Jerome Bruner argued in 1964 that concepts must be traversed in three stages: enactive (bodily action), iconic (image), symbolic (language and notation). The order is not a preference — it is a developmental logic. Symbols that arrive before their sensorimotor grounding are thin; they may produce correct test performance while leaving the concept unrooted.

Maria Montessori, working fifty years before anyone had the vocabulary of embodied cognition, designed learning materials that implement Bruner’s sequence with unusual precision. The Golden Bead cube for “one thousand” is about the size of a large fist and weighs roughly one kilogram. You cannot represent “one thousand” on a tablet screen in a way that competes with carrying that weight across a room ten times.

This post is about what embodied cognition research tells us, why Montessori implements it correctly, and what we are giving up when we substitute glass surfaces for physical materials.

Bruner’s Three Modes

Jerome Bruner proposed in a 1964 paper and the subsequent book Toward a Theory of Instruction (Bruner, 1964; 1966) that knowledge is represented in three distinct, developmentally ordered modes:

Enactive: Knowledge encoded in action patterns. You know how to ride a bicycle; you cannot fully describe it in words; the knowledge is in your body. An infant knows what “cup” means because she has grasped cups hundreds of times — before she has the word.

Iconic: Knowledge encoded in images or perceptual representations. You can visualise the route without navigating it. You recognize a melody without playing it.

Symbolic: Knowledge encoded in language or other arbitrary symbol systems. The numeral “7” has no visual resemblance to seven objects. Its meaning is purely conventional and rule-governed.

The developmental sequence matters. A child who acquires a symbol before the underlying enactive and iconic representations are established has a label without a referent. She can produce the word or numeral correctly — and her understanding of it is correspondingly brittle. Transfer to novel contexts is poor; the concept does not generalise.

This is not a fringe view. It is the core claim of embodied cognition research, which has spent thirty years producing experimental evidence for it.

What Embodied Cognition Actually Shows

Lawrence Barsalou’s 2008 review in Annual Review of Psychology is the canonical synthesis (Barsalou, 2008). The central claim: cognition is not implemented in an abstract, modality-free computational system separate from the body. Perception, action, and interoception are constitutive of — not merely scaffolding for — conceptual thought. When you think about “lifting,” the motor cortex activates. When you think about “rough texture,” the somatosensory cortex activates. Concepts are grounded in the sensorimotor systems through which they were originally experienced.

This has a direct pedagogical implication. If mathematical concepts are represented using perceptual-motor simulation systems, then the quality of that simulation depends on the richness of the founding sensorimotor experience. A child who has handled physical objects of different weights has richer representational resources for arithmetic and measurement than one whose entire numerical experience has occurred on a flat, weightless, textureless glass surface.

Arthur Glenberg and colleagues tested this experimentally. In a 2004 study, first- and second-graders read short texts describing farm scenes (Glenberg et al., 2004). Children who physically moved toy objects (horse, barn, fence) to enact the described events showed dramatically better comprehension and inference performance than children who merely read and re-read the passages. The effect size approached two standard deviations in some conditions. Children who imagined moving the objects also improved, but less than those who actually moved them. The physical action was not decorative. It was causally relevant to understanding.

Glenberg extended this logic to arithmetic word problems (Glenberg, 2008). Children who physically manipulated objects while working through problems were better at identifying what was relevant and computing correct answers. The enactive engagement was improving not just memory of the text but mathematical reasoning.

Montessori Got There First

Maria Montessori opened the Casa dei Bambini on 6 January 1907 in a San Lorenzo tenement in Rome, enrolling approximately fifty children aged two to seven. She had no Barsalou. She had no Glenberg. She had children, materials, and the patience to watch what happened when children were allowed to choose their own work.

What she built was a pedagogical system that implements the Bruner sequence without exception.

The Golden Bead Material is the canonical example. Units: single glass beads. Tens: ten beads wired into a bar. Hundreds: ten bars wired into a flat square. Thousands: ten squares wired into a cube. The child can hold a unit bead between two fingers. She needs two hands to lift the thousand cube. The physical weight scales with place value. She experiences — proprioceptively — that “one thousand” is categorically heavier and larger than “one hundred” before she has seen the numeral or heard the word “thousands place.”

The Knobbed Cylinder Blocks illustrate a different principle. Four wooden blocks, each containing ten cylinders varying in height, diameter, or both. The child removes all cylinders and replaces them. If any cylinder goes into the wrong socket, the remaining cylinders will not all fit. The task cannot be completed incorrectly and left that way. Error control is mechanical, built into the material. The teacher need not intervene. The child corrects herself, alone, through the physical feedback of the materials.

Montessori called this controllo dell’errore — control of error. It is one of her most important insights: if the feedback is physical, the child internalises the standard rather than depending on external evaluation. The authority is in the material, not in the adult’s judgment.

The evidence that this works has accumulated across more than a century. Angeline Lillard and Nicole Else-Quest published a landmark study in Science in 2006, using a lottery-based design: children who had won a lottery to attend public Montessori schools compared with those who had not (Lillard & Else-Quest, 2006). Montessori five-year-olds showed significantly higher letter-word identification, phonological decoding, and applied mathematical problem-solving. The lottery controlled for family self-selection.

A 2025 national randomised controlled trial — 588 children across 24 public Montessori schools, with lottery-based assignment — found significant advantages in reading, short-term memory, executive function, and social understanding at the end of kindergarten, with effect sizes exceeding 0.2 SD (Lillard et al., 2025). These are not small effects for field-based school research. And the costs per child were lower than conventional programmes.

Korczak and the Right to Make Mistakes

Janusz Korczak ran an orphanage in Warsaw and wrote How to Love a Child in 1919 (Korczak, 1919) and The Child’s Right to Respect in 1929 (Korczak, 1929). His central argument was that children are not pre-adults — they are persons with full moral status and a right to their own experience, including the experience of making mistakes.

In August 1942 German soldiers came to his orphanage. Korczak was offered false papers, safe houses, multiple escape routes arranged by friends and admirers. He refused each time. He led approximately 192 children and staff to the Umschlagplatz and did not return.

I mention Korczak not as an appeal to emotion but because his argument is structurally connected to Montessori’s. If a child has moral status, she has the right to encounter the actual consequences of her choices — including physical ones. A material that makes incorrect placement physically impossible before the child has had the experience of trying and correcting is a different kind of education from a screen that prevents error altogether through invisible software constraints, or one that simply supplies the correct answer.

Error is information. Physical error is particularly rich information. Taking it away is not protection — it is impoverishment.

Buber: What a Screen Cannot Offer

Martin Buber’s essay “Education,” delivered as an address in 1925 and published in Between Man and Man (Buber, 1947), argues that genuine education requires what he calls an I-Thou relation: an encounter in which the other is met as a whole, irreducible subject, not an object to be managed.

A touchscreen is the paradigmatic I-It relation. It is smooth, frictionless, optimised for engagement, responsive to exactly the touch it was designed to respond to. There is no otherness, no resistance, no genuine encounter. The screen does not push back. The Knobbed Cylinder Block does — literally. If you try to force a cylinder into the wrong socket, the material resists. That resistance is not a flaw in the pedagogical design; it is the pedagogical design.

Buber also introduced the concept of Umfassung — inclusion — by which a teacher must simultaneously stand at their own pole of the educational encounter and imaginatively experience the pupil’s side. A screen cannot do this. It has no pole. Its responsiveness is a simulation of attention, not attention itself. Turkle’s later phrase — “simulated empathy is not empathy” — is the same argument in a different register.

The Tablet Problem

The educational technology industry has produced an enormous quantity of “educational apps” for young children. The research is beginning to catch up.

Kathy Hirsh-Pasek and colleagues identified four pillars that distinguish educational from merely entertaining digital content: active engagement, depth of engagement, meaningful learning, and social interactivity (Hirsh-Pasek et al., 2015). Reviewing commercially available apps, they found that most fail on three or four of these criteria. They produce interactions in the shallow sense — tapping, swiping — without the kind of self-directed, goal-oriented, socially-embedded activity that drives genuine cognitive development.

A 2021 meta-analysis of 36 intervention studies found that educational apps produced meaningful gains when measured by researcher-developed instruments targeting constrained skills (letter naming, counting), but small to negligible effects on standardised achievement tests (Kim et al., 2021). The apps teach what they teach. Transfer is limited.

By contrast, a 2023 scoping review of 102 studies found that physical manipulatives — block building, shape sorting, paper folding, figurine play — showed consistent benefits across mathematics, literacy, and science that transferred to standardised measures (Byrne et al., 2023).

The fundamental problem is haptic. A 2024 review of haptic technology in learning found that force feedback and texture information substantially improve spatial reasoning, interest, and analytical ability (Hatira & Sarac, 2024). Standard capacitive touchscreens — every tablet your child has encountered — provide no force feedback and no texture differentiation. Every object, regardless of its symbolic “weight” or “size,” feels identical under the fingertip.

The Golden Bead thousand cube weighs approximately one kilogram. You cannot represent that experience on a tablet. The symbol arrives without the sensation, and Bruner’s sequence is violated from the first tap.

What We Should Ask

The question is not whether tablets have educational uses — they clearly do, particularly for older children working at the iconic and symbolic levels, and for content where direct physical manipulation is impossible or dangerous. The question is whether we are using them in developmental contexts where the enactive stage has not yet been established.

A child who has carried the thousand cube across a room, stacked the hundreds into the square, and felt the weight difference in her hands has a different representation of place value from one who has tapped numerals on a flat screen. Both may perform identically on a constrained test tomorrow. Ask them a transfer question in six months and the difference will appear.

We are teaching children to operate symbols before giving them the physical experiences that make those symbols mean anything. The result is not ignorance — the children can tap the correct numeral — but brittleness. The concept is a label, not a root.

Montessori knew this. Bruner formalised it. The haptics literature is now confirming it experimentally. The difficult question is why we are still buying flat glass rectangles for classrooms when a box of wooden cylinders costs less and works better.

References

• Bruner, J. S. (1964). The course of cognitive growth. American Psychologist, 19(1), 1–15.
• Bruner, J. S. (1966). Toward a Theory of Instruction. Harvard University Press (Belknap Press).
• Barsalou, L. W. (2008). Grounded cognition. Annual Review of Psychology, 59, 617–645. DOI: 10.1146/annurev.psych.59.103006.093639
• Glenberg, A. M., Gutierrez, T., Levin, J. R., Japuntich, S., & Kaschak, M. P. (2004). Activity and imagined activity can enhance young children’s reading comprehension. Journal of Educational Psychology, 96(3), 424–436. DOI: 10.1037/0022-0663.96.3.424
• Glenberg, A. M. (2008). Embodiment for education. In P. Calvo & A. Gomila (Eds.), Handbook of Cognitive Science: An Embodied Approach (pp. 355–371). Elsevier.
• Lillard, A. S., & Else-Quest, N. (2006). The early years: Evaluating Montessori education. Science, 313(5795), 1893–1894. DOI: 10.1126/science.1132362
• Lillard, A. S., Loeb, D., Berg, J., Escueta, M., Manship, K., Hauser, A., & Daggett, E. D. (2025). A national randomized controlled trial of the impact of public Montessori preschool at the end of kindergarten. Proceedings of the National Academy of Sciences, 122(43). DOI: 10.1073/pnas.2506130122
• Korczak, J. (1919). Jak kochać dziecko [How to Love a Child]. Warsaw.
• Korczak, J. (1929). Prawo dziecka do szacunku [The Child’s Right to Respect]. Warsaw.
• Buber, M. (1947). Between Man and Man (trans. R. G. Smith). Kegan Paul. (Original German publication 1947; contains “Education,” address delivered 1925, and “The Education of Character,” address delivered 1939.)
• Hirsh-Pasek, K., Zosh, J. M., Golinkoff, R. M., Gray, J. H., Robb, M. B., & Kaufman, J. (2015). Putting education in “educational” apps: Lessons from the science of learning. Psychological Science in the Public Interest, 16(1), 3–34. DOI: 10.1177/1529100615569721
• Kim, J. S., Gilbert, J., Yu, Q., & Gale, C. (2021). Measures matter: A meta-analysis of the effects of educational apps on preschool to grade 3 children’s literacy and math skills. AERA Open, 7. DOI: 10.1177/23328584211004183
• Byrne, E. M., Jensen, H., Thomsen, B. S., & Ramchandani, P. G. (2023). Educational interventions involving physical manipulatives for improving children’s learning and development: A scoping review. Review of Education, 11(2), e3400. DOI: 10.1002/rev3.3400
• Hatira, A., & Sarac, M. (2024). Touch to learn: A review of haptic technology’s impact on skill development and enhancing learning abilities for children. Advanced Intelligent Systems, 6. DOI: 10.1002/aisy.202300731

Changelog

• 2026-02-03: Changed “lottery-based quasi-experimental design” to “lottery-based design” for Lillard & Else-Quest (2006). A lottery provides genuine random assignment; “quasi-experimental” implies the absence of randomisation, which is the opposite of what the lottery design achieved.

The Occasion

“Next level?” The question mark is doing real work. The framing HfMT chose for the day was appropriately provisional: not a declaration that AI has already transformed music education, but an invitation to ask whether, in what direction, and at what cost.

The invitations that reach me for events like this tend to come with one of two framings. The first is enthusiasm: AI is coming, we need to get ahead of it, here are tools your students are already using. The second is anxiety: AI is coming, it threatens everything we do, we need to protect students from it. Both framings are understandable. Neither is adequate to the curriculum question, which is slower-moving and more structural than either suggests.

I prepared three sets of handouts. The first covered data protection — the least glamorous topic in AI education, and the one that most directly determines what can legally be deployed in a university setting. The second covered AI tools for students: what exists, what it does, and what critical thinking skills you need to use it without being used by it. The third covered AI for instructors: where it helps, where it flatters, and where it makes things worse.

This post does not recapitulate the handouts. It addresses the question I kept returning to across all three workshops: what does this change about what a music student needs to learn?

What the Technology Actually Is

My physics training left me professionally uncomfortable with hand-waving — including my own. Before discussing curriculum implications, it is worth being specific about what these tools are.

The dominant paradigm in current AI — responsible for ChatGPT, for Whisper, for Suno.AI, for Google Magenta, for the large language models whose outputs are now visible everywhere — is the transformer architecture (Vaswani et al., 2017). A transformer is a neural network that processes sequences by computing, for each element, a weighted attention over all other elements. The attention weights are learned from data. The result is a model that can capture long-range dependencies in sequences — text, audio, musical notes — without the recurrence that made earlier architectures difficult to train at scale.

What this means practically: these models are trained on very large corpora, they learn statistical regularities, and they generate outputs that are statistically consistent with their training distribution. They are not reasoning from first principles. They do not “know” music theory the way a student who has internalised harmonic function knows it. They have learned, from enormous quantities of text and audio, what tends to follow what. For many tasks this is sufficient. For tasks that require understanding of underlying structure, it is not — and the failure modes are characteristic rather than random.

BERT (Devlin et al., 2018) showed that pre-training on large corpora and fine-tuning on specific tasks produces models that outperform task-specific architectures on a wide range of benchmarks. The same transfer-learning paradigm has spread to audio (Whisper pre-trains on 680,000 hours of labelled audio), to music generation (Magenta’s transformer-based models produce melodically coherent sequences), and to multimodal domains. The technology is mature, improving, and available to students now. Knowing what it is — not just what it produces — is the starting point for any sensible curriculum discussion about it.

The Data Protection Constraint

Before any discussion of pedagogical benefit, there is a legal boundary that most AI-in-education discussions skip over. In Germany, and in the EU more broadly, the deployment of AI tools in a university setting is governed by the GDPR (DSGVO, Regulation 2016/679) and, at state level in NRW, by the DSG NRW. The constraints are not abstract: they determine which tools can be used for which purposes with which students.

The core principle is data minimisation: only data necessary for a specific, documented purpose may be collected or processed. When a student uses a commercial AI tool to get feedback on a composition exercise and enters text that could identify them or their institution, that data may be stored, processed, and used for model improvement by an operator whose servers are outside the EU. Whether such transfers remain legally valid under GDPR after the Schrems II ruling (Court of Justice of the EU, 2020) is contested — and “contested” is not a position in which an institution can comfortably require students to use a tool.

The practical upshot for curriculum design is this: AI tools running on EU servers with documented processing agreements can be integrated into formal coursework. Commercial tools whose terms specify US-based processing and model training on user data cannot be required of students. They can be discussed and demonstrated, but making them mandatory puts students in a position where they must choose between their privacy and their grade.

This is not a reason to avoid AI in teaching. It is a reason to be honest about the regulatory landscape, to distinguish clearly between tools you can require and tools you can recommend, and to make data protection literacy part of what students learn. The skill of reading a terms-of-service document and identifying the data flows it describes is not a legal skill — it is a general literacy skill that matters for every digital tool a music professional will use.

What Changes for Students

The question I was asked most often across the three workshops was some version of: “If AI can already do X, should students still learn X?”

The question is less simple than it appears, and the answer is not uniform across skills.

Skills where automation reduces the required production threshold do exist. A student who spends weeks mastering advanced music engraving tools for score production, when AI can generate a usable first draft from a much simpler description, has arguably spent time that could have been better allocated elsewhere. Not because the underlying skill is worthless — it is not — but because the threshold of competence required to produce a working output has dropped. The student’s time might be more valuable spent on something that has not been automated.

Skills where automation creates new requirements are more interesting. Transcription is a useful example. Automatic speech recognition — using models like Whisper for spoken-word transcription, or specialised models for audio-to-score music transcription — is now accurate enough to produce usable first drafts from audio. This does not eliminate the need for transcription skill in a music student. It changes it. A student who cannot evaluate the output of an automatic transcription — who cannot hear where the model has made characteristic errors, who does not have an internalised sense of what a correct transcription looks like — is unable to use the tool productively. The required skill has shifted from production to evaluation. This is not a lesser skill; it is a different one, and it is not automatically acquired alongside the ability to run the tool.

Skills that automation cannot replace are those that depend on embodied, situated, relational knowledge: stage presence, real-time improvisation, the subtle negotiation of musical meaning in ensemble, the pedagogical relationship between teacher and student. These are not beyond AI in principle. They are far beyond it in practice, and the gap is not closing as quickly as the generative AI discourse sometimes suggests.

The curriculum implication is not “teach less” or simply “teach differently.” It is: be explicit about which category each skill falls into, and design assessment accordingly. An assignment that asks students to produce something AI can produce is now testing something different from what it was testing two years ago — not necessarily nothing, but something different. The rubric should reflect that.

What Changes for Instructors

The same three-category analysis applies symmetrically to teaching.

Routine task automation is genuinely useful. Generating first drafts of worksheets, producing exercises at different difficulty levels, transcribing a recorded lesson for later analysis — these are tasks where AI can save meaningful time without compromising the pedagogical judgment required to make use of the output. Holmes et al. (2019) identify feedback generation as one of the clearer wins for AI in education: systems that provide immediate, targeted feedback at a scale that human instructors cannot match. A transcription model listening to a student practice and flagging rhythmic inconsistencies does not replace a teacher. It extends the feedback loop beyond the lesson hour.

Content generation with limits is where AI is most seductive and most dangerous. A model like ChatGPT can produce a reading list on any topic, a summary of any debate in the literature, a set of discussion questions for any text. The outputs are fluent, plausible, and frequently wrong in ways that are difficult to detect without domain expertise. Jobin et al. (2019) and Mittelstadt et al. (2016) both document the broader concern with AI opacity and accountability: when a model produces a confident-sounding claim, the burden of verification falls on the user. An instructor who outsources the construction of course materials to a model, and who lacks enough domain knowledge to catch the errors, is not saving time — they are transferring risk to their students.

Hallucinations — outputs that are plausible in form but false in content — are not bugs in the usual sense. They are a structural consequence of how generative models work. A model trained to predict likely next tokens will produce the most statistically plausible continuation, not the most accurate one. For music education, where historical facts, composer attributions, and music-theoretic claims need to be correct, this matters. The model’s fluency is not evidence of its accuracy.

Personalisation is the most-cited promise of AI in education (Luckin et al., 2016; Roll & Wylie, 2016) and the hardest to evaluate in practice. The argument is that AI can adapt instructional content to individual learners' needs in real time, producing one-to-one tutoring at scale. The evidence in formal educational settings is more mixed than the boosters suggest. What is clear is that personalisation at scale requires data — and extensive data about individual students’ learning trajectories raises the same data protection concerns already discussed, in more acute form.

The Music-Specific Question

I want to be direct about something that came up repeatedly across the day and that the general AI-in-education literature handles badly: music education is not generic.

The skills involved — listening, performing, interpreting, composing, improvising — have a phenomenological and embodied dimension that does not map cleanly onto the text-prediction paradigm that most current AI systems instantiate. Suno.AI can generate a stylistically convincing chord progression in the manner of a named composer. It cannot explain why the progression is convincing in the way a student who has internalised tonal function can explain it. Google Magenta can generate a continuation of a melodic fragment that is locally coherent. It cannot navigate the structural expectations of a sonata form with the intentionality that a performer brings to interpreting one.

This is not a criticism of these tools. It is a description of what they are. The curriculum implication is that music education must be clear about what it is teaching: the product — a score, a performance, a composition — or the process and understanding of which the product is evidence. Where assessment focuses on the product, AI creates an obvious challenge. Where it focuses on demonstrable process and understanding — including the ability to critically evaluate AI-generated outputs — it creates new opportunities.

The more interesting question is whether AI tools can make musical process more visible and discussable. A composition student who uses a generative model, notices that the output is harmonically correct but rhythmically inert, and can articulate why it is inert — and then revise it accordingly — has demonstrated more sophisticated musical understanding than a student who produces the same output without any generative assistance. The tool does not lower the standard; it shifts where the standard is applied.

There is an analogy in music theory pedagogy. The availability of notation software that can play back a student’s harmony exercise and flag parallel fifths changed what ear training and harmony teaching emphasise — but it did not make harmony teaching obsolete. It changed the floor (students can check mechanical correctness automatically) and raised the ceiling (more class time can be spent on voice-leading logic and expressive intention). AI tools are a larger version of the same displacement: the floor rises, the ceiling rises with it, and the pedagogical question is always what you are doing between the two.

Copyright and Academic Integrity

Two issues that crossed all three workshops and deserve direct treatment.

On copyright: the training data of generative music models includes copyrighted recordings and scores, the legal status of which is actively litigated in multiple jurisdictions. When Suno.AI generates a piece “in the style of” a named composer, it is drawing on patterns extracted from that composer’s work — work that is under copyright in the case of living or recently deceased composers. The output is not a direct copy, but neither is the relationship to the training data legally settled. Music students who use these tools in professional contexts should know that they are working in a legally uncertain space, and institutions should not pretend otherwise.

On academic integrity: the issue is not that students might use AI to cheat — they will, some of them, and they have always found ways to cheat with whatever tools were available. The issue is that current AI policies at many institutions are incoherent: prohibiting AI use in assessment while providing no clear guidance on what counts as AI use, and assigning tasks where AI assistance is undetectable and arguably appropriate. The more useful approach is to design tasks where AI assistance is either irrelevant (because the task requires live performance or real-time demonstration) or visible and assessed (because the task explicitly includes reflection on how AI was used and to what effect).

Three Things I Came Away With

After a full day of workshops, discussions, and the conversations that happen in the corridors between sessions, I left with three positions that feel more settled than they did in the morning.

First: the data protection question is not separable from the pedagogical question. Any serious curriculum discussion of AI in music education has to start with what can legally be deployed, not with what would be useful if constraints were not a factor. The constraints are a factor.

Second: the skill most urgently needed — in students and in instructors — is not AI literacy in the sense of knowing which tool to use for which task. It is the critical capacity to evaluate AI-generated outputs: to notice what is wrong, to understand why it is wrong, and to correct it. This requires domain expertise first. You cannot critically evaluate an AI-generated harmonic analysis if you do not understand harmonic analysis. The tools do not lower the bar for domain knowledge. They raise the bar for its critical application.

Third: the curriculum question is not “how do we accommodate AI?” It is “what are we actually trying to teach, and does the answer change when AI can produce the visible output of that process?” Answering that honestly, skill by skill, for a full music programme, is slow work. It cannot be done at a one-day event. But a one-day event, if it is well-designed, can start the conversation in the right place.

HfMT’s Thementag started it in the right place.

References

• Devlin, J., Chang, M.-W., Lee, K., & Toutanova, K. (2018). BERT: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805. https://arxiv.org/abs/1810.04805

• Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press. https://www.deeplearningbook.org

• Holmes, W., Bialik, M., & Fadel, C. (2019). Artificial Intelligence in Education: Promises and Implications for Teaching and Learning. Center for Curriculum Redesign.

• Jobin, A., Ienca, M., & Vayena, E. (2019). The global landscape of AI ethics guidelines. Nature Machine Intelligence, 1, 389–399. https://doi.org/10.1038/s42256-019-0088-2

• LeCun, Y., Bengio, Y., & Hinton, G. (2015). Deep learning. Nature, 521(7553), 436–444. https://doi.org/10.1038/nature14539

• Luckin, R., Holmes, W., Griffiths, M., & Forcier, L. B. (2016). Intelligence Unleashed: An Argument for AI in Education. Pearson.

• Mittelstadt, B. D., Allo, P., Taddeo, M., Wachter, S., & Floridi, L. (2016). The ethics of algorithms: Mapping the debate. Big Data & Society, 3(2). https://doi.org/10.1177/2053951716679679

• Roll, I., & Wylie, R. (2016). Evolution and revolution in artificial intelligence in education. International Journal of Artificial Intelligence in Education, 26(2), 582–599. https://doi.org/10.1007/s40593-016-0110-3

• Russell, S., & Norvig, P. (2020). Artificial Intelligence: A Modern Approach (4th ed.). Pearson.

• Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, Ł., & Polosukhin, I. (2017). Attention is all you need. Advances in Neural Information Processing Systems, 30. https://arxiv.org/abs/1706.03762

Based on a manuscript with colleagues from the RAPP Lab. Not yet peer-reviewed.

The Gap Nobody Talks About

There is a version of the NMP success story that stops too early. It goes: we installed LoLa, measured the latency, it came in at 9.5 ms to Vienna, the musicians played together across 745 km, it worked. Success.

What this story skips is the classroom after the demo. The student who can follow a setup checklist perfectly and still has no idea what to do musically when the connection is stable. The ensemble that gets a clean signal running and then plays exactly the same repertoire in exactly the same way they would in a co-present rehearsal, fighting the latency instead of working with it, frustrated when it does not feel right. The assessment rubric that checks off “maintained stable connection” and “completed the performance” and has nothing to say about everything that actually constitutes musical learning in a networked context.

The gap between technical feasibility and educational transformation is the subject of this post. Closing it turns out to require a different kind of curriculum design than most conservatoires have tried.

What Gets Taught Versus What Needs to Be Learned

The default curricular response to NMP has been to treat it as a technical skill with an artistic application. Students learn to configure an audio interface, manage routing, establish a LoLa connection, and then — implicitly — go do music. The technical content gets staged as a prerequisite to the “real” work.

This ordering is wrong in a specific way. Technical setup work is genuinely necessary, but making it a prerequisite treats the relationship between technology and musical practice as sequential rather than recursive. In practice, the interesting musical problems only become visible through the technical ones. A student does not understand why buffer size matters until they have felt the difference between a 5 ms and a 40 ms offset in a coordination-intensive passage. A student does not develop an opinion about audio routing configurations until they have experienced a rehearsal collapse caused by a routing error they could have prevented.

The RAPP Lab’s recurring insight across several years of module iterations at HfMT Köln was more direct: once learners can establish a stable connection, the harder challenge is developing artistic, collaborative and reflective strategies for making music together apart. Technical fluency is a foundation, not a destination.

The Curriculum We Ended Up With

It took several cycles to get there. The early format was weekend workshops — open, exploratory, no formal assessment, primarily for advanced students who self-selected in. These were useful precisely because they were informal: they revealed quickly how technical and musical questions become inextricable once you are actually playing, and they gave us evidence about where students got stuck that we would not have found from a needs analysis.

Over time, elements of those workshops were developed into recurring curriculum-embedded modules, which then fed into independent study projects and eventually into external collaborations and performances. The trajectory mattered: moving from a one-off event to something longitudinal meant that knowledge built across cohorts rather than resetting every time.

The module structure that emerged has three interlocking elements:

Progressive task design. Early sessions are tightly scoped: specific technical-musical exercises, limited repertoire, well-defined success criteria. Later sessions move toward open-ended projects, student-led rehearsal planning, and eventually cross-institutional partnerships where variables are genuinely outside anyone’s control. The point of the early constraints is not to make things easier — it is to create conditions where students can notice what they are doing rather than just surviving.

Journals and debriefs. Students kept individual reflective journals throughout modules, documenting not just what happened but how they responded to it — technical problems, musical decisions, moments of coordination failure and recovery, questions they could not answer at the time. Group debriefs after each rehearsal then turned those individual threads into collective knowledge: comparing strategies, naming the problems that came up repeatedly, developing shared language for rehearsal coordination.

The debrief is the part of this model that I think gets undervalued. It is not just reflection — it is curriculum production. Strategies that emerged from one cohort’s debriefs became documented starting points for subsequent cohorts. Knowledge accumulated rather than evaporating when the semester ended.

Portfolio assessment. Rather than assessing primarily on a final performance, students assembled portfolios that could include curated journal excerpts, rehearsal documentation, reflective syntheses, and accounts of how their thinking changed. The question being assessed was not “did you play the concert” but “can you articulate why you made the decisions you made, and what you would do differently.”

What Students Actually Learn (When the Curriculum Works)

Four outcomes recurred across the RAPP Lab iterations, consistently enough to be worth naming:

1. Technical agency

This is different from technical competence. Competence means you can follow a procedure. Agency means you understand the procedure well enough to deviate from it intelligently when something goes wrong — to diagnose what failed, generate a hypothesis about why, and try something different.

The shift happened when students stopped treating technical problems as interruptions to the music and started treating them as information about the system they were working inside. A dropout is not just an annoyance; it is evidence about where the failure occurred. Getting to that reframe took, on average, several weeks of structured reflection. It did not happen from reading documentation.

2. Adaptive improvisation

Latency changes what real-time musical coordination can mean. You cannot rely on the same multimodal cues — breath, gesture, shared acoustics — that make co-present ensemble playing feel intuitive. You have to develop explicit cueing systems, turn-taking conventions, contingency plans for when the connection degrades mid-performance.

What we observed was that this constraint generated a specific kind of musical creativity. Students improvised not just with musical material but with rehearsal organisation itself — inventing systems, testing them, discarding the ones that did not work, documenting the ones that did. Some of the most musically interesting moments in the modules came from sessions where the technology was behaving badly and students had to make it work anyway.

There is research on “productive failure” — deliberately designing tasks that exceed students’ current control, because the struggle and recovery produces deeper learning than smooth execution (Kapur 2016). NMP turns out to be a natural context for this, not by design but because the network does not cooperate on schedule.

3. Collaborative communication

Co-present rehearsal relies heavily on implicit communication: the physical space makes many things legible without anyone having to say them. In a networked rehearsal, the spatial and gestural channel is degraded or absent. Students had to make explicit what would normally be implicit — articulating coordination strategies, naming the problems they were experiencing rather than hoping the ensemble would notice, developing a vocabulary for talking about timing and latency as musical parameters.

This turned out to generalize. Students who had worked through several networked rehearsal cycles were noticeably better at explicit musical communication in co-present settings too, because they had been forced to develop the vocabulary in a context where it was necessary.

4. Reflective identity

The students who got the most out of the modules were the ones who stopped waiting for the conditions to improve and started working with the conditions as they were. Latency as a compositional constraint rather than a defect to be routed around. Uncertainty as an artistic condition rather than a technical failure.

The journal entries where this shift is most visible are not the ones that describe what the student did. They are the ones that describe a change in how the student understands their own practice — who they are as a musician in relation to an environment they cannot fully control. That is a different kind of outcome than anything a timing metric captures.

The Assessment Problem

The hardest part of all of this to translate into institutional language is assessment. The conservatoire has well-developed frameworks for evaluating performances. It has much weaker frameworks for evaluating the learning that happens before and between and underneath performances.

Checklist rubrics — was the connection stable, was the latency within acceptable range, did the performance complete — are useful for safety and reliability. They are poor evidence for whether a student has developed the capacity to work reflectively and artistically in a mediated ensemble environment. A student who achieved a stable connection by following instructions exactly and a student who achieved it by diagnosing a routing error mid-session look identical on a checklist. They have had very different learning experiences.

Portfolio assessment addresses this by making the reasoning visible. When a student can explain why they chose a particular buffer configuration given the specific network characteristics of that session, how that choice affected the musical phrasing in the piece they were rehearsing, and what they would change next time — that is evidence of something real. It is also harder to assess than a timing log, which is probably why most programmes avoid it.

The argument is not that quantitative indicators are useless. It is that they function better as scaffolding for reflective judgement than as the primary evidence of learning. Mixed assessment ecologies — technical logs plus journals plus portfolio syntheses — are more honest about what is actually happening educationally.

What This Does Not Solve

The model described here depends on teaching staff who can facilitate reflective dialogue, curate knowledge across cohorts, and participate in iterative curriculum redesign. That is a specific professional competence that is not automatically present in a conservatoire staffed primarily by performing musicians. The training and support structures needed to develop it are an open question this paper does not fully answer.

The curriculum is also not portable as-is. The RAPP Lab model emerged in a specific institutional context — HfMT Köln, specific partner network, specific funding structure, specific cohort of students. The four outcomes and the general pedagogical logic may transfer; the specific formats will need adaptation. Any institution that tries to implement this without going through at least one cycle of their own iterative development is likely to end up with a checklist version of something that works only when it is a living process.

And the technology keeps moving. LoLa is a mature platform but the ecosystem around it — network configurations, operating system support, hardware lifecycles — changes faster than curriculum documentation. Building responsiveness into the curriculum itself, rather than treating it as a fixed syllabus, is the structural answer. Easier to recommend than to institutionalise.

References

Barrett, H. C. (2007). Researching electronic portfolios and learner engagement. Journal of Adolescent & Adult Literacy, 50(6), 436–449.

Borgdorff, H. (2012). The Conflict of the Faculties. Leiden University Press.

The Design-Based Research Collective (2003). Design-based research: An emerging paradigm for educational inquiry. Educational Researcher, 32(1), 5–8.

Kapur, M. (2016). Examining productive failure, productive success, unproductive failure, and unproductive success in learning. Educational Psychologist, 51(2), 289–299. https://doi.org/10.1080/00461520.2016.1155457

Lave, J. & Wenger, E. (1991). Situated Learning. Cambridge University Press.

Sadler, D. R. (2009). Indeterminacy in the use of preset criteria for assessment and grading. Assessment & Evaluation in Higher Education, 34(2), 159–179. https://doi.org/10.1080/02602930801956059

Schön, D. A. (1983). The Reflective Practitioner. Basic Books.

Wenger, E. (1998). Communities of Practice. Cambridge University Press. https://doi.org/10.1017/CBO9780511803932

Changelog

• 2026-01-20: Updated the Sadler (2009) reference title to “Indeterminacy in the use of preset criteria for assessment and grading,” matching the journal article at this DOI. Updated the Kapur (2016) reference to the full published title: “Examining productive failure, productive success, unproductive failure, and unproductive success in learning.”

This post extends the latency analysis in Latency in Networked Music Performance with the dynamical systems framework that underlies it.

Two Clocks on a Board

The first documented observation of coupled-oscillator synchronisation was made not by a musician but by a physicist. In 1665, Christiaan Huygens, confined to bed with illness, was watching two pendulum clocks mounted on the same wooden beam. Over the course of the night, the pendulums had synchronised into anti-phase oscillation — swinging in opposite directions in exact unison. He reported it to his father:

“I have noticed a remarkable effect which no-one has observed before… two clocks on the same board always end up in mutual synchrony.”

The mechanism was mechanical coupling through the beam. Each pendulum’s swing imparted a small impulse to the wood; the other pendulum felt this as a perturbation to its rhythm. Small perturbations, accumulated over hours, drove the clocks into a shared frequency and a fixed phase relationship.

This is the prototype of every ensemble synchronisation problem. Each musician is a clock. The acoustic environment — the air in the room, the reflected sound from the walls, the vibrations through the stage floor — is the wooden beam.

The Kuramoto Model

Yoshiki Kuramoto formalised the mathematics of coupled oscillators in 1975, motivated by biological synchronisation problems: firefly flashing, circadian rhythms, cardiac pacemakers. His model considers $N$ oscillators, each with a phase $\theta_i(t)$ evolving according to:

The first term, $\omega_i$, is the oscillator’s natural frequency — the tempo it would maintain in isolation. These are drawn from a distribution $g(\omega)$, which in a real ensemble reflects the spread of individual preferred tempos among the players. The second term is the coupling: each oscillator is attracted toward the phases of all others, with strength $K/N$. The factor $1/N$ keeps the total coupling intensive (independent of ensemble size) as $N$ grows large.

Musically: $\theta_i$ is the phase of musician $i$’s internal pulse at a given moment, $\omega_i$ is their preferred tempo if playing alone, and $K$ is the coupling strength — how much they adjust their tempo in response to what they hear from the others.

The Order Parameter and the Phase Transition

To measure the degree of synchronisation, Kuramoto introduced the complex order parameter:

where $r(t) \in [0, 1]$ is the coherence of the ensemble and $\psi(t)$ is the collective mean phase. When $r = 0$, the phases are uniformly spread around the unit circle — the ensemble is incoherent. When $r = 1$, all phases coincide — perfect synchrony. In a live ensemble, $r$ is a direct measure of rhythmic cohesion, though of course not one you can read off a score.

Substituting the order parameter into the equation of motion:

Each oscillator now interacts only with the mean-field quantities $r$ and $\psi$, not with every other oscillator individually. The coupling pulls each musician toward the collective mean phase with a force proportional to both $K$ (how attentively they listen) and $r$ (how coherent the group already is).

This mean-field form reveals the essential physics. For small $K$, oscillators with widely differing $\omega_i$ cannot follow the mean field — they drift at their own frequencies, and $r \approx 0$. At a critical coupling strength $K_c$, a macroscopic fraction of oscillators suddenly locks to a shared frequency, and $r$ begins to grow continuously from zero. For a unimodal, symmetric frequency distribution $g(\omega)$ with density $g(\bar\omega)$ at the mean:

Above $K_c$, the coherence grows as:

This is a second-order (continuous) phase transition — the same mathematical structure as a ferromagnet approaching the Curie temperature, where spontaneous magnetisation appears continuously above a critical coupling. The musical ensemble and the magnetic material belong to the same universality class, governed by the same mean-field exponent $\frac{1}{2}$.

Above $K_c$, the fraction of oscillators that are locked (synchronised to the mean-field frequency) can be computed explicitly. An oscillator with natural frequency $\omega_i$ locks to the mean field if $|\omega_i - \bar\omega| \leq Kr$. For a Lorentzian distribution $g(\omega) = \frac{\gamma/\pi}{(\omega - \bar\omega)^2 + \gamma^2}$, this yields:

which is the exact self-consistency equation for the Kuramoto model with Lorentzian frequency spread (Strogatz, 2000).

The physical reading is direct: whether an ensemble locks into a shared pulse or drifts apart is a threshold phenomenon. A group of musicians with similar preferred tempos has a peaked $g(\bar\omega)$, giving a low $K_c$ — they synchronise easily with minimal attentive listening. A group with widely varying individual tempos needs stronger, more sustained coupling to cross the threshold. This is not a matter of musical discipline; it is a material property of the ensemble.

Concert Hall Applause: Neda et al. (2000)

The Kuramoto model is not only a theoretical construction. Neda et al. (2000) applied it to concert hall applause — one of the most direct real-world demonstrations of coupled-oscillator dynamics in a musical context.

They recorded applause in Romanian and Hungarian theaters and found that audiences spontaneously alternate between two distinct states. In the incoherent regime, each audience member claps at their own preferred rate (typically 2–3 Hz). Through acoustic coupling — each person hears the room-averaged sound and adjusts their clapping — the audience gradually synchronises to a shared, slower frequency (around 1.5 Hz): the synchronised regime.

The transitions between the two regimes are quantitatively consistent with the Kuramoto phase transition: the emergence of synchrony corresponds to $K$ crossing $K_c$ as people progressively pay more attention to the collective sound. Furthermore, Neda et al. document a characteristic phenomenon when synchrony breaks down: individual clapping frequency approximately doubles as audience members attempt to re-establish coherence. This frequency-doubling — a feature of nonlinear oscillator systems near instability — is exactly what the delayed response of coupling near $K_c$ predicts.

The paper is a useful pedagogical artefact: every music student has experienced concert hall applause, and hearing that it undergoes a physically measurable phase transition makes the connection between physics and musical experience concrete.

Latency and the Limits of Networked Ensemble Performance

In standard acoustic ensemble playing, the coupling delay is the propagation time for sound to cross the ensemble: at $343\ \text{m/s}$, across a ten-metre stage, roughly 30 ms. This is why orchestral seating is arranged with attention to who needs to hear whom first.

In networked music performance (NMP), the coupling delay $\tau$ is much larger: tens to hundreds of milliseconds depending on geographic distance and network infrastructure. The Kuramoto model generalises naturally to include this delay:

Each musician hears the others’ phases as they were $\tau$ seconds ago, not as they are now.

In a synchronised state where all oscillators share the collective frequency $\bar\omega$ and phase $\psi(t) = \bar\omega t$, the delayed phase signal is $\psi(t - \tau) = \bar\omega t - \bar\omega\tau$. The effective coupling force contains a factor $\cos(\bar\omega\tau)$: the delay introduces a phase shift that reduces the useful component of the coupling. The critical coupling with delay is therefore:

As $\tau$ increases, $K_c(\tau)$ grows: synchronisation requires progressively stronger coupling (more attentive adjustment) to compensate for the information lag. The denominator $\cos(\bar\omega\tau)$ reaches zero when $\bar\omega\tau = \pi/2$. At this point $K_c(\tau) \to \infty$: no finite coupling strength can maintain synchrony. The critical delay is:

For an ensemble performing at 120 BPM, the beat frequency is $\bar\omega = 2\pi \times 2\ \text{Hz} = 4\pi\ \text{rad/s}$:

This is a remarkably clean result. The Kuramoto model with delay predicts that ensemble synchronisation collapses at around 125 ms one-way delay for a standard performance tempo. The empirical literature on NMP — from LoLa deployments across European conservatories to controlled latency studies in the lab — consistently finds that rhythmic coherence degrades noticeably above 50–80 ms and becomes essentially unworkable above 100–150 ms one-way. The model and the data agree.

The derivation also shows why faster tempos are harder in NMP: $\tau_c \propto 1/\bar\omega$, so doubling the tempo halves the tolerable latency. An ensemble performing at 240 BPM in a distributed setting faces a theoretical ceiling of 62 ms — which rules out transcontinental performance for most repertoire.

Brains in Sync: EEG Hyperscanning

The Kuramoto framework has recently been applied at a neural level. EEG hyperscanning — simultaneous EEG recording from multiple participants during a shared musical activity — has shown that musicians performing together exhibit inter-brain synchronisation: coherent cortical oscillations at the frequency of the music are measurable between players (Lindenberger et al., 2009; Müller et al., 2013). The phase coupling between brains during joint performance is significantly higher than during solo performance and higher than for musicians playing simultaneously but without acoustic coupling.

This suggests that the Kuramoto coupling operates at two levels: the acoustic (each musician hears the other and adjusts physical timing) and the neural (each musician’s cortical oscillators entrain to the shared musical pulse). The question of which level is primary — whether neural synchrony causes or follows from acoustic synchrony — remains open.

A 2023 review by Demos and Palmer argues that pairwise Kuramoto-type coupling is insufficient to capture full ensemble dynamics. Group-level effects — the differentiation between leader and follower roles, the emergence of collective timing that no individual would produce alone — require nonlinear dynamical frameworks that go beyond mean-field averaging. The model that adequately describes a string quartet may need to be richer than the one that describes a population of identical fireflies.

What This Means for Teaching

The Kuramoto model reframes standard rehearsal intuitions in physical terms.

“Listen more” translates to “increase your effective coupling constant $K$.” A musician who plays without attending to others has set $K \approx 0$ and will drift freely according to their own $\omega_i$. Listening — actively adjusting tempo in response to what you hear — is not metaphorical. It is the physical mechanism of coupling, and its effect is to pull you toward the mean phase $\psi$ with a force $Kr\sin(\psi - \theta_i)$.

“Our tempos are too different” is a claim about $g(\bar\omega)$ and therefore about $K_c$. A group with a wide spread of natural tempos needs more and stronger listening to synchronise. This is not a moral failing but a parameter; it suggests that ensemble warm-up time or explicit tempo negotiation before a performance serves to reduce the spread of natural frequencies before the coupling has to do all the work.

Latency as a rehearsal experiment can be made explicit. Artificially delaying the acoustic return to one musician in an ensemble — via headphone monitoring with variable delay — allows students to experience directly how the coordination degrades as $\tau$ increases toward $\tau_c$. They feel the system approaching the phase transition without the theoretical framework, but the framework makes the experience interpretable afterward.

The click track replaces peer-to-peer Kuramoto coupling with an external forcing term: each musician locks to a shared reference with fixed $\omega$ rather than adjusting dynamically to the group mean. This eliminates the phase transition but also eliminates the adaptive dynamics — the micro-timing fluctuations and expressive rubato — that characterise live ensemble playing. It is a pedagogically important distinction, even if studios routinely make the pragmatic choice.

References

• Demos, A. P., & Palmer, C. (2023). Social and nonlinear dynamics unite: Musical group synchrony. Trends in Cognitive Sciences, 27(11), 1008–1018. https://doi.org/10.1016/j.tics.2023.08.005

• Huygens, C. (1665). Letter to his father Constantijn Huygens, 26 February 1665. In Œuvres complètes de Christiaan Huygens, Vol. 5, p. 243. Martinus Nijhoff, 1893.

• Kuramoto, Y. (1975). Self-entrainment of a population of coupled non-linear oscillators. In H. Araki (Ed.), International Symposium on Mathematical Problems in Theoretical Physics (Lecture Notes in Physics, Vol. 39, pp. 420–422). Springer.

• Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence. Springer.

• Lindenberger, U., Li, S.-C., Gruber, W., & Müller, V. (2009). Brains swinging in concert: Cortical phase synchronization while playing guitar. BMC Neuroscience, 10, 22. https://doi.org/10.1186/1471-2202-10-22

• Müller, V., Sänger, J., & Lindenberger, U. (2013). Intra- and inter-brain synchronization during musical improvisation on the guitar. PLOS ONE, 8(9), e73852. https://doi.org/10.1371/journal.pone.0073852

• Neda, Z., Ravasz, E., Vicsek, T., Brechet, Y., & Barabási, A.-L. (2000). Physics of the rhythmic applause. Physical Review E, 61(6), 6987–6992. https://doi.org/10.1103/PhysRevE.61.6987

• Strogatz, S. H. (2000). From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D: Nonlinear Phenomena, 143(1–4), 1–20. https://doi.org/10.1016/S0167-2789(00)00094-4

• Strogatz, S. H. (2003). Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life. Hyperion.

Changelog

• 2026-01-14: Updated the author list for the Demos (2023) Trends in Cognitive Sciences reference to the published two authors (Demos & Palmer). The five names previously listed were from a different Demos paper.
• 2026-01-14: Changed “period-doubling” to “frequency-doubling.” When the clapping frequency doubles, the period halves; “frequency-doubling” is the precise term in this context.