and hot damn: We are both 'less than correct' ....
An excerpt from : Microtones and Scale Temperments - History Part 2 ...
http://www.cdss.fsnet.co.uk/temper/no-frames/history2.htm Various other people had their two cents worth, so to speak, in order to find ways of making ensemble playing and multi-part music in general easier. In particular, it had been noted that people had a natural tendency to sing intervals that were closer to being pure rather than the intervals within the 3-limit, 2:1 and 3:2. Also, there were considerable difficulties in fitting these intervals within the limits of a keyboard and make them serve the requirements of the musical form. One major contribution to the theories of temperament was Giuseppe Zarlino, the Maître de Chapelle of St. Marks in Venice, who in 1560 proposed inverting the Major and Minor tones of the upper group in Aristoxenus's scale to relieve the monotony of having two identically tuned halves of the scale. This gives us the following intervals:
C D E F G A B C'
Zarlino 0 204 386 498 702 884 1088 1200
This is, effectively, the puretone scaling. He then suggested reducing every fifth by two sevenths of a comma in order to lose the comma amongst the rest of the intervals. This method of redistributing commas was further refined by Francis Salinas, a blind musician and Professor of Music in Naples, among others. Here we have the start of Meantone temperament, the purpose of which is to redistribute the intervals such that the principle ones, such as fourths and fifths remain fairly true and others retuned so as to still fulfil their function within the scale. The most common way of achieving this is to flatten the first four fifths C-G, G-D, D-A, A-E reducing them enough to produce a true third - this also has the effect of removing the difference between the Major and Minor tones producing a "Mean" whole tone between the two, hence the name. Thus the Major thirds and the Minor sixths are true whereas the fifths are a little flat.
The problem with this is that only keys that have few accidentals sound OK, others less so. This allows the use of the first six major keys in the cycle of fifths and the first three minor and also allows a certain degree of modulation from key to key.One of the most problematical intervals is the fifth G#-D#. It is way too sharp and its inversion too flat - this is known historically as the Wolf Tone, so called because the mistuning was reminiscent of the howling of wolves - there are other wolves but this one is the most disturbing. The Wolf Tones are probably the main reason that composers of the period avoided using keys with a large number of accidentals. For example, Mozart rarely, if ever, composed any works in Db, F#, Ab and B Major or C#, Eb, F, F# and G# Minor as these keys make wolf tones stick out like a sore thumb. Curiously he also avoided B Minor which is all the more odd when you consider that his favourite key was D Major, closely followed by C and Bb Major<5>.
It is worth remembering at this point that the main reason for all this tomfoolery is the burgeoning development of the keyboard. Because there is a physical limitation on the number of keys which can be used to play notes, some means had to be found to permit the tonalities demanded by developments in polyphonic music to work within this limitation <6>. Incidentally, a good orchestra will effectively play in puretone temperament but will constantly adjust its intonation so as to achieve the most concordant sound (remember that concordancy or discordancy is judged solely by its perceived effect on the ear) but with the pianoforte <7> becoming a more dominant force in Western composition and composers seeking to explore this new tonal palette, some means had to be found to facilitate their requirements.
Both Zarlino and Salinas knew about equal temperament but disliked the severe mistuning inherent in the thirds and sixths but, like the moving hand of Omar, progress moves ever on. A French monk called Marin Mersenne was probably the first person to calculate the equal tempered semitone, the basis for equal temperament, in around 1620 although some people accredit Simon Stevin, an organ tuner at the workshops of Andreas Werckmeister with the discovery somewhat earlier in 1608. There is also evidence to suggest that a Chinese gentleman by the name of Chu Tsai-yu worked it out several years before its calculation in the West. Since the guiding principle for equal temperament is the redistribution of the comma amongst all the intervals of the scale and not just certain ones as most variations on meantone temperament seek to do, the best way to do this is to find the twelfth root of 2, i.e., that number when multiplied by itself twelve times equals 2: this number is 1.059463094; this interval ratio is the equal tempered semitone. Composers now had, at least in theory, complete freedom to modulate to any key without hearing wolf tones - but at a price.
It has been suggested that J.S. Bach wrote "The Well-Tempered Clavier" for equal temperament but this is erroneous. Research by the American musicologist John Barnes in the Seventies shows that what Bach probably used was a variation on meantone temperament devised independently by Francescantonio Vallotti and Thomas Young. It is almost certain that Bach knew of the existence of equal temperament but would have never used it himself as it would have been impractical to tune a clavichord this way since its pitch alters depending on how hard the keys are struck; in extreme circumstances, the pitch can vary by up to a Minor 3rd.
-snip-
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