About the Arca musarithmica

• Who built the Arca and why?
• How did the Arca work?
• Was the Arca a computer?
• Who used the Arca?
• Other implementations and research

Who Built the Arca and Why?

The Arca musarithmica or box of music-numbers is a computation device invented by Athanasius Kircher. Kircher (1602–1680) was a German-born Jesuit priest and polymath in Rome who also wrote books about optics, magnetism, geology, and Egyptian hieroglyphics and corresponded with a global network of scientists, scholars, and missionaries. In book VIII of his treatise Musurgia universalis (Rome, 1650), Kircher promised that the device could automatically generate musical settings for four-voice chorus of any given text, in any language. The Arca could generate music in several styles, from a simple homophonic texture like a hymn to a complex fugal texture, all by combining preset permutations of musical patterns. Though the Arca did not have an automatic mechanism, a user could produce coherent output solely by following Kircher’s rules, without knowing anything about music.

The ark holds a set of data tables, which together with a list of rules for how to use the tables, amounts to a computation device for composing music. Given input in the form of a marked text in Latin and a small set of simple choices about the musical setting, a user who knows nothing about music can use the ark to generate unique, never-before-heard musical settings of that text for four-voice (SATB) chorus.

Kircher presented this as an embodiment of mathematical permutations. Like many of his seventeenth-century contemporaries, he understood music as sounding number, an embodiment of the mathematical order inherent in God’s created universe, and a reflection of the perfection of God himself. Out of the infinite variety of possible combinations of notes and rhythms, the user of the ark selects a specific set of permutations and thus imitates God in his original act of creation.

The combination of structured data and completely prescribed algorithms makes Kircher’s device a computational system, with some of the characteristics that would later characterize modern computers, though of course Kircher had no way to automate his system and certainly did not theorize it fully. This implementation, written in the pure-functional programming language Haskell, models the entire ark as a single mathematical function: it takes one series of symbols, representing an input text and the choice of musical parameters, and transforms those into another set of symbols that represents musical notation. Though others have implemented portions of the ark and related systems (see the bibliography), to my knowledge this is the first version to be fully automatic and to be capable of setting texts of any length and type.

The program and its output make it possible to reflect on what Kircher’s project meant: What does his artificial system of composition tell us about his and his contemporaries’ concepts of both music and computation? How much musical knowledge is embodied in this device? What is missing? And what purposes would such a device have served in its time?

How Did the Arca Work?

1. Kircher expects you to prepare a text in Latin or another language, with words divided into syllables, marked long or short. You can set texts in several Classical meters (like Anacreontic or Sapphic), but you can also set irregular verse and prose texts.
   • In this web app, we provide a set of pre-marked texts.
   • With the command-line version, you can create your own texts that might even have multiple sections in different styles and moods!
2. Next, Kircher wants you to choose several aspects of the music setting you want the ark to create:
   • The style and texture of the music:
     1. you can opt for simple, note-against-note counterpoint (like a hymn),
     2. or florid, melismatic counterpoint (like a motet, anthem, or even fugue).
   • The mood or character of the text and its setting, which determines the choice of one of twelve musical tones or church keys (toni ecclesiastici). The tones are not the same as modern keys, nor do they match up exactly with the medieval modes used to classify plainchant. They seem to be a hybrid of the polyphonic tones used by keyboardists to introduce and accompany the eight traditional chant psalm tones, with the twelve-mode system introduced by the earlier theorists Glarean and Zarlino. (see Barnett 2002).

   • The musical meter (technically, mensuration) of the setting.
     1. In the simple style, you can choose
        1. duple meter (cut C),
        2. triple minor (C3, three minims/half notes per metrical unit),
        3. or triple major (cut C3, three semibreves/whole notes per metrical unit).

        Some duple-meter permutations in the ark have a minim pulse (C) and others have a semibreve pulse (cut C).

     2. In florid style, everything is in duple meter (C).
3. If you are operating the ark by hand, you use these input parameters to select the proper tables.
   • Syntagma: There are three syntagmata or large divisions in the ark. For simple style, you select from Syntagma I; for florid, use Syntagma II.

     Kircher provides a third syntagma as well, but because it would be much more difficult if not impossible to make completely automatic, we have omitted it from this implementation. (An exercise for the reader!)

   • Pinax: Each syntagma containes a set of rods or pinakes (singular, pinax). You select the pinax based on the meter of the text.
   • Column: Each pinax contain several columns of data. The column is either a free choice, or is based on the order of lines in the poem.
   • Voice and rhythm permutations: Each column includes two parts: sets of numbers representing musical pitches (we call these voice permutations or vperms), and sets of musical notes representing rhythmic durations (rhythm permutations or rperms).
     • In syntagma I (simple counterpoint), the user is supposed to freely choose one set of vperms and one set of rperms. All the musical voices use the same rhythms.
       • In syntagma I, there are (generally) three sets of rhythm permutation, one for each type of musical meter you can use.
     • In syntagma II (florid counterpoint), there is a set of rperms that matches up with each vperm, so that there is a specified rhythm for each of the SATB voices. This makes independent contrapuntal voices possible.
4. Writing the music down: Next you lay out what Kircher calls the palimpsest phontacticum, or what the rest of us call music paper. You draw out four staves and put an appropriate combination of clefs for the four voices. These determine the octave and range of the musical output for each voice. You also add a flat in the key signature if your tone is in cantus mollis according to the table, and add the appropriate mensuration sign.
5. Matching pitches, rhythms, and syllables: For each phrase of text, you take the pitch numbers and look them up in the table for the given tone in order to get note names. You pair these note names with the rhythmic values, or just put rests when those are given. You write these values on the staff, making certain adjustments as needed.
6. Kircher describes a variety of adjustments, some of which are easier to automate (that is, are more completely prescribed) than others:
   1. If the notes go out of range for that voice (too far above or below the staff), you may shift them up or down by octave as needed.
   2. The tone tables specify that you may add sharps or flats on certain notes according to musica ficta rules.
   3. You may also do more complicated things like swapping the notes for adjacent voices, which require your own judgment.

Was the Arca musarithmica a computer?

Kircher’s device does not have an automatic mechanism, and requires a human operator. But the operator does not need to do anything except retrieve data from the tables in the Arca and transform it according to reasonably precise rules. Kircher intended his device for the total musical novice, a Tyro who knew nothing about music except the rules he provided. So although the device is not automatic, the system is automatic in the sense that no additional knowledge or discretion is needed beyond what is built in to the device and the specified algorithm for operating it.

In order for a digital computer to step in as the operator, only a few adjustments need to be made to fill gaps and inconsistencies in Kircher’s system. The rest of the work in writing the software just consisted of taking a system designed for a human operator who can manipulate graphic symbols on paper and translating it to the kinds of instructions needed by the digital automaton. The modern computer is the ultimate know-nothing, as it can only do exactly what it is programmed to do, and in the end its processing all boils down to manipulating binary numbers. The implementation in Haskell, a pure-functional programming language, demonstrates that the Arca can be modeled as a single mathematical function, transforming a set of inputs into a set of outputs.

For a more in-depth study of the Arca and this implementation, see the article, Athanasius Kircher’s Arca musarithmica (1650) as a Computational System.

Who Used the Arca?

After describing the Arca, Kircher says he not only built one but that he and other musicians had been using it to generate complex music, and gives an example of a composition produced with the Arca. He also says there is a secret, customized version of the Arca with considerably greater capabilities, which he is reserving for the inspection of trusted associates. No other trace survives of this Arca 2.0, but there is evidence that other people used the Arca as specified in the Musurgia universalis.

Three physical implementations of the ark are known to have survived in the UK and the rest of Europe—in Cambridge, Wolfenbüttel, and Florence. These look like elaborate recipe boxes filled with slats, each of which is covered with a copy of Kircher’s tables of musarithmi or music-numbers. No one has yet found a surviving piece of music generated with the ark.

I found evidence for a previously unknown copy of the Arca in Puebla, Mexico, which is the first known implementation in the New World. Buried within an obscure manuscript miscellany filled with mathematical calculations, tax records, and experiments in universal chronology and trigonometry, there is a copy of a core selection of Kircher’s Arca tables. The parts copied out would be sufficient to operate the device, even if one did not actually build a box to put them in. The author was probably a university mathematics professor in Puebla, writing around 1690, and the array of interests in the manuscript collection show the kind of intellectual climate in which Kircher’s work was read. The poet and nun Sor Juana Inés de la Cruz described herself as Kircherizing, when she followed Kircher in following chains of connections between elements of creation in the pursuit of universal knowledge. She and the copyist of the Puebla Arca were part of a circle of such Kircher enthusiasts in seventeenth-century New Spain. I also found another source, a digest of the Arca system from late eighteenth-century Madrid, that demonstrates the longevity of these interests through the period of the Scientific Revolution and into the Enlightenment—though numerous mistakes also suggest that the author struggled to understand Kircher’s system and likely viewed it as merely a curiosity, rather than a link in a chain leading to divine truth.

For a more in-depth study of the reception of Kircher’s device in the New World, see the article, Kircherizers and Trisectors: Athanasius Kircher’s Automatic Computing Device in the Spanish Empire, published in the journal Anuario musical in 2022.

Other Implementations and Related Projects