The Well-Tempered Scale

[Edella]

Pythagoras discovered the ratios of frequencies that make up the western musical scale.Early classical musicians used the Pythagoras system for tuning. As I understand,the instruments would sound lovely in one key and simply horrible in the others.

Somewhere in the beginning 18th century the well tempered method of tuning started being used,most notably by Bach with the Well-Tempered Klavier.All modern music today uses the well tempered method of tuning.The way I see it, well tempered tuning deviates from the purely mathmatical Pythagoras system(A=440khz,an octave higher 880,etc) and is a compromise in tuning where everything sounds pleasing,but certainly not a precise science.

Why does well tempered tuning sound more pleasing?Why would tuning in a purerly mathmatical way,i.e each note equidistant from adjacent notes,not work in all keys?This system of tuning seems to apply only to stringed instruments.

I wasn't sure where to put this question,the answer for all I know may have more to do with the nature of hearing,or something else not related to physics or math.Feel free to move it.

[Qfwfq]

Well, we don't have a specific forum for music but it is akin to math and acoustics is certainly physics, further more these are pleasant topics!

Yes, the natural (not tempered) scale is exact for the one tonic pitch but it is off for others. this is quite a concern for performers of music that was composed for a natural scale, which is best performed exactly so. With a tuning different from that intended by the composer, one doesn't hear quite the effect he had in mind and in some cases it can remove all the aesthetic subtlety. The trouble is that it isn't easy to completely re-tune a harpsichord or similar instruments, it isn't practical to do so on stage between pieces, so care is needed in chosing which pieces to perform in the same concert. Performers may sometimes tune for a compromise between things they plan to play (not an easy task) but the tempered scale is simply too much of a compromise for some early baroque works, especially those based on alternations between harmonies and discordances.

An exact diatonic scale is tuned for by making the fifth to have 3/2 the frequency of the tonic and then tuning other notes in sequence as 3/2 or 3/4 of the previous, until the next note would be slightly more than twice the tonic. At this point, an exactly double frequency is used and the next octave is tuned. This gives a scale with some exactly perfect harmonies. The trouble is that, if you start from any of the notes after the tonic, and make the exact scale for that one, you get one that has no pitches exactly equal to the first scale excepting the note you started the second one with. This means that transposition doesn't work, wihtout re-tuning.

The tempered scale is a geometric progression of frequencies up through the octaves, if sharps are included. This means that the semitones are exactly equal, so transposition doesn't change intervals, but no interval is really a perfect harmonic. The fifth is slightly less than 3/2 (1,4983070768766814987992807320298... instead of 1.5) the tonic. This compromise came to be preferred as baroque composers increasingly wanted transposition to be free and flexible because the styles were somewhat changing and, from later baroque onward, composers continued to reason in this way, e. g. with the interval between do and fa being exactly the same as that between mi and la, and that between fa and la sharp. It's more practical for the composer, just count the semitones and you're done, but gone are some of the effects used in earlier styles.

In short, scores are best performed with the tuning the composer had in mind.

[Tormod]

Great post, Q!

Originally Posted by Qfwfq

In short, scores are best performed with the tuning the composer had in mind.

In an audible sense, perhaps, this might be correct but I am not so sure.

I'd argue (as a musician) that scores sound best in whatever key an arranger chooses to use, as long as the music is played well. Often music sounds best when NOT performed with the tuning the composer had in mind.

Today, when a lot of music is composed on computers using samples and loops, transposing on the fly is no longer an art and most pieces can be played in any key - in fact, many modern composers pretty much dismisses any concept of key.

Things evolve in the music business, too. For example, the chamber tone (which is now A=440 Hz), used to be about 440 when Beethoven composed his music. And many of the instruments that were used in, say, the early 1800s, are no longer around or were swapped out for something that sounded better. And for all practical purposes, most performers don't get to play Bach's organ concertos on a church organ, or his lute concertos on a lute (they use a guitar instead, which wasn't invented yet).

[Qfwfq]

Hmmm, I guess I've long acquired an interest in philological performances. Some things from the days of Monteverdi or Frescobaldi are just not meant for tempered scale. Buy a record of Koopman on organ, you can see that he choses which church'es organ to play each piece on.