You wrote, "your choices were these: you could limit your harpsichord playing to certain keys, or you could have a bewilderingly complex instrument that you would spend half your life tuning."
May I respectfully suggest that an alternative option exist now, which did not exist then?
First, please note that the tunings that you are describing are all extended meantone tunings (more or less), derived from stacking perfect fifths (P5's) that have been more-or-less tempered away from their just ratio (702 cents). If you widen the P5 to 720 cents, you get 5-tone equal temperament (5-tet). If you narrow the P5 to 686 cents, you get 7-tet. In between, you get the historical meantone tunings and today's 12-tet (P5=700). They are all defined by the same three elements:
1. Their period of repetition, the perfect octave (P8) which maps the 2nd partial to the P8.
2. Their generator, the tempered P5, which maps the 3rd partial to the P5.
3. Their comma sequence, which starts with the syntonic comma, which maps the 5th partial to the M3.(If the first comma in the sequence were the schisma, then the 5th partial would be mapped to the d4, defining the schismatic temperament.)
This combination of a P8 period, P5 generator, and comma sequence that starts with the syntonic comma, is called the "syntonic temperament" in "Dynamic Tonality" (see Wikipedia for both quoted terms).
An "isomorphic keyboard" is a keyboard that is generated by the same period, generator, and comma sequence as the temperament with which it is said to be isomorphic.
Isomorphic keyboards had not yet been discovered before the West standardized on 12-tet. Had they been known then, then the course of musical history might have been quite different.
With brings me back to your quote. A keyboard that is isomorphic with the syntonic temperament is brain-dead simple. It has the same fingering in every octave of every key of every tuning of the syntonic temperament. 12-tet, 1/4 meantone, 32-tet, 53-tet, infinity-tet -- same fingering. One can even bias the mapping of pitches to notes to support irregular tunings such as well-temperaments.
Therefore, IF ISOMORPHIC KEYBOARDS HAD EXISTED during the period that you describe, then they would have had a shot at becoming the de facto standard keyboard instrument subsequently.
Today, isomorphic keyboards DO exist. Anyone who is serious about exploring the frontiers of tonality should check them out (and Dynamic Tonality, which requires an isomorphic keyboard).
You wrote, "your choices were these: you could limit your harpsichord playing to certain keys, or you could have a bewilderingly complex instrument that you would spend half your life tuning."
May I respectfully suggest that an alternative option exist now, which did not exist then?
First, please note that the tunings that you are describing are all extended meantone tunings (more or less), derived from stacking perfect fifths (P5's) that have been more-or-less tempered away from their just ratio (702 cents). If you widen the P5 to 720 cents, you get 5-tone equal temperament (5-tet). If you narrow the P5 to 686 cents, you get 7-tet. In between, you get the historical meantone tunings and today's 12-tet (P5=700). They are all defined by the same three elements: 1. Their period of repetition, the perfect octave (P8) which maps the 2nd partial to the P8. 2. Their generator, the tempered P5, which maps the 3rd partial to the P5. 3. Their comma sequence, which starts with the syntonic comma, which maps the 5th partial to the M3.(If the first comma in the sequence were the schisma, then the 5th partial would be mapped to the d4, defining the schismatic temperament.)
This combination of a P8 period, P5 generator, and comma sequence that starts with the syntonic comma, is called the "syntonic temperament" in "Dynamic Tonality" (see Wikipedia for both quoted terms).
An "isomorphic keyboard" is a keyboard that is generated by the same period, generator, and comma sequence as the temperament with which it is said to be isomorphic.
Isomorphic keyboards had not yet been discovered before the West standardized on 12-tet. Had they been known then, then the course of musical history might have been quite different.
With brings me back to your quote. A keyboard that is isomorphic with the syntonic temperament is brain-dead simple. It has the same fingering in every octave of every key of every tuning of the syntonic temperament. 12-tet, 1/4 meantone, 32-tet, 53-tet, infinity-tet -- same fingering. One can even bias the mapping of pitches to notes to support irregular tunings such as well-temperaments.
Therefore, IF ISOMORPHIC KEYBOARDS HAD EXISTED during the period that you describe, then they would have had a shot at becoming the de facto standard keyboard instrument subsequently.
Today, isomorphic keyboards DO exist. Anyone who is serious about exploring the frontiers of tonality should check them out (and Dynamic Tonality, which requires an isomorphic keyboard).
Respectfully,
Jim Plamondon