EulerBeats Enigma

Enigma Originals

EulerBeats Enigma originals are ultra scarce NFTs and are capped at 27 - completing the other side of the EulerBeats record. The token seed was generated on-chain at the time of creation. The creator’s wallet address and the block hash were part of the inputs into the seed and token generation process. The art + music generation script used the seeds to create the associated art + music for the 27 Enigma tokens. Just like in the Genesis release the project team was not able to preview the art + beat before the tokens were minted. 25 Enigma Original tokens will be sold as an OpenSea auction with the highest winning bid determining the price of each token. The Originals will be auctioned off using an English-style auction running for 48 hours with 5-minute extensions if there are new bids within the last 5 minutes of the auction.The token floor will be set at 2.71Ξ. One of the remaining 27 is set aside to donate to a DAO and the other is set aside for the EulerBeats Project team.

Burning Enigma Prints

The pricing of the Enigma prints and the redemption value of the burned print are set on the bonding curve described below. The burn prices vary based on the current supply. A print token burn yields 84% of the cost of printing the nth print. Burning reduces the current supply to n - 1.

Enigma Bonding Curve

The Genesis and Enigma releases are two sides of the Euler record that make one whole. Each side is an eternal experiment of an economic model and ownership. As a hat tip to all of the people supporting from the sideline because they were priced out of the Genesis print sale, we have allowed for a longer flatter bonding curve to get more fans into the community. In addition, the bonding curve for Genesis was designed for price discovery and thus makes it practically impossible at current ETH price levels to collect all prints for a single Original. The Enigma release is a nod to the EulerBeats community: it’s instrumented to make collecting of all prints of a given original a practical goal. The Enigma release bonding curve allows for more prints to be affordable starting at .1 ETH and cuts off the total print supply before the exponential curve becomes too steep.

It’s now practically possible to collect and hold the whole set of prints for a given token. This can open up new possibilities on the secondary marketplaces as well as create interesting game theory experiments for the people who hold the prints. This is one of the fundamental differences between the Genesis and the Enigma sides of the Euler record.

There is a bonding curve that prices the print for each original based on the function:

f(x) = a(b+x) + cx + d

where x denotes the current print supply+1 and

a = 1.2

b = -140

c = 0.1

d = 0

84% of the print price is stored in a reserve to refund burned prints. The remaining 8% is sent to the Original Holders and 8% goes to the EulerBeats project.

To further explain how the Enigma release introduces a bonding curve that makes it practical to buy all prints for a token we started the print price at 0.1 ETH and the print price increases in about 0.1 ETH increments through the 110th print token. On the 111th through the 137th print range the price step is slightly higher, but no more than 0.2 ETH. After the 146th print price, the exponential curve accelerates at a higher pace, but the total print supply is cut off at 160 prints per Original to make collecting the last print an achievable goal. The print price of the last (160th) print is ~54 ETH for Enigma while it’s 721.037 ETH for the last (119th) Genesis.

Enigma Art + Music Generation

Just like EulerBeats Genesis, the art and music generation of every Enigma original is guaranteed to be unique. The art is based on Euler's totient function Phi (ɸ) that counts the positive integers up to a given integer n that are relatively prime to n. For Enigma, instead of a square matrix, we’re representing the function in a radial grid arrangement, where cells are filled with color if the row and column the cell represents are co-prime to each other.

This diagram illustrates how a 4x4 matrix would be represented as a radial grid.

The art has the following traits

  • Grid Size: Ranges from a 7x7 to a 12x12 radial grid.

  • Diagonal Lever: This is a combined lever that shifts both the horizontal (X) and vertical (Y) viewing window of the typical Euler's totient Phi (ɸ) graph.

  • Horizontal Lever: This lever shifts the viewing window of the typical Euler's totient Phi (ɸ) graph horizontally in addition to the diagonal lever.

  • These levers help to produce interesting view patterns and also eliminates some edge conditions like an empty area.

  • Rings Width: This determines the width or thickness of each ring inside the radial grid. Each ring could have the same width as the others (less common) or different widths, in ascending order, descending order, or randomized.

  • Columns Width: This determines the width or thickness of each column in the radial grid. Each column could have the same width as the others (less common) or different widths, in an ascending order starting from the center, descending order, or randomized.

  • Rotation: This determines the way each ring rotates when the beat is playing. All of the rings could rotate together with a clockwise motion (this is slightly more common). They could rotate together with a counterclockwise motion. Even rings rotate in one direction and odd rings the other way around. The center half in one direction and the outer half in the opposite direction, and last of all, they could all rotate in different ways in a randomized manner.

  • Color Distribution: This determines how the different colors are arranged. They could be distributed by rings/rows or by columns.

  • Color Palette: This determines the colors used on the cells. Each palette has 4 colors.

  • Colors Used: Determines the number of colors that form the palette used on the artwork. It can go from 2 to all 4 colors.

The coprimes matrix is used as input to generate the rhythmic base. The first bottom rows and last columns are assigned to different percussion instruments. The dynamic accent comes from the count of coprimes in each column and row (a partial totient function of sorts) at every beat, so beat intensity is higher for prime numbers. The token seed is used to randomly generate chord progressions and melodies using our simple but effective music theory engine.

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