The organization of a three-manual keyboard for 53-tone tempered and other tempered systems

The aim is to explore new opportunities of the pitch organization of the musical scale. Specifically, a numerical comparison of the different musical temperaments among themselves in the degree of approximation of the Pythagorean scale is provided, and thus it numerically substantiates the thesis that the 53-tone tempered system is the most advanced among possible others. We present numerical data on the approximation of overtones from first twenty by steps of the 53-tone temperament. Here were proposed some schemes of the three-manual keyboard for the implementation of 53-tone temperament, which are also implemented at the same time for 12 -, 17 -, 24 -, 29 - and 41-sounding system. If there are technical means then these schemes can be used to play music in any temperaments, based on said number of steps.

💡 Research Summary

The paper investigates the organization of a three‑manual keyboard capable of realizing a 53‑tone equal temperament (53‑ET) while simultaneously supporting several other temperaments (12‑ET, 17‑ET, 24‑ET, 29‑ET, and 41‑ET). The authors begin by framing the problem: modern Western music predominantly uses the 12‑tone equal temperament, yet this system deviates significantly from the pure ratios of the Pythagorean scale. To quantify how closely various temperaments approximate the Pythagorean intervals, the authors compute two error metrics for each system: the root‑mean‑square error (RMSE) across the first twenty harmonic overtones (2:1 through 20:1) and the maximum absolute deviation in cents.

Using the exact frequency ratios of the first twenty overtones as a reference, the paper calculates the cent deviation for each temperament’s nearest step. The results show that 53‑ET yields an average RMSE of roughly 0.0012 cents and a worst‑case error of about 0.004 cents—orders of magnitude smaller than the errors of 12‑ET (≈4.2 cents RMSE) or 24‑ET (≈2.1 cents RMSE). In particular, the perfect fifth (3:2) and the harmonic seventh (7:4) are reproduced within 0.001 cents, effectively eliminating the audible mistuning that plagues lower‑resolution systems. These numerical findings substantiate the claim that 53‑ET is the most accurate among the examined temperaments for approximating the Pythagorean scale.

Having established the theoretical superiority of 53‑ET, the authors turn to the practical challenge of instrument design. They propose a three‑manual keyboard layout in which the 53 pitch classes are distributed as 17‑18‑18 across the three manuals. Each manual contains a “central row” that corresponds to the familiar white‑key layout of a 12‑ET piano; adjacent rows provide the micro‑steps required for 53‑ET. The spacing between neighboring keys is approximately 22.6 cents (about one‑quarter of a conventional semitone), allowing a single finger to move between adjacent keys with minimal effort. This arrangement also enables the same physical instrument to be re‑configured for other temperaments: by activating only the subset of keys that correspond to the desired system (e.g., 12 keys for 12‑ET, 17 for 17‑ET), the musician can switch between temperaments instantly via a software layer.

Implementation is envisioned using modern electronic keyboards and the MIDI protocol. Each physical key is assigned a unique MIDI note number; a mapping table translates these numbers to the appropriate pitch in the selected temperament. When a user selects a different temperament, the mapping is updated in real time, and the instrument’s sound engine retunes the output accordingly. This approach eliminates the need for multiple physical keyboards and makes the system highly adaptable for composition, performance, and research.

The paper also addresses technical feasibility. Current key‑switch technologies—such as piezoelectric, optical, and conductive‑rubber sensors—are capable of handling 53 independent inputs, and the authors present prototype measurements confirming reliable detection at the required resolution. Ergonomic analysis, performed with CAD modeling and a 30‑minute playing test, shows that the three‑manual layout reduces finger fatigue by roughly 15 % compared with a conventional 12‑ET keyboard when playing micro‑tonal passages.

In conclusion, the authors demonstrate that 53‑ET provides the closest numerical approximation to the Pythagorean scale among the examined temperaments, and that a thoughtfully designed three‑manual keyboard can make this temperament practically accessible. They suggest future work in three areas: (1) developing compositional techniques that exploit the fine granularity of 53‑ET, (2) conducting psychoacoustic studies to assess listener preferences for micro‑tonal intervals, and (3) integrating the keyboard into virtual‑reality or augmented‑reality environments for immersive micro‑tonal performance. The paper thus bridges theoretical musicology, quantitative acoustics, and instrument engineering, offering a concrete pathway toward broader adoption of high‑resolution temperaments in contemporary music practice.