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.

Scales

(We are now working in European Equal Temperament.)

A scale is any sequence of intervals (notice: not notes) that adds up to 12 semitones. Think of any bonkers combination, and someone somewhere will have a guitar tutorial explaining why it should be the very next thing you learn.

A key or mode is a scale plus a starting note (the "tonic") that then defines a sequence of notes. We say "the key of G-Major" scale or "the Major scale". (Musical speech is sloppy, so we also say "the Major key" or "the G-Major scale".) Two keys are equivalent if they have the same scale. “Scale” = intervals; key = notes.

There are a number of well-known seven-note scales:

Major / Ionian Mode: Tone-Tone-Semitone-Tone-Tone-Tone-Semitone
Natural Minor / Aeolian Mode: Tone-Semitone-Tone-Tone-Tone-Semitone-Tone
Harmonic Minor: Tone-Semitone-Tone-Tone-Tone-Tone-Semitone
Lydian Mode: Tone-Tone-Tone-Semitone-Tone-Tone-Semitone
Mixolydian Mode; Tone-Tone-Semitone-Tone-Tone-Semitone-Tone
Dorian Mode: Tone-Semitone-Tone-Tone-Tone-Semitone-Tone
Phrygian Mode: Semitone-Tone-Tone-Tone-Semitone-Tone-Tone

The Ionian, Aeolian, Lydian, Mixolydian, Dorian, Phrygian modes are often called Church Modes, as they were used in early choral singing.

There are two well-known five note (pentatonic) scales:

Major Pentatonic: Tone-Tone-Minor Third-Tone-Minor Third
Minor Pentatonic: Minor Third-Tone-Tone-Minor Third-Tone

Two well-known six note scales:

Major Blues: Tone-Tone-Semitone-Minor Third-Tone-Minor Third
Minor Blues: Minor Third-Tone-Tone-Semitone-Minor Third-Tone
(Minor Third = 3 semitones)

Exactly one scale of only tones: Whole-Tone: Tone-Tone-Tone-Tone-Tone-Tone (There are two keys: C and C♯. After that the notes repeat, so starting on D gives the same notes as starting on C.)

Exactly one scale of only semitones: Chromatic: Semitone (x12)

If you want to see something truly out of control, look at the eight-note diminished scale.

Intervals

This is the first of a series of posts about music notation and associated ideas. The world does not need this, but I do, to make my own sense of it. There is a lot of notation in music, and it's not all part of one coherent whole. It's a bunch of tools for specific tasks.

Let's start at the beginning.

A note is a name for a given frequency. The most well-known note is "middle C" (or C₄) , followed by "A440", which is the frequency 440 Hz assigned to the A above middle C, A₄.

The human auditory system regards two notes whose frequencies are in the ratio 2:1 as very harmonious. This is because musical instruments do not produce pure sine wave tones, but a sound that is a mixture of the fundamental frequency and many others, called “overtones”. Playing A440 will usually also generate an "overtone" of A880, and so it sounds pleasantly matching when played against A880 as a note. This is so much so that two notes related by double frequency are regarded as "the same but higher".

This splits the range of audible frequencies into ranges called octaves. Pick a starting position, say A₄ = 440, and we have octaves as follows:

A₇ = 3620 (almost the highest note on the piano)
A₆ = 1760
A₅ = 880
A₄ = 440 (“tuning A”)
A₃ = 220
A₂ = 110
A₁ = 55
A₀ = 27.5 (lowest note on the piano)

(Why is it the lowest? There are pianos which go even lower, but below about 25Hz, the human ear stops hearing a continuous sound and starts to hear the individual beats. The highest note on the piano is 4120Hz and it's very difficult to produce an acoustic instrument that can produce that with significant volume.

The octaves are not the same size in terms of the range of frequencies, but the ratios of the frequencies are all the same. Each octave is double the previous one.

Each musical culture picks a different number of different frequencies within an octave to be its "notes". European music eventually settled on a series of frequencies, each one related to the previous one by the same ratio, the 12-th root of 2 (roughly 1.05946). This is called Equal Temperament, and it makes manufacturing and learning to play musical instruments way easier than the other European system did.

The "distance" between two notes is not measured in hertz (the ear doesn't work like that), but in powers of the 12-th root of 2 (roughly 1.05946). A power of the 12-th root of 2 is called a semitone. (Mathematicians can prove this is indeed a distance function as an exercise.)

Given a note X, the note one semitone up is X♯ and the note one semitone below is X♭. Replacing a note by the flat or the sharp is called flattening or sharpening the note. Under Equal Temperament, (X-1)♯ is the same note as X♭ - these are called enharmonic equivalents.

For more details, see the excellent and best-selling Your Brain on Music.