London From Shooters Hill

 The Met Office changed its mind about Tuesday being sunny and decided Monday was going to be, so at the last moment Sis and I set out for Falconwood and points towards Greenwich. We found ourselves at the top of Shooters Hill - a high point on the old A2 - and saw this view over London. I may go back with a telephoto lens, but until then, cropping will have to do. Open up the original and zoom in on it. There aren’t many places where the whole length of the town, from Canary Wharf to Westminster appears in one panorama.

Guitarists and Triadic Chords

Not only can we not play full-fledged triadic extended chords on the guitar, but we can’t always play the notes of a special-effects chord in triadic order. (We can always do that on a piano.) So guitarists jumble the notes around the fretboard until they find a combination they can play and call that the “D♭13”. If they can’t manage that, they drop one of the notes and try again. This is why guitar chord books will show two different chords in different positions under the same name with nary a word of explanation.

These variations are called “voicings” of the chord. Because that sounds like they meant it.

What does it mean to say you’re playing B♭minor 7♯9 on the guitar, if there are three ways of doing it and each of them has a different set of notes in a different sequence (from the sixth to the first string) and each voicing may be a third or more higher than the next?

It means two things: first that the triadic naming convention rapidly becomes unwieldy above sevenths; second, that you’re hip to the tricks of the trade you have the musical taste a) not to make a fuss about the ambiguities, and b) to know which of the “voicing” of an extended chord fits which situations.

It’s even worse than that. Many guitar chords are “voiced” across all six strings. So we can strum an accompaniment easily.

The strummer’s F-major chord in the first position is F-C-F-A-C-F. An F-major triad is F-A-C - in any key. The six-string chord has the 6-4 inversion (C-F-A), the fifth (F-A-C), and the sixth inversion (A-C-F). Two are major fifths (C-F-A, F-A-C) and a minor fifth (A-C-F). All in one chord. It’s triadic sludge. So are all the other cowboy chords (so-called because they can be strummed across all six strings in the first position).

Classical guitarists don’t strum, so this mess does not happen in classical music.

So does this mean (non-classical) guitarists are doing something wrong, or does it mean that conventional triadic harmony is not the best way of understanding what kind of harmonic contributions the guitar can make?

I would say the latter.

There’s a story about Joni Mitchell working with Tom Scott. She’s playing piano and Scott - a fearsome jazz session musician - is in the recording booth. Joni plays one of those “Joni Mitchell” chords, and Scott hits the mic and asks “Is that an A-flat 4th diminished 6th” (or some other such). Joni looks at him, then at her fingers on the keys, and then back again.

“Tom. Ignorance is bliss.”

I’d suggest that guitarist-harmony / chords is more bliss than book. The better songwriters find their “odd chords” by experiment as much as theory. The theorists then rave about so-and-so’s use of a minor ninth sus-2 (or whatever) as if so-and-so thought about it, when in fact, it’s a chord that results when playing something fairly ordinary and moving one finger forward a fret and another backward a fret. Experimenting. And do-able on stage.

Every guitarist has to learn the cowboy chords in all the shapes. And the sevenths and major sevenths, and the sus-2’s and sus-4’s. After that, it depends on what genre they are aiming for. As for learning lots of arpeggios? Only be-boppers do that, and be-bop dominates (non-classical) music teaching, because it has rules. (Of course, classical guitarists learn arpeggios, but that’s because a) Bach, and b) treating a sequence of notes across the strings as an arpeggio - and therefore a “chord shape” - is a way of learning the piece.)

Extended Chords

Next up are the chords made up of four notes. These are seventh chords, because four notes each a third apart are a seventh apart from top to bottom. (Weird interval arithmetic again.) In D-major these are:

D-F♯-A-C♯ I₇ 

E-G-B-D ii₇ 

F♯-A-C♯-E iiI₇ 

G-B-D-F♯ IV₇ 

A-C♯-E-G V₇ 

B-D-F♯-A vi₇ 

C♯-E-G-B vii₇

Look at the cyclic permutations of (say) ii₇ (E-G-B-D). These are: G-B-D-E; B-D-E-G; and D-E-G-B. These are a sixth wide, and have two notes next to each other, the D-E. The first is major, the second diminished (two semitones), and the third can be described as a sus2 (D-E-G) with an add 7 (B).

For a long time classical musicians stopped at seventh chords, with an occasional foray into a ninth as a stunt. Jazz musicians, however, started with sevenths and worked upwards, notionally to five note chords (ninths), six note chords (elevenths) and seven note chords (thirteenths). A jazz pianist or guitarist thinks nothing of playing D♭13, which is

Furthermore we can shift the fifths, sevenths, ninths and so on, up a sharp or down a flat, to get truly wonderful monstrosities such as D♭13: D♭(1)-F(3)-A♭(5)-C (7)-E&flat(9)-G♭(11)-B♭(13).

We can’t play the full-fledged Triadic D♭13 on the guitar, or with a string quartet, and it would need some skilful orchestration to be heard if played by an orchestra. Even if we could, we would only do so very rarely. It’s a mess. As are full-fledged elevenths.

Composers and songwriters know that chords extending above sevenths are a special effect. (Hindemith says as much in an aside in his book on Harmony.) They may want, say, the effect of the root and the eleventh (fourth an octave up), with the third to indicate that the chord is “really” a minor, and a flat (aka dominant) seventh, because that flavours the chord, but the fifth and the ninth don’t do anything musically useful, and are just clutter. So they write the root, third, seventh and eleventh, and everyone calls it an “eleventh chord”.

Suppose we want the special effect of a sharp nine against the eleventh? Write the root, sharp nine and eleventh. How about the third? Sharp nines are flat thirds an octave up, and that sounds messy, so let’s leave out the third. Dominant sevenths are a special effect of their own that will distract from the one we want, so let’s leave out the seventh. Let’s put in the fifth so that the chord doesn’t sound too thin. So that’s root, fifth, sharp nine and eleventh. Which is also called an eleventh chord, strictly an “eleven sharp nine” chord.

Nobody plays or writes full-fledged triadic extended chords. They play or write random carefully-chosen groupings of different notes spreading over two octaves. (Playing the same note an octave apart doesn’t “extend” the chord.) And no group of instrumentalists does this more than guitarists - and any other ensemble with less than five players.

Chords and Triadic Harmony

A tune is made up of a sequence of intervals.

Chords provide a background against which the melody is set.

Western chords start with Triadic harmony and get more complicated from there.

We start with… Triads. A Triad is a three-note chord. The simplest are fifths: the base note, the one a third in the scale above it, and the one a third above that. The triadic fifths of C-major are 

C-E-G      I
D-F-A      ii
E-G-B      iii
F-A-C      IV
G-B-D     V
A-C-E     vi
B-D-F     vii^o

Lower case indicates minor chords, upper case indicates major chords, the 'o' indicates a diminished chord. Minor chords have three semitones at the bottom (D-E-F is Tone-Semitone), and two Tones at the top (F-G-A is Tone-Tone). Major chords are the other way round. Diminished chords have two sets of three semitones (B-D-F is Semitone-Tone-Tone-Semitone). Augmented chords have two sets of four semitones (C-E-G♯ is Tone-Tone-Tone-Tone).

What happens if we play (say) E-G-C (in that order on the piano)? Now the interval at the bottom is minor, not major.

Flipping the notes of those triads around, we get the so-called Neapolitan Sixth chords

E-G-C I ₆ (minor)
F-A-D II ₆ (major)
G-B-E III ₆ (major)
A-C-F IV ₆ (minor / sort of diminished-ish)
B-D-G V ₆ (minor)
C-E-A VI ₆ (major)
D-F-B vii₆ (minor)

Flip once more, we get the 6-4 triads

G-C-E I ₄⁶ (major)
A-D-F II ₄⁶ (minor)
B-E-G III ₄⁶ (minor)
C-F-A IV ₄⁶ (major)
D-G-B V ₄⁶ (major)
E-A-C VI ₄⁶ (minor)
F-B-D vii ₄⁶ (minor / sort of diminished-ish)

The 6-4 triads get their major or minor flavour from the top of the triad, rather than the bottom, as with the fifth and Neapolitan sixth triads.

All these chords have the property that adding another note a third above the top one just produces the bass note an octave higher. A-D-F goes to A-D-F-A. This is because in the weird arithmetic of notes, a sixth plus a third is an eighth. So these inversions are a Triadic dead-end - though we can add whatever note we want to any of them, and later on, we will.

The idea of the root of a triad was invented to explain why it is that C-E-G, E-G-C and G-C-E are all I chords in C even though they have different bass (bottom) notes. The root of a chord is the note that would be in the bass, if it was re-arranged as a series of ascending triads, filling in any missing notes and allowing for modifications.

Simple enough, surely?

Bassists play the root note, so the rest of us don’t have to.

Classical harmony theory loves these inverted triads, jazzers barely know they exist.

Roman Numeral Notation

Most music is written in one of the twelve major scales, and the Major scale has a pragmatically-central position in a (western) musician's technique.

Because all twelve major scales have the same intervals, anything we say about the musical properties of one scale will apply to any of the others. The Roman Numeral notation lets us do this: it abstracts out the tonic note, but fixes the Major scale.

I (tonic, first) the note that names the key

♯I / ♭II (sharp first, flat second) 

II (second)

 ♯I / ♭III (sharp second / flat third) 

III (third) 

♯III / ♭IV (sharp third / flat fourth) 

IV (fourth) 

V (fifth) 

♯V / ♭VI (sharp fifth / flat sixth) 

VI (sixth) 

♯VI / ♭VII (sharp sixth / flat seventh) 

VII (seventh) leading tone to the...

I an octave above the start

Counting the semitones from the tonic, these are the same names (without the adjectives like “perfect’) as the musical intervals defined in the previous post.

All the other (equal temperament) scales can be described in terms of this one:

Natural Minor / Aeolian Mode: I-II-♭III-IV-V-♭VI-♭VII
Major Blues: I-II-♭III-III-V-VI
Whole-Tone: I-II-III-♯IV-♯V♯VI

(The ability to recite any other scale or mode in terms of "sharp this, flat that" with utter fluency is an essential skill of any academic or jazz nerd. I'm not sure how much it helps, but it sounds impressive.)

Health Report

Regular readers will remember that about ten or so months ago I was having pains in my right shoulder and arm. I thought this was caused by bad posture playing guitar, but it turned out to be the bad posture of some of my neck vertebrae. Smart readers went long osteopathy and were not disappointed. 

I had a reasonably pain-free autumn and was okay until the end of December when I must have done Something Stupid which set the pains off again. I’m not getting the fizzing and buzzing down my arms, but I am getting persistent aches in my shoulder and neck, which are turning out to be so distracting that I can’t really focus on anything for long. I’m swallowing ibuprofen with intermittent paracetemol when needed, because the second time around a pain is much less bearable. 

I am long osteopathy again. With luck that will work, and isn’t a sign that my vertebrae have got worse.

In the meantime, I will carry on with the music posts. The real world looks way too shaky right now and I can’t focus on it.

Interval Names

(This is the first of two slightly dry posts on naming conventions.)

The intervals of European Equal Temperament scales are defined by counting the number of semitones between the notes and applying the following names (see here https://en.wikipedia.org/wiki/Interval_(music) for a longer discussion, including diminished and augmented intervals)

0 Unison P1
1 Minor second m2
2 Major Second M2
3 Minor third m3
4 Major third M3
5 Perfect fourth P4
6 Augmented fourth A4 / Diminished fifth D5
7 Perfect fifth P5
8 Minor sixth m6 / Augmented 5 A5
9 Major sixth M6
10 Minor seventh m7
11 Major seventh M7
12 Octave P8

The numbers 1,2,3... in the names are given by the number of lines and spaces ("staff positions") between the notes on the familiar five-bar stave. That method of counting notes will work for any scale with any number of notes in it.

C-F is... Tone(D)-Tone(E)-Semitone(F) = 5 semitones = Perfect fourth.

D-F is three semitones = Minor Third (D-E-F - D is on a line, E is in a space, and F is on a line, so an m3)

B-G♯ is Semitone(C)-Tone(D)-Tone(E)-Semitone(F)-Tone(G)-Semitone(G♯) = 9 semitones = Major sixth (G♯ is the sixth note in B-Major).

A♭ - E is Semitone(A)-Tone(B)-Semitone( C)-Tone(D)-Tone(E) = 8 semitones = Minor sixth (E♭ is the fifth note in A♭ and F is the sixth)

(You can use any method you like to count the semitones. This is my method at the moment.)

Since the number of semitones between any two notes is independent of the scale or key, interval names are independent of the underlying key or scale, since it depends only on the number of semitones. The same holds for staff positions, so the names of the intervals are also independent of the key or scale.