This post is a continuation of my explainer on the difference between F-sharp and G-flat. To sum that one up: in our present-day standard tuning system, F-sharp and G-flat sound the same; the only difference between them is notational. In historical tuning systems, however, they sounded quite different. Tuning is hard!
In this post, I address the deeper mystery of the notes C-flat and B-sharp. Are these real notes?
In 12-TET, C-flat sounds the same as B, while B-sharp sounds the same as C. However, note names became standardized during a period in Western European history when tuning systems were based on just intonation, and in just intonation, all of these notes sound different from each other. Very different! C-flat is sharper than B, and B-sharp is flatter than C. Listen for yourself.
Let’s derive these notes from the harmonics of C, and the harmonics of other notes that contain C. First, here’s how I got B.
- C’s third harmonic produces G.
- G’s fifth harmonic produces B.
Next, here’s C-flat.
- A-flat’s fifth harmonic produces C.
- F-flat’s fifth harmonic produces A-flat.
- F-flat’s third harmonic produces C-flat.
And here’s how I got B-sharp.
- C’s fifth harmonic produces E.
- E’s fifth harmonic produces G-sharp.
- G-sharp’s fifth harmonic produces B-sharp.
Here’s the family tree of these notes plus a few related ones.
Let’s compare these notes to their closest 12-TET equivalents. I used equal-tempered middle C at 261.626 Hz as my starting pitch. Given
- Just intonation B is about a tenth of a semitone flatter than equal-tempered B
- Just intonation C-flat is about a third of a semitone sharper than equal-tempered B
- Just intonation B-sharp is about two fifths of a semitone flatter than equal-tempered C
Why should you care? In the key of C-sharp minor, the leading tone is not C, it’s B-sharp. That’s not an unusual key! You don’t encounter C-flat quite as often, but it’s the fourth in G-flat major and the third in A-flat minor. I always felt slightly insane seeing B-sharp or C-flat (or E-sharp or F-flat for that matter) on a score, and I feel better knowing their historical origin.
Fun fact: there are many businesses out there named B Sharp, mostly entertainment companies and barbershops. I guess the owners don’t watch the Simpsons.
There are not many businesses named C Flat. Maybe my friends at F-Flat Books should jump on it.