Accidental Pandemonium


(entry for 11/29/2024)



One of the first frustrations any new music student has to contend with is the issue of ‘accidentals.’

If the student is learning piano, they have to absorb the notion of black notes having two names, a sharp one and a flat one, and if they are learning guitar or other fretted instrument, they have to absorb the idea that some frets named for the first seven letters of the alphabet have other frets between them that have names that are not just simple letters, but have other words attached to them, while other letter-named frets have no ‘extra’ frets between them. 

For example, on the guitar’s A string, the first fret is the one for the note A-sharp (or B-flat) and the next fret is for the note B. But the one after that is for the note C, with no other fret between the B and the C. (The next fret up from C is for C-sharp or D-flat.)

The piano student finds all this quite a bit easier, because they can see that there is no black note between B and C, nor is there one between E and F.


(note that there is no black note between E and F or between B and C)

But the guitar student and the piano student may both very well reasonably ask “Why?”

To answer this, we need to go back momentarily to the concept of ‘modes.’ Specifically, to the Ionian Mode. And we also need to introduce the concept of ‘steps’ of a scale. Let’s take the Ionian Mode as it is played or sung when starting and ending on the note we call ‘C.’ (Or, in Latin-based languages, ‘Do.’) The Ionian Mode scale (or the C-major scale, as we now call it), each letter represents a ‘degree’ of a scale. The home note (in this case ‘C’) is the first degree of the scale. As we go up the scale, each subsequent step up is the next number higher of degree.

Here’s a chart that may make this concept a bit easier to understand:

    Degrees of C major scale:

    name of note     degree of scale

        C                               1

        D                               2

        E                                3

        F                                4

        G                               5

        A                                6

        B                                7

        C                                1   (the eighth degree is the same as the 1st)

One might think that the distance between any two degrees of a scale would be the same.

One would be wrong!

The distance between the 1st and 2nd degrees IS the same as the distance between the 2nd and 3rd degrees, but then the pattern breaks down. The distance between the 3rd and 4th degrees is only half a big as those first two differences.

To be more precise, just as you multiply a particular note’s vibration frequency by 1.5 to get a note a fifth higher (see the previous post, on Temperament), you multiply it by 1.125 to get the larger difference between degrees. And you to get the smaller difference, you multiply by 1.0625.

The number of decimal points in all this get to be so many that it becomes pointless to try to keep track of all of them, so we’ll go to a different method of keeping track. One we’ve hinted at already, in earlier posts.

The larger difference, that is, the one between the first and second degrees or the one between the second and third degrees, is called a Whole Step. (In the chart below, we will abbreviate that as the initial ‘W.’)

The difference that’s half that big, that is, the one between the 3rd and 4th degrees, is called a Half Step, for which we will use the abbreviation ‘H.’

For ANY major scale, in ANY key, the sequence of whole steps and half steps, going up the scale, looks like this.


    Degrees:

    1St - 2nd            W

    2nd - 3rd            W

    3rd - 4th                  H

    4th - 5th              W

    5th - 6th              W

    6th - 7th              W

    7th - 8th (1st)      H


To repeat, the ratio to go a Whole Step is 1.125. (That is, you multiply the pitch you’re starting from by that ratio to get the note a whole step higher.)

The ratio for a Half Step is 1.0625.

Note: these ratios are for ‘pure’ tones. The ratios for tempered tones are slightly smaller and involve unending decimal places! (We’re not going there!)

There are two more things to remember before we begin explaining why sharps and flats and other accidentals can get quite confusing:

1. The note names in a scale cannot double any letters (except the first and last degrees) and cannot skip any letters. That’s why the seventh degree in the D major scale has to be C-sharp, and not D-flat. If you called it D-flat then you’ve skipped the name C, and the D was already ‘taken’ by the first degree of the scale. Two violations for the price of one!

2. In constructing a triad, you must skip one letter between each pair, and you cannot have two or more letters skipped in the note names of the chord. For example an A-major triad must be spelled ‘A’-’C-sharp’-’E.’ You cannot spell it ‘A’-’D-flat’-E, because then you haven’t skipped a letter between the D and the E, and you’ve skipped two letters between the A and the D. (You’ve skipped the B and the C.) Again, two violations.

When we’re in simple keys, this is pretty easy. The rules are easy to follow and ‘make sense.’ But when we get lots of sharps or lots of flats, it doesn’t stay so easy any more!

Let’s say you’re in the key of C-sharp major (seven sharps). How do you build the Tonic triad? If you play the correct-sounding notes on a keyboard, it looks like you’re playing C-sharp, F, and G-sharp. However, that can’t be right, because you’re not skipping a letter between F and G-sharp, and you’re skipping two letters between C-sharp and F. So the middle letter has to be some sort of E. But it’s not E-natural, it’s higher than that. So it must be E-sharp! Even though it’s not a black note, it’s a sharp!

The same thing happens when we play a G-sharp major triad. The middle note must be a B-sharp, even though it’s white.

Now let’s go back to that C-sharp major triad. Suppose we call that a D-flat major triad (key signature of five flats). Now we can call the middle note F, because that obeys the naming rule. But what if we want to make the major triad into a minor one? It now looks as though we’re playing D-flat, E, and G-flat. But, as you already know, that’s illegal. The middle note must be some kind of F. Therefore it’s F-flat.

In a similar way, the white note just to the left of C can be called C-flat instead of B, if the naming rules demand it.

Which is all a very long way of saying that just as there are seven letter names available, so there are seven flats available (though two of them are white notes) and seven sharps available (though, again, two of them are white notes).

You can even have double sharps and double flats. The white note that we call F can also be called G-double-flat when needed, and the note we call D can be C-double-sharp when needed. Here’s what those two notes look like on a treble-clef music staff.

The other accidental, and one we really haven’t talked about, is the Natural. Look up at the head of this post. The third example there is of a natural.

One of the things that’s confusing at first about naturals, is that they can cancel anything. They can cancel a sharp or a flat in a key signature. They can also cancel any accidental: any sharp or flat, or even any double-flat or double-sharp. (By the way, double sharps and double flats occur only as accidentals. They never occur in a key signature.)


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