intervals on the mandolin

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

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

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

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

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

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

So now let's look at this interval:

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

How about this one?

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

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

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

a little bit of math: 7 note scales

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

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

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

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

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

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

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

sus4 chords

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

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

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

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

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

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

Hopefully these will add something to your palette...

some scale relationships ii

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

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

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

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

some scale relationships

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

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

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

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

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

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

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.