# teaching machines

## Why 12?

We looked previously at how an octave—or doubling, as we called it—is partitioned non-linearly into intermediate tones. But we didn’t ultimately decide how many intermediate tones there should be. The pioneers of Western music converged on a 12-partition. But why not 8? Or 10? Or 11? Or 13? Ultimately, we want our instruments to sound […]

## Droplets

A year ago some students and I started designing programming languages for generating 2D vector art. I called my language Twoville, which pays homage to Seymour Papert’s metaphor of learning math in Mathland just as we learn French in France. The language is a place to learn about and forge 2D shapes. After the semester […]

## Twelve Steps

Music theory is a daunting subject, built out of centuries of mathematical analysis and aristocratic pride. Sometimes I think I’d be better off reinventing it than trying to learn it at this late stage of its maturity. Here’s how I might get started. One day I’d be absently fooling around with a bit of wire, […]

## Parametric Puzzle

Check out these these parametric equations: $$\begin{array}{rll}x &=& \cos v \cdot \cos u \\y &=& \sin v \\z &=& \cos v \cdot \sin u\end{array}$$ Do you know what they do? They are most assuredly not magic. Here, let’s rename the variables, and you can try again: \begin{array}{rll}x &=& \cos \textit{latitude} \cdot \cos \textit{longitude} \\y &=& […]

## Intervals in Deltaphone

For a few years now, schools and industry have been telling kids that they can code. They say the jobs are plentiful, and the salary is enviable—the workforce is waiting. The nobler agents of educational reform will also tell our kids that programming is a creative exercise that will make them better thinkers in other […]

## Flatcaps in Libigl

Madeup’s dowel solidifier has one job: thicken a sequence of line segments into a solid. But what if the sequence isn’t a polyline, but rather a branching structure like a tree or a fork? One could model each branch as a separate dowel and hope that nobody looks too closely at the joints, but that’s […]

## STEM in Education 2018 Workshop

Welcome to the notes for the Computational Making with Madeup workshop at STEM in Education 2018. It’s my hope that you read the abstract for this workshop, and you consent to our stated goals: In this interactive hands-on workshop, participants will learn to build models using Madeup and gain skills and resources they can use […]

## Stepping Up in Deltaphone

Recently I added chords to Deltaphone, which required me to rethink how I was structuring the blocks to play notes. Previously, the play block came in two forms. There was a form for an absolutely-specified note, like “play C natural in the fourth octave.” And there was a form for a relatively-specified note, like “play […]

## Kookaburra in Deltaphone

Though we have been in Australia for several months, we have hardly seen any of its unique wildlife. Fearing that we’d return home with no distinctly Australian experiences, we set out to find some animals. Given the number of news reports I’ve heard about kangaroos injuring humans, we decided to take it slow. We headed […]

## Sensing Shapes

In the late 1600s, William Molyneaux posed a question to a friend: Suppose a man born blind, and now adult, and taught by his touch to distinguish between a cube and a sphere of the same metal, and nighly of the same bigness, so as to tell, when he felt one and the other, which […]