Musical notes

A friend and I were talking, recently, about musical notation, and I decided to put some of this down here, to make it easier to refer to. For readers who already know this stuff, it'll seem oversimplified and silly. For those who don't, it doubtless will be insufficient to really follow it. So please forgive me if this isn't the best blog entry in a while, but, well, some of them are diamonds, and some are just rocks.

My friend is French, and the French refer to musical notes differently from how we English-speakers do, so we started with a problem in communication. In an attempt to sort that out, I'll start by talking about what we call a C Major scale. We refer to the notes of the scale using the letters A thru G, and a C Major scale starts with C: C-D-E-F-G-A-B-C. The French terms for those notes are “do” (or “ut”), “ré”, “mi”, “fa”, “sol”, “la”, and “si”, in that order, so the “ut majeur” scale goes do-ré-mi-fa-sol-la-si-do. For the record, the German system is similar to the English one, except what we call “B” they call “H” — when the Germans say “B” they mean what we call “B flat” (no, I don't know why).

Let's look at that graphically. Music today (and for a long time) is written on a staff of five lines. Piano and orchestral music uses two staves of five lines each, and there are three notes in the gap between them, but we're going to skip that here and just talk about the treble clef. When we put the treble clef symbol, which looks like a stylized "g" (and is also called the “G clef”), at the beginning of the staff, we indicate which notes appear where. The bottom line of the staff becomes an E (French mi) — specifically, the E above “middle C”, the C that's approximately in the middle of a piano keyboard. The C Major scale starting at middle C then looks like this:

But what is that scale? Let's look at the mathematics of it for a moment. Because of the way our ears/brains work, we perceive a relationship between an oscillating frequency and the one that's double (or half) that frequency. We take the range from one frequency to its double and call it an octave, and we call the two notes at the end of any octave by the same name. So in the C Major scale above, the two “C” notes are an octave apart, and the second represents a tone at twice the frequency of the first.

To put it in an absolute perspective, the tuning standard for modern western music is (usually) “A-440”, which means that the A above middle C (the A in the scale above) has a frequency of 440 Hz. So the A below middle C has a frequency of 220 Hz.

Now — and here's the trick — we divide each octave into 12 approximately equal intervals (in the typical western “twelve-tone equal tempered” tuning system), and we call the interval from one of those tones to another a half step. It turns out that the sequence that sounds most natural and pleasing to us goes like this: base, full step, full step, half step, full step, full step, full step, half step. We call that sequence a major scale, and so the C Major scale goes like this:

C	D	E	F	G	A	B	C
base	full	full	half	full	full	full	half

What that means is that the interval from, say, D to E is a full step, but the interval from E to F (and from B to C) is only a half step. It also means that there are five tones that have no names yet.

When we go to the half-step between two notes that are a full step apart, we say that it's the lower note sharp (in French, dièse) or the higher note flat (in French, bémol). So the note between D and E is either D sharp (written D#) or E flat (written, approximately, Eb (the flat symbol isn't exactly like a lower-case “b”, but it's close enough). Which way we refer to it depends on context, and is way beyond the scope of this ramble.

Now, it turns out that we don't have to start our scales with C, but we do have to maintain the same sequence of intervals. We refer to the musical key by the note that starts the scale. In the key of C Major, the scale starts with C, and in the key of F Major (fa majeur in French), the scale starts with F (minor keys have a different interval sequence; we won't get into them here). But the F Major scale is not simply F-G-A-B-C-D-E-F, because that doesn't have the right interval sequence:

F	G	A	B	C	D	E	F
base	full	full	full	half	full	full	half

The two intervals in italics are reversed. To fix that, instead of using B in the F Major scale, we use B flat:

F	G	A	Bb	C	D	E	F
base	full	full	half	full	full	full	half

The way we indicate that graphically, and tell the musicians about it, is by putting sharp or flat symbols at the beginning of the staff, just after the clef symbol, to tell the reader which notes are modified. Here's a B-flat major scale:

Note the two flat symbols at the beginning, which show that all B and E notes should be played one half step flat, as B flat and E flat. That's called the key signature, and the two flats tell us that we're in the key of B flat Major (si-bémol majeur in French, B-Dur in German). We also say, for instance, that a work in B flat major is in two flats, because there are two flats in the key signature — to steal a line from Victor Borge, the composer wrote it in two flats because he had to move apartments that summer.

There's one more wrinkle to all this stuff about scales and what we call the notes. When we're talking about specific notes and keys and such, the French syllables have absolute meanings: “do” is C, “fa” is F, and so on. But when one sings a scale, one always sings “do-ré-mi-fa-sol-la-si-do”, regardless of what key one is singing in. In that case, the syllables are relative, and “do” refers to the first note of the scale. That bit also caused some communication difficulty in our conversation.

Oh, and a final, tiny wrinklette: we do use those syllables in English, but only in that last sense, in singing scales — so for us, F always refers to a specific note, while “fa” always refers to the fourth note of the scale in whatever key we're in. And we use “so” and “ti”, not “sol” and “si”.

Do, a deer, a female deer
Re, a drop of golden sun
Mi, a name I call myself
Fa, a long long way to run
So, a needle pulling thread
La, a note to follow “so”
Ti, a drink with jam and bread
That will bring us back to “do”

-- Oscar Hammerstein