I used to go to record stores–back when there still were record stores–with the sense that my life could change depending what I stumbled across.  One such moment was discovering cantu a tenore, a style of vocal music from Sardinia, which I first heard at a listening station in Tower Records in 1996: the album was S’amore ‘e Mama by Tenores di Bitti.  Of course, it wasn’t so much the particulars of my life that changed in hearing this (or any other powerful recording), as it was my ear, and my sense of possibility.  The guttural sonorities of cantu a tenore were unlike anything I had experienced in music before, although they reminded me of sounds I had heard in the natural world, in particular the bleating of sheep and the lowing of cows.  Sardinian singers evoke these ambient animal sounds in a way that is startling and beautiful.  I call it “startling” because, to a mainstream listener (I’ll punt on trying to define “mainstream” here) the sounds of livestock probably occupy a separate mental category from “music,” and if one is familiar with the loose concept of bel canto singing (an operatic style favoring smoothness or legato, lightness, and agility) the bleating of sheep could be seen as its diametrical opposite.  How interesting to consider that Italy, which gave us bel canto, also gave us cantu a tenore.  I call it “beautiful” not only because of the visceral appeal of its sonic “mass,” but also because the voices sustain a sort of rhythmic, melodic, and timbral contrast that gives me the same rush I get from more formal contrapuntal music: at times it’s like hearing a bit of Bach somehow waft up from the grounds of a farm.

After ’96 I would always check for the “Sardinia” section whenever I stopped in a record store, and if I found such a section, it was usually stocked with the recording I already owned.  Through the sort of physical-world-sleuthing that was necessary in those days, I did gradually expand my collection to five or six albums, but at some point I had to concede that there just wasn’t much to do as an American fan of Sardinian polyphony: no new releases, no local performances, and practically no one in my circle of acquaintance who had heard of it.  So I stopped checking for “news.”  But just the other day some fortuitous web browsing led me the best Sardinian-polyphony news I’ve come across in a decade.   There’s a group of American singers, Tenores de Aterúe, who have taken the adventurous step of studying and performing traditional cantu a tenore, and their footage on YouTube sounds great.  (Where’s your website, guys?)  In reading a little about their background, I enjoyed the story of a 2008 ephiphany, when one member who had dreamed of learning Sardinian music found a video of Tenores di Bitti demonstrating how each voice part in cantu a tenore sounds by itself, and then in ensemble.  This video became a sort of Rosetta stone that exposed the workings of the music and opened the path for a group of experienced but non-Sardinian musicians to actually learn it.  I encourage you check out Tenores de Aterúe on YouTube and see if they might be performing near you (I’ll be doing the same).  Here, I wanted to post that revealing Tenores di Bitti video, as it’s probably the best short introduction to cantu a tenore one can find (replete with charming features like the onlooker who appears in the background near 3:50, and of course, some great singing).
When I sent it to my classical voice teacher, she emailed back: “This just blew my mind.”  Specifically, she was amazed at how these musicians create — in a way that appears relaxed and free from vocal strain — a completely different set of sounds from those we work to produce in “mainstream” Western vocal practice.

Prasanna mentioned to me that there are 34,776 theoretically possible janya ragas in the South Indian melakarta system and he asked if I could see how that number arises. I looked into the question and thought I’d post the details here for anyone who’s interested.

Recall that a janya raga is one that is somehow derived from one of the 72 melakarta ragas. The 34,776 figure refers to one specific class of janya ragas: those that are created by omitting notes from the parent. In forming such “varja” ragas, we can omit up to two notes from the arohanam (ascent), the avarohanam (descent), or both. (And it’s permissible to omit different notes in the ascent from those we omit in the descent.) We can’t apply other processes of derivation like reordering notes or borrowing notes from other ragas: these processes lead to many more possibilities!

So where does the number 34,776 come from?  Well, if we decide to omit one note from the arohanam of a melakarta raga, there are 6 ways to do it: any note besides sa is fair game for omission. If we’re going to omit two notes, there are (6 choose 2) = 15 possibilities. And of course, if we omit no notes, there’s only 1 way to do that. This gives 6+15+1=22 options. The same 22 options exist for the avarohanam, giving 22*22=484 ways of omitting notes from a parent raga to create a janya. Except, we don’t want to count the case where no notes are omitted in the arohanam and the avarohaman both, because this leaves us with the original raga. So, the total number of janya possibilities for each melakarta raga is actually 484-1 = 483. Multiplying this by the total number of melakarta ragas, we get 483*72 = 34,776. But there’s a catch…

The process of omitting notes from two distinct parent ragas can give us two janya ragas with the same notes. For example, we can get S R2 G3 P D2 (Mohanam), by omitting M1 and N2 from S R2 G3 M1 P D2 N2 (Harikambhoji), or by omitting M2 and N3 from S R2 G3 M2 P D2 N3 (Kalyani). In Western terms, one would say you can get 1 2 3 5 6 (major pentatonic) by omitting 4 and b7 from 1 2 3 4 5 6 b7 (Mixolydian) or by omitting #4 and 7 from 1 2 3 #4 5 6 7 (Lydian). So, the 34,776 figure contains many janya ragas that actually have identical swaras. Now, if the parent raga (including its mood, characteristic phrases, ornamentation patterns, and additional swaras) is kept in mind when performing the derived raga, then two ragas derived from different parents might be perceived as distinct even though the derived ragas happen to have the same notes. In this way of looking at raga derviation, 34,776 is a plausible theoretical count. However, if we’re only interested in janya possibilities that are distinct in their notes, and we’re not considering attachments to parent ragas, then 34,776 is an overcount.

Enumerating the janya possibilities that are note-wise distinct is more complicated. The rest of the post describes an approach I came up with when I first started thinking about the problem.  After I published the post, I received a comment from Narayana Santhanam pointing out that the technique of generating functions from enumerative combinatorics is another, more compact approach to counting janyas; and, in searching further, I found a 2002 paper by K. Balasubramanian that uses this approach.  Finally, I learned that P. Sriram and V. N. Jambunathan had investigated this question and published similar findings in a 1991 paper in the Journal of the Madras Music Academy.  (See the notes and comments at the end of this post for more details.)  Although the approach I present here is more verbose than some of the others, I hope it will be interesting to readers looking for insight into how all the possibilities arise.

In what follows, I assume R2=G1, R3=G2, D2=N1, and D3=N2 even though one could say those swaras would be performed with different ornamentation and/or intonation.  In this count, for example, an arohanam of S R2 M1 P D2 S will be considered equivalent to S G1 M1 P D2 as since we treat R2=G1.

To organize our count, we’ll consider how many notes occur in the union of the ascent and the descent of a janya raga. This union might have 5, 6 or 7 notes.

If the union contains 5 notes, then the ascent and descent must contain those same 5 notes. To find the number of janya ragas of this kind, we need to consider all the ways of choosing 4 notes from 11 possibilities without violating the melakarta rules. Note that we’re choosing 4 notes out of 11, not 5 out of 12, because one note, sa, is mandated. Unfortunately, we can’t use a simple formula for (11 choose 4) because this would count “illegal” cases where there are two ma’s, or where there are more than two notes betwen sa and ma.  One way to count only the legal possibilities is to divide the scale above sa into four sections. The first section contains ri and ga. The second section contains ma. The third section contains pa. The fourth section contains da and ni. Now there are 6 ways of filling the first section (i.e. ri and ga can be R1/G1, R1/G2, R1/G3, R2/G2, R2/G3, or R3/G3), two ways of filling the second section (i.e. ma can be M1 or M2), one way of filling third section (pa is always Pa), and six ways of filling the fourth section.  (This is how we get 6*2*1*6 = 72 melakarta ragas.) If we omit some notes, then one or more of the scale sections will no longer be full. (In particular, if the first section has only one note in it, there are 4 possibilities for its identity: R1, R2=G1, R3=G2, or G3.) If we write 2|1|1|2 to indicate a scale with all sections full, then here are the possibilities distributions of sections sizes for a five-note derived scale where the sections are not all full: 2|1|1|0, 2|0|1|1, 2|0|0|2, 2|1|0|1, 1|1|1|1, 1|0|1|2, 1|1|0|2, 0|1|1|2. Taking this list and substituting the sections sizes with the number of ways of filling each section to the designated size, we get the following count: 6*2*1*1 + 6*1*1*4 + 6*1*1*6 + 6*2*1*4 + 4*2*1*4 + 4*1*1*6 + 4*2*1*6 + 1*2*1*6 = 236.

Now if the union of the ascent and descent contains 6 notes, the respective sizes of the ascent/descent might be 6/6, 5/6, 6/5, or 5/5. The 6/6 case is similar to above: we know the ascent and descent use the same six notes, and we need to consider the number of ways of choosing 5 out of 11 notes without violating the melakarta rules. Following the logic above, the possible distributions of section sizes are 1|1|1|2, 2|0|1|2, 2|1|0|2, and 2|1|1|1. This gives 4*2*1*6 + 6*1*1*6 + 6*2*1*6 + 6*2*1*4 = 204 possibilities. In the 5/6 or 6/5 case, we can see that side containing 5 notes must be a proper subset of the side containing 6. The number of possibilities here can be calculated as (# ways of choosing 5 out of 11 notes legally)*(# ways of removing one of those 5 notes to create the smaller side) = 204*5 = 1020. Since we can make the smaller side the ascent or the descent, we multiply the last figure by two, giving 2040. Finally, in the 5/5 case, we can see that the ascent and descent must share precisely 4 notes including sa (each side contains a note that’s not in the other, creating a union of size 6). The number of possibilities can be calculated as (# ways of choosing 5 out of 11 notes legally, to create the 6-note union including Sa)*(# ways of picking 3 notes from those 5 to be shared by the ascent and descent)*(# ways of picking one of the remaining 2 notes to be in the ascent, while the other goes in the descent) = 204*(5 choose 3)*2 = 4080.

The last situation is where the union of the ascent and descent contains 7 notes. Here, the respective sizes of the ascent/descent might be 5/5, 6/5, 5/6, 6/6, 5/7, 7/5, 6/7, or 7/6.  In the 5/5 case, the intersection must contain 3 notes. The number of possibilities is: (# of ways of choosing 6 out of 11 notes following the melakarta rules, to create a 7-note parent raga including Sa)*(# ways of choosing 2 out of those 6, to create a 3-note intersection including Sa)*(# ways of picking two from outside the intersection to be in the ascent, while the other 2 go in the descent) = 72*(6 choose 2)*(4 choose 2) = 72*15*6 = 6480. In the 5/6 or 6/5 case, the intersection is size 4, and we can choose 3 out of six notes to form it. By similar logic to the previous case, the number of possibilities is 72*(6 choose 3)*(3 choose 2) = 72*20*3 = 4320 for each case by itself.  In the 6/6 case the intersection must be size 5, and the number of possibilities is 72*(6 choose 4)*2 = 72*15*2 = 2160. In the 5/7 and 7/5 cases, the smaller side of the scale must be a proper subset of the larger one, and we can form the smaller side by omitting two out of six notes from the larger. This gives 72*(6 choose 2) = 1080 possibilities for each case. And in the 6/7 and 7/6 cases, the smaller side again must be a proper subset of the larger one, so the number of possibilities is 72*(6 choose 1) = 432 for each case.

Here’s a recap of the cases we considered and the possibilities that exist in each case:

Union size 5.  Case 5/5: 236 possibilities.

Union size 6.  Case 6/6: 204 possibilities. Cases 5/6 and 6/5: 2040 possibilities together. Case 5/5: 4080 possibilities.

Union size 7.  Case 5/5: 6480 possibilities. Cases 5/6 and 6/5: 8640 possibilities together. Case 6/6: 2160 possibilities. Cases 5/7 and 7/5: 2160 possibilities together. Cases 6/7 and 7/6: 864 possibilities together.

The grand total is 236+204+2040+4080+6480+8640+2160+2160+864= 26864.

Now, how can we be sure that 26864 correct? In fact, I made many calculation errors as I was working through this the first time, leaving me in a state of great suspicion.  However, after I spotted and corrected my mistakes, I gained further confidence by writing a Groovy script that generates all distinct janya possibilities by brute force, and also yields… 26864!  (Warning: this is script-style code, meant for one-time use and not designed to be maintainable or easy to read, but it gets the job done.)

// this script is written in Groovy 2.1.0

// generate all melakarta ragas, representing each
// raga as a list of 11 ones and zeros
melakartas = []
for (riGa in [1,1,0,0].permutations()) {
    for (ma in [0,1].permutations()) {
        for (daNi in [1,1,0,0].permutations()) {
            melakartas.add(riGa + ma + [1] + daNi)
        }
    }
}
assert melakartas.size()==72

// generate all ways of deleting up to two notes from
// the ascent and descent of a raga
deletions = ([0..5,0..5].combinations()).collect( {it as Set} ) as Set
deletions.add([])
deletionPairs = [deletions,deletions].combinations()
deletionPairs.remove([[],[]])

// generate all possible janyas as ascending/descending scale pairs,
// by applying all possible deletions to each melakarta raga;
// keep the janyas in a set so that identical janyas will
// not be double-counted
janyas = [] as Set
for (mela in melakartas) {
    def noteIndices = (0..(mela.size()-1)).findAll({mela[it]==1})
    for (deletionPair in deletionPairs) {
        def ascending = mela.clone()
        def descending = mela.clone()
        deletionPair[0].each { ascending[noteIndices[it]]=0 }
        deletionPair[1].each { descending[noteIndices[it]]=0 }
        janyas.add([ascending, descending])
    }
}

println "Number of Distinct Janyas: ${janyas.size()}"

Continue reading →

This clip is the outline or “seed” of a composition I’m working on.  The seed is nothing but a sequence of ascending and descending scales over a fixed bass, arranged in a cycle that returns to the starting point. I arrived at it through a fairly theoretical process (see my other posts with geometric diagrams of the octave to get a sense for the kind of stuff I’ve been up to). But for me, it’s one of those cases where an essentially mathematical idea, translated into sound, produces something that’s eerily compelling. For my ear this pattern is prism-like: it seems to unlock a whole spectrum of colors and moods, and at times it has held me in a trance. I’m posting it here for two reasons: first, even though it’s just a bare pattern, I hope you might find it interesting to listen to as a sort of musical meditation. Second, I’ve found that in many of my creative projects, when I get excited about an idea, I form great ambitions around it and keep it to myself until I feel I’ve done it justice. This has led to many cases in my life where I’ve poured effort and passion into a project but never released it because my goals were too big for me. So I’m trying a different approach, which is to share things early, especially when I come upon something that strikes me as a gem–something that I might otherwise get lost in polishing endlessly. So here is “Charupriya Cycle,” played on my steel string guitar a few hours after the idea came together the other day:

In this post I’d like to review some interesting properties of the familiar major and minor pentatonic scales, and then consider how to create a pentatonic version of the melodic minor scale.

Major Pentatonic as a Scale Complement

One of the neat the things about the standard major pentatonic scale, which uses degrees {1, 2, 3, 5, 6}, is that it can be seen as the complement of a full 7-note major scale: it contains all the notes that some major scale excludes.  For example, the black keys on the piano — all the notes that are not in C major — form Gb major pentatonic.  In general, if you take all the notes that a major scale excludes, you’ll have the major pentatonic scale rooted at its tritone.  We can visualize this with a clock-style diagram of the octave.  Here, the numbers are not hours but degrees of the major scale, and the red dots represent notes outside the scale.  In the leftmost diagram, the polygon formed by connecting the red dots represents the pentatonic scale rooted at #4 of the parent scale.  Transposing the scale up a semitone (rotating the polygon 30 degrees clockwise) gives the major pentatonic rooted at 5 of the parent scale.  Transposing it down a semitone gives the major pentatonic rooted at 4 of the parent scale.  And transposing it by a tritone (rotating it 180 degrees) gives the pentatonic rooted at 1 of the parent scale.  Notice how the pentatonics rooted at 1, 4, and 5 exist entirely within the parent scale, which is to say that none of the polygon vertices fall on any of the red dots.  The green circle indicates the root of the pentatonic scale, and the dotted line inside the polygon is its axis of symmetry.

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Major Vs. Minor Pentatonic: Comparing Definitions

If the major pentatonic is built from degrees {1, 2, 3, 5, 6} it’s natural to wonder why we define the minor pentatonic as {1, b3, 4, 5, b7}.  Why not simply flatten 3 and 6 in the major pentatonic formula so that it fits inside a minor scale: {1, 2, b3, 5, b6}?  I don’t know the history of how the major and minor pentatonics evolved in various cultures over time, but I will comment on some abstract reasons why {1, b3, 4, 5, b7} might be more versatile as a scale than {1, 2, b3, 5, b6}.  First of all, it’s convenient to define our minor pentatonic so that it turns out as a mode of our major pentatonic.  This lets us shift from major to relative minor while staying in the same pool of notes.  Indeed C-major pentatonic (C D E G A) and the standard A-minor pentatonic (A C D E G) are modes of each other, or reorderings of the same set of notes.  But if we were to use the formula {1, 2, b3, 5, b6} to get an alternative A-minor pentatonic, it would be A B C E F, which doesn’t fully overlap with C-major pentatonic.  Another notable thing about the standard major and minor pentatonics are that they have no semitones, no tritones, and lots of perfect fourths and fifths, making them very “user-friendly” scales.  The same is not true of {1, 2, b3, 5, b6}, which has two semitones (2-b3 and 5-b6), one tritone (2-b6), and fewer perfect fourths and fifths.  This alternate scale might still be great to make music with, but it lacks the many of the convenient properties of the standard minor pentatonic.

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Melodic Minor Pentatonic

With these observations in mind, let’s look at the question of how to create a pentatonic scale from the 7-note jazz melodic minor scale.  In sacrificing two notes, we want to preserve as much of the character of melodic minor as possible.  So, which two notes can we afford to lose?  We should probably keep 1 and 5!  To have a minor sound, we need to keep b3.  To have something different from a plain old minor scale, we need to keep at least one of 6 or 7.  Now, as far as I know, there isn’t one universally accepted way to make the remaining choices; that’s to say, the idea of a melodic minor pentatonic hasn’t been standardized in the same way that the major and natural minor pentatonics have been.  I’ve seen melodic minor pentatonic defined alternatively as {1, 2, b3, 5, 6}, as {1, b3, 4, 5, 7}, and as {1, b3, 4, 5, 6}.  What I’d like to do here is explore some interesting qualities of that last definition, {1, b3, 4, 5, 6}.  First of all, it’s the only one of the lot that has no semitones.  In fact it’s a member of one of only three possible pentatonic families with that property (see my post on anhemitonic pentatonics).  Second, it turns out to be a complement of the 7-note melodic minor scale, in the same way that the major pentatonic is a complement of the 7-note major scale!  And third, you can actually transpose the whole thing up a whole-step while remaining inside the 7-note melodic minor parent scale.  To experiment with this, start by playing C-melodic-minor: C D Eb F G A B. Now make up some phrases using {1, b3, 4, 5, 6} or C Eb F G A.  Now transpose your phrases up a whole-step, so you’re using the same {1, b3, 4, 5, 6} pattern rooted at D.  This will give you the notes D F G A B.  The nice thing is that after this whole-step transposition, you’re still playing inside the parent scale of C-melodic minor.  Now try doing that with the other versions of the melodic minor pentatonic and let me know how it works out.

These diagrams show how the {1, b3, 4, 5, 6} pentatonic pattern originates as the complement of a parent melodic minor scale.  If we take the excluded notes of the parent scale (i.e. the red dots) and transpose them down a semitone, we get our {1, b3, 4, 5, 6} pentatonic rooted at 1 of the parent scale.  If we transpose the excluded notes up a semitone, we get the same pentatonic pattern rooted at 2, and still fitting entirely inside the parent.

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Mix and Match

Notice that {1, b3, 4, 5, 6} is also a subset of the Dorian mode of the major scale, so we could call it a “Dorian pentatonic” as well as a “melodic minor pentatonic.”  There is a difference in how it functions in those two context though.  If you’re thinking of 7-note Dorian as your parent scale, you can only form this pentatonic on the Dorian root: any attempt to transpose it will put you outside the Dorian note set.  However, if you’re thinking of 7-note melodic minor as the parent, then as we just saw, you can transpose this pentatonic between 1 and 2.  It’s also worth noting that the standard minor pentatonic fits inside the full melodic minor scale — you can play it on 2 — but you can’t transpose it anywhere else.  For example, if C-melodic-minor is your parent scale, you can play D minor pentatonic (D F G A C) and remain inside the parent.

In the diagram on the left below, you’ll see how the melodic minor pentatonic (as we’ve defined it) fits inside a major scale.  Here the pentatonic root is at position 2 of the major scale (or at 1 of the major scale’s Dorian mode).  In the diagram on the right, you’ll see how the standard minor pentatonic pattern fits inside a melodic minor scale.  Again, the pentatonic root is positioned at 2.  If we think of the same shape as representing a major pentatonic, its root would positioned at 4 of the enclosing scale.