The Unsolvable Problem - John Dunn, Consultant, Ambertec, P.E., P.C.

Some problems have no solution, absolutely no solution at all. The following is my dilettante's look at one such problem.

In music, in a tempered chromatic scale of Do-Do#-Re-Re#-Mi-Fa-Fa#-Sol-Sol#-La-La#-Ti-Do, there are twelve intervals between all of the notes including the sharps and flats and this can be called a twelve-tone scale. (For the sake of this discussion, we can let Do-sharp equal Re-flat etc.)

We start at some particular frequency as the "root" note which we call "Do" and then each note going up the scale has a frequency of the note just below itself multiplied by the twelfth root of two which is an irrational number, but is approximately 1.059463.

The intervals between these tones are called half-tones. Thus Do to Do# is a half-tone while the interval from Do to Re is two half-tones and we call that a whole-tone. In the eight notes we sing when we go Do-Re-Mi-Fa-Sol-La-Ti-Do, the intervals between Mi and Fa and between Ti and Do are half-tones while all the other intervals are whole tones.

When you do the aforesaid multiplication twelve times over, the twelfth time you do so, you arrive at a frequency which is exactly double that of the root note. This is the same as having gone up one octave.

 
Also, when you sing Do-Re-Mi-Fa-Sol-La-Ti-Do, Re is the second note you sing, Mi is the third note you sing, Fa is the fourth note you sing, Sol is the fifth note you sing and so on.

Observe that the frequency ratio of the fifth to the root in the tempered scale is shown as 1.498307 which is the ratio for a "tempered fifth". This is almost a ratio of 1.5 which would be a "perfect fifth". The harmony between Do and Sol sounds pretty good for the tempered fifth, but that Do and Sol harmony sounds better for the perfect fifth.

The unsolvable problem is that if you set Do and Sol in the perfect fifth ratio, then the ratios of those two notes to the other notes go bad and many otherwise pleasant harmonies sound awful. There are similar frequency ratio issues for tempered versus perfect of the other notes as well.

This conundrum is part of what is called "Pythagoras' Comma". In this usage, the word "comma" does not refer to a punctuation mark. This word "comma" means an unsolvable problem.

Some suggested reading on this can be found at http://www.justonic.com/pythagoras.html as well as at other web sites.

Comments

[John Dunn]

That would be one really versatile, very high-Q bandpass filter indeed, Peter. It would probably want to be a digital filter.

I haven't looked in many years at any such thing, bit I do recall that Linear Technology had a line of digital filter ICs. Their applications people could perhaps offer some guidance.

[Jonathan Cahm]

Do the calculations on a 432hz instead of 440hz and magically youll have answers...

[John Dunn]

Not so, Jonathan. The frequency ratio issue problem, also called Pythagoras' comma, holds for any chosen note "do". Also, A440 at 440 Hz has been the standard middle-A tone frequency being broadcast by NBS/NIST since 1936 on station WWV.