the triads of the major scale

Here is a C major scale:

What we're going to do now is build triads on top of each of those scale degrees (except for the 8th scale degree because it's the same as the 1st and will give us the same chord). We will do this by using only notes from the C major scale, that is with no sharps or flats. What we get is the following:

The chord built on C gives us a C major triad, the chord built on D gives us a D minor triad, the one on E gives us E minor, F gives us F major, G gives us G major and A gives us A minor. Lastly the chord built up from B gives us a B diminished triad.

So you might wonder, why the Roman numerals in the chart? These numerals are actually very helpful, because they give us a more general way to talk about the chords in any major scale/key. Since all major keys are built up from the same recipe (a sequence of WWHWWWH steps) all the chords built up from the individual degrees will likewise occur in a predictable pattern.

Put more simply: In D major the chord built on the D (I) gives us a D major triad. In B flat major a chord built on Bb (the I) gives us a Bb major triad. In F sharp major the chord we get on C# is C# major, the V chord, just as we got a G major chord in C major. 

Here's a quick way to think about it:

I, IV, V are Major triads 

II, III, VI are Minor triads

VII is a diminished triad.

This is important information because if you find a chord progression like the following:

Gmaj | Amaj | Cmaj | Dmaj 

You know right away that the entire thing is NOT in one single key. Why? because there are FOUR DIFFERENT major chords, and in any given major key there are only three. 

If on the other hand we had only the following chords:

Gmaj | Cmaj | Dmaj 

we would be dealing with the I, IV and V of G major. 

How about these chords:

Emaj | Dmaj | Amaj

Why is the above key not E major? If it were in E major the VII should be a D# diminished chord, but here we have D major, a bVII with respect to E major. 

Why is it not in the key of D? If the above were in D major the II chord should be E minor, but here we have E major. 

Are all of the above chords found in the key of A major? YES! E major is the V chord, D major is the IV and A is the I. 

For some more practice check out the following post.

triadic inversion another way

Today's discussion is about inverting triads. What is generally meant when inverting triads is the following, done with C major:

That is to say, what we're really dealing with here is a re-ordering of the notes.

The concept of inversion as applied to melodic lines, however, has more to do with the actual meaning of inverting, i.e. turning upside down/placing in an opposite order (like a mirror). For example the following little line

will invert (diatonically, that is not adding any sharps or flats) to:

Now, what if we apply the same idea to chords? Something interesting will happen. We'll invert C major three times, first with C as the axis of symmetry, or mirror line:

The chord we end up with is an F major chord. Now let's use G as the axis:

Now we've produced a G major triad. So the interesting point here is that simply by inverting a triad (let's say melodically) we end up with the IV (subdominant) and V (dominant) chords. Not only that, these inversions have given us all the notes of the key of C major.

Oh, and the last way to invert the triad, with E as axis, produces...

...yeah, we get the same chord right back.

These are, by the way, diatonic inversions. Next time we'll examine what happens when we invert our intervals strictly.

intervals on the mandolin

I bought a mandolin a couple of months ago and though I haven't exactly tamed it I have made a little progress in investigating its fretboard layout. In this post we're going to examine how intervals lay and look on the mandolin fretboard. In the next post we'll talk about the concept of inversion and a way of conceptualizing that using the mandolin and bass fretboards.

For starters: the mandolin is generally tuned exactly like a violin. Starting with g3 (the g right below middle c on the piano, aka c4), the next strings are tuned in perfect 5ths ascending giving us altogether g3, d4, a4, e5, or simply g, d, a, e. N.B. The mandolin actually has eight strings, but they are tuned in unison pairs and are played conceptually as if there are only 4 different strings.

Before going further it will be important to read what was said about intervals in this previous post, at least the first four paragraphs. We'll be referring to the chart found there enough that I'll put it up here again:

So, again, for the mandolin (or really any stringed, fretted instrument born of music that has 12 pitches in an octave) we can easily ascertain the name of any interval on a single string simply by looking at the number of frets spanned and finding that number in the chart above in the left hand column. The number to its immediate right will be the name of the interval.

Some simple examples: what is the interval from an open string to the 5th fret? We probably won't need the calculator for this one: 5 - 0 = 5. Consulting the chart gives us P4, the perfect fourth. How about the interval from the 4th to 8th fret? 8 -4 = 4, and the chart says that that is a M3 (major third).

OK, so now when we branch out from one string to another we'll apply this same principle, i.e. we'll simply calculate the number of frets away the two notes in question are and consult the chart. To begin with let's recall that the mandolin is tuned in perfect fifths. How many frets is that? If we locate P5 in the above chart we'll notice it's equivalent to 7 frets. So going from one open string up to the next one on the mandolin is the same as going up 7 frets on the initial open string. Generalized this means that if we have notes on the same fret but on adjacent strings they are a P5 apart, such as:

So now let's look at this interval:

To determine the interval name we proceed in the same manner as before. We've gone across one string (= 7 frets) and up 2 more frets, giving us a total of (7 +2 =) 9 frets. On our chart we see that this is a M6 (major sixth).

How about this one?

Here we've gone across one string and back 4 frets. When we move backwards (lower on the neck) we simply subtract (or add negative numbers, if you like). So we have 7 - 4 = 3, a m3 (minor third) on the chart.

Let's now work the other way around: we'll select an interval and then figure out how it should look on the fretboard. And what better interval to select than the octave (P8)? That's a total of 12 frets according to the chart above. We could do that on the mandolin on 2 strings (or even on a single string) because the frets aren't all that large. but why don't we do it on three strings? Travelling laterally across 2 strings is the same as (7 + 7 =) 14 frets. That's 2 more frets than we need, so all we have to do is go down 2 frets and we should have our octave:

And that's really all there is to it. One thing worth doing: if you're a guitarist/bassist coming to the mandolin you might set your fingers on the instrument in familiar ways and see what the new intervals are. Mandolinists taking up the guitar/bass could do the same thing. It can take the mind a little while to straighten out all of this information as old patterns end up producing new sounds (it's certainly taking my brain a lot longer than expected!) Doing this will lead us into the next post concerning inversion...

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.