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!

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.

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.

music as patterns i

This is one of two series I want to start on this blog (look for the other "mystery" series to appear shortly!), viz. the investigation of music as patterns.

Let me just discursively throw out some ways in which patterns are a part of music:

VIBRATIONAL (from a simple vibrating sine wave to complex multi-timbrel occurrences, the vibrating ear drum, and so on)
RHYTHMIC (organization of sound even irrespective of pitch)
FORMAL (melodic shapes, harmonic structures, harmonic progressions, divisions of a piece of music into common forms -- sonata, song, aba -- scale structures, fingering patterns on particular instruments)

Also patterns may be grouped into those that are PERCEPTIBLE and those that are more CONCEPTUAL. A melodic phrase is an example of the former whereas the graphic representation of a square wave producing the pitch B4 is an example of the latter. Of course a melodic phrase notated is more conceptual but still perceptible, so perhaps another category of VISUAL needs to be added.

At any rate future posts in this series will start to examine some of these issues and others having to deal with emotion, entrainment and the like.

the enigmatic scale and even some more extended triads

I made a point in the posts about extended triads of keeping only to diminished, perfect and augmented fifths because those were the only ones I could imagine happening in scales. Not so! I recently looked at the enigmatic scale and found that we need to be more inclusive still. Here is the scale along with its triads:


As you can see both the II and V chords go beyond the parameters which I naively set. Since in this scale there are three minor seconds/half steps in a row (A#-B-C-Db) we end up with doubly diminished and doubly augmented fifths. If we consider that each fifth can have four types of 3rd (diminished, minor, major and augmented) then we will end up with 20 (count them twenty) triad types! (As far as symbols go I just simply stacked either plus signs or minus signs to indicate doubly diminished.)

Once again: give the triads a go...the whole point is that hopefully they aid you in expanding your vocabulary.

By the way, check out the wikipedia article on this scale here. You can discover its history (which includes Verdi) and that it actually has a different descending form.

See also extended triads iii, extended triads ii, extended triads i, basic triad types.

extended triad types iii

In this post we're going to look at extended triads in the way in which some might believe we should have done at the beginning: analytically or simply mathematically. The reason we didn't begin this way is because I wanted to demonstrate that there are real (and not just conceptual/theoretical) reasons why one needs the idea of extended triads, viz. the fact that they are produced naturally in certain scales.

So let's consider triads this way: chords consisting of a root, 3rd and 5th (as opposed to a harmonic structure that has just any 3 tones). As for 3rds we will allow diminished, minor, major and augmented; and for 5ths we'll allow diminished, perfect and augmented. The reason for "will allow" is that these intervals are all found naturally occurring in scales (such as the whole-half diminished and ionian flat-2, etc). We aren't going to examine triads which are comprised of a quadruply diminished 3rd and a quintuply augmented 5th (which triad would produce a third lower than the root and the like -- this could be interesting in a modern physics/mathematical kind of way but it's not extremely useful to us at present.

So based on our options of intervals we can simplify the triad families based on the 5th: there are only three kinds (diminished, perfect and augmented). Each family will have 4 types of third (diminished, minor, major and augmented). You won't have to resort to your calculator to figure, then, that there are, all in all, 12 triads. Here's a 1000 words:


The 3rds in blue denote that that chord is a basic triad type (diminished, minor, major or augmented) -- for example, the min 3rd in the diminished 5th family is blue because that chord is your garden variety diminished triad. And another 1000 words follow here with actual instantiations of all of the chord types:



Finally some issues of nomenclature. A M3P5 chord is simply called a major chord, so all chords containing a major or minor 3rd could be likewise appelled (a major diminished chord for M3dim5). Also since perfect 5ths are somewhat privileged we could drop that part of the name when it occurs: a diminished 3rd chord, an augmented 3rd chord.

Lastly -- and as promised -- let's look at the °3dim chord again. It has another name and is actually an extremely common chord in common practice "classical" music. Here it is in first inversion, showing its outer voice resolution:



Yes, that's right! An It+6 chord! For our purposes, then, the It+6 chord in root position is simply an It°3 chord. More on augmented 6th chords to come.

Play through these chords: some of these triads will undoubtedly be (or become) sonically interesting to you...

see also extended triads i, extended triads ii, basic triads

extended triad types ii

In this post we're going to examine the whole-half diminished scale. It is, like the whole tone scale, a symmetric arrangement of tones, and its name is the formula of the arrangement (i.e. a whole tone followed by a half tone followed by a whole tone...). This scale is different from many, many others in that it has more than 7 tones, in fact it is octatonic (8 toned). Here it is along with its triads:



Since it's composed of more than 7 tones some problems of nomenclature occur. For instance, at some point a scale tone (degree) has to be duplicated. Above we've made it F and F#, though it could've been D and D# (which would've produced an E#, then F#), etc. A massive ramification of this is that there is not a unique, one-to-one correspondence between scale degrees and triads -- some triads will manifest in 2 different forms. Above it is II, IV and VII (because they each contain F, which scale degree also exists as F#).

In this scale there are two triads which are "extended": the I and III. But these extended triads are a bit different from the ones which occurred in the previous post, and that's because they are enharmonically equivalent to basic triads. Put another way, the first two extended triads we encountered have no sonic equivalents, whereas these two that hail from the whole-half diminished scale do. Here they are:



The I chord sounds like a 1st inversion Ab major triad; the III chord sounds like a 2nd inversion G# minor one. But even though this is the case it's still important to be able to analyze these triads as extended types, if only for consistency's sake (most likely from a performer's point of view). But there is precedent for this sort of thing. Consider the Fr+6 chord: it is enharmonically equivalent to a dominant 7th chord, but its function is quite different in common practice harmony (that is its resolution is different from that of a dominant 7th chord).

So, back to the chords at hand: the I chord can be called a minor augmented (min Aug) and the III chord a diminished 3rd augmented (°3 Aug).

extended triad types

Over the next 3 posts we're going to be discussing extended triad types, i.e. triads in addition to the 4 basic types of major, minor, augmented and diminished (which topic was covered in this post). Perhaps unexpectedly this investigation will lead us into taking up the topic of augmented 6th chords.

If we only concern ourselves with the major and minor scales (both melodic and harmonic), and even if we throw in the harmonic major as well, we'll never encounter any triads beyond the 4 basic ones already mentioned. But this isn't always the case. Take for example the Double Harmonic scale and the triads which form from the scale degrees:


There are major triads (I and II), minor triads (III and IV) and augmented triads (VI) here, but look at the V and VII chords. They don't fit the pattern of the 4 basic types. The issue is that there isn't any ready-made term for either of these, so let's simply name them according to their intervals (which are some form of 3rd and 5th). The V chord is composed of a major 3rd and a diminished 5th, so we could name it a major diminished triad. Likewise the VII chord reveals a structure of a diminished 3rd and a diminished 5th: let's nominate this one a diminished 3rd diminished. And in abbreviated format: maj dim (V) and °3 dim (VII). Here's how they look built on C:



Why are there these different triad types lurking in this scale? Because scale degress 7, 1 and 2 are 2 consecutive minor 2nd intervals, which adds up to a diminished 3rd (and not a full minor 3rd found in the major and minor scales). This accounts for the double flat in the °3 dim triad built on C.

So now armed with this new knowledge you can analyze all of the following scales and you'll find that they contain the 4 basic triad types plus these 2 new "extended" triads: Neapolitan Major, Neapolitan Minor, Double Harmonic (as above), Ionian flat-2 and the Whole Tone Leading. Next time we're going to delve into the whole-half diminished scale and find even more triad types...

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

what chord is it?

Let's assume that you've played some sort of harmonic structure and you want to know (for some reason or another) what's the name of this thing I just played? We've discussed before that context has a lot to do with a chord's naming, but for now let's say that we're just dealing with one chord that sounds really cool and we need some kind of name for it. Here's a method which can get you in the game:

1. Write out all the chord tones and remove any duplicated tones.

2. See if there are any triads present (might have to look enharmonically). If no triads
go to step 4.
(a) If there's just one triad then this is likely your chord: go to step 3.
(b) If there is more than one triad pick the one that makes sense to you and go to step 3.

3. If there are any remaining tones they will relate in one of three ways, as:
(a) extensions
(b) additions
(c) suspensions

4. If no triads:
(a) Is one implied (e.g. a major 3rd could imply a major triad)
(b) Is it a power chord?
(c) Is it a stacked interval (stacked 5ths)
(d) Is it a tone cluster?
(e) Does the harmonic structure correspond to/imply any mode?

Needless to say for the above to work we have to have some sort of knowledge of basic triads and extensions, etc. Let's take a few examples and see what happens.

Example 1: f, g, a, c#.
Are there triads present? Yes: f, a, c# is an augmented triad.
Any remaining tones? Yes: g. This relates to f as a 2nd or 9th. Since there's no 7th present (which would be some sort of e) let's call this an F aug add 9 (or F+5 add 9).

Also example 1 can be viewed like this: f might actually be e# enharmonically spelled. That means we would have an A augmented triad: a, c#, e# (f). The remaining g is simply the 7th, so the chord could also be named: A7+5. (This kind of enharmonic spelling is quite common in music, classical or otherwise as music is really a guide for performers and not analyzers.)

Example 2: f#, g, b. Any triads? No. Any triad implied? Yes: g and b can easily give the sense of a G major triad (the perfect 5th is not needed for the ear to hear the "complete" chord). Then f# is simply the 7th which gives us a Gmaj7.

Example 3: bb, a, c, b. Again, no triads, buuut...a and c are enough for an A minor triad. The remaining bs are 9ths: we could call this structure A min add 9 add b9. Clearly, though, this is a form of a chord cluster, which will never yield very willingly to a nomenclature born of tonal music.

Also these 2 rules will help from time to time:
(1) If you ever have both a major 3rd and a minor 3rd treat the major 3rd as the actual 3rd and the minor 3rd as a(n enharmonically epelled) sharp 9th. E.g. the tones e, g#, b, d, g can be viewed as an E7#9 (g = f double sharp).
(2) Similarly in cases where you have a perfect 5th and a diminished 5th the perfect 5th is the real fifth and the diminished 5th can be seen as a sharp 11: c, e, gb, g, b = C maj7 #11 (gb = f#).

If the above is of only the slightest help then it will have served its purpose: the sound of the chord (and its emotive evocations) is the most important thing; the importance of naming it lies somewhere between a distant second and not completely a worthless endeavor.

See also: triads, seventh chords, slash chords, sixth chords.

transforming the circle of fifths

Yesterday we looked at the circle of fifths (COF) and regarded how some scales and intervals looked when displayed on that circle. But as interesting as the COF is, it isn't the only way to display all 12 pitch classes: another intuitive way to do so would be to arrange the pitches chromatically. Here's how a circle of minor 2nds (COM2) looks:


It might not be obvious at first glance, but there's an interesting relationship between the COM2 and the COF. To start to see it let's look at how the whole tone scale -- or major 2nds -- looks when displayed on the COM2:

And let's recall how the same collection of tones looks on the COF:

They're the same shape! The two circles differ only by the tones not a part of this collection, and they differ in a very systematic way. The easiest way to see the pattern is like this:

that is, the tones not a part of the major 2nd collection simply swap places with their tritone.

Another way -- a more arithmetical way -- to understand this transformation is to use a formula:

1. Convert all the tones to numbers as in 12-tone music (C=0, Db=1, D=2 and so on).
2. Multiply all tones by 7 mod 12 (i.e. multiply by 7 and if the number is 12 or bigger subtract 12: if it is still bigger subtract 12 again, and so on until the number in question is less than 12).

Some examples:

C [0 M7 mod 12 = 0 x 7 = 0 =] C
Db [1 M7 mod 12 = 1 x 7 = 7 =] G
D [2 M7 mod 12 = 2 x 7 = 14 -12 = 2 =] D
A [8 M7 mod 12 = 8 x 7 = 56 (mod12) = 8 =] Eb

This multiplicative operation is a way of extending the basic operations of 12 tone music (retrograde, inversion, inversion retrograde), but here we have a graphic display.

Lastly we can see how some scales look when displayed on the COM2. Here's a major scale:
which is still a symmetric shape, though far different from the one it takes in the COF.

And here's how perfect 5ths look:

This shape is more visible evidence of a strong correspondence between the two circles under question...if this shape is mapped onto a COF one then has a chromatic scale:

circle of fifths and scales

We touched on the circle of fifths before in this post. The circle of fifths is just that: notes seperated by a fifth (C - G -D - A, etc) and arranged in a circle because (in equal tempered tuning, at any rate) the fifths lead back to the starting point after all 12 tones have been accounted for (...Eb - Bb - F - C).

The circle of fifths (henceforth COF) has many uses, but one I've been playing around with lately is examining how scales look -- i.e. what shapes they take when the collections of notes are joined one to another as in the sequence of a scale. E.g. here's a whole tone scale:

And a chromatic scale:

Not surprisingly the 2 scales above take symmetric shapes when graphically displayed (the scales are symmetric in terms of their construction: comprised of 1/2 steps or whole steps). Surprising -- to me, at any rate -- is that when the major scale is displayed it also forms a symmetric shape:

As does the melodic minor.

Some asymmetric shapes: the harmonic minor:

and the neapolitan minor:

You can do these on your own, of course. Some other symmetric scales: the neapolitan major, the major pentatonic, the double harmonic. Some asymmetric ones: the hungarian minor and the harmonic major.

Lastly we can observe intervals -- actually we've already done this with the chromatic and whole-tone scales (min2 and Maj2 respectively). Here in one diagram are min3, Maj3 and P5:

The remaining intervals can all be found simply by going the other way round: C to Ab can be seen as a Maj3 down or as a min6 up. C to F is a P5 down or a P4 up, and so on...

creepy chords addendum

I was reading through Ponce's fabulous Guitar Sonata no. 3 last night and realized that in measures 39 and 40 (of the first movement) there is the same chord progression as mentioned in the post on creepy chords. Ponce is in Bb, but the chords qualities are the same: Major 7th +5 and a Minor(Major) 7th:



In this context I wouldn't say that the chords are "creepy", though they are tense. More creepy -- or at least menacing -- are the chords immediately preceding the 2 above (actually starting in measure 35: the structure is repeated):


Here we're really only dealing with one chord per measure: Ponce simply "flips" the outer voices (the 7th and root exchange places) giving him a very nice stepwise-ascending bass line.

And lest we think that these are just contemporary harmonic structures, or something born of jazz, keep in mind that this piece was written in 1927 -- and it's probably a safe bet that Ponce didn't invent this all on his own, so one might be able to trace it back at least to Debussy (maybe even Liszt???)...worthy of investigation.

intervals on the guitar ii

So now we come to the part of tuning the guitar that makes life interesting...the B string. In our last post we conquered the first four strings of the guitar (which are the same as the bass except that the bass being a bass has its strings an octave lower). Check out this diagram: I've made the B string red here to call attention to it:



Under each string is listed the number of frets that that string is distant from its neighbor to the left. If you wanna check this for accuracy consider the following: the low E and high E strings are 2 octaves apart (E2 and E4 respectively). Now look at the numbers: if we simply add up them up we get 5 + 5 + 5 + 4 + 5 = 24 frets. What's the interval corresponding to 24 frets? It's 2 x 12, which is the same as 2 octaves.

So let's take an example: what is the following interval?

We've traveled down the neck 2 frets and laterally 1: but this lateral move has brought us onto the B string. Checking our chart up above we see that that means we've moved the equivalent of 4 frets down the neck. Consequently our interval is (2 + 4 = 6 frets which we know from the last post's first chart is) a tritone. You'll see this show up in A-form dominant chords.

Here's one more:


In this case we've moved laterally 2 strings, one of which is the B string, and backwards 1 fret. The B string is 4 frets, the E is 5 and our "backwards" move is -1, and all of that adds up to 8 frets, a minor 6th.

That really wraps up our discussion. One loose end, though: we've always considered our intervals from lowest to highest, but this may not be the order in which they occur melodically. Just keep in mind that from C up to G is the exact same as from G down to C (a perfect 5th). You can always measure from the bottom up even if you want to know the interval starting from the higher note.