Accomodating temperamental characteristics

And if we change our tuning, that music would no longer be playable as it was intended to be heard, right? Equal temperament - the bland, equal spacing of the 12 pitches of the octave - is pretty much a 20th-century phenomenon. After all, the music of Bach, Mozart, Beethoven, et al, was written for 12 equally-spaced pitches to the octave, right?Eight pitches have virtually perfect major thirds on them - all except Db, F#, Ab, and B, whose major thirds are all about 427 cents.A third of 427 cents sounds like this: WAWAWAWAWAWAWAWAWA...!!! (Trust me on this.) All of the fifths are about 696 cents except for one, that on Ab, which is 737 cents and sounds terrible.A pure perfect fifth is a 3 to 2 frequency ratio; if A pure perfect fifth should be 702 cents wide, which is just about 7/12 of an octave; our current equal-tempered tuning accomodates perfect fifths (at 700 cents) within 2 cents, which is closer than most people can distinguish, but the thirds (at 400 cents) are way off, and form audible beats that are ugly once you're sensitized to hear them. There was no one invariable meantone tuning; before the 20th century, tuning was an art, not a science, and each tuner had his own method of tuning according to his own taste.The following is a chart of what was initially the most common form of meantone, called 1/4-comma meantone, first documented by Pietro Aaron in 1523, though he didn't draw it out to all twelve pitches: A major third and perfect fifth on the same pitch, of course, make up a major triad, the most common chord in European music from 1500 to 1900 - the meantone era.(If you'd like this explained in more detail, visit my Just Intonation Explained page.) The size of a pure 5:4 major third is 386.3 cents, a cent being one 1200th of an octave, or one 100th of a half-step.

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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).

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.

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?

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