In 75 words or less, how does Bachs temperament eliminate the seemingly random harshness we encounter in all the other familiar unequal temperaments?
The problem is enharmonically misspelled notes where the intonation and tonal music disagree. The other temperaments sound rough whenever Ab, Db, Gb, or Cb are in the bass; or those or D#, A#, E#, or B# in the treble. This is egregious in all the regular “meantone” layouts, and in the French/Italian “ordinary” temperaments, and Werckmeister and Kellner. Others such as “Vallotti” and the Neidhardts merely ameliorate the problem through compromises, without solving it. All this math, and additional musical analysis in Bach’s repertoire, is covered thoroughly in the May 2005 half of the Early Music article. I ask that any more probing mathematical questions be held until the people asking them have read and thoroughly digested that second part, and (even more importantly) have listened to Bach’s and other temperaments hands-on at a harpsichord. There is no substitute for playing through the repertoire–and listening very closely in hands-on thoroughbass improvisation–to experience this c
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