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. 

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!

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.

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.

the neapolitan minor

A plug for one of my favorite scales. You can think of this as a Phrygian with a sharp 7 (i.e. a major 7th) or you can think of it as a harmonic minor with a flat 2. Here it is built on C:


Harmonically there are some interesting chords. Look at the V7 chord: it's a free-roaming dominant 7 b5 chord. Look at it in 2nd inversion...that's right, it's a French augmented sixth chord (which is also analyzable as a Fr 4/3). Here is the G7b5 and the Fr+6 with its common resolution (the +6 opening up to a P8):


There are two extended triads: the V (a maj-dim, aka the Italian augmented 6th in root position) and the VII (dim3-dim).

As for the modes they're really interesting:

[mode i: phrygian #7]
mode ii: lydian #6
mode iii: mixolydian #5 (mixolydian augmented)
mode iv: aeolian #4 (lydian-aeolian)
mode v: locrian #3
mode vi: ionian #2
mode vii: (dorian #1) locrian bb3, bb7

And here's an actual use of the aeolian #4 found in measure 11 of Manuel Ponce's fantastic Sonata III:



And for you guitarists here is the scale in five positions comparable to the five positions of major, melodic minor and harmonic minor:



Learn it, love it, live it...

hidden patterns: the harmonic major

There are different ways of learning modes: one way is to look at (say) a C major scale and then see that it contains the D dorian, E phrygian, F lydian and so on. Another way to approach learning modes would be to keep the same tonic: C ionian, C dorian, C phrygian, etc. In this method we're actually moving through many (7 in fact) different keys: C major, Bb major, Ab major... If we do this with the harmonic minor modes something very interesting happens...but first let's take a look at the major and then melodic minor scales.

This table shows the major scale modes starting on C as well as the parent mode whence they come:



If we examine the roots of the parent scales we'll notice that -- when rearranged -- they form an A-flat major scale: Ab Bb C Db Eb F G.

Let's do the same thing with the melodic minor modes:


In this case the roots of the parent scales form a B-flat melodic minor scale (Bb C Db Eb F G A).

Up to this point there isn't much surprising: the roots of the parent scales both spell out major and melodic minor scales. However this situation changes when we subject the harmonic minor modes to the same investigation:


Here the roots of the parent scales form a (drum roll please...) harmonic major scale! F harmonic major to be exact: F G A Bb C Db E. And in a way this gives a nice rationale for the harmonic major: even if we'd never heard of nor conceived of it there it is, buried in the harmonic minor in a sense. File this one under how wonderously strange nature is...

See also: modes

exploring modes via pentatonics

OK, so if you've been reading this blog I understand that at this point you're rolling your eyes and emitting some form of loud "arrgh!" at the thought of yet another post dealing with pentatonic scales. Well, that's just how cool and useful I think they are, so here goes.

Let's say you have a Dmin7 chord that you're going to play over (or even write a melody over, etc). By using pentatonic scales you can elicit the colors of certain modes, and can do so by using your own and perhaps already copious supply of pentatonic licks.

The pentatonics that can easily be used are the following: start with the pentatonic with with the same root, in this case D minor. We can use the pentatonics which are 2 "clicks" both clock and counter-clockwise on the circle of fifths:

C | G | D | A | E

Each pentatonic scale, when combined with the underlying chord, corresponds to one or more modes. For example, if we take an E minor pentatonic scale (e, g, a, b, d) and play that over a Dmin7 chord (d, f, a, c) our resulting conglomeration of tones will be:

d, e, f, g, a, b, c

aka the dorian mode.

Starting with the pentatonic 2 clicks to the left and moving to the right (or clockwise on the circle of fifths) we can generalize the mode relationships as:


The reason that there can be more than one mode hinted at is because not all 7 tones of a scale are present in those situations. E.g. if we play a D minor pentatonic (d, f, g, a, c) over a Dmin7 chord (d, f, a, c) we only have five tones:

d, f, g, a, c

and without knowing what the 2nd (some kinda e) and 6th (some kinda b) are we can't tell what the mode is with complete precision. For instance, if there were an eb and a b natural we would end up with a complete dorian flat-2 (or flat-9), the second mode of the c melodic minor scale.

That is how the minor pentatonics will work over minor chords. In some future post we'll explore how they work over major chords.

the many flavors of 7th chords

Seventh chords are triads that have had a third added to their uppermost member, the 5th (a triad contains some form of 1, 3 and 5). The third will 'skip' a note if you're thinking alphabetically, meaning that it skips a scale degree -- there the 6th -- and so we end up with a tone that is a 7th above the root of the original triad.

OK, my apologies for such a convoluted definition. Since a picture is worth a thousand words here's a C major seventh (the gray notes are not part of the chord):


Looking at this it's clear that a seventh chord can also be seen as a triad with the interval of a 7th added to the root of that triad. Regarding this approach the first thing to keep in mind is that there are two main intervals of a 7th: major and minor. (There's actually another one that comes into play and we'll talk about it in a minute.) So if we recall that there are 4 basic triads we should be able to produce (4 triads x 2 types of sevenths =) 8 types of 7th chords.

So the major triad gives us a major 7th and a dominant 7th.

The minor triad: a minor/major 7th and a minor 7th.

The augmented: a major 7th +5 and a dominant 7th +5

The diminished...well, let's talk about this one for a second. Here we really need to look at the first way (of stacking thirds) of constructing sevenths to see how these come about. We could slap a major 7th interval onto the root of a diminished triad but in practice this doesn't happen: it's enharmonically equivalent to a B major triad with a b9. But the real reason that it doesn't "happen" is that seventh chords developed historically from voice leading and from functional harmonic concerns. The diminished triad is a chord that came about as a VII chord, and in neither the major nor the minor modes will we get a VII chord that is diminished and has a major seventh.

So what chords do we get? If we add a minor seventh to the triad we get a (classically appelled) half-diminished chord. On modern charts you're most likely to see this called a minor 7 b5.

Then there is a fully diminished seventh chord: it results from stacking minor thirds up from the root. In the key of C we could start on B and get:
b, d, f, a-flat.
And if we added another minor 3rd we'd be back to b. (If we examine the sort of 7th here we don't have a major or a minor variety: in this case it's a diminished 7th.)

This chord is unique because any of it's tones could be the root (and because of this in 19th century music it was used a pivot chord in modulations). Mo' on dat in a later post...

There is one more seventh that is not a basic triad with an added third. It kind of results from whole tone harmony, or it can be seen as an analogue of the dominant 7th +5:
the dominant 7th b5:

You won't really see a C-E-Gb chord floating around out there all on it's own, but this seventh chord is not uncommon at all (and not just in jazz: it comes up in Granados' Valses Poeticos if memory serves).

Play the sevenths, use them, they're beautiful...

repurposing pentatonics pars ii

In our former post on pentatonic scales we discussed how the minor pentatonic (specifically the A minor pentatonic) functioned over its minor and relative major chord/key areas. Today we're going to pick up that thread and see where else a minor pentatonic can be used. Keep in mind that I'm going to be dealing with the A minor pentatonic in these examples, but if it's easier for you to think C major pentatonic just substitute that whenever I use A minor.

There are 4 other keys where the notes of the A minor pentatonic are found. The first one is C major, which was implicit in our last post. The notes of A minor are in bold:

C major scale = c d e f g a b c

There are 2 other major scales where these notes are found:

G major: g a b c d e f# g
F major: f g a bb c d e f

And one melodic minor scale:

G melodic minor: g a bb c d e f# g

We can generalize our information as follows: the minor pentatonic scale is found in it's parent major scale and the major scales located to the immediate right and left of this scale on the circle of fifths.


(Clicking on the image will take you to the great site whence the image came.)

Other examples: F minor pentatonic is found in Ab major as well as Db and Eb major; C# minor pentatonic is found in E major, A major and B major.

The minor pentatonic is also found in the melodic minor scale one whole step below the minor pentatonic's root (F minor would be found in Eb melodic minor).

And since the minor pentatonic can be found in 3 major scales and 1 melodic minor scale, it will consequently be found in any of those derived modes. So A minor pentatonic is found in E phrygian, G dorian, C lydian b7, D mixolydian and so on.

And when we say 'found in' we mean 'can be used where...(blank) is used'.

So bust out your pentatonic licks and ideas and see how they sound repurposed in these key/mode areas.

a modal question

So here's a question about modes not often asked as far as I can tell. As a bit of prologue let's assume that we all know a bit about modes: their structures and the sorts of chords (triads/sevenths) that they produce. Armed with such knowledge we could easily address the following question: what is the mode of the following progression?

G7 | Fmaj7/G | G7 | Fmaj7/G | (repeated)

This is fairly unambiguous: it's G mixolydian. An analytic play-by-play might go like this (here the paths are many, but this one). The notes of the chords in question are:

G7: g, b, d, f
Fmaj7/G: f, a, c, e (with a g in the bass)

Now we put these notes in order to see if they form a scale. We could start anywhere, but since we notice that G is prominent (it's the bass note in both chords) let's put it first. We then get:

g, a, b, c, d, e, f

We now have a 7-note scale. Since there are no sharps or flats we can easily conclude that it must be some kind of mode of C major. G is the 5th tone of C major, and the mode on that tone is the mixolydian. (We could also have deduced mixolydian from the tones themselves simply by analyzing their relationship: 1 2 3 4 5 6 b7).

OK, so far so good. But here's the question part: What if the chord progression is changed just slightly to this:

G7 | Fmaj7 | G7 | Fmaj7 | (repeated)

That is we no longer have a grounding G bass note -- instead we have 2 different chords. If they're both of equal length (and consequently equally prominent) what is the mode now? Can we really safely say that it's G mixolydian? It seems that it might just as likely be F lydian. And really the first or last chords may not give any aid in determining: the first chord might likely lead to the real tonal area later, and the last chord might produce some kind of unresolved, "hanging" effect.

Again, we could have a chord progression as above that does emphasize one modal area, by rhythm or perhaps even the melody. But my main point is that in no way would we be on sure footing in certain circumstances when attempting to answer the question of mode.

And, by the way, why would we ask such a question? Because if we're improvising we have to have some way of dealing with the music at hand. And this goes to my over-arching view of practicality. If someone were to say, well, given the 2nd chord progression above I'd just play a C major scale, I don't see how that's a real problem. Also if one were to say that there's no single mode and that s/he would switch between mixolydian and lydian, also not a problem (though if the tune were up-tempo it might be difficult to manage 'switching').

The question is also interesting from an historical point of view: in music around Mozart's time an ambiguous key center would have to be deliberate (like in a developmental section) or it would just be bad music because projecting a key center was the name of the game. In our time that's just not the case: ambiguity abounds and we can bask in it, but also have to deal with its implications...

repurposing pentatonics

For a lot of musicians (especially guitarists) pentatonic scales and licks are among the first things learned improvisationally. But of course pentatonics go way beyond this: they're used everywhere from real American sounding tunes like Amazing Grace to jazz/blues/rock/country/bluegrass to 20th century classical music.

So here's the thing to consider first: just like any other scale pentatonics have modes. That is the collection of notes that make them up can be started (and ended) at different points. Take for example this collection of tones:

c d e g a c d e g a c d ...

If we start this at c we get:

c d e g a c, aka the major pentatonic scale.

If we start at a we get:

a c d e g a, the minor pentatonic scale.

There are 3 other modes, too, but the major and minor are the most common.

Now, as a guitarist I'm just naturally inclined to consider the minor pentatonic as the basic one. Especially from a playing point of view. And this is the first main point of this post: any minor pentatonic lick can be used over its relative major key area. It just simply becomes a major pentatonic in those instances. And the reason for this is that the mode isn't really determined by where we start and stop and what we emphasize in a scale but by what is emphasized harmonically. Put yet another way: playing an A minor pentatonic lick over a C major chord results in a major pentatonic lick sound.

In that vein let's end this post with some numeric thinking which will become extremely helpful when you start to use pentatonics (especially in later posts). When you play your pentatonic licks try to think of how the tones relate to the underlying harmony. Consider these situations again:

A minor triad:
a = 1
c = b3
d = 4
e = 5
g = b7

C major triad:
a = 6
c = 1
d = 2
e = 3
g = 5

The main advantage to a numeric understanding is that it generalizes the information so that you can understand not how just a specific pentatonic relates to a certain chord but how they relate in all cases.

In the next pentatonic we'll see a whole array of other areas where pentatonics can easily be used.

minor key signatures misleading

You don't have to think about it too hard to realize that key signatures are really major-key oriented affairs. They do in fact and 'to a T' accurately describe the makeup of a major key: C major has no sharps or flats, F-sharp major has six sharps (all but B) and A-flat major has four flats and so on.

But when it comes to the minor keys it's a whole different story. Well, if we restrict ourselves to the natural minor (aeolian) keys then it isn't a different story. But historically, even in Renaissance music where they did use the 'Church modes', sharps and flats entered into cadences. So much so that G# really did become a part of A minor. And melodically so did F# (hence the name melodic minor).

So, in effect, if we're talking about the melodic minor the key signature of its relative major is likely to be 'off' by 2 accidentals: A minor's key signature has no sharps or flats, but A melodic minor has 2 sharps; C minor's key signature has 3 flats but C melodic minor has only one flat (Eb).

Now, none of this may be an issue if you're playing Bach, etc (though in the older Baroque music the key signatures really hadn't been worked out, exactly because of the issue we're discussing), but it is an issue if you think key signatures when you consider modes from an improvisational/compositional point of view. For example: C melodic minor and E minor might seem quite distant by key signatures (3 flats and 1 sharp respectively), but in fact there's much more overlap: the only accidentals are Eb and F#.

Consider also the difference between a melodic minor key and its parallel major. A melodic minor, for example, only differs from A major by 1 tone: the 3rd (in this case C). Obviously if you think of scales/modes/keys from a numeric skeletal point of view then this won't ever be a problem. But we're generally taught the key signature approach at first, so in a sense there's a bit of unlearning to do.

tetrachords i

I mentioned tetrachords in passing before. Tetra in Greek means four, and chordos means string, but as in a lyre (or harp) string, so we can also read this as note. We won't go into Greek music theory here (where the 2 outside parts of the tetrachord were constant while the middle two changed): suffice to say that a tetrachord is 4 contiguous notes. I'd say it's a perfect 4th, but we're going to use the tetrachord as a way of analyzing parts of modes/scales, and in the instance of the Lydian mode (say) it's an augmented 4th.

A major scale (or any 7-note scale) can be seen as 2 disjunct (not overlapping) tetrachords:

In this case both tetrachords are a P4 in span, and are exactly the same in terms of intervals:
W, W, H, or M2, M2, m2. Let's call this the major tetrachord.
A major scale therefore = major tetrachord + major tetrachord.

If we examine a melodic minor scale we see these 2 tetrachords:

These tetrachords are different: the lower one is M2, m2, M2. Let's call this the minor tetrachord. The upper tetrachord is the same as in the major scale. So we can say that a melodic minor scale is a minor tetrachord + major tetrachord.

If we stick with just these 2 types of tetrachords let's see what other combinations we can can come up with.

Try a minor tetrachord + minor tetrachord:
Look familiar? This is the Dorian mode.


How about a major tetrachord + minor tetrachord:
This is the Mixolydian mode.

Let's look at one more scale today: the natural minor/aeolian.
Here the lower tetrachord is the familiar minor tetrachord but the upper one is different. Based on the Phrygian mode (with it's lowered 2nd and 3rd) let's call this the phrygian tetrachord.
So a natural minor/aeolian scale is a minor tetrachord + phrygian tetrachord.

In fact the Phrygian mode can be seen as 2 phrygian tetrachords:

By adding this third type of tetrachord let's see what scales we can generate. Play these so that you can hear them:

major tetrachord + phrygian tetrachord = Mixolydian b6 (mode V of a melodic minor scale).

phrygian tetrachord + major tetrachord = Neapolitan Major

phrygian tetrachord + minor tetrachord = Dorian b2 (mode II of a melodic minor scale).

So from just 3 different tetrachords we've been able to come up with 9 different modes. In fact I discussed this with a physicist friend of mine who informed me that the formula for determining how n objects can be arranged into a pair is simply n². We could, therefore, arrange 6 different tetrachords into 36 different scales!

I bring up tetrachords because since they're small they're easier to deal with conceptually. Also, at least for guitarists a tetrachord is easy to finger, and therefore it's easy to string them together. Try playing a major tetrachord starting on C then play a major tetrachord starting on G: you get the major scale (as above) and it's super easy to play.

Next tetrachord session we'll look at the Harmonic Minor and Lydian modes to see what tetrachords they possess and what we can generate then with all of our various tetrachords.

harmonic major modes

We've already discussed the Harmonic Major mode, but I want to cover its modes just because they're so cool.

The Harmonic Major is basically an Ionian b6. All the other modes will fall into this pattern (i.e. a familiar mode name plus one alteration).

ionian b6	1	2	3	4	5	b6	7	1	2	3	4	5	b6	7
dorian b5		1	2	b3	4	b5	6	b7
phrygian b4			1	b2	b3	b4	5	b6	b7
lydian b3				1	2	b3	#4	5	6	7
mixolydian b2					1	b2	3	4	5	6	b7
aeolian b1						b1	2	b3	4	5	b6	b7
locrian bb7							1	b2	b3	b4	b5	b6	bb7

The Locrian bb7 might seem weird. Since we've only added one alteration why is there a double flat??? The reason is that the 7 in the Locrian is already a b7, so if you slap on another flat you get the bb7.

Also, as detailed elsewhere, the b1 isn't so useful practically. If we re-analyze the aeolian b1 we'll see that it's construction is thus:

1 #2 3 #4 #5 6 7

The #4 is a hallmark of the Lydian mode. And we've already encountered the Lydian #5 -- Lydian Augmented -- in the melodic minor modes (it's mode 3). So this can easily be named
Lydian Augmented #2.

a multitude of names ii : harmonic minor modes

Having gone over some salient features of the melodic minor and its modes it's time to have that same fun with the harmonic minor and its modes.

The harmonic minor differs from the major scale by two notes: it has a b3 and a b6. The b3 is in common with the aeolian (natural minor) and melodic minor. What the harmonic minor has that is different and very unique is the interval between the b6 and 7, an augmented 2nd. This has a decidedly Middle Eastern flavor, and in fact is found in scales of Asian musics (Arabic, Indian...).

When we named the modes of the melodic minor scale we tried to keep to a pattern that utilized the name of the parallel mode in the major scale with one alteration, e.g. the mode starting on the 4th degree of the melodic minor would have to be some sort of Lydian, and and in fact all of its intervals are the same except that this mode of the melodic minor has a b7, so it's called a Lydian b7.

With the harmonic minor modes the situation is a little different: we could keep to the same pattern, but as was also detailed in the melodic minor post, what we need as players/composers is a practical and easy way to deal with this information. When there are 2 alterations it can be easier just to see what mode it's closest to in the major scale modes and then add an alteration.

Here's a tabulation:

mode 1: Harmonic Minor aka Aeolian #7
1 2 b3 4 5 b6 7

mode 2: Locrian #6 (instead of Dorian b2 b5)
1 b2 b3 4 b5 6 b7

mode 3: Ionian #5 aka Ionian Augmented
1 2 3 4 #5 6 7

mode 4: Dorian #4
1 2 b3 #4 5 6 b7

mode 5: Phrygian #3 aka Phrygian Dominant aka Spanish Phrygian (instead of Mixolydian b2 b6)
1 b2 3 4 5 b6 b7

mode 6: Lydian #2 (instead of Aeolian b1 b5)
1 #2 3 #4 5 6 7

mode 7: Superlocrian bb7 aka Altered bb7
1 b2 b3 b4 b5 b6 bb7

These modes are all super-cool, and the Spanish Phrygian is common (if you play through the Spanish Phrygian you'll understand why it's so named -- and if you play classical/flamenco guitar you'll already be quite familiar with it). These modes aren't used so much in jazz^{FN 1}, but definitely in other styles. In fact Satie makes use of them in his Gnossiennes -- especially check out no. 2 (written in 1893, hardly a modern phenomenon).

There is something cool that happens if we examine the Ionian #5. This is a case like the melodic minor: it differs from the major scale by only one tone. If we use this as a basis and compare the modes with the major scale we get the following correspondences:

major scale modes	harmonic minor scale modes
Ionian	Ionian #5
Dorian	Dorian #4
Phrygian	Phrygian #3
Lydian	Lydian #2
Mixolydian	Mixolydian #1
Aeolian	Aeolian #7 (harmonic minor)
Locrian	Locrian #6

Again, the Mixolydian #1 isn't so useful, and when we compare it to another mode that's close we get the Superlocrian, and so Superlocrian bb7.

Now, what I'm about to say is blasphemous to some, buuuut since we need the absolute easiest way as players/composers (musicians) to deal with information sometimes it's easiest to forget the mode names altogether and simply think about the harmonic minor (though this could apply to any mode) over a certain tone. The Spanish Phrygian is a good example: you can think of a Phrygian with a sharp-3 or simply think of a harmonic minor scale up a perfect fourth -- e.g. E Spanish Phrygian could be conceived of as A Harmonic Minor over E. In fact you could probably argue that this is the way a lot of Spanish music was written -- at least in the 19th century: not in a modal framework but in a tonal one.

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1. See The Jazz Theory Book by Mark Levine, pps. 476 - 478.

a multitude of names i : melodic minor modes

It really is a good idea to know (and not have to think about) the modes of the major scale if you're interested in learning the modes of other scales. The reason is that the modes of other scales will tend to resemble the modes of the major scale to some degree, and that can be used as a basis for describing the mode in question.

Let's focus on the melodic minor scale. It differs from the major scale by only one note, the 3rd (which is a minor 3rd in the minor scale, needless to say). Sometimes it's even described as an ionian flat-3. And if you think about it, since there's only one difference between the melodic minor and the major scale all of the modes of the melodic minor should be able to be described like ionian flat-3, i.e. like their parallel mode in the major scale with one alteration. In fact this is the easiest way, but there is one troublesome spot. Here are the modes tabulated:

ionian b3	1	2	b3	4	5	6	7	1	2	b3	4	5	6	7
locrian b4							1	b2	b3	b4	b5	b6	b7
aeolian b5						1	2	b3	4	b5	b6	b7
mixolydian b6					1	2	3	4	5	b6	7
lydian b7				1	2	3	#4	5	6	b7
phrygian b1			b1	b2	b3	4	5	b6	b7
dorian b2		1	b2	b3	4	5	6	b7

The problem is with the phrygian b1. The issue is that a flat-1 is kind of weird, because the 1 is usually the basis for determining the rest of the scale, and only when you know ahead of time (as in the table above) does it make sense to talk about a flat-1. Put another way: if you know you're dealing with a phrygian mode that has an alteration, and you've determined that the alteration is the 1, then a flat-1 designation might make sense. But in actual practice we don't usually consider that the 1 is flat -- we're usually trying to figure out how the rest of the mode relates to that tone. If we take our phrygian b1 and reexamine it without any bias regarding what its name should be, its structure looks like this:

1 2 3 #4 #5 6 7

If we look at the modes of the major scale we'll see that the the closest one is lydian (with its characteristic sharp-4). So it might be better to name this mode (and in fact it usually is named) Lydian #5, or Lydian Augmented.

So this taxonomy may or may not sit well with certain people who would prefer to see a 1 to 1 correspondence between the modes of the two scales in question. As a player, however, or a composer we simply need the best practical way of dealing with what's at hand.

And as far as taxonomy is concerned here are some of the other names of the modes discussed:

Melodic Minor aka Ionian b3 aka Jazz Minor
Lydian b7 aka Lydian Dominant aka Overtone
Locrian b4 aka Superlocrian aka Altered

Next time we discuss modes we'll focus in on the modes of the harmonic minor. Same idea but there are 2 (count them two) different notes with respect to the major scale...

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1. Or just the ascending portion of the melodic minor scale if you're used to classical music. The descending is the same as the natural minor or aeolian.

the harmonic major

Since I just discussed the major key a bit here's a plug for a related mode: the harmonic major. Compared to the major scale it's construction is:
1 2 3 4 5 b6 7
the b6 being the only difference from the major scale. C harmonic major looks like this:

At first glance this might seem like a mode of F melodic minor, but the melodic minor doesn't have a #4, so this is a different animal altogether. ^{FN 1}

The name comes from the fact that the upper tetrachord (4 notes) of this scale is the same as that of a harmonic minor with the same root. In this particular case: G Ab B C. The bottom tetrachord is the same as a major scale, hence "harmonic major".^{FN 2}

That's just one application. Others are taking triads or other harmonic structures from the scale and building tunes/pieces from them, and exploiting those new nuances of mood.

And another plug here: for names (and structures) of many, many modes visit this site.
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1. Though we could insist that a melodic minor sharp-4 is the parent scale and that this is consequently a derived mode.

2. Actually analysing with respect to tetrachords is a topic in its own right. It has the advantage of being able to quickly isolate where differences occur between modes. Another meaningful example is the melodic minor itself: it's often called an Ionian flat-3. This because the upper tetrachord is the same as a major (ionian) scale, and when the bottom two tetrachords are compared the differnce found is that the melodica minor has that b3.

the moods of modes: the major key

The major key is usually described as bright, cheerful and happy, etc. (in contradistinction with the minor key which is somber, sad and dark). This is only a shorthand, though. Whenever I hear or play through Sor's Study no. 16, op. 60 I'm amazed at how poignant and teary-eyed it can make one:

And another Sor example that comes to mind readily is his Bagatelle no. 3 (from op. 43), marked Cantabile:


It would be hard to characterize this as cheerful, happy or bright. And while it is certainly very pretty there is an undeniable gravitas which makes the piece fairly somber.

Carcassi's study no. 3 is another great example, and there are many, many other examples to be found in the guitar repertoire of the 19th century, not to mention the entire classical repertoire of that period, or of other periods and styles.

Of course I point this out not for taxonomic purposes or any sort of academic reason, but because noticing this point, i.e. that the major key is multidimensional with respect to mood, and that by extension so are all the other modes, we can ourselves become more creative as improvisers and composers.