Piano tunings - Pianotech controversy

This is moderately lomg, but I believe it to be well worth reading if
you have any interest in alternative keyboard tunings.

>}There is good
>}solid controversy on the Pianotech list right now, concerning ET vs WT vs
>MT,
>}(nobody championing Pythagorean or Just, at the moment).
>
>There are two other types of tuning for a 12-note keyboard that one should be
>aware of. I'm not sure what the contreversy or "championing" entails; if
>we're talking about music for which MT is even an option, that's a limited
>portion of the piano repertoire anyway. So I'm assuming this argument is more
>along the lines of interesting or favorite tunings.
>
>
>The first tuning I want to talk about was used on keyboard instruments in the
>late medieval period. Known as schismatic tuning, this was really nothing
>more than Pythagorean tuning, but instead of tuning 11 pure fifths from, say,
>Eb to G#, one would tune 11 pure fifths from, say, A to Cx (C double-sharp).
>Then, when one thinks one is playing in C major, one is actually starting on
>B#, and the following scale results:
>
>0 204 384 498 702 882 1086 1200 (cents)
>
>Compare this with one version of the major scale in just intonation:
>
>1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
>0 204 386 498 702 884 1088 1200
>
>Clearly the correspondence is close. The scales that look like G major and F
>major resemble other versions of the just major scale. Late medieval
>composers wanted to take advantage of the consonant thirds and sixths that
>characterized the new art, but did not want to stray too far from Pythagorean
>principles. Clearly this was one way, though one which did not offer the full
>triadic possibilities, even in one key, that later tunings such as meantone
>and ET provided.
>
>
>The second tuning I want to talk about is, as far as I know, and invention of
>mine. As you may have guessed, it is a subset of 22-tone equal temperament.
>Rather that approach the subject as I do in my paper, I will introduce
>7-limit harmony in another way.
>
>In MT, the two German augmented sixth chords (usually Bb D F G# and Eb G Bb
>C#) have a particularly concordant quality, resembling the seventh chords
>sometimes encountered in barbershop singing. The two incomplete French
>augmented sixth chords (Bb E G# and Eb A C#) also have a unique resonance.
>Huygens, in advocating meantone tempermant, pointed out (back in the 17th
>century, I believe) that these latter chords are used in compositions and in
>meantone are tuned very nearly 1/7:1/5:1/4. This may have been the first
>attempt to use the number 7 in explaining musical practice. The former chords
>are tuned very nearly 4:5:6:7. These are what Partch would call 7-limit
>otonal tetrads, while the latter chords are subsets of 7-limit utonal
>tetrads. The complete 7-limit utonal tetrads, in what I consider to be "root
>postion" (because of the perfect fifths) are C# E G# Bb and F# A C# Eb. In
>MT, the chords in question are tuned
>
>0 386 697 965 (otonal)
>
>0 310 697 931 (utonal)
>
>Compare with just intonation:
>
>1/1 5/4 3/2 7/4 (otonal)
>0 386 702 969
>
>1/1 6/5 3/2 12/7 (utonal)
>0 316 702 933
>
>Clearly the correspondence is quite close. However, with only two of each
>kind of tetrad available, there is not much 7-limit music one can make with
>MT on a piano keyboard.
>
>One of the main findings of my paper is that 22-tone equal temperament is a
>good way to make small sets of pitches contain large numbers of tolerable
>7-limit tetrads. One example can be tuned on a piano as follows: The
>intervals E-F and B-C are both tuned to 1/22 octave; the others are tuned to
>2/22 octave. Now, using the traditional keyboard notation, the list of otonal
>tetrads is
>
>Ab C Eb F#
>Eb G Bb C#
>D F# A C
>A C# E G
>E G# B D
>
>and the list of utonal tetrads is
>
>C# E G# Bb
>G# B C# Eb
>F Ab C D
>C Eb G A
>G Bb D E
>
>Notice that if the "third" of the chord is a white key, the others are black,
>and vice versa. This makes it very easy to find these chords when playing
>around with this tuning.
>
>How close is the correspondence to just intonation? These chords are tuned:
>
>0 382 709 982 (otonal)
>
>0 327 709 927 (utonal)
>
>Compare with just intonation again:
>
>1/1 5/4 3/2 7/4 (otonal)
>0 386 702 969
>
>1/1 6/5 3/2 12/7 (utonal)
>0 316 702 933
>
>Though not as good as meantone, clearly the approximation to just intonation
>is still good, and the payoff is a 2.5-fold increase in the number of tetrads
>to play with.
>
>The integrity of the diatonic scale is destroyed by this tuning, but my paper
>suggests other scales that could take its place. These scales have 10 notes
>instead of 7, so with only 12 notes on the keyboard, only a few modulations
>will be possible before one needs more keys.
>
>Incidentally, there is one historical tuning with 10 semitones of one size
>and 2 (E-F and B-C) of another. This is the tuning of Grammateus, where the
>white keys are in meantone, and the black keys divide the whole tones exactly
>in half.
>
>I would love to hear people's impressions (or better yet, music) of these
>piano tunings.

As something of an aside to this discussion, I have long thought that
movie music is be a great avenue for microtonal composers to get their
ideas to the proverbial masses.