Ting's PowerPoint Presentation "Scalable CPU Architecture" - PDF
Bach Organ Recital
by Dr. C. H. Ting
The Art of Fugue, BWV 1080 1748-50 AD
The Art of Fugue
Bach began composing the Art of Fugue in 1748 or 1749 and continued to work on it in 1750, the last year of his life. He saw the first eleven fugues through the engraving process, but died before its completion. His sons hastly published it with all his last works they found, which included four canons, an earlier version of fugue 10, two clavier arrangements of fugue 12, and an unfinished triple fugue in which the theme of the last fugue spells B-A-C-H, and a beautiful choral prelude "Wenn wir in hoechsten Noeten sind" dictated to his son-in-law Altnikol.
Most Bach scholars agreed that the Art of Fugue consists of fifteen fugues on one and the same theme. In it we meet with simple, double, and triple fugues, fugues built upon the theme altered either in melody or rhythm, fugues with strettos, with the answer inverted, both in notes of the same value and in diminution and augmentation, fugues in double counterpoint, in the octave, tenth, and twelfth, and lastly in which all three or four parts are inverted. These are forms from the very simplest up to the hardest that is conceivable, such as even Bach himself never produced before in his life.
The inner structure of this work is in a sequence of grand, majestic groups. The first group consists of the four opening fugues. The first fugue seems like the solemn repose of a winter's night and the theme is steeped in this feeling. The second fugue contrasts with the first by its somewhat more animated counterpoint. The third has the theme altered by inversion, and here the theme resembles the chorale melody in "Aus tiefer Not schrei' ich zu dir." It reveals a deep and yearning character of its own. It is continued in the fourth fugue on broader lines, rising from bar 61 onwards to a striking degree of power.
The second group consists of the fifth, sixth, and seventh fugues. In contrast to the first group, this group is worked out on a modified form of the theme, the effect depending on the rhythm. It exhibits the theme in different kinds of time, combined with itself in double counterpoint. The animation thus produced is increased by Bach in the sixth fugue by the use of the rhythms of the French overture style. In the seventh fugue, which introduces the theme in its natural time, diminished and augmented simultaneously, the animation verges on restlessness. The progressive development of the second group lies chiefly in these external qualities.
In the third group (fugues 8-11) external and internal elements combine to attain the climax. The principal subject is now associated with independent and contrasting themes. The eighth fugue begins at once with a subject of this kind, which glides in with stealthy, snake-like windings, and is full of peculiar individuality both in rhythm and in melody. After it has been thoroughly worked out, a second and very agitated theme, not less important in rhythm and melody, accompanies it. The strange little taps increase to hammering beats, and to animated violence of unrest. It is not until the original quiet movement has been restored that the chief subject enters and is treated fugally. The other theme now join in with it, and they work together to produce a progressive intensity of effect in a triple fugue in three voices with only 188 bars.
The four part fugue which follows has only one counter-subject which, owing to its being treated in counterpoint at the twelfth, does duty for two. Setting out with a mighty spring the counter-subject goes on without stop or stay, while against it the chief subject goes grandly and solemnly on in augmentation. In the tenth fugue, which is written in counterpoint at the tenth, there is again only one counter-subject. Its mild and flowing character is felt as a repose after what has gone before, and it prepares us for the full appreciation of the last fugue. The three themes of the eighth fugue are worked out again, but in four parts. Their expression intensifies to the last degree and entirely exhausts their harmonic capabilities. This concise and cyclic group reveals the master in that gloomy grandeur which we seek in vain in any other composer.
The fourth group consists of the last two pair of fugues. Bach's last intention with respect to the arrangement and order of the pieces, we only know up to the eleventh fugue. Nevertheless, it may be confidently assumed that of the last two pairs, those in three parts were intended to precede those in four. Each of a pair is note for note an exact inversion of the other, as if we were reading it in a mirror. Here again Bach soars above every technical difficulty. The pair in three parts are bright and animated from the beginning to the end, as if it were a pure accident that one of them happened to be the reflection of the other. Bach must have felt the purest pride in them. He arranged them for two claviers, adding a fourth obbligato part so that both claviers should be fully occupied. In the pair in four parts, the solemn repose of the opening theme returns upon us, but sublimated to a feeling which is best expressed by the words "a cold grandeur."
In reality the Art of Fugue is a single gigantic fugue in fifteen sections. The separate sections bear a relation to one another, and only through one another can they be perfectly understood. For instance, in the seventh fugue, where the theme is combined with its inversions, and in the natural, diminished, and augmented forms, it will scarcely be possible so much as to understand the relations of the parts to each other, unless the ear has been prepared for the task by the combinations that have gone before, and has become thoroughly familiar with the chief subjects. This preparation along will enable us to see the full justification of such ingeniously intricate forms.
The text is adopted from Johann Sabastian Bach by Philipp Spitta and J. S. Bach by Albert Schweitzer.
The Computer Organ
This computer organ was designed, built, and programmed by Dr. C. H. Ting to experiment music making using an IBM PC personal computer. His goal was a simple, low cost, and self contained system, which can be used to enter music score and play back his favorite organ music by J. S. Bach. In spite of the simplicity of this computer organ, the tonal quality is remarkably similar to an early pipe organ. With six channels of output, it is capable of reproducing faithfully all the organ pieces written by Bach and his Baroque contemporaries.
When Bach was complimented upon his organ playing, he replied: "There is nothing remarkable about it. All you have to do is to hit the right key at the right time and the music plays itself." Hitting the right key at the right time, that's what this computer organ does the best. Once that is done, Bach seems to take good care of his music by himself.
The most interesting feature of this computer organ is that different voices in a counterpoint composition can be assigned to different speakers which can be separated by long distances in a large hall. It is thus possible to spatially separate the voices in a polyphonic music and fill the hall with voices coming to the listeners from many directions. Separating the voices this way may help the listeners to appreciate better Bach's organ music.
In order to make the computer play polyphonic music, Dr. Ting developed a music description language which he used to code a large collection of Bach's organ works. This language consists of rules to construct sequences of chords and to string them together to form a playable piece. Each chord is specified by the notes in it, the assignment of notes to channels, and the duration of the chord. Measures are thus defined in terms of chord progressions, phrases in terms of measures, and whole pieces in terms of phrases.
This music description language is based upon the very powerful programming language 'Forth' which has been widely used for machine control and instrument automation. Many attributes of Forth, such as fast execution speed, effective memory utilization, ease in high level construction, and interactive interface between user and computer, make it possible to encode large and complicated music for the computer to compile and play back.
This implementation was a software simulator of 6 binary oscillators, driven by a master clock of 4.41 MHz. Each oscillator is loaded a 16 bit count, and the counter is decremented on the rising edge of the master oscillator. When a counter reaches 0, the same 16 bit count is reloaded, and the 1 bit output of the counter is flipped. Thus an oscillator can produce a continuous stream of 50% duty-cycle square wave. The outputs of 6 oscillators are summed to become the output wave. The sum of the oscillator outputs is thus from 0 to 6. It is then shifted left 4 bits to fill the eight bit .WAV data word.
To reduce computational load, the outputs of oscilators are sampled at 44.1 KHz, and therefore conform to the specifications of an 8 bit mono waveform, playable on most sound producing systems.
The source code of the fugues consists of a sequence of chords. Each chord specifies up to six voices, and a playing duration.