a little bit of math: 7 note scales

I've been considering scales and modes lately, and have been wondering how many possibilities there are out there. I started writing out some lists (based on the major/Ionian scale, such as 1234b567, 1#2345b6b7, etc). At a certain point, however, I started to consider using any combination of 7 notes from the total chromatic of 12. Here writing out by hand started to become futile, so I wondered how to go about determining the actual number of possibilities. So here's a little math about that.

If we are concerned with 7 notes from a total of 12, and are not concerned about order -- we're looking for a set of tones, not a melodic sequence -- then what we want to find is known mathematically as combinations. There's a simple formula for determining them which is shown in the following image (which image was swiped from wikipedia -- thanks, guys!!!):

(In case you're not familiar with it, that ! doesn't indicate a loud, demanding or angry number: it's a factorial. 4! = 4 x 3 x 2 x 1 = 24. It's better if your calculator has a factorial button, because 12! =

479,001,600...best to do that in one keystroke!)

In our case n = 12 and k = 7. If you work through the equation you'll see that 12 tones taken 7 at a time can be arranged 792 different ways! (That exclamation is not a factorial). Some of these modes will be quite strange beasts from a typical scale point of view: c, c#, d, d#, e, f, g# is not the most common mode around. But if we want to know the exact, finite number then here we have it.

And here's something interesting, too, very, very interesting: if we want to know how many pentatonic scales there are we will find that there are 792, the exact number of septatonic scales (start to work it out and you'll see why). Hexatonic scales, by the way, produce the highest number of combinations: 924.

So if you're wondering if there are any more modes/scales out there to investigate the answer is most probably YES!

sus4 chords

Superimposing triads over a given harmonic structure is a well-known and -documented phenomenon. I personally love hearing a D major triad over an E minor harmony. And by triads usually meant are the famed major, minor, augmented and diminished. But we shouldn't overlook sus4 chords (or sus2 chords: we'll talk about that, too) as possibilities. As a refresher: a Csus4 chord is comprised of the notes c, f and g, and generalized a sus4 chord is made up of a root, P4 and P5. To a certain extent they can have a "cold" sound as there is no third, major or minor, and are found natively in quartal/quintal harmony.

So as far as use goes there's the obvious: wherever you want! Also here are some conventional usages:

Root of sus4 chord matches root of harmonic chord (e.g. Absus4 over Abmaj7; Esus4 over Emin).

Sus4 chords come from the harmony of a scale implied by the harmonic chord. For example take Dmi7. In a certain context this could be a dorian chord, meaning that we're dealing with a C major scale. In the case of major scales sus4 chords can be built on the 1, 2, 3, 5 and 6 scale degress (yup, you guessed it: a major pentatonic scale!). Concretely: over Dmi7 we could use Csus4, Dsus4, Esus4, Gsus4 and Asus4. Over melodic minor there are less: take sus4 chords built on the 1, 2 and 5 scale degrees. Basically we just have to check the scale tones against those of the sus4 chords and we'll be good.

OK, mention was made of sus2 chords: whassup with them? Let's examine the following 2 chords: Asus4 and Dsus2:
Asus4: a, d, e  
Dsus2: d, e, a
Yeah, the same notes. So we can generalize the situation as: a sus4 chord is the same collection of tones as a sus2 a perfect 4th higher.

As far as that goes, let's look at these notes again, but now starting with e as the root: e, a, d. This can be seen as an E7sus4 without the 5th. So a sus4 chord can be used as a 7sus4 the root of which is a perfect 5th higher.

Hopefully these will add something to your palette...

some scale relationships ii

Following up on what we discussed yesterday I'd like to offer a variant upon that approach.

It's all fine to see how scales can be linked in a chain, each "link" being one accidental away from the ones before and after it. But it might be that you're familiar with certain modes, but not so much with the parent scales whence they hail. For example tons of musicians know about the overtone scale but not all realize that it's a mode of the melodic minor.

So, in today's diagram what we've done is to look at the modes of the major/ionian scale and see how one -- the lydian -- relates to other lydian modes.

In this case we've tracked through the lydian flat-7 (aka lydian dominant) to arrive at the lydian dominant augmented (lydian b7#5). Please note that bi-directional arrows indicate a scale-mode relationship, while the uni-directional arrows indicate scales that are distant by one accidental. The other way of saying what this diagram is hoping to express is that if you conceptualize your modes in this fashion (lydian b7, lydian b6, lydian #2, ...) you are still obviously framing your mode/scale understanding as we outlined yesterday.

some scale relationships

One way to ponder and categorize scales is to organize them so that a new scale is described as an old one with one modification. For example, the melodic minor scale can be viewed as a major scale with a flat 3; the harmonic minor can be conceptualized as a melodic minor with a flat 6. The following image describes several scales this way, taking the major/ionian scale as primary:

The box for the whole-tone leading has been made a different color because it doesn't strictly involve only one change (but it is deducible by a series of changes starting from an augmented (ionian sharp-5) then to a lydian augmented).

(By the way the above image was made with Open Office Draw: a great and free program!)

The modes of these parent scales haven't been included, though not doing so is to a certain extent a taxonomic bias. For instance I had at first included the scale/mode ionian #2, as it's only one deviation from the major scale. But upon reflection it turns out that it is a mode of the neapolitan minor, a scale which is already quite well known. Consequently I decided against the inclusion of the ionian #2, though an interesting and extremely complex chart could be generated by including such modes and showing their relationship(s) to other scales.

A chart like this also tells use at a fairly quick glance just how far scales are from one another. For instance the doulbe harmonic scale is just one note different or one "scale away" from the harmonic major; the neapolitan minor is three scales away from the major/ionian.

Of course there are a myriad scales out there, but this beginning should at least get the mind working with a view towards simplifying that array -- "well begun is half done", after all.

permutations

Lately I've been examining how very little musical material can generate vast amounts of music. Think about all the tonal music that basically elaborates a I - V - I relationship.

To get some of this flavor let's take 3 notes (a, b, and c) and put them into sixteenth note "slots". Let's also stipulate -- at first -- that we can only duplicate one note (i.e. we have to use all of the tones). Here's what we start to get:

aabc abca bcaa
aacb acba cbaa
abac acab
baca caba
baac caab

bbac bacb acbb
bbca bcab cabb
babc bcba
abcb cbab
abbc cbba

ccab cabc abcc
ccba cbac bacc
cacb cbca
acbc bcac
accb cbba

So here we get 36 different little motives from 3 notes distributed over 4 note-slots. We could augment our rule to allow the duplication of 2 notes (thereby not using all three notes). Here's a little of what we get:

aabb abba bbaa
abab baba

aacc acca...

bbcc bccb...

That's 15 more motives or cells. Also let's allow a triplication of notes:

aaab abaa baaa
bbba babb abbb
bbbc bcbb cbbb
cccb cbcc bccc
ccca cacc accc
aaac acaa caaa

There's 18. And lastly let's allow a quadruplication:

aaaa bbbb cccc

which adds 3 more cells. All in all this totals 72 different motive-cells.

And this is just a surface scratching. We could further define some rules for our rhythms: take for example
aaaa.
This could be 4 sixteenth notes, but we could also combine them into larger units, such as:
one 16th and a dotted eighth,
one 16th, an eightn and a 16th,
a dotted eighth and a sixteenth,
2 eighth notes,
one quarter note.

Obviously our cell-motives will increase dramatically when this "rule" is applied across the board.

Why so possibly obsessive about this sort of thing? Well in improvisation and composition we're always looking for ways to make what we do more organic. Just this most basic surface examination shows that there is A LOT of material waiting to be made out of very little building blocks (similar to how electrons, protons and neutrons combine to form over a hundred different elements). Anyway if you're ever bored or just un-inspired take up this sort of exercise and see where it leads you.

what key is it in?

This question of what key something is in is one that comes up often, and the reasons for it being asked can range from the academic to the very practical -- it is in the spirit of the latter that we will offer up an answer.

The no.1 reason we might want to determine the key of a tune/piece or section thereof is for improvisational purposes: it's hard (though not impossible) to improvise without knowing the key. In certain cases this will be ambiguous, which means more leeway for the improviser; at other times there will be only one key.

So, let's define a key as the parent scale of all the harmonic/melodic structures in a given instance. That might be a rather convoluted way of stating something very simple. Here are some examples.

A favorite: Knocking On Heaven's Door by Bob Dylan. The chords:

      Gmaj | Dmaj | Amin | Amin | Gmaj | Dmaj| Cmaj| Cmaj| (repeat to infinity)

The key here (according to our definition) is fairly unambiguous: G major. Major keys give us 3 major chords and 3 minor chords. In G major those chords are specifically: Gmaj, Cmaj, Dmaj; Amin, Bmin, Emin. All of the chords of the tune number among those just enumerated, so there we have it.

Here's a slightly more involved one: House of the Rising Sun.

      Amin | Cmaj | Dmaj | Fmaj | Amin | Cmaj | E7 | E7 |
      Amin | Cmaj | Dmaj | Fmaj | Amin | E7 | Amin | Amin |

We have 4 major chords (analysing E7 as such) which tells us right away that we're going beyond the chords found amongst our normal major keys. In this case Amin going to E7 is telling us that this in in A minor. Now there are 3 different minor keys:

     1. Natural (same as its relative major)
     2. Melodic
     3. Harmonic

One way of looking at this would be to say that this song is in A natural minor (i.e. C major) whenever the chords are Amin, Cmaj, or Fmaj. When we encounter Dmaj it's probably really in A melodic minor (the natural 6 gives us the F#) but it might be easiest to think of it as Gmajor (D mixolydian). The E7 is either melodic or harmonic minor.

How about a chord progression like this:

     Emaj7 | Bmaj add b9 | Amin | AminMaj7 |

There are some possibilities here, but all of these chords come from E harmonic major, though you might conceptualize/hear it as shifting from E major to A minor.

Of course there are other indicators that you might already be aware of / be doing: II - V is more or less subsumed by our definition, but it is a distinct and very prevalent pattern to be on the lookout for.

Keep one thing in mind: this is a practical way of understanding the concept of key. Take the following example:

      Dmin | Cmaj | Dmin | Dmin |

According to our method this is in C major, though really C doesn't seem to the tonic but instead D does (that is the progression is in D dorian). However have no fear: as far as improvising goes you'll still be on solid ground if you're thinking C major -- though knowing the major key's derived modes is a good idea.

For the above mentioned "way" to work of course we need to know some basic scales (and where to look for those that we might not know) and their triads, and all of us can always learn more of these.

aura lee caged

If you have a guitar method book like Mel Bay's or Alfred's sitting around and you feel like you've learned the notes in open position (or maybe not even those) and you'd like to expand your knowledge of notes over the entire neck try the following. Take a simple tune such as "Aura Lee" -- perhaps better known as Elvis's "Love Me Tender" -- and play it in as many of the 5 traditional major scale patterns (CAGED) as possible.

Here is what the first 4 bars of "Aura Lee" will look like as found throughout the CAGED system:

(E₀ means the E pattern in open position, E₁₂ is the E pattern at the 12th fret.)

So in this case the tune can be played in six different positions. And more generally speaking we can note that if the open G string on the guitar is the lowest note in a first position melody that same melody will be able to be played in all of these same patterns. If we only had notes on the E and B strings we would have even more possibilities; if an open D is in the mix less.

And since there's an interest in this blog about patterns in music, let's examine how the notes relate to each other across contiguous patterns. If we examine the penultimate measure we'll see a regularly occurring interlocking/overlapping-ness:

This is a beginning: we could also explore this tune as found throughout 3 note per string scales, too (or even 4-note/string if you're so inclined), which might be the subject of a blog down the line.