Tuning continua and keyboard layouts

@article{Milne2008TuningCA,
  title={Tuning continua and keyboard layouts},
  author={Andrew J. Milne and William A. Sethares and James Plamondon},
  journal={Journal of Mathematics and Music},
  year={2008},
  volume={2},
  pages={1 - 19}
}
Previous work has demonstrated the existence of keyboard layouts capable of maintaining consistent fingerings across a parametrized family of tunings. This paper describes the general principles underlying layouts that are invariant in both transposition and tuning. Straightforward computational methods for determining appropriate bases for a regular temperament are given in terms of a row-reduced matrix for the temperament-mapping. A concrete description of the range over which consistent… 
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References

SHOWING 1-10 OF 56 REFERENCES
The structure of recognizable diatonic tunings
TLDR
In a comprehensive work with important implications for tuning theory and musicology, Easley Blackwood establishes a mathematical basis for the family of diatonic tunings generated by combinations of perfect fifths and octaves.
Isomorphic Controllers and Dynamic Tuning: Invariant Fingering over a Tuning Continuum
TLDR
This paper presents Isomorphic Controllers and Dynamic Tuning: Invariant Fingering over a Tuning Continuum, which aims to demonstrate the power ofHaskell's inequality.
Facts and Counterfacts Mathematical Contributions to Music-theoretical Knowledge
The present article argues in favor of a discipline of Mathematical Music Theory (see [16], [18] for earlier attempts). By reviewing and re-interpretating known results, we draw further conclusions
Aspects of Well-Formed Scales
Pentatonic, diatonic, and chromatic scales share the same underlying structure, that of the well-formed scale. Well-formedness is defined in terms of a relationship between the order in which a
Tuning, Timbre, Spectrum, Scale
The Octave Is Dead ... Long Live the Octave.- The Science of Sound.- Sound on Sound.- Musical Scales.- Consonance and Dissonance of Harmonic Sounds.- Related Spectra and Scales.- A Bell, A Rock, A
Coherence and sameness in well-formed and pairwise well-formed scales
Abstract A common theme running through many of the scale studies in recent years is a concern for the distribution of intervals and pitch classes. The question of good distribution becomes
Lecture 1 Introduction
In this course we will consider mathematical objects known as lattices. What is a lattice? It is a set of points inn-dimensional space with a periodic structure, such as the one illustrated in Figure
Available Online at www
Titrimetric using N-bromosuccinimide and cyclic voltammetric methods were used to determine the concentration of ascorbic acid in green amaranth (Amaranth candatus) and irish potato (Ipomoea batata).
Invariant Fingerings Across a Tuning Continuum. Submitted to the Computer Music Journal. Preliminary version available for review purposes at www.cae.wisc.edu/∼sethares/InvariantFingering.pdf
  • 2007
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