How Many Arpeggios Are There? Really?

How Many Arpeggios Are There? Really?


My colleague and friend, Samantha Coates, was wondering aloud on Facebook the other day just how many arpeggios there were. Not just 12 major and 12 minor arpeggios, but what if you counted all the inversions and included dominant 7ths and diminished 7ths, and all the permutations of arpeggios there might be?

Sam’s original count reached 132, and she then experimented with how long it would take to play them all: 11 minutes and 30 seconds. But then the concept of articulations was raised, and then the idea of major 7ths, minor 7ths and minor-major 7ths… At which point I began to wonder if we weren’t already well over the 1000 mark in terms of all the different arpeggios there might be.

So I broke it down, and this is how it went:

First up, we’re talking 12 keys X 3 inversions (root, 1st, 2nd) X 2 qualities (major/minor – I’m leaving out augmented and diminished because they get covered in the 7th chords below) in similar motion. That’s 72.

If we practice them contrary motion we’ve immediately doubled them. 144.

If we practice them hands a 5th/6th apart we all of a sudden get another 72. So that’s 216.

And if we practice them contrary motion, starting a 3rd/4th apart we get another 72, so that’s 288.

Then we do 12 keys X 4 inversions (root, 1st, 2nd, 3rd) X 7 qualities (diminished 7th, dominant 7th, major 7th, minor-major 7th, minor 7th, half diminished, augmented 7th) X 2 motions (similar/contrary) and that adds another 672. So we’re now up to 970.

If we practice each of these 7th chord arpeggios (there are 84 of them) in each inversion (4 of them) a 10th (or 9th/11th, as the case may be ) apart, then we get an additional 336. So, 1306 total so far.

But wait. Of course: diminished arpeggios tessellate. So, in fact we need to remove from the count all the repetitions of tessellating arpeggios. That’s ALL the inversions other than root of the diminished 7ths, so we need to subtract 72.


Cool number.

I’d love to leave it there, but what if we then practiced the 7th chord arpeggios in contrary motion starting a 3rd/2nd apart? There’s another 336 *minus* our tessellating dim 7ths (36), so that’s 1534.

Of course, this is assuming that both hands are using the same articulation.

If we perform the arpeggios with just four basic articulation variants: both hands legato, both hands staccato, one hand legato the other staccato, then swap which hand is which, then we end up with a total of 6136 different arpeggios to practice.

And it’s perfectly appropriate to want more complicated articulation differentiations than this: both hands in two-note slurs, for example, or both hands in three-note slurs, or in a two-note slur followed by two staccato notes, or two-note slurs in one hand, staccato or legato in the other… Each new variant, if applied to all the arpeggios, sees us adding 1534 to our total. And the same articulation can be performed with different underlying metrical shapings, so that needs to be taken into account also. Let’s go with these options, without worrying about metrical shapings other than 4 note groupings:

  • legato
  • staccato
  • RH legato/LH staccato
  • RH staccato/LH legato
  • two-note slurs
    • as above, but the two-note slur occuring on the off beat
  • RH two-note slurs/LH legato
    • off-beat version
  • RH two-note slurs/LH staccato
    • off-beat version
  • RH legato/LH two-note slurs
    • off-beat version
  • RH staccato/LH two-note slurs
    • off-beat version
  • three-note slurs
  • RH three-note slurs/LH legato
  • RH three-note slurs/LH staccato
  • RH three-note slurs/LH two-note slurs
  • RH legato/LH three-note slurs
  • RH staccato/LH three-note slurs
  • RH two-note slurs/LH three-note slurs
  • two-note slur + two staccato notes (2+2, for short)
    • displace the pattern by one note (the two-note slur *ends* on the first note of the arpeggio)
    • displace by two notes (the arpeggio begins with the two staccato notes)
    • displace by three notes (the arpeggio begins with one staccato note followed by the two-note slur)
  • RH 2+2/LH legato
    • + 3 displacements
  • RH 2+2/LH staccato
    • + 3 displacements
  • RH 2+2/LH two-note slurs
    • + 3 displacements
  • RH legato/LH 2+2
    • + 3 displacements
  • RH staccato/LH 2+2
    • + 3 displacements
  • RH two-note slurs/LH 2+2
    • + 3 displacements
  • three-note slur + 1 staccato note (3+1 for short)
    • all variants as for 2+2 articulation above (total: 24)

So that’s 73 articulation variants times 1534 kinds of arpeggios. 111,982.

And that’s before we add in rhythmic variation. Let’s conservatively estimate we have 4 of these variants we’d like students to master: LH 2 against RH 3; LH 3 against RH 2 – hands start 2 octaves apart; dotted rhythm one hand, straight rhythm the other. And let’s only apply this to the first 13 articulations listed above. That’s 4 rhythms X 13 articulations X 1534 arpeggios. 79768. Plus the 111,982.


ANY variation of dynamic contour doubles this number. So, say we practice

  • piano,
  • forte,
  • RH piano/LH forte
  • RH piano/LH forte
  • with a crescendo and diminuendo,
  • with a diminuendo and a crescendo


Now we reach the best calculation of them all: if it takes Sam 11 minutes and 30 seconds to play through 132 arpeggios, anyone want to figure out how long it will take to play through this million-odd?

69 days, 14 hours and 33 minutes. Without a break.

Putting it another way: 41 and three quarters of 40-hour working weeks. If Sam started tomorrow she’d be knocking off at lunchtime on December 21.

You’re all very welcome.

P.S. Samantha Coates is the brains behind the very wonderful ScaleBlitzer app. I promise it won’t make you practice arpeggios from now til Christmas. :)

Major Harmonic Revisited

Major Harmonic Revisited

The last scale-of-the-day I blogged about (back on February 20) was the Major-Harmonic scale, and when I wrote my post about this particular pattern I found myself with little good to say about it (much to my own surprise). I complained about the clichéd cadence that this pattern allowed, and surmised that it may well have been the first scale to which I was impelled to give a thumbs down.

This negative assessment was no doubt impacted on quite considerably by the fact that that weekend I was supposed to get my first 8 hour sleep since 2006 (pregnancy, newborn, toddler who doesn’t sleep through) and thanks to noisy hotel neighbours it just didn’t happen.

But I think maybe more germane to my disdainful summary was that I was only thinking about this pattern in its C incarnation. This is an important point, because I know full well that the physical sensation of any pattern changes from one semitone to the next, and these physical changes impact on one’s imaginative interaction with that pattern.

On the weekend, in the throes of giving a presentation about P Plate Piano, I realised that one of the pieces I’ve composed for that series, “Hickory Dickory” uses this major-harmonic pattern (no other scale pattern fits), and “Hickory Dickory” is far from the hackneyed composition my assessment of the scale would suggest.  But then, it’s not on C.  It’s on F sharp.

Here’s the difference.

When we think of the Major-Harmonic scale on C it’s just the usual sequence of white notes interrupted just once, and by A flat. Once you have that particular geographical scene firmly pictured (along with the chord possibilities that immediately spring to mind) now move to the next image: the same pattern on F sharp.

The group of three black notes, followed by the start of the B minor scale (B C sharp D) and the raised 7th, E sharp, so that the pattern is black-black-black-white-black-white-white-black, with the gap between the two consecutive white notes being an augmented 2nd. And an augmented 2nd created by two white notes is always going to register somewhere in your pianistic mind as being just another way of spelling a minor 3rd (even though you definitely experience the interval in the scale pattern as a 2nd with an augmented quality).

“Hickory Dickory” mobiles these black note/white note patterns thus: the left hand only ever plays the three black note group, and the right hand only ever plays the three white notes of this major harmonic pattern (B, D and E sharp).  But of course, this is educational piano music for students in their first year or so of tuition, so the white notes are notated B D F, and in the context of this piece (with no 5th degree of the scale present) it definitely creates the impression of a diminished chord.

Here’s the first line of this piece:

Most of the piece is spent exploring this one run (across the full length of the piano keyboard), and beginner students find the piece both easy to learn and engaging to practice and perform.

The lesson for me here (besides being warned off reducing every pattern to an on C version) is that when working with new patterns don’t be constrained by the diatonic idea of using triads that skip every second note!  The major-harmonic pattern produces this wonderful, shimmering contrast between major melody and diminished chord, and if I hadn’t been fiddling about on the black notes I would have missed it altogether.

Scale of the Day: Major-Harmonic

Scale of the Day: Major-Harmonic

This week let’s look at what I regard as being a kind of reverse of the mystery scale from the previous scale-of-the-day post.   Just as in our ‘mystery scale’ this is a major scale with a change made to only one note, but whereas last time we raised the 5th, this time we are lowering the 6th: same pitch, different degree, and very different end result.

Here it is on C:

It’s called Major-Harmonic in a fairly obvious way, the tail of the harmonic has been attached to the head and torso of the major pattern, and here’s our hybrid.

Being, to our diatonic ears, a hybrid, one might unthinkingly assume that this scale is a curiosity, rather than descriptive of real life music-making.

But take a listen to the chords this pattern makes:

The most significant change from the major scale triads is the chord IV is now chord iv – yep, it’s minor. [And along with that we have a diminished ii and an augmented VI.] This minor chord ivand diminished chord ii are what we need to make one of the most used cadences of the 20th century:

I’ve heard this referred to as the Hollywood cadence, and that label really sums it up!

The chord progression

or even more commonly

is a staple of the popular song – just hearing these chords moving from one to the other (especially with a soft electronic piano sound) evokes a torrent of lyric clichés just aching to attach themselves to this progression.

And once you have these clichés in your head, it’s hard to think what else this scale is good for.

Am I being too harsh? Dismissing the pattern as being of limited and prescribed value? Dissing the emotional content as pre-fabricated and predictable?

Well, one disclaimer: the scale is rarely strong enough to sustain an entire composition – one section of a song or piece might utilise this pattern for a while, but relief is needed, and it nearly always comes in the form of both a modulation to a new tonal centre and a new scale pattern.

Does this observation reflect how things need to be, or simply that composers have found this a convenient quick-fix for whatever emotional short-comings their compositions may have had? Are there pieces out there that do something with this scale beyond exploiting the tug of that minor chord iv?

I’d love to think this scale is the basis of something more profound than generic unrequited love songs!

Scale of the Day #5: A Mystery Scale

Scale of the Day #5: A Mystery Scale

Before you read another word try playing this week’s scale-of-the-day through on your instrument:

Written on C it’s clear that this is the major scale with a single note adjusted a semitone higher.  It’s that it’s the 5th degree that has been adjusted that leads to the tonic chord having an augmented quality.

I’m not sure how I would go about composing with this scale, or how I would create a sense of genuine C tonality. Our western ears are attuned to the perfect 5th defining harmonic spaces, so this scale is challenging simply because that 5th degree does not create that expected consonance.

But even more challenging is the fact that the 5th is augmented. When we hear these two pitches out of context we assume (and believe deeply) that we are hearing a minor 6th, and we further project into this harmonic outline either a 2nd inversion of a minor chord or a 1st inversion of a major one.

It’s only when that additional note is positioned plumb between the bottom and top pitches that we hear the shape as expressing an augmented 5th (and an augmented chord). And when we hear that augmented quality, we rarely experience it as a point of conclusion or even pause; augmented-ness  suggests being on the way to somewhere else.

On the other hand, this scale presents us with some interesting relationships: the first and last 4 notes of the scale separate out into two contrasting contours. The first is simply the default position of the major scale, while the second is the shape we associate with the 7th, 1st, 2nd and 3rd notes of the melodic scale (that curly part of the pattern where two semitones get as close as they can to each other).

And the chords we can create with this scale are intense and wonderful.  Beyond moving between the I augmented and the IV major chords, and between the I augmented and the V diminished (an interesting exploration of how our diatonic expectations are stymied), we can create spectacular and fascinating complex chords. For example:

Where is this scale fun to play on the piano?  I like it a lot on E and B!

These are examples of what I call pseudo-symmetry: the pattern of white and black note is symmetrical, but the physical spacing is definitely not! But it’s a great-feeling scale in contrary motion because of that white/black pattern.

It’s also pseudo-symmetrical on F, but because the 4th and 5th degrees of the scale are black notes we end up having to play this scale with the thumbs on a note other than the tonic, and that feels very bewildering after a lifetime of diatonic white tonic scales. But the pseudo-symmetrical version on B flat is a delight (for the same reasons).

Finally, now we’ve explored all these positions of the scale, let’s acknowledge exactly what this scale is: the harmonic minor starting on the 3rd degree. In jazz parlance it’s known as Ionian #5, and if you’re up with your terminology that does actually express quite a lot (it’s a major scale with the 5th raised, therefore creating an augmented tonic).

But I’d love to get some feedback as to what this scale makes you feel, what this scale communicates. Scales that create a tonic with an augmented or diminished 5th are in a different category to those with that expected perfect 5th outline, and I’d love to come up with a name for these scales that don’t have the kind of tonic we are used to coming home to. Suggestions?

Scale of the Day #4: The Phrygian Mode

Scale of the Day #4: The Phrygian Mode

This week we’re looking at one of the modes of the major scale, and as all the modes of the major scale have a long tradition of being named in western music theory we won’t need to get worried about what it ‘ought’ to be called – that flag was planted long ago.

While the Phrygian mode and last week’s scale both have the 2nd and 7th flattened they end up sounding nothing alike, and that’s because the Phrygian mode has another two notes flattened (the 3rd and the 6th), while last week’s scale had one note raised (the 4th), with the result that three notes in the pattern are not shared.

The Phrygian mode is actually only one note different to the natural minor scale, but that flattened 2nd has such an unexpected aspect to it that we tend to hear it as vastly different from the familiar natural minor pattern.

Here is the Phrygian mode starting on C:

Notice how in this permutation the pattern is symmetrical – white-black-black-white-white-black-black-white.  This makes it a joy to play in contrary motion, and is one of my contrary favourites, without a doubt.

There are two super-easy ways of finding this mode.  The first is to simply play all the white notes from E to E.  The other is to play F and C and then add all the black notes.  (I find this second method more fun).

I used this Phrygian on F in my Grade 1 standard piano piece Milli-Molli-Mandi-pede from the Faber/Trinity publication Creepy Crawlies (also available in Australia in the Hal Leonard publication Getting to Grade One New Mix). Here’s how that piece starts:

This ends up sounding reasonably ‘normal’ because the theme is so focussed on the F and the C, the same perfect 5th you would expect to hear in a major, melodic minor or harmonic minor scale.  So the Phrygian marker note (the flattened 2nd) doesn’t stand out so much as it is rapidly passed through as the melody moves between the dominant and tonic notes.

But try playing through the triads of the Phrygian mode.  The pattern goes like this:

i II III iv v° VI vii


Or, in notation, on C:

This combination creates a quite positive spin on the extremely minor pattern, as the minor key chord rises to two Major chords. And while in a major mode the chord a 3rd above and a 3rd below the key note are minor, in the Phrygian mode the situation is reversed, creating a background harmonic context of contentedness, even if the key chord is melancholic. On the other hand, the semitone distance between the tonic and the 2nd note of the scale creates a high degree of tension (maybe it’s just heightened alertness, or then maybe it’s just a kind of unease) which works against the contented vibe major chords normally communicate.

I described the flattened 2nd as imparting a sensual atmosphere to last week’s scale, but here that impact is mitigated by the 3rd also being flattened – or maybe it’s just that the flattened 2nd sensuality feels more everyday without the augmented 2nd interval following.

How do you respond to the Phrygian mode?  Does it have any predictable emotional connotations to you?  And do you have any favourite pieces of music in this most interesting of tonic-minor modes?

Scale of the Day #3

Scale of the Day #3

This scale, starting on D as shown above, is my favourite scale of all time. It feels unbelievably wonderful under the hand – three black notes in a row, four white notes in a row. And as good as it is to play in similar motion hands an octave apart, just wait til you try it thirds or sixths apart (magic), or contrary motion (strange and wonderful).

What makes this my favourite? Well, it has all my favourite features: the raised 4th of the Lydian mode which communicates curiosity and optimism; the flattened 7th of the Mixolydian mode which communicates a lack of tension and a trusting approach to life; and then to top it off we have the flattened 2nd which imbues any scale with exoticism and sensuality.

How could anyone not like this scale?

If you’ve been following the scale of the day you will notice that this is almost the same as last week’s Simpsons Scale  – the only change is that flattened 2nd.  But this one change means that the pattern is no longer from the pitch class of major modes, or of melodic ascending modes, and it doesn’t belong to the harmonic minor pattern either. This scale is one of the modes of the pattern I call the melodic diminished. Keeping the same notes we have in the scale shown at the start of this post, let’s just start on a different note:

You can quickly see that this is the melodic ascending pattern, only the 5th note (E) has been flattened, which creates a diminished tonic chord, hence my naming of this pattern ‘melodic diminished’.

So one option for giving a name to my favourite scale of all time is to call it “Melodic Diminished on the 4th degree”, but this approach to labelling is forensic rather than evocative, creating little incentive for the newcomer to make the acquaintance of this scale.

Naming is a powerful thing.  When we know the name of something, our ability to know the thing itself is transformed. Naming is about classifying a thing, making a judgement as to how it works and what it does.  So choosing to give an obtuse and derivative title to a scale implies that the scale’s meanings are equally obtuse and derivative.

So, what should we call this pattern? And what names has it been given in traditions outside of classical Western music theory?

Do you know any pieces of music that feature this pattern? And do you enjoy the sound and the feel of this pattern as much as I do?!

Scale of the Day #2: The Simpsons Scale

Scale of the Day #2: The Simpsons Scale

Rather than alter the original post (which would make the comments below somewhat hard to follow) I will leave it as is, but point out that “The Simpsons Scale” certainly does have a name within the jazz tradition, the Lydian-Dominant (just as last week’s scale has a name within the tradition of South Indian classical music, “Mayamalavagowla”), so in reality when I call this scale “The Simpsons Scale” I am boldly naming what hundreds of thousands in the world of jazz have named before. (And note that this scale has a name in the South Indian tradition  [Mouli’s comments below]).

Now this scale isn’t actually called “The Simpsons Scale”, but since it isn’t actually called anything [in western theory] I have decided to boldly name what no one has named before.

In reality the Simpsons scale is the melodic ascending pattern starting on the 4th degree, but it happens to be the pitch pattern used for the tonic harmonies in the theme music to The Simpsons, so I decree that the scale henceforth be known as…..

Here’s the pattern, in F (because the pattern of white/black notes is identical to that of G Major):And here in C (so it is easy to see at a glance which notes have been altered, and by how much, from a ‘neutral’ major pattern):

And here is how the theme from the Simpsons goes (in seriously truncated form, so that all the notes of the pattern as evident):

This sounds like the Lydian mode, to a casual listener, because the raised 4th is the predominant note in the melodic sequence, while the Mixolydian marker, the flattened 7th, only makes an appearance as the theme wraps up at the every end .  But a careful listener will notice that this is the only kind of 7th note that occurs in the harmony also.

Yes, this pattern has the Lydian and the Mixolydian marker notes, so it’s a kind of Hyperlydian, succeeding in doing both the fundamentally major modes at once.  It’s a sensationally modern take on major, sounding quirky but smart, and full of a very contemporary energy.

I used this “Simpsons Scale” as the basis for my trumpet composition, Go-Goanna, published by Faber Music in their Fingerprints series, and now an ABRSM exam piece (Grade 4). But while Danny Elfman creates the feeling that we are flickering between C Major and D Major, in Go-Goanna the melody is shaped so that it feels like an alternation between C Major and G minor (in transposition), with the G minor leading note (F sharp) as part of the equation.  It’s interesting to me that this same scale produces two equally successful harmonic partnerships from its triads.

It takes a while to become accustomed to playing this scale, obviously to the ear, which is expecting neither the raised 4th or flattened 7th, but more especially to the fingers, who simply refuse to believe that a major-sounding pattern has its two semitones positioned so close to one another.  This is why I included the scale pattern in F – one’s fingers can be tricked into playing this correctly quite swiftly if one focuses on playing that G Major pattern that we know so well, but hearing this brand new pattern! As it turns out, starting on G is a similar proposition: play the white/black note pattern of F Major and you’ll get it first try.  Starting on F sharp can be quite rewarding also, as one can concentrate on playing the C (really B sharp) and the E around the two black note group.

Have a play, and then have your say.  How do you like it, and what does it make you feel?  And is this name, The Simpsons Scale, really the right one??!