That's the whole problem of keyboard tuning, where you're limited to 12 steps per octave.Where do you put the gaps in your chains of perfect major thirds?So meantone tuning gives us eight usable major triads: on C, D, Eb, E, F, G, A, and Bb.
Meantone tuning appeared sometime around the late 15th century, and was used widely through the early 18th century.
(Return to top menu) Let's start with Europe's most successful tuning, if endurance can be equated with success.
The fifths would sound better at 702 cents, but at 696 or 697 you don't really notice the difference, especially if the chord is filled in with that perfect major third to smooth over the discrepancy.
This is where the practice originated in European music of never having an open fifth sounding by itself without a third filling it in: the spare perfect fifth isn't quite consonant, and that fact becomes obvious if the third isn't there.
For example, if C# vibrates at 550 cycles per second, E should vibrate at 660. None of the minor thirds in this meantone are quite that wide, but most of them are 310 cents, which is, pardon the expression, close enough for jazz.
(Actually, a narrow 7/6 minor third, often used by La Monte Young, is 266.8 cents, invitingly close to that 269; but 7/6 is an interval that was never recognized by European theory, though used in jazz and Arabic music among others.) Therefore the minor triads on C, C#, D, E, F#, G, A, and B are acceptable.If you want to use I, IV, and V chords in your piece, you can write in the keys of C, D, F, G, A, or Bb major.If you're writing in A major, you can't go to the V/V chord (B major), because it sounds awful. Conclusion Just Intonation - a whole other subject Those who attack equal temperament, the tuning of our modern pianos - as I do on my Just Intonation Explained page - seem to be attacking the great European musical tradition itself.In fact, it survived in pockets of resistance, especially in the tuning of English organs, all the way through the 19th century. The generating principle behind meantone was that it was more important to preserve the consonance of the major thirds (C to E, F to A, G to B) than it was to preserve the purity of the perfect fifths (C to G, F to C, G to D).Eight pitches have virtually perfect major thirds on them - all except Db, F#, Ab, and B, whose major thirds are all about 427 cents.