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.

Ear Training

One of the many skills academically-trained musicians have is being able to identify an interval - the distance between two notes. There are twelve in an octave, from the minor second - 6% increase in frequency) - to the octave - a 100% increase in frequency.

There is of course an app for that. Several. I tried Earpeggio, which offers a wide range of tests. I passed the test of identifying which of two intervals was greater, and I can reliably spot a unison (same note, no difference) and an octave.

You’d think anyone could tell the difference between a minor third and a major sixth, seeing as how they are different ends of the octave, but nope. Major thirds went unidentified. If I’d been guessing, I would have got about two out of the twenty examples right, so even 50% isn’t awful. I noticed that as soon as I had two succeeding intervals close together, I was much more accurate, since I was relying on the memory of the previous note. But an interval on its own… ouch.

However, I’ve never done this before, so it’s not hopeless.

My quick foray into identifying chords was much less impressive.

It’s a neat thing to do when you have twenty minutes to spare in a quiet place, or with headphones.

Happy New Year and A Prosperous 2024 To You All

 

Nothing says "Happy New Year" like three abandoned boats on Dungeness beach.