Making use of music theory, group theory, and category theory
From Musical Actions of Dihedral Groups
Abstract:
The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.
Summary:
This paper connects the twelve musical tones to elements in the dihedral group of order 24 (the symmetries of a regular dodecagon). The translation from pitch classes to integers modulo 12 allows for the modeling of musical works using abstract algebra. The first action on major and minor chords described in the paper is based on the musical techniques of transposition and inversion. A transposition moves a sequence of pitches up or down and an inversion reflects a melody about a fixed axis. The other action arises from the P, L, and R operations of the 19th-century music theorist Hugo Riemann. It is through these operations that the dihedral group of order 24 acts on the set of major and minor triads. The paper also describes how the P, L, and R operations have beautiful geometric presentations in terms of graphs. In particular the authors describe a connection between the PLR-group and chord progressions in Beethoven’s 9th Symphony, which leads to a proof that the PLR-group is dihedral. Another musical example is Pachelbel’s Canon in D. In summary, the paper gives a very pretty explanation of what we commonly hear in tonal music in terms of elementary group theory.
VOICE ARRAY | Rafael Lozano-Hemmer
As a participant speaks into an intercom, their voice is automatically translated into flashes of light and then this unique blinking pattern is stored as a loop in the first light of the array. Each new recording pushes all previous recordings one position down and gradually one can hear the cumulative sound of the 288 previous recordings. The voice that was pushed out of the array can then be heard by itself.
(via lefuckinsoleil)
![Nanotube Paint Can Spot Structural Defects and Alert Authorities Before Damage Occurs
By Rebecca Boyle
A new paint made of power plant waste and carbon nanotubes can automatically detect structural faults, alerting authorities before damage occurs. It could be a cheaper, easier way to monitor facilities like bridges, mines and even wind turbines.
It’s made from aligned carbon nanotubes, which can carry an electric current, and fly ash, which is a byproduct of coal burning. The paint can be sprayed onto any surface, and electrodes are attached to it, according to developers at the University of Strathclyde in Glasgow. If the nanotubes bend, their conductivity will change, which will be detected by the electrodes. Small wireless transmitters placed throughout the structure would receive data from the electrodes. If they detect a change in conductivity, this would be considered a sign of a defect in the structure. Then the system could conceivably alert the company or government body responsible for maintaining said structure.
This would be much cheaper and simpler than current monitoring methods, Strathclyde scientists said — currently, wind turbine foundations are inspected visually, and bridges and tunnels only have monitoring networks in certain areas, not throughout the whole structure. Early defect detection could be cheaper to repair, not to mention safer.
A network of electrode-embedded nanotubes doesn’t sound inexpensive, but the researchers say it would be cheap — one percent the cost of alternative inspection methods — in part because of the fly ash component. Fly ash is a byproduct of coal combustion and it’s generally stored at power plants and landfills or it’s recycled. The nanotube paint could be one new use for it. It also lends the paint some added durability, which means it could last in a wide range of environmental conditions.
For now, the electrode transmitters would be powered by batteries, but other designs could incorporate solar panels, piezoelectrics or other energy-harvesting technology, the researchers say. Strathclyde Ph.D candidate David McGahon and civil engineering professor Mohamed Saafi have built a prototype and it was shown to be effective, according to a Strathclyde news release. They plan to carry out larger-scale tests in Glasgow in the future.
[via Science Daily]](http://27.media.tumblr.com/tumblr_lz12kviiAi1qa32ieo1_500.jpg)
Nanotube Paint Can Spot Structural Defects and Alert Authorities Before Damage Occurs
A new paint made of power plant waste and carbon nanotubes can automatically detect structural faults, alerting authorities before damage occurs. It could be a cheaper, easier way to monitor facilities like bridges, mines and even wind turbines.
It’s made from aligned carbon nanotubes, which can carry an electric current, and fly ash, which is a byproduct of coal burning. The paint can be sprayed onto any surface, and electrodes are attached to it, according to developers at the University of Strathclyde in Glasgow. If the nanotubes bend, their conductivity will change, which will be detected by the electrodes. Small wireless transmitters placed throughout the structure would receive data from the electrodes. If they detect a change in conductivity, this would be considered a sign of a defect in the structure. Then the system could conceivably alert the company or government body responsible for maintaining said structure.
This would be much cheaper and simpler than current monitoring methods, Strathclyde scientists said — currently, wind turbine foundations are inspected visually, and bridges and tunnels only have monitoring networks in certain areas, not throughout the whole structure. Early defect detection could be cheaper to repair, not to mention safer.
A network of electrode-embedded nanotubes doesn’t sound inexpensive, but the researchers say it would be cheap — one percent the cost of alternative inspection methods — in part because of the fly ash component. Fly ash is a byproduct of coal combustion and it’s generally stored at power plants and landfills or it’s recycled. The nanotube paint could be one new use for it. It also lends the paint some added durability, which means it could last in a wide range of environmental conditions.
For now, the electrode transmitters would be powered by batteries, but other designs could incorporate solar panels, piezoelectrics or other energy-harvesting technology, the researchers say. Strathclyde Ph.D candidate David McGahon and civil engineering professor Mohamed Saafi have built a prototype and it was shown to be effective, according to a Strathclyde news release. They plan to carry out larger-scale tests in Glasgow in the future.
[via Science Daily]
Auxin Flux Canalisation is a an algorithmic process developed by Adam Runions at the University of Calgary Algoritmic Botany group to model the morphogenesis of leaf venation. The key to the process is a simulation of the distribution and flux of auxin -a plant hormone- whose distribution contributes to many aspects of cell growth, division and specialization. The transport of auxin through a plant coordinates response to external stimuli in a way that does not require a central command system.
The beauty of the algorithm is -like flocking and other bottom-up algorithms- in the iteration of a very few simple steps.Nervous System have put together the most clear and concise description in explaining the generation of their beautiful Hyphae Lamp series. Go watch their video, it’s a great example of how an algorithm is the best representation of its own process.
via codequotidien
(via montycantsin)
Peter Jellitsch’s work is awesome!! And it’s all HAND DRAWN!! You all really need to check out his flickr page!!
