Bifurcations: Splitting ragas into ‘poorvang + uttarang’ formulas

 

Bifurcated ragas: A concise way to summarise the ragascape using a simple set of ‘poorvang + uttarang’ formulas, derived from the full set of 96 sequences. Get in touch with thoughts on the system!

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—SHORTLIST OF SEQUENCES—
• Concept: ‘Poorvang + uttarang’ splits •
Step 1: Deriving all 96 sequences
Step 2: All sequences + raga matches
Step 3: Selecting a ‘useful shortlist’
Step 4: The 16 ‘universal sequences’
• Further discussion & global parallels •

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—Why split ragas in half?—

The idea of ‘splitting the octave in half‘ appears across the world – from the ‘djinns‘ of Middle Eastern maqam to the ‘Greater Perfect System‘ of the Ancient Greeks. Likewise, it is common to conceptualise ragas in terms of their ‘poorvang + uttarang’ split: plenty of jod ragas are named this way (e.g. Nat Bhairav: Nat’s SRGm + Bhairav’s PdNS), and most other scales can be summarised in similar fashion (e.g. Hemavati = ‘Madhuvanti’s SRgM + Kafi’s PDnS‘). However, it has always surprised me that this area of raga analysis is comparatively ‘under-theorised’.

 

In contrast to the complex multi-century history of thaat/mela classifications (not to mention the various permutations of merukhand swara-sequencing systems), the idea of summarising ragas by splitting them in half occupies little space in the shastras. So I thought I’d try and ‘formalise’ this zone a little – prompted by late-2025 discussions with regular collaborator Ashish Dha around his recent work on the same topic, as well as the fantastic Rafi Kharis’ formulation of a similar ‘tetrachordal’ system for Western jazz, folk, and classical scales.

 

In the hope of bringing some conceptual clarity, the article below presents a tentative list of ‘useful combinations’, selected from the full derived sets of all possible poorvangs (=64) and uttarangs (=32), as well as some further discussion of how other global traditions treat similar ideas around ‘scale bifurcation’. There’s nothing particularly new here – but it seemed valuable to set out a basic mathematical formulation of this area, hopefully as a first step to catalysing better research in the future. Feedback welcome!

 

Poorvang | Uttarang

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—Sequences: useful shortlist—
[brackets = semitones] Click on swaras for more raga matches, and see below for a full explanation of the derivation and selection processes

—’Universal sequences’ (named after ragas where the poorvang and uttarang share the same ‘semitone shape’: explained below):

Other useful poorvang sequences:

Other useful uttarang sequences:

(also browse the full list of 96 sequences)

—Using the sequences—

The main intention of this list is to empower the process of analysing and summarising ragas in terms of their ‘poorvang+uttarang’ split. For example, Vachaspati‘s swara-set is produced by the ‘Yaman+Kafi‘ sequences (SRGM+PDnS), and Charukeshi‘s by ‘Bilawal+Bhairavi‘ (SRGm+PdnS) – while Sehera is shaped like ‘Yaman+Malkauns‘ (SRGM+dnS), and Deepavali like ‘Bihag+Marwa‘ (SRGmM+DNS).

 

The vast majority of ragas can be neatly described using such formulas – including almost all prachalits, as well as most rare and strange forms too (e.g. Adbhut Kalyan = ‘Bhupali+Marwa‘: SRG+DNS). It also highlights connections between ‘half-prakriti‘ ragas (e.g. compare and contrast the 9 ragas which share Gunkali‘s distinctive PdS uttarang), as well as offering deeper geometric insights via identifying a set of ‘universal patterns‘, derived from ragas which share the same ‘semitone shape’ in both poorvang and uttarang (e.g. the ‘Bhairav tetrachord‘ always has semitone jumps ‘1-3-1‘, whether low as SrGm or high as PdNS; thus Bhairav‘s formula is just ‘Bhairav+Bhairav‘).

 

—How not to use them: This system is designed to be flexible, and the list above is just my selection of the most useful sequences from the full list of 96 mathematical possibilities. Use it and adapt it to your own musical situation – if you prefer to associate the pattern ‘Sgm‘ with Malkauns rather than Dhani, then call it the ‘Malkauns poorvang’ instead. Similarly, be cautious not to ‘over-write’ a raga’s actual high-low derivation (e.g. Lalita Sohini = ‘Lalit+Marwa‘; but its DNS uttarang is not of the Marwa-ang – so call it ‘Lalit+Sohini’!). Always bear in mind that this is just ‘scalar analysis’, which ignores srutis and does not inherently tell us anything about a raga’s phrasal movements. As ever, send in your feedback!

 

  • Listen: poorvang & uttarang (S-rR-gG-mM | P-dD-nN-S):

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—Step 1: Deriving all possibilities—

Given the swara axioms of Hindustani music, we can formulate a list of all possible poorvang and uttarang shapes. As we will see, many of these are barely used in the actual ragascape – but we should begin by defining the entire set. For one thing, the real-world absence of certain shapes can teach us about raga’s broader geometric tendencies…

 

Notably, the poorvang space is larger than the uttarang space (SrRgGmM vs. PdDnNS), doubling its combinational possibilities (7 swara positions vs. 6: technically, we’re not splitting the saptak into literal halves). Given that Sa is fixed, and every other position is effectively a binary ‘on/off’ variable (i.e. komal re is either present or not present), the poorvang offers 2^6=64 sequences, while the uttarang offers 2^5=32.

 

Browse the full sets below, sub-separated by number of swaras (‘|’) – and see the next section for detail and discussion of raga matches. And for full calculations, see the Ragatable (n.b. I actually used a slightly different calculation method, to ease the generation of neat text-strings).

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—Poorvang: all 64 possibilities—
  • S | S-r; S-R; S-g; S-G; S-m; S-M | S-r-g; S-r-G; S-R-g; S-R-G; S-r-m; S-r-M; S-R-m; S-R-M; S-g-m; S-g-M; S-G-m; S-G-M; S-rR; S-gG; S-mM | S-r-g-m; S-r-g-M; S-r-G-m; S-r-G-M; S-R-g-m; S-R-g-M; S-R-G-m; S-R-G-M; S-rR-g; S-rR-G; S-r-gG; S-R-gG; S-r-mM; S-R-mM; S-rR-m; S-rR-M; S-g-mM; S-G-mM; S-gG-m; S-gG-M | S-rR-g-m; S-rR-g-M; S-rR-G-m; S-rR-G-M; S-r-gG-m; S-r-gG-M; S-R-gG-m; S-R-gG-M; S-r-g-mM; S-r-G-mM; S-R-g-mM; S-R-G-mM; S-rR-gG; S-rR-mM; S-gG-mM | S-rR-gG-m; S-rR-gG-M; S-rR-g-mM; S-rR-G-mM; S-r-gG-mM; S-R-gG-mM | S-rR-gG-mM
—Uttarang: all 32 possibilities—
  • S | P-S; d-S; D-S; n-S; N-S | P-d-S; P-D-S; P-n-S; P-N-S; d-n-S; d-N-S; D-n-S; D-N-S; dD-S; nN-S | P-d-n-S; P-d-N-S; P-D-n-S; P-D-N-S; P-dD-S; P-nN-S; d-nN-S; D-nN-S; dD-n-S; dD-N-S | P-dD-n-S; P-dD-N-S; P-d-nN-S; P-D-nN-S; dD-nN-S | P-dD-nN-S

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—Step 2: All sequences & raga matches—

Below is a list of known raga matches for each poorvang and uttarang possibility. Often, the choice of name is obvious, but some cases warranted further deliberation: see discussion in the shortlisting section below, particularly around the aim for ‘unambiguous patterns‘ (e.g. the ‘Bhairav tetrachord’ has the same semitonal shape in both poorvang and uttarang: SrGm and PdNS both have jumps of ‘1-3-1‘).

 

Click on the sequences themselves to bring up a list of all matches from the Megalist of 365+ ragas [square-bracketed figures indicate the total count]. For sequences with no Megalist matches, I’ve also searched through the Masterlist of 1000+ swara-sets (these matches, if present, are named in brackets). Brief analysis of the most and least prevalent combinations is below – and free to disagree with my decisions!

 

—Poorvang: all possibilities—

1 swara:

  • S: [none found]

2 swaras:

3 swaras:

4 swaras:

5 swaras:

6 swaras:

7 swaras:

—Uttarang: all possibilities—

1 swara:

  • S: [none found]

2 swaras:

3 swaras:

4 swaras:

5 swaras:

6 swaras:

—Quick highlights: In poorvang, the most popular combinations are SRGm (46 ragas) and SRgm (34) – and in uttarang, the winners are PDNS and PDnNS (both 59) followed by PDnS (44). From the 3-swara combinations, the most prevalent poorvang options are SRm (17) and Sgm (10), and in uttarang PnS (17) plus PDS and DNS (both 14).

 

At the other extreme, 14 combinations appear to be entirely absent from the modern Hindustani ragascape, with no Masterlist matches: 9 in poorvang (S; SM; SrR; SmM; SrRg; SrRM; SrRgM; SrRgG; SrRgGM) and 5 in uttarang (S; dDS; nNS; dDNS; dDnNS). A further 6 sequences only turn up in a single known raga, all from the poorvang space: Jaldhar Sarang‘s SR; Vibhasini‘s SrgG; Jait‘s SrRG; Kanakangi‘s SrRm; Madhusurja‘s SrRmM; and Desi Todi‘s SrRgmM. I wonder what intrepid raga artists could do with these seldom-heard sequences?

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—Step 3: Selecting a ‘useful shortlist’—

From the 96 possibilities above, we can now select a shortlist of the most useful sequences. After all, the overall goal is to ease the task of raga summarisation – so our results should be pared down to only those which are intuitively useful…

 

—Omit: All with no prominent raga matches: If a sequence isn’t used in actual raga performance, there is no need to feature it in a summarisation system – and the same applies if it only matches very rare ragas. This removes most of the 96 – including all of the 2-swara sequences, and most of those with double-swaras.

 

—Include: All with 4 generic swaras (i.e. one each of ‘Sa-Re-Ga-Ma’ / ‘Pa-Dha-Ni-Sa’): Including these forms means that any 7-swara sampurna scale can be summarised (see the 32 thaat system for this idea in context) – as well as being required for a majority of shadav ragas (e.g. Parameshwari = ‘Bhairavi‘s Srgm + Rageshri‘s DnS’). 

 

—Include: Most with 3 generic swaras (i.e. no doubles, and only one generic varjit: e.g. ‘Sa-Re-Ma’, ‘Pa-Ni-Sa’): Most of these sequences match with at least fairly well-known ragas – and many also form ‘unambiguous patterns’ (see below). Although we may not want to include all options (e.g. no prachalit raga shares Madhukauns‘ SgM poorvang, despite rare gems such as Harikauns and Tivrakauns…).

 

—Include: Other useful sequences: It makes intuitive sense to include a few prominent double-swara sequences. Based on the raga-match counts above, the ones that jump out most are S-gG-m (11: e.g. Jog, Jogkauns, Nandkauns, Mohankauns), S-r-G-mM (24: e.g. Lalit, Poorvi, Bhatiyar, Lalita Gauri), S-R-G-mM (21: e.g. Bihag, Nand, Shuddha Sarang), P-D-nN-S (59: e.g. Jaijaiwanti, Desh, Miyan ki Malhar, Alhaiya Bilawal). Now, we have our provisional set – see the list here:

—Sequences: useful shortlist—

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—Step 4: The 16 ‘universal sequences’—

You may have noticed some of the sequences in my suggested name list above share the same name (e.g. the ‘Bilawal tetrachord’ = SRGm or PDNS). Counterintuitively, the aim of this is actually to reduce ambiguity: in all such cases, the poorvang and uttarang segments share the exact same pattern of ‘semitone jumps’ (in Bilawal’s case, ‘2-2-1‘ from Sa and Pa) – meaning that the term ‘Bilawal tetrachord’ is unambiguous. To find the full set of these matches, we essentially have to answer the question ‘how many ragas repeat their poorvang from Pa‘?

 

Given that the poorvang space is one semitone wider than the uttarang space, we must exclude all poorvang sequences which contain tivra Ma – and the necessity of including Sa in both halves means that ma and Pa must always be present in theirs (i.e. ‘poorvang Sa becomes uttarang Pa, and uttarang Sa becomes poorvang ma’). This leaves us with 4 ‘variable’ swara-spaces in each half (rRgG | dDnN) – which in turn gives us a total of 2^4=16 congruent sequences. Unsurprisingly, most of them match with well-known ragas – after all, these self-referential scale forms offer inherent geometric intrigue, allowing for nuanced low-to-high reflection of melodic phrase-shapes. Semitone jumps are in [square brackets].

 

2-swara:

  • [1-5] S-m | P-S: [none found]

3-swara:

4-swara:

5-swara:

6-swara:

To zoom in a little, most matches with prominent ragas comprise sequences which feature no double-swaras (all 8 of these cases have notable matches) – giving credence to our criteria above, which already include all 8 of these combinations. In any case, I think that geometric clarity is obscured by denser, more complex scale forms (while Bhairav may be dominated by its poorvang-uttarang congruences, does it really matter that non-Ma forms of Khat fulfil this property too?). Make your own mind up via browsing my list of ‘Pa-repeating’ ragas…

 

Naturally, many lines of inquiry still lie untouched. For example: what insights could be gained by dividing the saptak space at a different point (e.g. DnNSrR; gGmMPd)? Or by slicing it into three neat segments (e.g. SrRg; GmMP; dDnN), instead of our current ‘pseudo-halves’? And what could be gleaned by the identification of ‘ma-repeating ragas‘ (i.e. those which repeat their poorvang pattern from shuddha ma rather than Pa)? Let me know what you come up with!

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—Further ideas & global parallels—
  • Merukhand: Learn about Amir Khan’s ‘168 swaramaila patterns’, formed by calculating all permutations of adjacent groups of 4 generic swaras (e.g. SRGM; SRMG; SGRM; SGMR; RSGM; RSMG…), derived from the full mathematical set of 5040 sequences – a core component of the great vocalist’s riyaz throughout his career.
  • Maqam: The main melodic framework of the Middle East is built around combining ‘djinns‘: building blocks of 4-5 adjacent notes which are stacked on top of each other to create a full ‘maqam‘ – often featuring distinctive microtonal tunings. Learn more in my writeup of Bayati (a maqam-inspired ‘quarter-tonal’ raga created by Dinkar Kaikini).
  • Tetrachords: Ancient Greek musical systems focused on combining distinct tetrachords (Greek for ‘four-notes’) – notably, the ‘Greater Perfect System‘, which flourished between the 5th and 4th centuries B.C.E., comprises four tetrachords (‘Hypaton’, ‘Meson’, ‘Diezeugmenon’, & ‘Hyperbolaion’) – read more in this overview from Dartmouth.

—Hindustani Raga Index—

An open-ended project seeking to bring North Indian raga closer to all who approach with open ears. Combines direct input from dozens of leading Hindustani artists with in-depth insights from music history, global theory, performance practice, cognitive science, and much more besides!

–Megalist (365+ ragas)–
• Search: Find your new favourite •
• Tags: Classifying the ragascape •
• Glossary: Raga jargon demystified •
• Murchanas: Swara-set rotations •
• Thaat: Bhatkhande’s base scales •
• Bifurcations: Poorvangs/uttarangs •
• Ragatable: Analytical connections •
• Tanpuras: Divine overtonal drones •
• Quotes: Musings from raga artists •
• Talas: Hindustani rhythm cycles •
• Instruments: Singing sculptures •
• Masterlist: 1000+ ragas profiled •
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George Howlett is a London-based musician, writer, and teacher (guitars, sitar, tabla, & santoor). Above all I seek to enthuse fellow sonic searchers, interconnecting fresh vibrations with the voices, cultures, and passions behind them. See Homepage for more, and hit me up for Lessons!

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