About EulerBeats

Imagine music as mathematical art on an NFT substrate. This is EulerBeats. The Euler Record is a limited edition set of algorithmically generated art + music based on the Euler number and the Euler's totient function. Genesis (27 NFTs) and Enigma (27 NFTs) are two sides of the Euler Record. The EulerBeats smart contract extends the ERC-1155 Ethereum multi-token standard to create non-fungible unique original tokens and a set of fungible print tokens for each original. Both releases share a common mathematical basis, but differ in art, music, and bonding curve structures. Each original NFT has a limited set of prints priced on a bonding curve.

The token is the art piece in itself. Everything needed to re-create each masterpiece is stored on-chain on Ethereum. EulerBeats is a careful and creative ensemble of math, art, music, royalties, instantly liquid tokens, and scarcity into a neat digital pack. It’s a first of its kind.


Diamonds are forever. EulerBeats are for eternity.

This art is designed to last a lifetime. The data and the script to re-generate the art and music are stored on the Ethereum blockchain. If the EulerBeats website is ever shut down, you can use the seed and the script to generate your art and the beat. We show how in a video in the FAQ section. This project joined the ranks of a handful of pioneering NFT projects that keep all data and logic required to construct the art and music permanently on Ethereum for eternity. The Treum team pushed the boundaries of the “on-chain” NFTs storage by placing the “recipe” for art, music, and economics into an Ethereum block. This is in contrast to many NFT projects that store the token media only on IPFS-like decentralized storage, or even in centralized cloud storage. Our article On-Chain Artwork NFTs provides an elegant explanation of the differences in media storage choices between different NFT projects.


Originals of the Euler Record are ultra scarce NFTs and are capped at 27 for both Enigma and Genesis. Originals birth a new art + beat based on the Euler's totient function. The token seed is generated on-chain. Nobody knows the look and sound of the token until it’s minted. The music and the artistic design of Genesis and Enigma are unique but they feel familiar.

Owning Originals

Buyers of Originals shouldn’t need any more incentive other than the historical significance and the uniqueness of the art + music EulerBeats NFTs. That said, all EulerBeats Original holders will receive 8% of the print price on each print sold of the original token they own. The royalties are paid automatically as they are programmed into the smart contract and cannot be changed. In addition to the royalties, holders of Originals have full commercial rights. Some Genesis original owners have already created remixes and collaborations with other projects.


EulerBeats prints are replicas of an Original and can be previewed by the buyer before purchase. The cap for the fungible print tokens is set to 119 for each Genesis original token and 160 for each Enigma original token.

Any collector can buy a print based on the price formula set on a bonding curve. As the number of prints in circulation grows for a particular original, the price of issuing its next print increases at an exponential rate.


A percentage of the print price gets instantly placed into a reserve in the smart contract. The reserve provides instant liquidity to a print holder who decides to “burn” a print. Genesis and Enigma have separate reserve pools. The reserve is not accessible by the project team and it’s only accessible by the smart contract to pay print holders when they burn their prints.


The bonding curve acts as a liquid market that allows a print holder to burn their print token, thus reducing the current print token supply of the associated Original, in exchange for receiving the burn price at that time. Thus, the burning process returns ETH to the print token burner from the reserve. The burn price is determined according to the formula set on the bonding curve. Genesis and Enigma have different burn price calculations (see below). It’s not possible to burn a print into an abyss by sending it to a 0x000… address. We are using an OpenZeppelin base contract, which does not allow ERC-1155 transfers to the zero address.

Bonding Curves

Both Genesis and Enigma use different bonding curves to calculate the print and burn prices. The print purchase price and the burn price paid back upon the burn of a print token are set on the exponential bonding curve determined by the current token supply for a given Original. The higher the token supply the higher the print and burn price, and vice versa. The EulerBeats bonding curves for Genesis and Enigma are transparent and not changeable since they are programmed into a smart contract on Ethereum blockchain. Additional details are below or post a question to the community on the Discord channel.

EulerBeats Story

Treum.io, a ConsenSys backed company incorporated in 2016, is one of the early leaders in the blockchain space. Treum team has a demonstrable track record for launching some epic firsts in blockchain including but not limited to implementing the first non-financial use case for tracking oil samples for BHP Billiton, Bait-to-plate tracking of yellowfin Tuna from Fiji to NY with WWF, live auction of tokenized Game worn Jerseys for Sacramento Kings. Our most recent project, Swether.io, which ran through December 2020, enabled crypto users to design, mint, and gift a swether token NFT and receive a sweatshirt with their very own custom design. After the Swether project, we continued to enhance the Treum tokenization platform to include a market price discovery mechanism using a bonding curve.

For the first project, we decided to build a bonding curve based on Euler's number (mathematical constant) and the totient function(ɸ) to celebrate the mathematician's work. From Calculus textbooks to compound interest to cryptography: you have experienced Euler’s work in theory and practice. We launched EulerBeats to commemorate Leonhard Euler's discovery of his totient function phi (ɸ). Leonhard Euler’s contributions to mathematics are well known and include geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory, and other areas of physics [Wikipedia].

Euler’s musical mathematics reflected his passion for sound. He looked to find and explain harmony in music with math. He believed it’s possible to quantify the “degree of agreeableness which might be translated as sweetness, charm, or tunefulness.” The relationship between math, physics, music and art is tightly intertwined. Fittingly we packaged math, art, music, royalties, instant liquidity, and scarcity into an NFT project dubbed as EulerBeats. Join the EulerBeats Discord to chat with the creators and collectors.

Collaborating with EulerBeats

Treum is an NFT platform. We will continue to work with artists, creators, and production houses entering the token economy space to accelerate their way to token creation, pricing and price discovery, and royalty distribution. Please reach out to us if you would like to explore a collaboration. Follow this project on Twitter @eulerbeats and the team @treum_io. Join the EulerBeats Discord to chat with the creators and collectors.


The treum.io team making the project a reality includes:


Joe Lubin for allowing us to buidl big and for the gift that keeps on giving #Ethereum. Jonny Howle, an amazing designer, music lover, and producer. Thank you for the beautiful and thoughtful initial design of the Genesis release.

For the many folks that took our calls and helped us think through the Genesis release of EulerBeats. They are ecosystem shapers, champions of the coolest projects, and best of all, friends. Corbin Page, Igor Lilic, Jordan Lyall, Priyanka Desai, Alanna Roazzi Laforet.

Lastly, to a few of the projects that helped inspire our thinking: Art Blocks, Neolastics, and Avastars.

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