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.

zhou xuan -- song of the four seasons

I just got a collection of songs by the fabulous and (in China) ultra-famous singer Zhou Xuan. Here's an example of her singing "Song of the Four Seasons" (hopefully hearing it will tempt you to seek out and listen to more!):



There's some info about here here at the wikipedia site.

And to give credit where credit is due: I had never heard of Zhou until I saw the film Electric Shadows (which, by the way, is the literal reading of the Chinese characters for the word 'movie(s)' which looks like 电影 / 電影).

ivy -- nothing but the sky

I'm comfortable enough with myself to admit that I watched the film Shanghai Kiss. Not a fabulous movie. In fact not even a decent movie, though I liked all the actors and the places (it's great to see Shanghai on film -- see below).

But one excellent thing about the film is that the tune "Nothing But the Sky" by Ivy is in it. Here's a link:




But to see/hear how it was used in the film check this one out:



The nearly sci-fi landscape of Shanghai along with that ambient, ultra-airy sounding voice is really an excellent match...

Oh, and here are the entire lyrics for the song:

Meet me tonight
Fifteen miles high
Nothing but the sky
Shining in your eyes...

rhythmic training by robert starer

A book I used a million years ago at the College-Conservatory of Music in Cincinnati (for a guitar sight reading class): Robert Starer's Rhythmic Training. I'm giving an amazon link here because you can look through some of the book there, but of course buy it at your favorite vendor of music scores...

Anyway it's an excellent book to go through from time to time, and especially if you're having particular difficulties (e.g. switching between triplets and sixteenths while keeping a steady pulse). The last few exercises are still brutal for me...

the fan man

I read this book last year and looooved it. The main character -- Horse Badorties -- is a hippie who's exclusively into medieval music. He lives in the Lower East Side and has a choir of young runaway girls and has them hold tiny battery-powered fans which emit to him a beautiful sound. A great, weird book. Horse's hatred of Puerto Rican music parallels my own detesting of Dominican music...



Click here to read what wikipedia has to say about the novel. If you read and enjoyed Confederacy of Dunces you'll dig this...highly recommended.

And if you do read it try to get the illustrated version (though it looks like the newer edition has a forward by Vonnegut which must be a great read).

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.

ai no tenshi

This is my favorite tune from Satoshi Kon's 1998 fabulous anime classic Perfect Blue, and it's called "Ai no tenshi" (The Angel of Love).



Also of potential interest is the following clip which shows the tune being recorded by the singers (which clip is on the dvd, btw):

highest f major chord?

Sor's Fantasy no. 2 (op. 7) is a spectacular piece. For those of you who have the Bream Baroque Guitar record and have heard the piece that way keep in mind: Bream only plays the Introduction. There's a whole theme and variations which follow, some of which are truly remarkable.

Remarkable in terms of just having to deal with a somewhat uncomfortable fingering examine the following passage from the B section of the 3rd variation:


Yes, that F chord is stratospheric! Yes on the electric guitar (or any guitar with a cutaway) this is not a real beast. But keep in mind: the lowest fret here is the 13th. I'd be tempted to introduce some rubato at this point just to accommodate this technical aspect.

By the way: you shredders should check out the 6th variation...

new ideas in a new year

If I were made dictator of the universe for a minute or two I'd like to change some aspects of conservatory training. These would be the additions/changes:

One semester (at least) of music writing and improvising. NOT taught by a "composer", especially not by an academic one. Here the goal is to reveal to musicians that if they can play an instrument they can improvise on that instrument. And also they can write music of very diverse styles for that instrument of any other.

One semester of "pop music" performance. Here we would, for example, play Let It Be. But the pianists would be given the chart on a cocktail napkin and NOT (I repeat NOT) permitted to write out any parts. The performance would have to be convincing, etc. Real world.

Basics of recording, multi-tracking, etc. Are there any musicians, anywhere, who at this point don't need recordings of their playing? Might as well be armed with some knowledge of how it's done...maybe even enough knowledge to do it one's self.

I would certainly toy with the idea of getting rid of music history. Maybe a course that contains the basic bullet points. Maybe in theory situate the concepts historically. There seem to be lots of musicians in the world who function just fine without knowing thing one about clausulae -- just as there are millions of musicians worldwide who know nothing of set theory and would much prefer to stab themselves repeatedly with rusty implements than learn anything about said subject.