what is a key? part iv -- caveat i

So up to this point we've been analyzing keys by looking at the group(s) of notes that we've found in pieces of music and giving the group(s) a name, so far basically a major key name. There are two caveats we need to be aware of now, and in this post we're going to look at the first one of them.

CAVEAT NUMBER ONE

Let's look now at the following progression:

Amin | Dmin | Emin| Amin |

Let's go through all the steps we did before to find the key of this progression. We'll look at the notes of the chords:

Amin = A, C, E

Dmin = D, F, A

Emin = E, G, B

Putting them all side by side gives us

A, C, E, D, F, A, E, G, B

And if we only leave the first instance of a note (and throw out the repetitions) we are left with

A, C, E, D, F, G, B

Finally while we put these notes in ascending order we might notice that there are no sharps or flats, which means that we have the notes of a C major scale:

C, D, E, F, G, A, B

And this is true. From an improvisational standpoint you could take your C major scale and blow away over these chords and you would be on firm ground. 

BUT if we had to describe the key of this piece to someone it might seem strange to say that it's in the key of C major. Perhaps they ask well, shouldn't there be a C major chord in the key of C?

That seems like a good point, and it's really true. When we want to describe the key of a piece as accurately as possible we should take note of the harmony. For instance a piece in the key of C major should probably somehow articulate "C-ness". In the progression above it seems most likely that the A minor chord is the tonic, or the harmonic base. If A is the harmonic base we could organize all of the notes above as

A, B, C, D, E, F, G

This is commonly referred to as the A natural minor scale, and it is also known as A aeolian, The above progression is best seen as being in A minor because that is the most pronounced chord. 

The A minor-ness of this progression could even be more pronounced. Let's switch Emin to E7:

Amin | Dmin | E7 | Amin |

The only difference from before is that E7 (and you could just go ahead and use plain ol' E maj) has  a G# where E minor had a G natural. 

E7 = E, G#, B, D

The scale then looks like this:

A, B, C, D, E, F, G#

This, too, has a name: the A harmonic minor scale.

The point of all of the foregoing is that simply identifying a group of notes is not really enough to determine a key: we really have to take into account the chords and see (actually hear) what the harmonic focal point is. 

And by the way this is why I think a lot of people get tripped up on modes. Simply telling someone that ABCDEFG is A aeolian is not enough information to make something sound like it's centered around A. You could play those group of notes over 

Cmaj | Fmaj | G7 | Cmaj |

and the result will never be that the progression sounds like A aeolian. It will sound in C major because of the chords. 

what is a key? part iii

Following on the heels of the last post we're going to talk a little bit about dominant 7th chords in this post. And as a way of getting into that let's look at the following common chord progression:

Cmaj | Fmaj | Cmaj | Cmaj |D7 | Gmaj |

To determine the key here we'll go through the same process as we did in the last post. First let's determine all of the notes of all of the chords:

Cmaj = C E G
Fmaj = F A C
D7 = D F# A C
Gmaj = G B D

Now let's put them all side by side:

C E G, F A C, D F# A C, G B D

and get rid of the repititions of notes:

C E G F A D F# B

And now we'll put them in an ascending order:

C D E F F# G A B

Well now we've got a strange situation: there are eight different notes here. All major scales (and minor scales, double harmonic, neapolitan, etc) contain only 7 different notes. If we are to assign this progression to a major key we'll have to get rid of one of these notes. How should we go about this? Let's look at the F, F#, G area. There are no major scales which contain this sequence of half steps, so we might be able to get rid of one of these notes.

If we get rid of the F# we're left with
C D E F G A B
the ol' C major scale. If, on the other hand, we get rid of the F natural, we are left with:
C D E F# G A B.
This has a name, C Lydian. But that is a mode of (or the same group of notes as) G major:
G A B C D E F#

So now if we're only examining the notes of the chord progression we're at somewhat of a stand off: is the key of the progression C major or G major? Or maybe it's both?

Actually it is both, but just not at the same time.

Let's look at the Fmaj and D7 chords. When we come to Fmaj it's safe to say that the note F# is not really being articulated; likewise when we reach D7 the F natural is not part of the chord (look at the notes of the chords above if this doesn't make sense). In other words when we have F in our chord progression we are in the key of C major, and when we have D7 we are in the key of G major.

Notice something else here, too. The D7, as related to G, is V chord (that is the note D is the fifth note up in the key of G major). This is the only place that this happens in the major scale: there is only one dominant 7th chord, and it is built on the 5th scale degree. This is very good information to know, because when we encounter dom7 chords we can check to see if we are in a key where that dom7 chord is the V chord.

When this is the case we say that the dominant seventh chord is functional, i.e. that it has a certain role that it is fulfilling, namely that of pronouncing the key.

Not all dominant seventh chords are funtional! The blues progression is the most famous example of this, but actually this chord (and any chord) might just be used for the sonic atmosphere which it produces. These are cases where the dom7 chord is non-functional.

And even when we have a live, functioning dom7 chord, it may not go to it's I chord. The most famous example is what is known as a deceptive cadence:

Cmaj | Fmaj | G7 | Amin |

Here G7 goes to Amin, the VI of C major, but it's still a part of functional harmony.

And here's a way to hear how dominant 7th chords really push acoustically to a new key.

Let's take a Cmaj triad and go to all the other diatonic chords in that key:

Cmaj | Dmin |
Cmaj | Emin |
Cmaj | Fmaj |
Cmaj | Gmaj |
Cmaj | Amin |
Cmaj | Bdim |

Now in between these chords we'll add a dominant of the second chord -- i.e. the second chord is considered the I and we'll add a V of that right before:

Cmaj | A7 | Dmin |
Cmaj | B7 | Emin |
Cmaj | C7 | Fmaj |
Cmaj | D7 | Gmaj |
Cmaj | E7 | Amin |
Cmaj | F#7 | Bdim |

This isn't really an academic exercise, either. All of these progressions are far from rare, and you're probably familiar with hearing them (with the exception of the last one).

So the main point of this post is that not all progressions are in a single key; and sometimes when this happens dom7 chords can be a helpful signpost in helping us to determine the key(s).

ENDNOTE: There are in fact eight note (octatonic) scales running around out there in the world, the most famous of which are bebop scales and the diminished scale. The reason they weren't considered above is because bebop and diminished scales aren't actual generators of harmonic progressions, they are scales which are used to play over existing harmonies (bebop dominant is used over dominant chords, for example, whereas the diminished scale can be used over diminished chords and dom7b9 chords).

what is a key? part ii

If you read part i of this series and found yourself thinking WTF??? have no despair: this post will (hopefully) be a bit more clear and get us nearer to understanding what a key is.

Let's take a common chord progression:

Cmaj | Amin | Dmin | G7 |

Let's examine the notes of all these chords:

Cmaj = C, E, G
Amin = A, C, E
Dmin = D, F, A
G7 = G, B, D, F

If we put all these notes together side by side we get the following:

C, E, G, A, C, E, D, F, A, G, B, D, F

And if we take out the repetitions of a given note we will obtain:

C, E, G, A, D, F, B

Lastly we could put them in ascending order:

C, D, E, F, G, A, B

This group of notes may look familiar: it's the C major scale. (You might have put the notes in order beginning with A, obtaining A, B, C, D, E, F, G, aka A aeolian, aka the A natural minor scale. Same group of notes. Improvisationally it won't matter how you think of it.)

This is a clear example of, or a very clear articulation of a C major scale. But the scale actually is just an abstract way of organizing notes. They aren't actually encountered in the scalar order in this progression. To make a distinction we say that the above progression is in the key of C major. Just keep in mind that a key and scale are really the same thing:

A scale is a way of presenting the notes in a very clear manner.
A key differs from a scale in that the notes can come in varying orders and combinations.

Next post will deal with dominant 7th chords and how they help to determine a key,,,

what is a key? part i

Today's post kicks off a series on the nature of keys.

I want to come at the question from a different angle: instead of defining what a key is first let's look at some music and try to describe what's going on melodically. In so doing perhaps we'll get a better understanding not only of what a key is but why they're very helpful descriptively.

Here's a transcription of the beginning of a traditional Thai piece called Javanese Suite. To hear a great performance of this by the ensemble Fong Naam click here.

If we had to describe the notes of this to someone (perhaps even to ourselves if we wanted to improvise our own embellishments) how might we start? Well we can see right away that there are a lot of repeating notes. Let's consider all the repeating notes as one note for the purposes of analysis (for example: there are 7 Ds in this musical passage: but for our purposes we'll just write down one D). Then let's put the notes in order from lowest to highest. Here's what we get:

We can now see that the entire passage is made up of only five different notes: G, B flat, C, D, and F. If we jumped in and improvised to this tune (which we can do with the YouTube link above) we should be safe if we stick to these notes. And by safe I mean that we won't be adding any "colors" to the piece that aren't already present.

[If you're so interested try playing along and adding some other notes like A, E, Fsharp and so on. They will objectively change the sonic nature of the tune. This is very different from saying that what is added would be 'good' or 'bad', because those judgments would depend upon a performing and listening community.]

So basically we can say that by knowing the notes that make up a piece we have a key to unlocking the door which might otherwise bar our entry.

And if we went through hundreds and even thousands of pieces we might start seeing patterns. And that is of course what has already happened for centuries and centuries. At this point we don't have to reinvent the wheel. There are handy names for the sets of notes we're going to encounter (and if there isn't there are still other ways to classify those sets, too!)

If we look at our five notes from above again we might recognize that they have a name: they form a G minor pentatonic scale. And this is the same group of notes as a B-flat major pentatonic scale:

Knowing what a pentatonic scale is and how to play one on your instrument(s) in any key would be crucial if you were going to play other music like this. For instance try playing along to Zhou Xuan's performance of Song of the Four Seasons. If you sing or play the melody you'll soon discover that it's from the F sharp major pentatonic scale. And if you had to transpose the song for some reason (like for certain instruments' tuning, or for a singer, etc) doing so shouldn't be too complicated if you know the pentatonic scale.

The former paragraph is getting at this main point: we might find hundreds of other tunes in G minor pentatonic, and the order of notes might be different melodically. It would be a severe pain if we couldn't recognize that:

D, F, D, G, D, C, B-flat
C, B-flat, C, D, C, B-flat, G

are both made up of the same group of notes. Knowing that they both are from the G minor pentatonic scale cuts down on what we have to categorize tremendously. See my post on permutations: it's obvious that we don't want to name every sequence of notes as they can become nearly infinite in number.

Now if you didn't play along with the recordings or if any of the above particulars are confusing don't be alarmed whatsoever! The main thing to take away from this post is this:

Knowing the notes that make up a piece of music is the key that allows us to enter into the music more fully. And being able to determine what pattern (if any) the notes are in is extremely important as it's less that we have to memorize. 

This is why it's a good idea to learn many different kinds of scales: they allow us to categorize the music we encounter. And the more scales that we know the more able we'll be able to understand various styles of music.

To be sure there are some complicating factors that arise (e.g. a lot of music isn't in one single key or sometimes it's ambiguous as to what the key actually is or sometimes the sets of notes used don't correspond to any scale) but fundamentally the process is always the same: discern the notes and see if they fall into pre-established patterns.

More to come!

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...