puer natus est nobis

Yes, today is Christmas and so accordingly here is offered a very old Christmas song: the Introitus Puer natus est nobis. You can google it yourself, of course, but the first one here has the plainchant notation as found in the Liber usualis.

But this has organ accompaniment (which seems to be the way it's done these days...I spent several days at the Abbey of Gethsemani and they did all of their plainchant not only in English but with organ also. I guess it does help the pitch not to deviate...) so the following one may be the way you're more used to hearing it:

Also this selection is found on In Dulci Jubilo's record Salve Feste Dies, and it's worth checking out because of their approach to the rhythm (the rhythmic values in "Gregorian" chant are highly controversial: it's known that there were different rhythmic values but what's not known is what they were).

In terra pax, let's hope...

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...

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...

noh -- live 100% improvised

Yes, back at Goodbye Blue Monday on Wednesday 1 December at 9pm:

NOH the 100% improvised spoken word free rock/blues band.

The lineup:

Mark Zebra Warshow: drums

Jesse Martin: bass

Charles Ramsey: guitar

Raymond Todd: spoken/vocals

That's at 1087 Broadway in Bushwick, Brooklyn, NYC...J train to Kosciuszko...FREE!

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