This is a program which will generate 3 midi files, using a few parameters. The three files encode the three movements of the concerto:

• Rondo
• Slow variations
• Fugue

These may be played in any order, although the order above seems best to me. What you hear are the 9 channels of the midi player, which sound quite bad on my computer. So this is basically conceptual art, as I cannot really control orchestration. Whatever comes out may be, if not beautiful, interesting as an automatic serial, thematic composition. The reason it exists is hidden in the mists of time... I make no claims about its pianistic or orchestral value, and if you don't like the result, try some other parameters.

The analysis, partly musical and partly algorithmic, follows.

• Rondo

  The rondo is built from joins of the various linear aspects of the series: orig+orig, orig+mirror, orig+crab, orig+table, mirror+orig, mirror+mirror, ... etc. There are two or three terms in the joins. These are then folded over several voices; the shortest result is the rondo refrain, the others will serve as episodes. Therefore, the music is based on certain polyphonic patterns, not on tunes. The durations, although irregular, are determined by pitch, so the folds which start with the same pitches will sound the same initially, and change only when a different linear aspect appears in the join.

  Consider, for instance, the following example: ( everywhere, obviously, 0=C 1=C# 2=D, etc. )
  

  K0 and K1 agree up to the A natural in the second bar, where the 2nd original form (resp. table form) enters. They will (hopefully) be perceived as variations. Note that K0 is a join folded over 3 voices, while K1 is folded over 4.

  In addition to these basic lines there are two additional voices playing the various linear aspects, in equal notes, in optimal harmony, and some doubling. The general harmony is refined by trills.

  The rondo ends with a coda.

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• Slow variations

  This is a form of my own invention, from ancient times, say 1970. Pick several transpositions of the 4 series forms, and fix the rhythms so that if the first selection has an E on the 3rd beat, the following one will also have an E on the 3rd beat. The more such matches, the better. The last selection should also match the first, e.g. both have a C# on the last beat. Then the selections can be played in order, cyclically, with some feeling of structure, because of the repeating matching tones.

  Notice that this is a purely monophonic device.

  Here is an example; the matching notes are shown in color. Each measure contains one series presentation, with its proper rhythm -- let's call this a satz. In this case, all satzes share an Ab in the same position; satzes 1,2,3 share an F#, satzes 3,4 an Eb, etc. Also, satz 2 is a crab-mirror transpose of satz 1, satz 4 a mirror transposition of satz 1, etc. These 4 measures could form a full satz cycle, as they are all linked by the Ab near the end.

  The movement opens with two presentations of the cycle of satzes, with matching tones emphasized:

  • during the first presentation of the satzes by various orchestra sections and solo instruments, the piano plays these matching tones in octaves in both hands;
  • during the second presentation, the piano plays the satzes, while the orchestra supplies tone color for the matching tone. These are characteristic instrumental sounds, e.g. an A could be played on open strings by cello, violins or violas; a C by flutes (the lowest available note) or open strings of violas and cellos.

  After these presentations, the satzes – by now reasonably familiar – are played in counterpoint by the piano and orchestra. In the following section, the piano plays the satz cycle in counterpoint with fillers, while the orchestra again supplies tone colors. In addition, cadences which occur in the counterpoint are emphasized, leading to the movement final close: an arpeggio presentation of the series on the piano solo, followed by the final cadence, piano and orchestra.

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• Fugue

  • fugue: subject, countersubject

    The fugue subject+countersubject is generated by examining various joins, looking for short motives in "near enough" transposition. For instance, in the following example, the six quarter notes form such a subject: F# A# B is transposed down by 2,3 semitones to Eb G A.

    

    The "near enough transposition" may be considered a variant of "tonal answer". Of course, the transposition may be exact, e.g. :

    

    These motives are emphasized by longer durations; the short notes following will form the countersubject, but the preceding or intervening short notes, if any, add variety to the subject. The results are easily recognizable – maybe too obvious ... As all the linear aspects are examined, mirror forms (or cancer or table forms) are created for each subject.

  • fugue: expositions

    The fugue consists of expositions of the subjects in two, three and four voices. The program optimizes the exposition, by looking for best harmony under various transpositions, augmentation, subject choice and time of entry. Each such exposition is presented twice: first "rough", then after dissonance smoothing by trills and cambiatas. The presentations are separated by gaps of 1-2 measures. Most of the trilling material is assigned to the soloist.

    Here is an example of the "rough" form, in two voices, of an exposition based on the first subject above:

    

    The first two entrances of the subject are based on the join mirror+cancer+table, the third is from the join cancer+orig; they still sound as transpositions of each other.

    

  • fugue: filling gaps with series

    Finally, wherever the structure built above is too thin (the rests between the expositions, parts of the score where only one voice is active) one or two new voices are added, consisting of the various linear aspects of the series, with uniform durations. These are optimized for best harmony, and substitute for the usual episodes in a fugue.

    The fugue, like the other parts of the concerto, ends with a coda, built on the four-parts presentation.

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• General properties

  • dodecaphonic

    1. The set (or series) is a specific ordering of all twelve notes of the semitonal scale.
    2. No note is repeated within the set, once another note has followed it. More precisely,
    assuming the series: eb bb f# f D C B a e c# g# g ,

    

    is correct, as a rhythmic setting of a single C, but
    

    

    is not allowed.
    3. The set may be stated in any of its "linear aspects":
    

    prime	= original
    inversion	= mirror
    retrograde	= crab
    retrograde inversion	= crab-mirror	= table

    
    

    4. The set in any of its four linear aspects may be started upon any degree of the semitonal scale.

    All of this is very standard tone row framework, from the 1920–30s . The way I see it, each note has to be justified : around it must appear a transposition of one of the linear aspects, in right order. "Around" means "in all the voices"; "right order" means that if note B follows note A in the row, B should not start before A is heard.

    Once the series – i.e. the linear aspect and its transposition – has beens selected, and a note in that series played, the following note is already determined, so the only question is where and when will it appear. Practically, this means one must decide how long the current note lasts and whether the following note will be sounded in the same voice or in another voice – see "folding". In either case, that note may be preceded by a rest. This is all the arbitrariness there is in serial composition, plus the rarer choice the next series, when the current series ends.

    Thus, there are very restricted development possibilities which makes for easy programming; the choices are constrained by the requirement that the result should also be thematic.

  • thematic

    The music should contain recognizable themes and motives, i.e. recognizable by hearing by me. The structure should be reasonably obvious, i.e. not the passacaglia in "Wozzeck", rather a baroque passacaglia. Besides, as a concerto it must have the contrasts solo/tutti and a virtuosity element – about which I have no idea, as I cannot play any instrument. Eventually, there is much repetition and theme passing between the piano and orchestra – I could not think of anything better. In order to produce variety, the themes are repeated in counterpoint – again I could not think of any other variation method consistent with dodecaphony. The main impediment is that there are no shorter motifs than the complete series.

    • contrapunctal

      Besides dodecaphonic "themes", i.e. sections in one or more voices, containing notes of arbitary durations, the counterpoint uses simpler "passages" consisting of series, joins and fillers. All the notes in the passage have the same duration; note also that the passage is usually quite long: at least 12 notes for a series, and maybe longer. These play a role similar to scales and arpeggios in a tonal composition – somewhat neutral material surrounding the themes.

    • generating themes:

      • folding

        An example of folding a sequence of notes:

        Bb B Ab F# D C# A F Eb C G E D Bb F# F C# ...

        over 3 voices:
        
        Note that the durations are random, and a note may start anywhere, except before a preceding note (i.e. a note which precedes it in the sequence). This is a polyphonic procedure; the resulting harmony may need correction.

      • joins of series

        

      • cyclic joins

        The join above can be turned into a cyclic join:
        The cell is 20 notes long: C# D Eb Bb C E Ab G F B A F# E Bb Ab G B Eb F C. After any number of such cells, C# D must appear, to complete the 12 notes of the last series.

      • fillers

        A filler is a 2-voice counterpoint of a join and a series. It may use augmentation, but, in a single voice, all notes have the same duration. E.g.:
        
        Voice 1 is a 23 note join, voice 2 an augmented series, in "2nd species" counterpoint.

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• Parameters

  • random seed

    During composition, the program will make numerous choices – some deterministic, e.g. based on optimization, and some random. The random choices are made using a pseudorandom generator initiated with the supplied random seed. In particular, the basic series – unique for the whole concerto – is determined using this random generator. However, the other parameters supplied will also influence the series – tritone is an obvious case. The choice of harmony may also be such that the full composition fails, and it will then be restarted with a different seed.

  • tritone

    if specified, the basic series may not contain tritones between adjacent notes

  • harmony

    One may select the harmony type to be used overall. The choices are:

    • default

      This is an approximation to basic triadic harmony. Thirds and sixths get very high grades, fifth and octaves somewhat lower grades, major seconds are bad, minor seconds worst... Chords are graded by averaging all the intervals they contain.

    • self

      This means that only groups of 1,2,3 and 4 notes from the series, transposed or inverted will be considered consonant.

      For instance, if the series is:

      
      it should be harmonized only by the chords:
      
      with their transpositions and inversions. I have not experimented with this option, mostly because it is difficult to check that the harmony has been actually optimized in the various counterpoints.

    • 345

      A code like this means that the intervals of 3,4 and 5 semitones and their inversions are consonant. Unisons/octaves are always consonant (according to Schoenberg they should be dissonant) .
      All the other intervals are dissonant. Chords are graded by averaging all the intervals they contain.
      

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  Harmony and dissonance

  All these examples presume "default" harmony, i.e. seconds and sevenths dissonant.

  • reduce dissonance using trills

    An ad-hoc method to lessen dissonance (i.e. chords containing intervals of 1 or 2 semitones):

    	becomes
    	becomes

    With a lot of good will, the trills may be considered a stylistic or virtuoso element.

  • reduce dissonance using cambiata

    Assume, e.g. that the counterpoint algorithm produces:

    

    This contains some harsh dissonances, B+C and E+F (also G+F). These can be fixed (in this particular case) by:
    • deleting an offending tone, if that tone appears in an immediately preceding chord. E.g. delete E in tenor, as there is an E in alt before it.

    • replacing it with a consonant tone, if that tone appears in an immediately preceding chord, or in the same chord. E.g. place an F instead of E (there is an F in the chord already)

      This is consistent with the "justification" requirement for serial music. The counterpoint above becomes:

      

      with the modified notes in color. This example is, of course, quite artificial, containing very tame progressions – fully dodecaphonic examples are much more dissonant, and sound truly harsher than their "fixes".

    Both procedures mix notes from the different voices, so they produce some relief from serial predictability.

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• Enhancements

  • Passages

    • using series, joins and fillers in counterpoint

      Some of the techniques employed – e.g. slow variations – are inherently monophonic. Some variety is obtained by adding series, joins and fillers, as "neutral" passage material. The resulting counterpoint is optimized for harmony.

    • filling gaps with series, joins and fillers

      This technique is used in all the parts of the concerto. For instatance, in the fugue, up to four voices play the subjects/countersubjects, and other up to two voices "thicken" the resulting structure by playing series. This is similar to the use of scales and arpeggios in tonal music.

  • Coda, ending in a full cadence if possible

    • coda, desiderata:

      The coda should be playable by the whole orchestra and piano; traditionally a piece ends with a big bang. It should not be too long, but related to the preceding material (development problem, again). In addition it should be somehow cadential, with a feeling of full close – whatever that may mean. I decided that a close is one of a list of melodic and harmonic cadences, as explained below; it is played molto rallentando, leading to rest.

    • cadences

      A cadence is a harmonization of one of the melodic cadences:

      • LA SI DO
      • RE SI DO
      • SI RE DO
      • SI DO
      • RE DO

             (all of the above in minor or Major scales)

      • MI RE DO – Major scale only
      • MIb RE DO – minor scale only

      SI/RE - DO should be harmonized as V/V7/VII7b - I/i with any fitting triad for LA/MI/MIb. The fuller the harmony, the better the cadence. Components may be missing, e.g. I/i may be reduced to tonic only, V7 played without third or fifth – and the grade of the cadence reduced accordingly.
      If none of the cadences above is available, in any key, then the music will halt just by repeating some consonant chord – in the worst case, a single note.

      Such cadences may appear by chance in the counterpoint – they are not part of the design. In fact, the common melodic cadences DO SI DO and DO RE DO may also be there, although a single melodic dodecaphonic line should not contain them. In any case, I ignore DO SI DO and DO RE DO.

      Notice that the cadence harmony is not influenced by the overall harmony chosen. A full close is a tonal/melodic full close.

    • coda, techniques

      Each of the three parts of the concerto ends with a coda as described above. The program identifies cadences near the end of the section – these are note groups in all the voices, that may be rearranged as melodic cadences with their proper harmony – then emphasizes such a cadence by slower rhythm and orchestration. The last cadence recognized is particularly embelished, as it will serve to conclude the section.

Note by note composition – details

Here is a schematic of the basic step of adding one note to the current score. One may take the pitch class of that note as given – assuming that we are in the middle of a given linear aspect with a given transposition.

Conceptually, it does not matter if the composition is automated or not. Practically, I am limited by the 9 voices of the music chip, and by the basic decision to use shorties (e.g. no duration below 1/16, and all durations multiples of 1/16). Then the situation is neatly represented as a table:

The white cells are all rests. The gray and red area shows all the notes already in the score. In red is the current note (last entered: it lasts 3 shorties and starts at shorty 7, voice 4). The green area is where the new note may enter; its right boundary is not defined, but its left boundary is clear: it can start at shorty 7 , or later.

The bordered region is a possible choice: voice 6, starts at shorty 8, lasts 5 shorties. Once this choice has been made, the note may be modified:

allowing repetitions, rests, subdivision of shorties and possibly different octaves, e.g.:

( with the shorty a dotted quarter ).

Now, the question arises: how exactly do I make the choice 'voice 6, starts at shorty 8, lasts 5 shorties'? Exactly is the operative word for automatic composition, because the program must be told what to do, without ambiguity. Of course, the choice may be random. But if I want some structure, or, if you prefer, organic development? I cannot implement – even if I wanted to – "the music should express yearning for Harriet Smithson" or "the music should express joy/despair at Napoleon's victory". So what is left?

One can require that the new notes must be consonant with some pre-existing notes. Consonant according to arbitrary rules of harmony, pre-existing notes a thorough bass, or some notes which appeared earlier in the score.

Here the blue cells represent the pre-existing counterpoint. It may be that "next" enters on shorty 8, because it is dissonant with some grey, red or blue notes on shorty 7, and ends on shorty 12 because it is dissonant with some blue notes on shorty 13. Presumably the "next note" is consonant with the blue, grey and red notes on shorties 8 to 12.

The procedure will reduce the arbitrariness (very little arbitrariness in dodecaphony), and connect sections among themselves. But, apart from this contrapunctal solution, I cannot think of any other non-random way of composing. On the other hand, it is a simple method of continuous development.