 to 2800 BC. I don't know if anyone has looked at the geometry of the finger holes to figure out the tuning inherent to the instrument, but that's the sort of scholarship that is theoretically available to us, and the sort which should supplant 24-EDO simplifications.
Anyway, I don't support 24-EDO analysis, but I'm also not a medieval Arabic scholar, or even a modern Arabic scholar, so a lot of this chapter consists of me taking modern data about maqamat in 24-EDO and trying to figure out better intervallic representations as much as I can, though this of course still doesn't give enough recognition to real Arabic music history.
In our chapter "Higher Rank EDO Generators", we learned that in order to analyze 24-EDO music intervallically, we need to use at least a 4-dimensional interval space. I will provide this rank-4 intervallic interpretations of 24-EDO scales and also give a little bit of "detempering" analysis - that is, looking at what just frequency ratios are related by tempering to 24-EDO scales, in order to get an idea of what frequency ratios the scales really should have.
One way to interpret Arabic scales that are nominally tuned to 24-EDO is to interpret the half-flat accidental as a lowering an unaccented pitch by a septimal super unison, Sp1, which is justly tuned to 36/35, and which 24-EDO tunes to one step, i.e. a frequency ratio of 2^(1/24). Likewise the less often used half-sharp accidental can be interpreted as raising a pitch by a septimal super unison. In 24-EDO, a septimal super minor second, Spm2, and a septimal sub major second, SbM2, are both tuned to 3-steps of the EDO, so we can call them both "neutral seconds", and this confluence of intervals also works for neutral thirds, sixths, and sevenths. Everything meets in the middle, just like how F# and Gb meet in the middle in 12-EDO. I might not like 24-EDO analysis, but I do freely admit that it's a very easy system for thinking about music with neutral intervals.
Almost all maqamat, at least as they're presented in 24-EDO, can be characterized using just the normal western pitches plus Ed and Bd. In principle, we can also stack the 24-EDO "quarter-tone accidentals" with the regular "half tone" accidentals of # and b that are familiar to us from rank-2 interval space. For example, in 24-EDO, a C half flat, Ct, is equivalent to D three-quarters flat, Dbd.
Since 12-EDO and 24-EDO both temper out the acute unison, you might wander what m2 and M2 we're deviating from by septimal accidentals to get a septimal neutral tone. The best I can tell you is that the Persian dastgah system, from which all these musical traditions were derived, itself came from ancient Greek music, which suggests a Pythagorean framework, and one of the oldest treatises we have on medieval intonation, the Kitab al-Musiqa al-Kabir of al-Farabi, uses lots of Pythagorean frequency ratios and very high-prime limit frequency ratios and I don't know that he ever uses regular 5-limit major and minor frequency ratios. On the other hand, Turkish music often uses 53-EDO instead of 24-EDO, which is a schismatic temperament that distinguishes among Pythagorean and just intervals, and indeed some 53-EDO analysis would suggest that Turkish neutral intervals are simply 5-limit just intervals instead of Pythagorean ones, although more precise just analysis from music theorists like Ozan Yarman cast doubt on this. Thus far Arabic maqamat, my analysis will mostly ignore the 5-limit accidentals, those of acuteness and gravity, and you may think of the analysis as working in a meantone framework. Perhaps unfortunately, 5-limit just intonation is the one true way in western music, and all of my analytical programs are based around it. For this level of analysis of Arabic maqamat, it might be useful to have programs that recognize a septimal extension of Pythagorean tuning, what the Xenharmonic microtonalists would call the "2 3 7 just intonation subgroup". But we'll also fare well enough without such specialized machinery. And eventually we will want factors of 5 and the entirety of rank-4 interval space (and higher), to analyze neutral tones like the frequency ratio 11/10 that was used in tetrachords by the great lutenist Mansour Zalzal, who even preceded al-Farabi.
I'm going to be real with you: if you use Pythagorean just tunings for the major and minor intervals, and tune Ed to 27/22 over C and Bd a perfect fifth higher to 81/44 over C, then you get a pretty decent tuning for Arabic maqamt that most people willa accept as good enough. But well try to do even better than that.
Now that we have provisionally established some notation for rank-4 intervals and hinted at an interpretation of 24-EDO in terms of rank-4 intervals, here's our first bit of data. These are my transcriptions of some Arabic maqam scales based on staff notation on 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, Ed, F, G, A, Bb, C, D] # Has Ed.
Jiharkah [0, 4, 8, 10, 14, 18, 21, 24] [F, G, A, Bb, C, D, Ed, F] # Has Ed.
Huzam [0, 3, 7, 9, 15, 17, 21, 24] [Ed, F, G, Ab, B, C, D, Ed]. # Has Ed.
Rahat al Arwah [0, 3, 7, 9, 15, 17, 21, 24] [Bd, C, D, Eb, F#, G, A, Bd] # Has Bd.
Saba [0, 3, 6, 8, 14, 16, 20, 24] [D, Ed, F, Gb, A, Bb, C, D] # Has Ed.
Rast [0, 4, 7, 10, 14, 18, 21, 24] [C, D, Ed, F, G, A, Bd, C] # Has Ed and Bd.
Husayni 'Ushayran [0, 3, 6, 10, 13, 16, 20, 24] [A, Bd, C, D, Ed, F, G, A] # Has Bd and Ed.
On each line, I show the name of the maqam, the number of steps of 24-EDO for each scale degree relative to the tonic, the pitch classes of the maqam using "d"s as a half-flat accidental, and finally a summary comment on the presence or absence of any notes with half-flat accidentals. Wikipedia also tells us that the maqamat 'Ajam, Nahawand, and Kurd are tonal, and correspond to western Major, Minor, and Phrygian modes. Wikipedia does not tell us the tonic pitches though. Maybe we'll figure them out from another data source.
So those are the scales in terms of pitch classes and tuned steps of 24-EDO. But what are the scales in terms of intervals? There's a one to one association between pitch classes and intervals over a root, like C natural, which is only slightly complicated by ignoring 5-limit accidentals. If there's a "d" on the pitch class, then the associated interval over C is a Sub-(whatever the interval over C would have been without the "d" accidental). And if there's a "t" on the pitch class, then the associated interval is a Super-(whatever it would have been otherwise). Here are three example pitch classes with their intervals over C:
Dbt <> Spm2 over C
C#t <> SpA1 over C
Dd <> SbM2 over C
Take a moment to think about them and verify for yourself that 24-EDO tunes all of them to 3 steps. The first two pitches are one step up from m2 or A1, which are both tuned to 1 step of 12-EDO, and thus 2 steps of 24-EDO. The third pitch is one step down from M2, which is tuned to 2 steps of 12-EDO and thus 4 steps of 24-EDO.
Here are some simple intervals that 24-EDO tunes to each of its steps:
^0: P1
^1: Sbm2 ~ Sp1
^2: m2
^3: SbM2 ~ Spm2
^4: M2
^5: Sbm3 ~ SpM2
^6: m3
^7: SbM3 ~ Spm3
^8: M3
^9: Sb4 ~ SpM3
^10: P4
^11: Sbd5 ~ Sp4
^12: A4 ~ d5
^13: Sb5 ~ SpA4
^14: P5
^15: Sbm6 ~ Sp5
^16: m6
^17: SbM6 ~ Spm6
^18: M6
^19: Sbm7 ~ SpM6
^20: m7
^21: SbM7 ~ Spm7
^22: M7
^23: Sb8 ~ SpM7
^24: P8
For every odd step, i.e. for every quarter tone, we have at least two simple options for naming. For example, step 1 of 24-EDO could be called a half-sharp unison or a half-flat minor second. I hope you'll find that the list above nicely shows off how rank-4 septimal intervals fill in the gaps between familiar intervals of 12-EDO.
From the pitch classes listed for each maqam above, we can start to see that Arabic maqamat use mostly half-flats in preference to half-sharps; there are only half-flat accidentals in in the scales presented so far, but that pattern won't continue to hold absolutely.
Curiously, none of the maqamat listed above use 19 steps of 24-EDO. This is weird because the septimal sub-minor seventh, Sbm7, is justly tuned to the reduced seventh harmonic, 7/4, and 24-EDO tunes Sbm7 to 19 steps. I think this is a little bit of an argument against the idea that maqam music in 24-EDO is secretly septimal under the hood. If maqamat were septimal (with 24-EDO being a convenient shorthand for collapsing the septimal half-sharp/half-flat distinction), then I'd expect that Arabic music would be using that 7th harmonic to good effect. But even if the maqamat intervals don't come from 7-limit just intonation, still rank-4 intervals are the minimal way to analyze 24-EDO intervallically, and the limited data we have does not yet license us to give better analysis.
If we naively infer rank-4 intervals for maqam scale degrees by looking at the pitch classes, then we get some acute and grave intervals:
Hijaz (Nahawand ending): P1, Acm2, M3, Ac4, P5, Acm6, m7, P8.
Nawa Athar: P1, M2, m3, A4, P5, m6, M7, P8.
Shad 'Araban: P1, m2, M3, P4, Gr5, m6, GrM7, P8.
Bayati: P1, SbAcM2, m3, Ac4, P5, Acm6, m7, P8.
Jiharkah: P1, AcM2, M3, Ac4, P5, M6, SbM7, P8.
Huzam: P1, Spm2, Spm3, Spd4, Sp5, Spm6, SpGrm7, P8.
Rahat al Arwah: P1, Spm2, SpGrm3, Spd4, SpGr5, Spm6, SpGrm7, P8.
Saba: P1, SbAcM2, m3, Acd4, P5, Acm6, m7, P8.
Rast: P1, M2, SbM3, P4, P5, M6, SbM7, P8.
Husayni 'Ushayran: P1, SbAcM2, m3, P4, Sb5, m6, m7, P8.
Here are the relative intervals between steps:
Hijaz (Nahawand ending): Acm2, A2, Acm2, M2, Acm2, M2, M2.
Nawa Athar: M2, Acm2, A2, Acm2, m2, AcA2, m2.
Shad 'Araban: m2, AcA2, m2, M2, Acm2, A2, Acm2.
Bayati: SbAcM2, Spm2, AcM2, M2, Acm2, M2, M2.
Jiharkah: AcM2, M2, Acm2, M2, M2, SbAcM2, Spm2.
Huzam: Spm2, AcM2, m2, AcA2, m2, M2, SbAcM2.
Rahat al Arwah: Spm2, M2, Acm2, A2, Acm2, M2, SbAcM2.
Saba: SbAcM2, Spm2, Acm2, A2, Acm2, M2, M2.
Rast: M2, SbAcM2, Spm2, AcM2, M2, SbAcM2, Spm2.
Husayni 'Ushayran: SbAcM2, Spm2, M2, SbAcM2, Spm2, AcM2, M2.
If we lop off the "acute" prefixes for a quick and dirty meantone analysis, then on the relative interval, then all of the relative intervals come from [m2, Spm2, SbM2, M2, A2]. Minor, neutral, neutral, major, and augmented, but all seconds of one kind of another.
An intervallic description of the scales allows us to freely transpose maqamat to other tonics, however in Arabic music, the scales are not actually transposed, e.g., part of the definition of Rast is that it starts on C. So far as maqam tones can be quantized down to 24-EDO, Arabic music uses a very limited pallet of notes, e.g. the only quarter tones we see above are B half-flat and E half-flat. Here's the full set of pitch classes from the Wikipedia transcriptions:
[C, D, Ed, Eb, F, F#, Gb, G, Ab, A, Bb, Bd, B]
although we'll expand this in time. You might notice that there's no E natural. A very limited pallet!
In practice, 24-EDO started out as more a notational convenience than a tuning system for actual use, and the tuning/intonation of things are instead hugely dependent on the region, the time period, the musician, and the maqam. Like, the D of Maqam Huzam might not be exactly the D in Maqam Rahat al Arwah, for all I know, and middle eastern musicians know - by feel or vibes or muscle memory - the tiny variations in frequency between them and can have discussions about which should be flatter or sharper by amounts that are basically imperceptible to untrained ears.
But "it's contextual and sometimes imperceptible to untrained ears" is not useful data that can be transmitted for teaching, so Arabic musicians notate things in 24-EDO as a start, and we'll do our best to learn both from their notation and to learn more about actually played musical intonation and its regional variation as we go.
In addition to this rank-4/septimal analysis, I have an interesting 17-limit interpretation of 24-EDO maqamat that we'll get to eventually. But this is a good start for now.
:: 24-EDO Ajnas
Maqamat are made out of smaller scale fragments called "ajnas" in plural, the singular being "jins". They commonly consist of four notes, in which case they are called tetrachords, and tetrachords almost always span an interval of a perfect fourth. Ajnas are not just scale fragments - they are modes themselves, in which musicians produce small melodic phrases - riffs, licks, motives, figures, whatever - before passing into other modes. In some traditions, sepcific transitions between ajnas are prescribed producing a large scale temporal structure to the muscial peace. In Arabic maqmat, these paths are called "sayir". But let's start with the scale fragments.
Here are some Arabic ajnas presented in steps of 24-EDO:
Sikah trichord: [3, 4]
'Ajam trichord: [4, 4]
Kurd tetrachord: [2, 4, 4]
Hijaz tetrachord: [2, 6, 2]
Saba tetrachord: [3, 3, 2]
Bayati tetrachord: [3, 3, 4]
Nahawand tetrachord: [4, 2, 4]
Rast tetrachord: [4, 3, 3]
Nikriz pentachord: [4, 2, 6, 2]
These are relative steps between scale degrees of each jins, so Jins Hijaz, written in relative steps as [2, 6, 2], could be written cumulatively to show that it has four notes at steps [0, 2, 8, 10].
Those ajnas are all taken from Wikipedia, although the Wikipedia page uses e.g. 1 "whole step" to decribe the size of a M2 whole tone, so I multiplied through by a factor of 4 to get 24-EDO commas. The Saba tetrachord only reaches 8 steps of 24-EDO, falling flat of the usual P4 at 10 steps of 24-EDO, but the other tetrachords reach it. The ajnas with odd-valued step sizes are microtonal, while the ajnas with all even-valued step sizes are representable in 12-EDO.