1. From Trevor and Víctor

We are composers Trevor Bača and Víctor Adán, creators of Abjad, and our earliest collaborative work dates back to shared undergraduate years in Austin. It was the mid- to late-90s and we found ourselves interested in ways of building up ever larger sets of musical materials in our scores, with ever greater amounts of musical information.

Our work then began with pitch formalization, creating materials in C and then writing the results as MIDI to hear what we’d created. Turns out that this is a fairly common gateway into materials generation for many composers, and so it was for us. Probably this was, and is, due to the ever present availability of MIDI and, to a lesser extent, CSound. But even back then it was clear to us to finding ways to embody other aspects of the musical score – from nested rhythms to the different approaches to the musical measure to the arbitrarily complex structures possible with overlapping musical voices – would require a wholly different level of consideration, and different development techniques as well.

As an example, consider flat lists of floating-point values. This basic data structure, together with the constant need some type of quantification or rounding, feeds much of most composers’ work with CSound, pd and the like. It is a good thing, therefore, that essentially all modern programming languages include tools for manipulating flat lists of floats out of the box, or in the standard library. But what happens when you want to think of pitch as something much more than integers for core values with, perhaps, floats for microtones? What if you want to work with pitches as fully-fledged objects? Objects capable of carrying arbitrarily large sets of attributes and values? Objects that might group together, first into sets, and then into larger assemblages, and then into still larger complexes of pitch information loaded, or even overloaded, with cross-relationships or textural implications? Carrying this surplus of information about pitch, or the potential uses of pitch, in data structures limited to, or centered around, the list-of-floats paradigm then becomes a burden.

And what of working with rhythms not only as offset values, as implied by the list-of-floats approach, but as arbitrarily nested, stretched, compressed and stacked sets of values, as allowed by the tupleting and measure structures of conventional score? A different approach is needed.

There was, and still is, no reason to believe that general purpose programming languages and development tools should come readily supplied with the objects and methods most suitable for composerly applications. And this means that the attributes of a domain-specific language that will best meet the needs of composes interested in working formally with the full complement of capabilities in traditional score remains an open question.

We continued our work in score formalization independenly until 2005, Trevor in a system that would come to be called Lascaux, and Víctor in a system dubbed Cuepatlahto. We experimented with C, Mathematica and Matlab as the core programming languages driving our systems before settling independently on Python, Víctor out of experiece at MIT, where he was working on his masters at the Media Lab with Berry Vercoe, and Trevor out of the working necessities of a professional developer and engineer.

We passed through indepedent experiences using Finale, Sibelius, Leland Smith’s SCORE, and even Adobe Illustrator as the notational rendering engines for Lascaux and Cuepatlahto. Through all of this, both systems were designed to tackle a shared set of problems. These included:

1. The difficulty involved in transcribing larger scale and highly parameterized gestures and textures into traditional Western notation.
2. The general inflexbility of closed, commercial music notation software packages.
3. The relative inability of objects on the printed page in conventional score to point to each other — or, indeed, to other objects or ideas outside the printed page — in ways rich enough to help capture, model and develop long-range, nonlocal relationships throughout our scores.

Afer collaborating on a joint paper describing the two systems, and after discussing collaborative design and implementation at length, both online and in weekends’ long review of our respective codebases, we decided to combine our efforts into a single, unified project. That project is now Abjad.

In our work on Abjad we strive to develop a powerful and flexible symbolic system. We picked the phrase ‘formalized score control’, or FSC, as a nod to Xenakis, who was so far ahead in so many ways, and also to highlight our primary project goal: to bring the full power of modern programming languages, and tools in mathematics, text processing, pattern recognition, and modular, iterative and incremental development to bear on all parts of the compositional process.

2. Why MIDI is not enough

Given that Abjad models written musical score, it might seem odd for MIDI to be even mentioned in this manual. Yet, until fairly recently, MIDI has played a role (sometimes tangential, other times fundamental) in a variety of software tools related to music notation and engraving.

2.1. A very brief overview of MIDI

MIDI (Musical Instrument Digital Interface) was first introduced in 1981 by Dave Smith, the founder of Sequential Circuits. The original purpose of MIDI was to allow the communication between different electronic musical instruments; more specifically, to allow one device to send control data to another device. Typical messages might be “note On” (play a note) “note Off” (turn off a note). A MIDI “note” message, for example, is composed of three bytes: the first byte (the Status byte) tells the device what kind of message this is (e.g. a Note On message). The second byte encodes key number (which key was pressed) and the third byte, velocity (how hard the key was pressed). It should be clear that a Note in this context means something very different than Note in the context of a traditional printed score. While the bias towards keyboard interfaces is clear in the definition of the MIDI Note control message, one can still give the MIDI note a more general use by reinterpreting “key number” as pitch and “velocity” as loudness, the usual perceptual correlates of these control changes as well as the most meaningful musical parameters in western music.

With the subsequent proliferation of music production software, the SMF (Standard Midi File) was introduced to allow the recording and storage of the control data from a MIDI stream. The SMF required a time stamp to keep track of when control messages took place. These are called “delta-times” in the SMF specification.

“The MTrk chunk type is where actual song data is stored. It is simply a stream of MIDI events (and non-MIDI events), preceded by delta-time values.”

In combination with the MIDI Note message, the addition of duration now allowed one to have a minimal but sufficient machine representation–a machine score–of music requiring only these parameters: duration, pitch and loudness. Such is the case of most piano music.

2.2. Limitations of MIDI from the point of view of score modeling

But, alas, there is much more information in a printed score that can not be practically encoded in a SMF. Common musical notions such as meter, clef, key signature, articulation, to name only a few, are ignored. A desire to include some of these concepts in MIDI is evident in the inclusion of some so called meta-events. From the SMF specification: ” specifies non-MIDI information useful to this format or to sequencers.” Examples of meta-events are Time Signature and Key Signature. In addition to the semantic elements just mentioned, there are also the typographical elements (such as line thickness, spacing, color, fonts, etc.) that all printed scores carry. This extra layer of information is completely absent in a SMF. However, from the point of view of encoding a printed score, the main limitation of MIDI is not the lack musical features or the absence of typographical data, but the assumption that musical durations, pitches and loudnesses can be each fully and efficiently encoded with integers or even fractions. In a printed score, this is not the case for any of them. MIDI encodes only magnitudes: time interval magnitudes, pitch interval magnitudes, velocity magnitudes. While these may be sufficient attributes for an automated piano performance, they are not all the attributes of notes in a printed score.

2.3. Written note durations vs. MIDI delta-times

Assume a fixed tempo has been set. Assume that all magnitudes are represented with (and limited to) rational numbers. A time interval magnitude d = 1/4 has an infinity of equivalent representations in terms of magnitude: d = 1/4 = 1/8 * 2 = 1/8 + 1/16 * 2 ... etc. So, for example, while equivalent in magnitude, these are not the same notated durations:

abjad> m1 = measuretools.AnonymousMeasure([Note("c'4")])
abjad> m2 = measuretools.AnonymousMeasure(Note(0, (1, 8)) * 2)
abjad> tietools.TieSpanner(m2)
abjad> m3 = measuretools.AnonymousMeasure([Note(0, (1, 8))] + Note(0, (1, 16)) * 2)
abjad> tietools.TieSpanner(m3)
abjad> r = stafftools.RhythmicStaff([m1, m2, m3])
abjad> iotools.write_expr_to_ly(r, 'MIDI-1')

2.4. Written note pitch vs. MIDI note-on

A similar thing happens with pitches. In MIDI, key (pitch) number 61 is a half tone above middle C. But how is this pitch to be notated? As a C sharp or a B flat?

abjad> m1 = measuretools.AnonymousMeasure([Note(1, (1, 4))])
abjad> m2 = measuretools.AnonymousMeasure([Note(('df', 4), (1, 4))])
abjad> r = Staff([m1, m2])
abjad> iotools.write_expr_to_ly(r, 'MIDI-2')

2.5. Conclusion

MIDI was not designed for score representation. MIDI is a simple communication protocol intended for real-time control. As such, it naturally lacks the adequate model to represent the full range of information found in printed scores.

3. Why LilyPond is right for Abjad

Early versions of Abjad wrote MIDI files for input to Finale and Sibelius. Later versions of Abjad wrote .pbx files for input into Leland Smith’s SCORE. Over time we found LilyPond superior to Finale, Sibelius and SCORE.

3.1. Nested tuplets works out of the box

LilyPond uses a single construct to nest tuplets arbitrarily:

\new stafftools.RhythmicStaff {
    \time 7/8
    \times 7/8 {
        c8.
        \times 7/5 { c16 c16 c16 c16 c16 }
        \times 3/5 { c8 c8 c8 c8 c8 }
    }
}

abjad> staff = stafftools.RhythmicStaff([Measure((7, 8), [])])
abjad> measure = staff[0]
abjad> measure.append(Note('c8.'))
abjad> measure.append(Tuplet(Fraction(7, 5), 5 * Note('c16')))
abjad> spannertools.BeamSpanner(measure[-1])
abjad> measure.append(Tuplet(Fraction(3, 5), 5 * Note('c8')))
abjad> spannertools.BeamSpanner(measure[-1])
abjad> Tuplet(Fraction(7, 8), measure.music)
abjad> staff.override.tuplet_bracket.bracket_visibility = True
abjad> staff.override.tuplet_bracket.padding = 1.6
abjad> show(staff)

LilyPond’s tuplet input syntax works the same as any other recursive construct.

3.2. Broken tuplets work out of the box

LilyPond engraves tupletted notes interrupted by nontupletted notes correctly:

\new Staff {
    \times 4/7 { c'16 c'16 c'16 c'16 }
    c'8 c'8
    \times 4/7 { c'16 c'16 c'16 }
}

abjad> t = Tuplet(Fraction(4, 7), Note(0, (1, 16)) * 4)
abjad> notes = Note(0, (1, 8)) * 2
abjad> u = Tuplet(Fraction(4, 7), Note(0, (1, 16)) * 3)
abjad> spannertools.BeamSpanner(t)
abjad> spannertools.BeamSpanner(notes)
abjad> spannertools.BeamSpanner(u)
abjad> measure = Measure((4, 8), [t] + notes + [u])
abjad> staff = stafftools.RhythmicStaff([measure])
abjad> show(staff)

3.3. Nonbinary meters work out of the box

The rhythm above rewrites with time signatures in place of tuplets:

\new Staff {
    \time 4/28 c'16 c'16 c'16 c'16 |
    \time 2/8  c'8  c'8 |
    \time 3/28 c'16 c'16 c'16 |
}

abjad> t = Measure((4, 28), Note(0, (1, 16)) * 4)
abjad> u = Measure((2, 8), Note(0, (1, 8)) * 2)
abjad> v = Measure((3, 28), Note(0, (1, 16)) * 3)
abjad> spannertools.BeamSpanner(t)
abjad> spannertools.BeamSpanner(u)
abjad> spannertools.BeamSpanner(v)
abjad> staff = stafftools.RhythmicStaff([t, u, v])
abjad> show(staff)

The time signatures 4/28 and 3/28 here have a denominator not equal to 4, 8, 16 or any other nonnegative integer power of two. Abjad calls such time signatures nonbinary meters and LilyPond engraves them correctly.

3.4. Lilypond models the musical measure correctly

Most engraving packages make the concept of the measure out to be more important than it should. We see evidence of this wherever an engraving package makes it difficult for either a long note or the notes of a tuplet to cross a barline. These difficulties come from working the idea of measure-as-container deep into object model of the package.

There is a competing way to model the musical measure that we might call the measure-as-background way of thinking about things. Western notation pratice started absent any concept of the barline, introduced the idea gradually, and has since retreated from the necessity of the convention. Engraving packages that pick out an understanding of the barline from the 18th or 19th centuries subscribe to the measure-as-container view of things and oversimplify the problem. One result of this is to render certain barline-crossing rhythmic figures either an inelegant hack or an outright impossibility. LilyPond eschews the measure-as-container model in favor of the measure-as-background model better able to handle both earlier and later notation practice.