**Foundational forms**: What are Bhatkhande’s ‘thaat scales’? How should they be applied to raga study? And how can we extend these classifications?

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###### —Bhatkhande’s Ten Thaat—

**Kalyan** • **Bilawal** • **Khamaj** • **Kafi** • **Asavari**

**Bhairavi** • **Bhairav** • **Marwa** • **Poorvi** • **Todi**

###### • **How to use the scales** •

###### —Further Forms—

###### • **Computing the total possible set** •

###### • **‘Dartboard’ of 32 sampurna scales** •

###### • **Why the ‘unfilled’ raga sequences?** •

###### • **Melakarta, merukhand, & more** •

**What are thaat?** The concept originates with theorist **V.N. Bhatkhande** (1860-1936), who proposed ten ‘reference scales’, each with **7 swaras**, designed to help classify the ragascape (*Thus, the common translation of ‘parent scale’ is misleading: the ragas long predate the scales!*).

Some ragas fit exactly into a particular thaat (e.g. **Yaman** & **Kalyan thaat**), or can be described neatly enough (e.g. **Vachaspati**: ‘**Kalyan thaat** **komal** **ni**’) – although many more defy easy summarisation (e.g. **Jog** & **Lalit**). Next: the scales themselves – and go deeper in **below**!

### • Kalyan thaat •

### S-R-G-M-P-D-N-S

###### 1-2-3-#4-5-6-7-8

###### (Lydian Mode)

**Exact**: **Yaman**, **Shuddha Kalyan**

**Enclosed**: Bhupali, Deshkar, Jait Kalyan, Adbhut Kalyan, Hansadhwani, Hindol, Pahadi, Kesari Kalyan, Malashree, Raj Kalyan, Shankara

### • Bilawal thaat •

### S-R-G-m-P-D-N-S

###### 1-2-3-4-5-6-7-8

###### (Major Scale)

**Exact**: **Bilawal**, **Tilak Kamod**, **Nat Kamod**, **Nat**, **Bihari**, **Dagori**, **Gagan Vihang**, **Gaud Malhar**, **Hem Bihag**, **Hemant**, **Maluha**, **Manjari Bihag**, **Swanadi**

**Enclosed**: Bhupali, Durga, Pahadi, Hansadhwani, Kaushik Dhwani, Bhinna Shadja, Nat Kamod, Shuddha Malhar, Jaldhar Kedar, Deshkar, Shankara, Rasaranjani, Jait Kalyan, Kesari Kalyan, Adbhut Kalyan, Bhavani, Malashree

### • Khamaj thaat •

### S-R-G-m-P-D-n-S

###### 1-2-3-4-5-6-b7-8

###### (Mixolydian Mode)

**Exact**: **Khamaj**, **Jhinjhoti**, **Gaoti**, **Kambhoji**, **Khambavati**, **Chaya Malhar**, **Savani**

**Enclosed**: Bhupali, Durga, Megh, Gorakh Kalyan, Kalavati, Jansammohini, Rageshri, Bhavani, Deshkar, Jait Kalyan, Durgawati, Pahadi, Madhumad Sarang, Kalashri, Malashree, Jaldhar Kedar, Shuddha Malhar, Narayani

### • Kafi thaat •

### S-R-g-m-P-D-n-S

###### 1-2-b3-4-5-6-b7-8

###### (Dorian Mode)

**Exact**: **Kafi**, **Bageshri**, **Bhimpalasi**, **Shahana**, **Desi**, **Dhanashree**, **Hussaini Kanada**, **Mudriki Kanada**, **Raisa Kanada**

**Enclosed**: Durga, Dhani, Megh, Abhogi, Shivranjani, Gorakh Kalyan, Bhavani, Durgawati, Gaudgiri Malhar, Madhumad Sarang, Jaldhar Kedar, Manavi, Shuddha Malhar, Narayani, Nayaki Kanada, Suha Kanada, Sundarkauns

### • Asavari thaat •

### S-R-g-m-P-d-n-S

###### 1-2-b3-4-5-b6-b7-8

###### (Aeolian Mode)

**Exact**: **Darbari**, **Kaunsi Kanada**, **Adana**, **Jaunpuri**, **Sampurna Malkauns**

**Enclosed**: Malkauns, Megh, Dhani, Gopika Basant, Gaudgiri Malhar, Madhumad Sarang, Nayaki Kanada, Pancham Malkauns, Shobhawari, Suha Kanada

### • Bhairavi thaat •

### S-r-g-m-P-d-n-S

###### 1-b2-b3-4-5-b6-b7-8

###### (Phrygian Mode)

**Exact**: **Bhairavi**, **Bilaskhani Todi**

**Enclosed**: Malkauns, Dhani, Bhupali Todi, Bairagi, Bairagi Todi, Saheli Todi, Gunkali, Gunakri, Gopika Basant, Lankeshri Kanada, Pancham Malkauns

### • Bhairav thaat •

### S-r-G-m-P-d-N-S

###### 1-b2-3-4-5-b6-7-8

###### (Double Harmonic Scale)

**Exact**: **Bhairav**, **Kalingada**, **Gauri**, **Prabhakali**

**Enclosed**: Jogiya, Bibhas, Devranjani, Gunakri, Gunkali, Bangal Bhairav, Zeelaf, Malashree, Triveni, Reva

### • Marwa thaat •

### S-r-G-M-P-D-N-S

###### 1-b2-3-#4-5-6-7-8

###### (Lydian b2)

**Exact**: **Puriya Kalyan**, **Shree Kalyan**, **Dakshinatya Basant**, **Baradi**

**Enclosed**: Marwa, Puriya, Sohini, Hindol, Hansa Narayani, Malavi, Malashree

### • Poorvi thaat •

### S-r-G-M-P-d-N-S

###### 1-b2-3-#4-5-b6-7-8

###### (Lydian b2 b6)

**Exact**: **Shree**, **Puriya Dhanashree**, **Basant**, **Gauri Basant**, **Jaitashree**, **Tankeshree**

**Enclosed**: Bibhas, Din ki Puriya, Hansa Narayani, Malashree, Reva, Triveni

**• Todi thaat •**

**S-r-g-M-P-d-N-S**

###### 1-b2-b3-#4-5-b6-7-8

###### (Lydian b2 b3 b6)

**Exact**: **Todi**, **Multani**, **Annapurna**

**Enclosed**: Bhupali Todi, Gujiri Todi

**• Using the Scales •**

*How can thaat help illuminate the ragascape?*

T…

**():**

*“Bhatkhande realised that [vs. Carnatic ragam], Hindustani ‘parent scales’ were far fewer in number. This was partly due to aesthetic convention, which frowned upon the frequent juxtaposition of pitches…only a semitone apart…He added a rule to deal with this: that each scale shall have just one note from the pairs [rR, gG, mM, dD, nN], to eliminate many of the half-tones…He called each of these scales a ‘thaat’ (‘manner, style’), and…by using well-reasoned inductive argument, he identified 10 thaats as being in common use, each named after an important raga…” (via Ramesh Gangoli)*

**—Search the Raga Index—**

**• Also see the RAGATABLE •**

**• Beyond Bhatkhande’s Thaat •**

*Which other ‘fundamental scales’ are possible?*

Then ten scales above – chosen by Bhatkhande for their classificatory utility – are only a fraction of the total potential set. So, how many are possible? As per his original formulation, all scales must be:

**Heptatonic**(=**7 specific swaras**)**Sampurna**(=all**7 sapta swara**)**In aroha order**(i.e. no**vakra**)**Achal Pa**(remains ‘**immovable**’)**Sruti-invariant**(no**microtones**)

Given these axioms, we can calculate our set size. While we have 7 swaras to arrange, 2 are immovable (**Sa**, as the root, and **Pa**, as per tradition). The remaining 5 each offer a ‘higher/lower’ binary (‘**komal**/**shuddha**‘ for **rR**–**gG**–**dD**–**nN**, and ‘**shuddha**/**tivra**‘ for **mM**). Following this through in ‘decision tree’ fashion equates to [2^{5}=] **32 scales**. See this process visualised on my ‘dartboard’ below!

**–Dartboard: 32 Sampurna Scales–**

*‘Decision tree’ of scales, arranged as per the Carnatic melakarta:*

*(e.g. #9: ‘SP+MnGDR’ = SRGMPDnS = Vachaspati)*

**–Wheels: 32 Sampurna Scales–**

*The ‘results’ of our ‘dartboard decision tree’:*

*(n.b. dotted lines indicate ‘mirror symmetry‘)*

**–Corresponding Ragas–**

**SRGMPDNS**:**Kalyan thaat****SrGMPDNS**:**Marwa thaat****SRGMPdNS**:**(‘Kalyan komal dha‘)****SrGMPdNS**:**Poorvi thaat****SRgMPDNS**:**Madhuvanti****SrgMPDNS**:**(‘Todi shuddha Dha‘)****SRgMPdNS**:**Simhendra Madhya.****SrgMPdNS**:**Todi thaat****SRGMPDnS**:**Vachaspati****SrGMPDnS**:**Rampriya****SRGMPdnS**:**(‘Charukeshi****tivra Ma****‘)****SrGMPdnS**:**(‘Shree komal ni‘)****SRgMPDnS**:**Madhukant****SrgMPDnS**:**(‘Ahiri tivra Ma‘)****SRgMPdnS**:**(‘Darbari tivra Ma‘)****SrgMPdnS**:**(‘Todi komal ni‘)****SRGmPDNS**:**Bilawal thaat****SrGmPDNS**:**Asa Bhairav****SRGmPdNS**:**Nat Bhairav****SrGmPdNS**:**Bhairav thaat****SRgmPDNS**:**Patdeep****SrgmPDNS**:**(‘Patdeep komal re‘)****SRgmPdNS**:**Kirwani****SrgmPdNS**:**(‘Bhairavi shuddha Ni‘)****SRGmPDnS**:**Khamaj thaat****SrGmPDnS**:**Ahir Bhairav****SRGmPdnS**:**Charukeshi****SrGmPdnS**:**Basant Mukhari****SRgmPDnS**:**Kafi thaat****SrgmPDnS**:**Ahiri****SRgmPdnS**:**Asavari thaat****SrgmPdnS**:**Bhairavi thaat**

**• Also see the MEGALIST (300+ ragas) •**

**• ‘Unfilled’ raga sequences •**

*Why are some scales ‘missing a raga’?*

Of the 32 scales above, only 23 clearly match with known ragas (as far as I can trace…). **So, why have the other 9 possibilities seemingly gone unused?** Which ragas are they closest to, or expressible in terms of? And where else do these sequences turn up across the wide world of music?

Explore these ‘**unfilled sampurna scales**‘ below, along with some brief analysis (mainly derived via search-querying my **Ragatable** spreadsheet, as well as just staring at the swara-wheels in search of quirks and familiar features). *n.b. I assume that at least a couple will already have matching ragas out there somewhere (…***share your insights**!)

*(Wheels: the 9 ‘unfilled’ scales)*

**–Unfilled #3 (SRGMPdNS)–**

**Expressible:**‘**Kalyan**komal dha’, ‘**Poorvi**shuddha re’, ‘**Nat Bhairav**tivra Ma’, ‘**Simhendra Madhyamam**shuddha Ga’ (or ‘**Yaman**‘s poorvang +**Todi**‘s uttarang’)**Reversed**:**S-r-G-mM-d-n-S**(=’**Ahir Lalit**komal dha’, or ‘**Basant Mukhari***komal pa*‘)**Murchanas**:**Asa Bhairav**(on Pa)**Equivalents**:**1-2-3-#4-5-b6-7-8**(=Carnatic**Latangi**)

**Analysis**: No ragas even appear to be only one ‘**double-swara**‘ away (i.e. those produced by ‘doubling’ any one of**#3**‘s**chal**swaras, e.g. ‘S-R-G…’ > ‘S-r**R**-G…’).

**–Unfilled #6 (SrgMPDNS)–**

**Expressible:**‘**Todi**shuddha Dha’, ‘**Puriya Kalyan**komal ga’, ‘**Madhuvanti**komal re’ (or ‘**Todi**‘s poorvang +**Yaman**‘s uttarang’)**Reversed**:**S-r-g-mM-D-N-S**(=’**Mangal Todi**shuddha Dha’, or ‘**Unfilled #22***komal pa*‘)**Murchanas**: (none found)

**Equivalents**:**1-b2-b3-#4-5-6-7-8**(=Carnatic**Suvarnangi**)

**Analysis**: Could be seen as a ‘shuddha Ni only’ version of the rare**Niranjani Todi**.

**–Unfilled #11 (SRGMPdnS)–**

**Expressible:**‘**Charukeshi**tivra Ma’, ‘**Vachaspati**komal dha’ (or ‘**Yaman**‘s poorvang +**Bhairavi**‘s uttarang’)**Murchanas**: ‘**Unfilled #22**‘ (on Pa)**Reversed**:**S-R-G-mM-d-n-S**(=’**Sehera**double Ma’, or ‘**Charukeshi***komal pa*‘)

**Equivalents**:**1-2-3-#4-5-b6-b7-8**(=Carnatic**Risabhapriya**)

**Analysis**: Like a ‘shuddha Ga only’ version of the fairly well-known**Dev Gandhar**– or a ‘poorvang-uttarang’ combination of either ‘**Yaman**–**Bhairavi**‘, or ‘**Vachaspati**–**Charukeshi**‘ (both of these raga-pairs are from the same murchana families: see the**Bilawal**&**Charukesi**sets).

**–Unfilled #12 (SrGMPdnS)–**

**Expressible:**‘**Poorvi**komal ni’, ‘**Basant Mukhari**tivra Ma’, ‘**Rampriya**komal dha’ (or ‘**Marwa**‘s poorvang +**Bhairavi**‘s uttarang’)**Reversed**:**S-R-G-mM-d-N-S**(=’**Lalit**shuddha Re’, or ‘**Nat Bhairav***komal pa*‘)**Murchanas**: (none found)

**Equivalents**:**1-b2-3-#4-5-b6-b7-8**(=Carnatic**Namanarayani**)

**Analysis**: No ragas even appear to be only one ‘**double-swara**‘ away (i.e. those produced by ‘doubling’ any one of**#12**‘s**chal**swaras). Also expressible as ‘**Malay Marutam**add tivra Ma’.

**–Unfilled #14 (SrgMPDnS)–**

**Expressible:**‘**Ahiri**tivra Ma’, ‘**Madhukant**shuddha Re’, ‘**Rampriya**komal ga’ (or ‘**Todi**‘s poorvang +**Kafi**‘s uttarang’)**Reversed**:**S-R-g-mM-D-N-S**(=’**Mangal Todi**shuddha Re/Dha’, or ‘**Patdeep***komal pa*‘)**Murchanas**: (none found)

**Equivalents**:**1-b2-b3-#4-5-6-b7-8**(=Carnatic**Shadvidamargini**)

**Analysis**: Could be seen as a ‘komal Ni only’ version of the rare**Niranjani Todi**, or alternatively ‘**Salagavarali**add tivra Ma’.

**–Unfilled #15 (SRgMPdnS)–**

**Expressible:**‘**Asavari**tivra Ma’, ‘**Madhukant**komal dha’, ‘**Simhendra Madhyamam**komal ni’ (or ‘**Madhuvanti**‘s poorvang +**Bhairavi**‘s uttarang’)**Reversed**:**S-R-G-mM-D-n-S**(=’**Ahir Lalit**shuddha Re’, or ‘**Khamaj***komal pa*‘)**Murchanas**: ‘**Unfilled #24**‘ (on Pa)

**Equivalents**:**1-2-b3-#4-5-b6-b7-8**(=Carnatic**Shanmukapriya**)

**Analysis**: No ‘double-swara proximates’

**–Unfilled #16 (SrgMPdnS)–**

**Expressible:**‘**Todi**komal ni’, ‘**Bhairavi**tivra Ma’ (or ‘**Todi**‘s poorvang +**Bhairavi**‘s uttarang’)**Reversed**:**S-R-G-mM-D-N-S**(=’**Lalit**shuddha Re/Dha’, or ‘**Bilawal***komal pa*‘)**Murchanas**:**Kedar**(on re),**Jogeshwari Pancham**(on ga),**Mangal Todi**(on Pa),**Lachari Kanada**(on dha)

**Equivalents**:**1-b2-b3-#4-5-b6-b7-8**(=Carnatic**Bhavapriya**)

**Analysis**: Can also be seen as ‘**Mangal Gujari**add Pa’ – or as the ‘final komal step’ in the implied ‘**Kalyan**>**Marwa**>**Poorvi**>**Todi**‘ chain – i.e. ‘start with all swaras set to their highest positions, then make one more of them komal each time, jumping in**ascending 5ths**from re’ (SRGMPDNS > S**r**GMPDNS > S**r**GMP**d**NS > S**rg**MP**d**NS > S**rg**MP**dn**S). These geometric proximities afford it a curious place in the raga universe: taking a heavyweight low-high split of ‘**Todi**–**Bhairavi**‘, while also being just a single swara-shift from both these ragas (…could it work as a ‘missing link’ between them? Either way, it is true that ‘**#16**is to**Yaman**as**Bhupali Todi**is to**Bhupali**‘, in the sense of the formers being ‘all-komal’ versions of the latters: so maybe**#16**should be called something like ‘**Kalyani Todi**‘?). It also offers several handy**murchanas**, as a ‘missing member’ of the**Kedar**set – while its reversal gives the ‘**Major Scale b5**‘.**Side quest: following the ‘chal chain’**: Given the ‘flattening pattern’ outlined above (‘flatten in order of ascending**generic 5ths**, starting from re’: i.e. ‘**R>D>G>N**‘), the next implied step (‘**N>M**‘) would lead to lowering the tivra Ma: producing ‘S**rgm**P**dn**S’, i.e. the ‘**all-komal**‘**Bhairavi**(and thus traversing both numerical extremities of the dartboard: #1.**Kalyan**, #32.**Bhairavi**). After this, the next upward 5th-jump is the**tritonal**‘**M>S**‘ – and, since Sa cannot be flattened, the resulting scale ends up ‘re-rooted’: as a ‘flat Sa’ just equates to raising all other swaras by the same amount: giving ‘**SRG**P**dDNS**‘ (‘**Shankara****double-Dha**‘, or ‘**Hansadhwani**add double-dha’). This ‘shifting circle’ is completed with a final step of ‘**S>P**‘, introducing a ‘komal Pa’, i.e. reintroducing the original**tivra Ma**(=’**SRGMdDNS**‘: or ‘**Raj Kalyan**double-dha’). [How long until you get back to the exact recurrence of the original scale form? Or if you do similarly with a different interval generator? Or go backwards around the swara wheel?**Let me know**…I’m already working on it!]

**–Unfilled #22 (SrgmPDNS)–**

**Expressible:**‘**Patdeep**komal re’, ‘**Ahiri**shuddha ni’, ‘**Asa Bhairav**komal ga’ (or ‘**Bhairavi**‘s poorvang +**Yaman**‘s uttarang’)**Reversed**:**S-r-g-m-P-D-N-S**(=itself! i.e. ‘**Palindromic**‘)**Murchanas**: ‘**Unfilled #11**‘ (on ma)

**Equivalents**:**1-b2-b3-4-5-6-7-8**(=Carnatic**Kokilapriya**)

**Analysis**: Perhaps the most surprising ragascape omission – given its robust ‘**SmP**triangle’, neatly**palindromic**nature, and**A-list****poorvang**–**uttarang**combination of ‘**Bhairavi**–**Yaman**‘ (=’**all-komal**low,**all-shuddha**high’). Also relatable to other famous palindromic forms: either ‘**Bhairav**komal ga shuddha Dha’ (i.e. shifting Bhairav’s G-d ‘one step closer to Sa’, thus retaining the original symmetry), or ‘**Kafi**komal re shuddha ni’ (the same idea, but moving R-n closer to Sa instead). Oddly, I can’t even identify any ragas which are only one ‘**double-swara**‘ away (i.e. produced by ‘doubling’ any one of**#22**‘s**chal**swaras, e.g. ‘S-r-g…’ > ‘S-r**R**-g…’). Ripe for exploration!

**–Unfilled #24 (SrgmPdNS)–**

**Expressible:**‘**Bhairavi**shuddha Ni’, ‘**Bhairav**komal ga’, ‘**Todi**shuddha Ma’, ‘**Kirwani**komal re’ (or ‘**Bhairavi**‘s poorvang +**Todi**‘s uttarang’)**Reversed**:**S-r-G-m-P-D-N-S**(=**Asa Bhairav**)**Murchanas**: ‘**Unfilled #15**‘ (on ma)**Equivalents**:**1-b2-b3-4-5-b6-7-8**(=Carnatic**Dhenuka**)

**Analysis**: Once again easily summarizable in terms of**ultra-famous**ragas, with a**poorvang**–**uttarang**of ‘**Bhairavi**–**Todi**‘ (as well as being a single swara-shift from**Bhairav**and**Kirwani**) – and, along with**#22**, the only ‘unfilled’ scale to have a**shuddha**ma. Also expressible as ‘**Jogiya**add komal ga’, ‘**Viyogavarali**add Pa’, or ‘**Devata Bhairav**with komal ga only’.

**–General Observations–**

*From the above, several trends emerge…*

**7 of the 9 scales take a tivra Ma**: perhaps unsurprising, given the general strengths of the ‘**S-m-P**‘ sangati (=’a perfect 5th above and below Sa’), beloved across global music for its palindromic solidity. The presence of ‘tivra Ma but no Pa’ presents particular challenges when it comes to melodic resolution…

**Only one has a familiar ‘reversal’**(**#24**with**Asa Bhairav**, which isn’t particularly well-known: although, as noted,**#22**is neatly palindromic). And, due to the prevalence of the ‘**MP**’ pairing, all but the final two scales are non-sampurna when reversed: as ‘**MP**’ becomes ‘**mM**with**no Pa**’ (hence extending my related raga suggestions to include ‘komal Pa’ shifts: in particular, the ‘**mM-n**’ sangati is vanishingly rare in the wild…I can only find two such ‘double-Ma, komal ni only’ ragas:**Ahir Lalit**&**Jaunpuri Todi**).

**• Melakarta, Merukhand, & More •**

*A couple of proximate theoretical constructs…*

**–The Carnatic Melakarta–**

As discussed above, my ‘**sampurna dartboard**‘ draws heavily from the design of the South Indian** ‘melakarta’: the predominant classification system used in Carnatic ragam**. The melakarta taxonomises 72 different **7-note** scales in similar ‘decision tree’ fashion – separating them by interval variant to form a dense-yet-logical ‘wheel’ structure:

###### —Melakarta Wheel: 72 Janaka Scales—

**How does the melakarta differ from its Hindustani equivalents?** As you can see, the wheel above is far more intricate than my **dartboard**, spanning more scales and many other curiosities. Most notably:

**The melakarta has 72 scales (vs. our 32)**: This is due to the broader note-naming flexibilities of the South Indian system. In Hindustani music, each of the 12**specific swara**positions is occupied by only one ‘note-letter’ (i.e. ‘**SrRgGmMPdDnN**‘) – whereas in Carnatic ragam, 4 of these positions can take either of two alternate names (**shuddha****Re**: ‘Ri/Ga’,**komal****ga**: ‘Ru/Gi’,**shuddha****Dha**: ‘Di/Na’, &**komal****ni**: ‘Du/Ni’). In other words, the Carnatic Ga & Dha each offer three positions (‘Ga/Gi/Gu’ & ‘Da/Di/Du’), vs. the Hindustani two (‘ga/Ga’ & ‘dha/Dha’):

**The melakarta includes ‘non-melodic’ cultural associations**: While the dartboard is merely an ‘intervallic decision tree’, the Carnatic equivalent also features specific symbolic identifiers, in the form of 12*chakram*divisions (see the**red segment**above), each linked to a range of ancient cultural associations:

**Indu**(“the moon…of which we have only one”)**Netra**(“the eyes, of which we have two”)**Agni**(“the three forms of divine fire”)**Veda**(“the four Vedas [Rig-, Yajur-, Sama-, & Atharva-]”)**Bana**(“the five arrows of Manmatha, God of Love”)**Rutu**(“the six*grishma*[seasons] of the Hindu calendar”)**Rishi**(“the*saptarishi*[seven great sages]“)**Vasu**(“the*ashtavasu*[eight fire deities] of Hinduism”)**Brahma**(“the nine [chosen] forms of the deity”)**Disi**(“the ten directions [N/S/E/W, NE/NW/SE/SW, up/down]”)**Rudra**(“the eleven deities of divine storm winds”)**Aditya**(“the twelve avatars of Vishnu’s Vamana incarnation”)

Each of these segments contains exactly 6 ragam (72/12=6), which, due to the wheel’s particular interval hierarchy, differ from each other only in their placements of **Dha** and **Ni** (3*2 positions=6 options).

**Logical naming system**:**Actual ‘parent scales’**:

**–Merukhand: Unique Sequences–**

Another North Indian swara-sequencing technique, known as ‘merukhand’ (or ‘khandmeru’: literally ‘divisional analysis’) – involves **deriving all possible ‘unique sequences’ from a given swara-set**. This is calculated via the ‘factorial’ of the swara count (denoted, with pleasing spark, as ‘**n!**‘). Merukhand’s axioms are fairly similar to those of the **dartboard**, except with the goal of denoting how many sequences are possible in *any* order (vs. just ascending sampurna forms). For example:

**3 swaras**: 3! = 3*2*1 =**6**(**SRG**,**SGR**,**RSG**,**RGS**,**GSR**,**GRS**)

This, however, involves an exponential increase in complexity:

**3 swaras**: 3! =**6****4 swaras**: 4!=**24****5 swaras**: 5! =**120****6 swaras**: 6! =**720****7 swaras**: 7! =**5040****8 swaras**: 8! =**40,320****9 swaras**: 9! =**362,880****10 swaras**: 10! =**3,628,800****11 swaras**: 11! =**39,916,800****12 swaras**: 12! =**479,001,600**

Thus, practical use of the system is typically limited to the seven **sapta swara** (…more akin to the **dartboard**). As per Amir Khan, perhaps the most prominent merukhand aficionado: “My father made me practise the 5040 **taan** patterns which are possible through permutation [of] the 7 notes…for 22 years…Even [with] **akaar**, you should be clear which note you are singing…Due to this practice, I can do any number of variations…”).

Khan adds: “Later, I realised that **out of these 5040 patterns, only 168 are useful**“. This figure (exactly 30 times smaller) is primarily because many intervals between different swara-pairs are of identical pitch-distance (e.g. **S>P**, **r-d**, **M-N** are all 7-semitone leaps, despite occupying different **specific** positions). In his own explanation, “For knowledge of **swaras**, and for **riyaz**, practice of **aroha**–**avroh** is the first step. In our system, there are 360 **alankars** [phrase-shapes] and 5040 **paltas** [sequences]. To remember all of them is very difficult, if not impossible. **Therefore, I have prepared 168 swara mailas [‘combinations’]**“.

Learn more about the Ustad’s thinking in the interview below (turn subtitles on), and go deeper into how he applied these ideas to raga via Ibrahim Ali’s excellent *Amirkhani Khayal*: e.g. “To maintain the importance of **nyas**…Khan only considered those permutations in which no harm was done to the raga if a pause was laid on the final note. In **Bageshri**, he applied **Pa** in both **aroha** and **avroh**, but didn’t pause on it. Similarly, maintaining the role of **komal** **re** in **Ahir Bhairav** when improvising with ‘**SrGm**’, using permutations of ‘**SGmr, SmGr, SrGm, SGrm, GmrS’…these phrases were applied very emotionally during alap**“.

**Merukhand Interview – Amir Khan (~1970s: subtitled):**

*“There is something called ‘ khandmeru’: it has 5040 taans, starting [SRGMPDNS] and ending [SNDPMGRS], with no repetitions. In my childhood, I knew them all by heart. This gave me a lot of help: now, I can go wherever I want to. But the entire set of 5040 is too much to remember. So I created a ‘simplified subset’ of 168 taans, chosen for their practical uses (I even gave them to Vilayat Khan, whether he agrees I don’t know!). They particularly help in vilambit, as I can apply the same sur in multiple ways with these ‘stock variations’, giving scope and variety…” (Amir Khan)*

**–Further Learnings–**

*Onward links for my fellow swara-sequencing explorers…*

**Vishnu Narayan Bhatkhande (1860-1936)**: In my reckoning, the ‘Chatur-pandit’ exerted more influence on actual performance practice than virtually any other global music theorist of the past couple of centuries. Learn about his life in the video below, as well as via Ramesh Gangoli, and also Sriram V in*The Hindu*(“He qualified in law, and set up practice at the High Court of Bombay….[later] he began to ponder over the fact that Hindustani music did not have a structured curriculum…He travelled far and wide across North India, collecting information about the way music was taught in the various**gharanas**. He then moved South, coming to Madras in 1904…).

**Contributions of V.N. Bhatkhande – CNC Sanskriti (2017):**

*“[Bhatkhande] lost both his wife and daughter after short illnesses…having a profound effect on him. He never married again, never gave further ‘hostages to fortune’, and devoted an increasing proportion of his mental life to musical contemplation. True to his promise to his father, he never sought to be a performer…His bent was more for acute observation, analysis, and synthesis. This, naturally, led him to musicology…” (Ramesh Gangoli)*

**Header images**: My haphazard geometric renderings of the 32 scales (*InDesign/Photoscape X*)