Inversions of Melakatas

The Carnatic melakatas form a system of 72 seven-note scales. What happens if we play them upside down?

By playing a scale upside down, I mean inverting it -- reversing its interval map. For example, the major scale's intervals are t, t, s, t, t, t, s. If we run those backwards we get s, t, t, t, s, t, t -- the Phrygian scale. So the major scale's inverse is a mode of the major scale. On the other hand, if we invert Harmonic Minor, which is t, s, t, t ,s, mT, s (where mT = minor third) we get s, mT, s, t, t, s, t, which is the melakata called Chakravakam, a mode of Harmonic Major. So sometimes a scale inverts to a mode of itself and sometimes it inverts to something new.

While a scale and its inversion obviously sound different, they have a lot of similarities. If the modes of a scale all live together in the same house, the modes of the scale's inversion live next door. Pitch class set theorists often don't even make a distinction between the two, although that's a bit extreme outside perhaps the most rigorous serial music. If you know a scale and its modes and you're looking for a new sound, but not too new, why not visit the neighbours?

The 72 melakatas have many structural features, which Indian music theorists have understood very well for a long time. This image, from Wikipedia, shows some of these (I won't explain it all, but it will be useful for reference):

The most important feature here for us is the division between suddha melakatas, which occupy the right-hand half from 12 noon to 6 o'clock, and the prati melakatas that occupy the left half from 6 o'clock to 12. A scale is suddha if it contains a natural 4 and prati if it contains a #4.

No melakata ever contains an altered fifth. When we invert a scale, the perfect 4 inverts to the perfect fifth, so if we start with something on the suddha side we should get another melakata. What's more, the perfect fifth, which is always present in the starting scale, inverts to a perfect fourth, so the inverse of a suddha melakata is another suddha melakata.

On the left-hand side, though, things are very different. The #4 inverts to a b5 (the same note) while the 5 inverts to a 4, so the prati melakatas always invert to scales with no fifth, that are therefore outside the melakata system. For this reason I suspect these scales haven't been investigated in depth by Indian music scholars, although I have no doubt they've been discovered and experimented with in the past.

Personally I don't make a strong distinction between scales that are modes of each other; Indian theorists, by the way, are very familiar with modes which they describe by a process called graha bhedam. But they (like most Western theorists) consider two scales to be different even if they're modes of each other. Like most people working in non-tonal music, though, I don't care as much about the difference.

It turns out that most of the prati melakatas invert to scales that are modes of other melakatas. I don't consider that very interesting (someone working in a slightly different way might well disagree). The ones that interest me are the ones that invert to something that's far enough outside the melakata system that you can't get back inside just by going to a mode.

Here's a quick list:

• Navanitam inverts to [0, 2, 3, 5, 6, 10, 11]
• Pavani inverts to [0, 1, 3, 5, 6, 10, 11]
• Raghupriya inverts to [0, 1, 2, 5, 6, 10, 11]
• Sucharitra inverts [0, 3, 4, 5, 6, 8, 9]

These are all very exotic! Not only are they unknown to European or Indian musicians, I suspect they're not recognized by any musical tradition. Yet they're inversions of melakatas, which means they live next door to scales that are in regular use.

Because none of these scales has a traditional name and I like giving things silly names, I'm going to call them "sinister" scales -- they're like the evil twins of the originals, their reflections in an enchanted mirror. I appreciate the word "sinister" has some, well, sinister history behind it but I can't think of a better one offhand (no pun intended).

Navanitam Sinister

This scale is 1 2 b3 4 b5 #6 7 or, in C, C D Eb F Gb A# B. So it's kind of a melodic minor scale that's having a weird day. You could think of it as Forte 4-3 at the 7 (B-C-D-Eb) with a Forte 4-8 joined to the bottom of it (F-Gb-A#-B) -- we looked at 4-3 recently and will probably have to do with same with 4-8 and their other friend, 4-7 at some point.

Here it is in the key of A, because I haven't worked out how to change that on Guitar Scientist yet:

This mode has its root on the top note of that sequence of semitones: b7-7-1. The mode that sits at the bottom of that sequence of semitones is very like Harmonic Major, but this time the natural 9 and flat 9 combine to make it all feel very odd indeed.

Pavani Sinister

This is the "phrygian" version of the previous scale -- it's the same but with a b2. I call this scale group Apeliotean. Here it is in A, for the same reason as before:

You can grab the mode built on the tritone (in C, that would be Gb) and get a very strange mixture of Lydian and Ionian -- you can certainly pull some jazz lines out of it if you don't mind sounding like a mollusc trying to play bebop. Or look at the one that starts off like a a nice, normal minor scale with C-D-Eb but then, like a drunk telling an anecdote in the pub, lurches into a chromatic puddle with G-Ab-A-Bb.

Raghupriya Sinister

This one is a monster -- it's a string of semitones interrupted by a jump. One of its modes is C Db D Eb E G Ab. This contains all of the C minor, C major and C augmented triads, combined with both the flat and natural ninth. It's very similar to a mode of Raghupriya, but they're not the same.

I'm not sure there's much I can tell you about this that isn't obvious from the diagram -- if you have a use for it, you probably already know. Since it contains a pair of minor triads (m9 arpeggios, in fact) separated by a semitone, it's related to the Lydian Minor system I described recently, but I'm not sure that helps much.

Sucharitra Sinister

The most characteristic mode here is probably 1 b3 b5 5 b6 6 7 -- or C Eb Gb G Ab A B. I've spelled it that way to emphasise how it looks to me: it's a diminished seventh with the natural 5, b13 and 7 added. The added notes can be thought of as a mM7 built on the b6 if that helps (this arpeggio overlaps with the diminished 7 at the root to give the whole scale):

That b6mM7 can also be played as a straight up b6M7 and that probably holds the most promising sounds here if you're looking for things that combine with tonal harmony. I think there may be some interesting melodic possibilities in this scale but they're certainly not straightforward to find.