Pure Data: Music and Mathematical Foundations

• Pure Data is a dual-use concept: it serves as a real-time tool for generating OSC messages in animated musical notation and as a minimal framework for constructing mathematical and computational objects from finite sequences.
• In computer music, Pure Data drives dynamic score transformations—such as continuous changes in position, color, and blur—by separating offline score preparation from online real-time control.
• In its foundational role, pure data redefines proof and computation by formulating all objects through finite sequences and codas, offering a unified approach to logic and algebra.

Searching arXiv for the supplied topic and ids to ground the article in current arXiv records. Pure Data designates at least two distinct research objects in contemporary literature. In computer music, Pure Data (Pd) is treated as a real-time composition and control environment that can drive animated notation rendered by INScore, with the score changing continuously during performance rather than remaining fixed (Calatayud, 2022). In recent foundations research, pure data denotes an axiomatic framework for mathematics and computing based on finite sequences of codas as the sole primitive, with logic, language, proof, and computation defined internally rather than assumed externally (Youssef, 2023). These usages are conceptually unrelated but structurally comparable in one respect: both reframe “data” as an active medium of transformation rather than a passive representation.

1. Terminological scope and principal usages

In the music-technology literature, Pure Data appears as Pd, a software environment used not primarily as an audio engine but as an event generator / control system, an algorithmic composition tool for notation, and an OSC sender / network client. In that role, Pd produces continuous control data—such as fades, movement, and color changes—and sends Open Sound Control messages to INScore, which updates graphical and notational parameters in real time (Calatayud, 2022).

In the foundations literature, pure data is defined very differently. The primitive notion is the finite sequence; a coda is a pair of data, and data is a finite sequence of codas. The framework proposes that all mathematical objects can be represented in this way, without primitive sets, primitive types, or external logical constants (Youssef, 2023). A subsequent development extends this program by introducing spaces as associative data and studying them through the semiring of their endomorphisms, with natural numbers, integers, rationals, boolean spaces, matrix algebras, Gaussian integers, quaternions, and integer octonions emerging as spaces or subspaces (Youssef, 19 Aug 2025).

A common misconception is that these two usages describe the same concept. They do not. In one case, Pure Data is a real-time graphical programming environment for music and notation control; in the other, pure data is a foundational ontology for mathematics and computing. The similarity is lexical, not theoretical.

2. Pure Data in computer music and animated notation

In the case study on animated notation, Pure Data is used mainly as the engine for sending continuous controller messages in OSC format to INScore, to be displayed on a screen. The system separates score preparation (offline) from real-time transformation (online). Offline, original string quartet fragments are segmented and rewritten in MuseScore, exported as MIDI, imported into NoteAbilityPro to generate GUIDO notation files, and then loaded by INScore into a scene. Online, Pure Data patches generate control streams, format them as OSC messages, send them over UDP, and thereby update notation parameters such as position, alpha, color, blur, rotation, and related graphical properties (Calatayud, 2022).

INScore is described as a system for “partitions musicales augmentées”, rendering GUIDO notation files and arbitrary graphical elements while supporting dynamic transformation of position, scale, rotation, color, alpha, blur, and shadow. Pd is combined with INScore because Pd provides flexible real-time data and algorithms, whereas INScore provides real-time visual notation. The result is animated notation / dynamic musicography, meaning notation whose visual and notational parameters change in real time during performance (Calatayud, 2022).

The paper distinguishes static computer scores, algorithmically generated scores, and animated notation / dynamic musicography. “Dynamic musicography” is defined as “all drawings that are annotated in a music score, and have the intention to be gestures (i.e. are intended to have a musical utterance)”. The emphasis is not on generating sound directly. Rather, the music is produced by manipulating what performers see, so that performer gesture is shaped by notation that morphs over time (Calatayud, 2022).

This suggests a shift in compositional control from sonic synthesis to visual prescription. A plausible implication is that Pd, in this context, functions less as a digital audio environment than as a notation controller whose outputs are interpreted by performers rather than by loudspeakers.

3. System architecture, OSC communication, and notation parameters

The Pd–INScore architecture is conceptually simple. Pure Data generates time-varying parameter trajectories, formats them with oscformat and a MobMuPlat-inspired oscformat-mobmuplat abstraction, and transmits them via netsend -u -b; netreceive -u -b also appears in the patch, indicating that Pd can receive OSC over UDP as well (Calatayud, 2022). The patch includes receive itl, receive itl_blur, receive itl_colorize, and receive itl_shadow, which indicate separate Pd channels for controlling different INScore features.

The paper explicitly lists several OSC format patterns: oscformat -f ssi, oscformat -f ssfiii, and oscformat -f ssiiiiiii. These correspond to different combinations of OSC strings, integers, and floats, matching INScore’s expected message signatures. A smaller connection patch implies a Pd abstraction conexiones that connects to localhost, likely using a fixed port such as 7001, on which INScore listens for OSC traffic (Calatayud, 2022).

INScore’s OSC interface follows an ITL (INScore Text Language) model, with addresses of the form /ITL/scene .... Although explicit full command strings are not printed, the paper makes clear that separate OSC message types control distinct score-object attributes: alpha, X and Y position, blur, shadow, color, scale, angle, and overlapping (Calatayud, 2022).

The following table summarizes the parameters explicitly enumerated in the case study.

Parameter	Visual effect	Stated musical interpretation
Alpha channel	Fade in or out	Visual crescendos/decrescendos
X position	Horizontal displacement	Change in onset or phasing
Y position	Vertical displacement	Mapping to a different pitch interval
Blur and shadow	Distortion and layering	Foreground/background reading
Color	Colorized fragments or noteheads	Dynamic or expressive mapping
Scaling	Larger or smaller fragments	Dynamic or expressive emphasis
Angle	Rotation around a point	Bent scrolling lines or arcs
Overlapping	Superposed fragments	Chordal aggregates or extended shapes

Timing is described as continuous rather than metrical. There is no strict tempo grid; Pd sends continuous controller messages, performers see the notation change in real time, and their responses are partially indeterminate, aligning with the Indeterminacy movement (Calatayud, 2022). The paper briefly formulates a mathematical distinction between notation transformation and spatial transformation, the latter admitting continuous transformations in displayed score space. While explicit formulas such as $x(t)$ or $\alpha(t)$ are not printed in the original article, the description is that Pd computes time-dependent values for such parameters and streams them to INScore (Calatayud, 2022).

4. Case study: string quartet fragments, scrolling lines, and readability

The musical case study is built from an old string quartet by the author, with four parts: Violin I, Violin II, Viola, and Violoncello. Fragments are selected from the original quartet, rescribed in MuseScore, exported as MIDI, converted in NoteAbilityPro to GUIDO, and loaded into an INScore scene as discrete objects (Calatayud, 2022).

For performance, the author adds four scrolling lines, each a different color and assigned to one instrumentalist. Each line moves across the fragment similarly to a playback cursor, but each performer reads at his own time. Because the lines can be bent or reshaped under Pd control, the setup can produce independent gestures and timing, including phasing or indeterminate alignment (Calatayud, 2022).

The principal “possibilities of new notation enhanced by Pure Data in combination with INScore” are fading, spatial motion, blur and shadow, color, scaling, angle, and overlapping. Each is paired with a short speculation about performance: X displacement may generate phasing texture between players; Y displacement may prompt performance of a fragment at a different interval; blur and shadow may create foreground/background reading or distortion; color may correspond to dynamics or timbral instructions; scale, angle, and overlap may introduce emphasis or deliberate ambiguity requiring performer decisions (Calatayud, 2022).

The paper also presents readability experiments, including continuous rotation and continuous blurring of fragments. These are described not as complete pieces but as tests aimed at asking “what can be performed with a series of continuous transformations of notation, and its mathematical possibilities for analysis.” In that sense, the Pd–INScore system is not only a compositional device but also a platform for notation research and pedagogy (Calatayud, 2022).

A recurring controversy in animated notation concerns whether deformation merely reduces legibility. The paper does not deny readability problems; rather, it frames them as part of the inquiry. Overlapping and extreme rotations can compromise readability, and some transformations—especially angle and Y-position-based solmization—require explicit dialogue between composer and performer to establish rules of interpretation (Calatayud, 2022).

5. Pure data as a foundation for mathematics and computing

The foundational usage of pure data begins from one primitive: finite sequences. A coda is a pair of data, written schematically as $A:B$, and data is a finite sequence of codas. The empty sequence of codas is written (), and the pairing of two empty sequences is written (:) (Youssef, 2023). The algebra of data has two operations: concatenation, written $A\ B$, and pairing, written $A:B$. Binding is specified so that colon binds from the right first and less strongly than concatenation; thus $A:B:C$ means $(A:(B:C))$, while $A:B\ C$ means $(A:(B\ C))$ (Youssef, 2023).

The paper characterizes this as “pure data” because it is data “made from finite sequences of nothing.” There are no primitive sets, no primitive types, and no primitive logical constants. Bits, bytes, numbers, functions, logical values, language expressions, spaces, and morphisms are all defined as data constructed via this algebra (Youssef, 2023).

Equality of data is defined relative to a context $\delta$, which is a partial function from codas to data and is extended to all data. The paper gives the equations

$\alpha(t)$0

and

$\alpha(t)$1

Within a context, data can be empty, invariant, atomic, true, false, or undecided, depending on how it behaves under evaluation (Youssef, 2023).

The entire system is governed by a single axiom, the Axiom of Definition:

The empty context is valid. If $\alpha(t)$2 is a valid context and $\alpha(t)$3 is a definition, and if no coda is in the domain of both $\alpha(t)$4 and $\alpha(t)$5, then the union of $\alpha(t)$6 and $\alpha(t)$7 is a valid context.

This axiom regulates how contexts grow by adding disjoint definitions. A definition is a particular partial function from codas to data whose domain codas share a fixed invariant tag. The example pass is defined schematically by $\alpha(t)$8 (Youssef, 2023).

One of the strongest claims of the framework is that proof, computation, and deduction are the same process. An evaluation sequence

$\alpha(t)$9

is simultaneously a proof of equality, a computation from input to output, and a deduction carried out by context-driven rewriting. Unlike type systems with a Curry–Howard correspondence, the paper states that proof and computation are literally the same process in coda (Youssef, 2023).

6. Internal logic, spaces, and emergent mathematical structures

The framework defines logic internally rather than axiomatically. Data is true if it is empty, false if it is atomic, and undecided otherwise. Operations such as equality, conditional if, coercion bool, and boolean connectives xor, and, and or are defined as data operations. On true/false cases they reproduce classical truth tables; when inputs are neither empty nor atomic they remain undecided (Youssef, 2023). The law of excluded middle therefore does not hold as a general logical axiom inside the system, because some data and their negations can both remain undecided (Youssef, 2023).

An internal language is introduced as another definition rather than a primitive syntax. Parsing rules include composition and pairing:

$A:B$0

$A:B$1

together with projection rules and syntactic sugar. The paper further states that any finite byte sequence $A:B$2 can be compiled into pure data via $A:B$3 and that every finite byte sequence is valid syntax (Youssef, 2023).

The later paper on Pure Data Spaces develops the theory of spaces. A space is defined as associative data: data $A:B$4 such that, for all data $A:B$5,

$A:B$6

If $A:B$7 is a space, then data of the form $A:B$8 are “in” that space, and the neutral data is $A:B$9 (Youssef, 19 Aug 2025). Endomorphisms of a space form a semiring, with multiplication given by data product and addition given by a space-specific sum operation (Youssef, 19 Aug 2025).

Within this framework, familiar objects are said to emerge from minimally combinatoric constructions. The space (is a) yields organic natural numbers, with one data a^n for each natural number and homomorphisms corresponding to multiplication by $A\ B$0 (Youssef, 19 Aug 2025). Subspaces of (is a b) yield structures isomorphic to integers and to $A\ B$1 natural number matrices, while other constructions yield Gaussian integers, quaternions, integer octonions, boolean spaces, and set-like semilattice spaces (Youssef, 19 Aug 2025).

The paper explicitly compares this role of spaces and morphisms to Category Theory, while also stressing key differences: in coda, composition is always defined as data composition; morphisms are determined by the space structures; and objects and morphisms are not ontologically separate kinds (Youssef, 2023). A plausible implication is that the framework seeks a more operational replacement for external syntax/semantics distinctions, grounding both algebra and computation in rewriting on finite sequence structures.

7. Applications, limitations, and conceptual significance

In the music-technology setting, Pure Data is presented as well suited to animated notation because it is open-source, free, based on graphical patching, and real-time, data-driven. Although originally designed to process audio information to obtain sound, the paper emphasizes that “there’s always been the possibility of generating only data.” This permits Pd to function as a notation controller and as a flexible environment for experimenting with new score behaviors without building custom software (Calatayud, 2022).

The paper also notes technical difficulties: OSC formatting was initially problematic because INScore documentation relied on a non-vanilla Pd object; the author instead used MobMuPlat’s formatting subpatch, and float-to-integer conversion required community assistance. Readability, performer comprehension, screen layout, and unquantified concerns about latency and reliability remain practical constraints (Calatayud, 2022). The conclusion suggests that as animated notation gains popularity, new grammatical and orthographic rules will emerge (Calatayud, 2022).

In the foundational setting, the authors present pure data as a highly minimal system with one primitive and one axiom, intended to replace external logical axioms, set-theoretic axioms, type-theoretic judgment rules, and language axioms with internal constructions (Youssef, 2023). The framework is further extended toward mathematical machine learning, where spaces are discovered through classification tasks, and toward treatments of Gödel, Berry, Curry, and Yablo paradoxes, which are classified as undecidable data rather than contradictions (Youssef, 2023). The later spaces paper emphasizes both the richness of emergent structure and unresolved issues, including search complexity, the limited development of a theory of morphisms between spaces, and uncertainty about how far the framework can model analysis, topology, or geometry (Youssef, 19 Aug 2025).

The conceptual significance of the term therefore differs by field. In one literature, Pure Data redefines the score as a process that unfolds and changes while being played (Calatayud, 2022). In the other, pure data redefines mathematics and computing as operations on finite sequences whose meanings arise from contexts and whose higher structures arise as spaces and endomorphism semirings (Youssef, 2023). What unites these otherwise unrelated usages is a shared emphasis on transformation, real-time or formal, as the locus of structure.